TEL AVIV UNIVERSITY Raymond and Beverly Sackler Faculty of Exact Sciences Blavatnik School of Computer Science

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1 TEL AVIV UNIVERSITY Raymond and Beverly Sackler Faculty of Exact Sciences Blavatnik School of Computer Science Algebraic Techniques in Combinatorial and Computational Geometry A thesis submitted toward a degree of Doctor of Philosophy by Noam Solomon This work was carried out under the supervision of Prof. Micha Sharir Submitted to the Senate of Tel-Aviv University September 2017

2 To Sapir, my love i

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4 Abstract In this thesis we study several problems in combinatorial geometry, mainly in incidence geometry. We study a variety of incidence questions that involve points and other geometric objects, starting with lines in three and four dimensions, moving to algebraic curves, and then also to algebraic surfaces in three dimensions. We develop and use several infrastructural tools in algebra and algebraic geometry for tackling these problems, tools that should also be useful for many other combinatorial problems too. We also apply these bounds in several other problems in combinatorial geometry. In the first part of the thesis we consider the problem of obtaining tight asymptotic bounds on the number of incidences between points and lines in higher dimensions, extending the foundational bound of Szemerédi and Trotter from 1983 for the planar case, and the more recent groundbreaking result of Guth and Katz (in 2010) for the three-dimensional case. The latter work introduced methods from advanced algebra and especially from algebraic geometry, which were not used in combinatorics before. This enabled Guth and Katz to (almost) settle the distinct distances problem of Erdős, of obtaining a lower bound for the number of distinct distances determined by any set of n points in the plane, a problem which stubbornly stood open for over 60 years, despite very bold attempts to solve it. The work of Guth and Katz has given significant added momentum to incidence geometry, making many problems, including those studied in this thesis, deemed hopeless before the breakthrough, amenable to the new techniques. We extend the study of point-line incidences to four dimensions, and then to points lying on two- and three-dimensional varieties. We also found a more elementary proof of the Guth- Katz point-line incidence bound in three dimensions. We also derive lower bounds for incidences between points and lines on a 3-dimensional quadratic surface in R 4, and obtain Ramsey-type results involving the contact graph between lines in R 3. In the second part of the thesis, we extend our study of point-line incidences to the study of incidences between points and algebraic curves in three and higher dimensions. One particular case of this study results in a new bound on the number of incidences between points and circles in R 3. We then study incidences between points and constant-degree algebraic surfaces in three diiii

5 mensions, such as planes, spheres, etc. As a result of this study, we obtain several improved bounds for the number of distinct and repeated distances in a set of points lying on a two-dimensional variety in R 3. iv

6 Table of Contents Acknowledgments vii 1: Introduction Background Our results Incidences between points and lines in R Incidences between points and lines in R 3, with applications to Ramseytype theorems for lines in R Improved bounds on incidences of lines with points on a variety Incidences with curves in three and higher dimensions Incidences with surfaces in three dimensions Algebraic Preliminaries Lines on varieties Generalized Bézout s theorem Generically finite morphisms and the Theorem of the Fibers Flecnode polynomials and ruled surfaces in three and four dimensions Flat points and the second fundamental form Finitely and infinitely ruled surfaces in four dimensions, and u-resultants 36 I Incidences between points and lines 39 2: Incidences between points and lines in R 4 41

7 3: Highly incidental patterns on a quadratic hypersurface in R 4 97 II Incidences between points and lines in R 3, with applications 105 4: Incidences between points and lines in R : Ramsey-type theorems for lines in 3-space 129 III Incidences between points and lines on varieties 147 6: Incidences between points and lines on two- and three-dimensional varieties 149 IV Incidences between points and curves and points and surfaces 191 7: Incidences with curves in R d 193 8: Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances 211 V Conclusion 249 9: Discussion Summary Future challenges References 257 vi

8 Acknowledgments I first want to express my deepest gratitude to Professor Micha Sharir, my advisor and friend. Micha has been an incredible source of inspiration to me, and a true mentor. I am both humbled and proud to have been his student, and will continue to collaborate with him in the future. Thank you so much Micha! I am also indebted to my supporting family, my mother Ma ayana, my father Arie, my brother Shay, my sister Michal, my sister-in-law Hagar, and my baby nephew Yoavi. Finally, I am so happy to dedicate this thesis to the love of my life, Sapir. Part of this research was carried out while I was visiting, and enjoying the warm hospitality and stimulating environment of, the Institute for Pure and Applied Mathematics (IPAM) at UCLA, in the spring of vii

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10 1 Introduction 1.1 Background Combinatorial geometry is a field that studies combinatorial problems that have some geometric aspect. It was pioneered and developed by Paul Erdős, starting at the beginning of the 20th century. While such problems (sometimes referred to as Erdős-type problems) are often easy to state, some of them are very difficult, have a deep underlying theory, and remain (or have remained) open for many decades. In the past decade the landscape of combinatorial geometry has considerably changed, due to two groundbreaking papers by Guth and Katz ([55] in 2008 and [56] in 2010). They have introduced reasonably simple techniques from algebraic geometry that facilitated successful solutions of several major problems in combinatorial geometry. Their first paper obtained a complete solution to the joints problem, a problem involving incidences between points and lines in three dimensions which has been proposed by Chazelle et al. [28] in The second Guth Katz paper presented a nearly complete solution to the classical problem of Erdős [43] on distinct distances in the plane, which was a major open problem since Both problems (especially the second one) have been extensively studied over the years, using more traditional, and progressively more complex methods of combinatorial geometry, but with only partial and incomplete results. This marriage of Algebraic Geometry with Combinatorial and Computational Geometry poses major challenges for both areas, including distilling existing machinery in algebraic geometry and developing new tools, geared towards the new client area, and the application of this toolbox to numerous basic problems in combinatorial and computational geometry. In this thesis, we develop additional bridges between the two disciplines, develop additional algebraic machinery, and apply this machinery to a successful solution of several problems in Combinatorial Geometry. We assume basic knowledge of Algebra and Algebraic Geometry, and whenever more advanced knowledge is needed, we will elaborate and give the relevant background. Most of this advanced machinery is presented in Section 1.3, as well as in the various chapters. 1

11 Review of developments preceding this thesis. The main topic of study in the thesis is incidence geometry in three, four, and higher dimensions. We first review the earlier developments in this field, and then go on to discuss our results and original contribution. In its simplest, original form, the joints problem, posed in [28] in 1992, is to obtain a sharp upper bound on the number of points that can be incident to at least three non-coplanar lines, in any set of n lines in three dimensions; these points are called joints. Simple constructions show that the number of joints can be Ω(n 3/2 ) and the goal was to obtain a matching upper bound. After 15 years of frustrating research, the best upper bound obtained using the traditional machinery, was O(n ) [46]. Then, in 2008, Guth and Katz [55] established the tight upper bound O(n 3/2 ), thus solving the problem completely. They used in the proof several reasonably simple tools from algebraic geometry, and we mention here two of them: (i) Given a set P of m points in R 3, one can find a trivariate polynomial f of degree D = O(m 1/3 ) that vanishes at all the points of P. (An appropriate generalization holds in any dimension d 1, except that the degree of the resulting polynomial drops to O(m 1/d ).) (ii) Given two trivariate polynomials f and g with no common factor, and with corresponding zero sets Z( f ), Z(g), the number of lines that are fully contained in Z( f ) Z(g) is at most deg( f ) deg(g); see Corollary (This can be regarded as an extended variant of Bézout s theorem (see, e.g., Section 1.3 and Fulton [49]).) Here is a brief, rough, and informal description of the analysis of Guth and Katz. Given a set L of n lines in R 3, they force, in a preliminary pruning and sampling step, most of the joints of L to lie on the zero set Z( f ) of a polynomial f of degree D cn 1/2, with a sufficiently small constant c, and then only consider lines of L that are also fully contained in Z( f ) (the other lines do not generate too many joints). Now a joint incident to three non-coplanar lines, all contained in Z( f ), must be a singular point of f, and lines that contain more than D such joints must consist exclusively of singular points (each of the three first-order derivatives of f must vanish identically on such a line). Lines that contain fewer than D joints contribute at most nd = O(n 3/2 ) joints, so they can be ignored. Now, assuming f to be irreducible, and applying the preceding result (ii) to f and one of its partial derivatives, say f x, we conclude that the number of such critical lines is at most D 2 c 2 n, which we can make smaller than, say, n/2. An inductive argument on n then completes the proof. The actual proof in [55] is more involved and technical. It has been greatly simplified in two subsequent papers by Kaplan, Sharir and Shustin [67] and by Quilodrán [86]; the bound has also been extended, in both papers, to any dimension d 3, where now a joint is a point incident to at least d lines, not all in a common hyperplane; the worst-case bound is Θ(n d/(d 1) ). Incidences. Although the joints problem might appear, on the face of it, just a curiosity, later developments, as being reviewed here, show that it is in fact a significant pillar in the study of incidences between points and lines, curves, hyperplanes or surfaces, as well as of several other related fields, both combinatorial and algorithmic. We briefly mention the highlights of this topic, 2

12 extensively studied during the past 30 years, and considered as one of the main active areas in combinatorial geometry; see Pach and Sharir [83] for a bit outdated but still relevant comprehensive survey. In the simplest form of the problem, we are given a set P of m (distinct) points and a set L of n (distinct) lines in the plane, and we wish to obtain sharp upper and lower bounds for the maximum possible value of I(P,L), the number of incidences between the points of P and the lines of L, where an incidence is a pair (p,l) P L with p l. The 1983 celebrated theorem of Szemerédi and Trotter [118] asserts that I(P,L) = O(m 2/3 n 2/3 + m + n), and that this bound is tight in the worst case. Many extensions of the problem have been studied, in which one considers, instead of lines, other curves in the plane (e.g., circles), or lines and other curves in higher dimensions, or planes, hyperplanes, or more general surfaces in higher dimensions. In most of these extensions, though, tight bounds on the maximum number of incidences are not known. Incidence problems, besides being a fascinating topic of study in its own right, show up in many applications in combinatorial geometry, including Erdős s famous repeated distances problem (see below), and many other problems concerning repeated patterns in a point set. Moreover, in general, there is a close connection between combinatorial and algorithmic questions in geometry, which has been a major and recurring theme during the past 30 years, because (i) sharp combinatorial bounds are needed to estimate the efficiency of algorithms that compute the relevant structures, and (ii) both types of studies tend to use the same or very similar tools. In the specific case at hand, incidences have deep links to numerous problems in computational geometry, with several common tools that they share (most notably, space partitioning techniques, discussed in detail below). As some simple illustrations, the Szemerédi Trotter bound is more or less the same as the best running time of an algorithm for performing n halfplane range queries on a set of m points in the plane, where the goal is to count the number of points in each halfplane, or for counting intersections between m red line segments and n blue line segments in the plane, as well as for many other algorithmic problems of a similar nature. In addition, connections between incidences and the continuous variants of the Kakeya problem [119] have been observed for some time, and have served as a major motivation for Guth and Katz (following an earlier breakthrough progress by Dvir [34] for the case of finite fields) to study the joints problem, as indicated by the title of their first paper Algebraic methods in discrete analogs of the Kakeya problem. If one considers incidences between m points and n lines in higher dimensions, say in d = 3 dimensions, the problem, on first sight, seems totally uninteresting. Indeed, one can project the points and lines onto some generic plane, observe that incidences are preserved in the projection, and apply the Szemerédi Trotter bound. Since the bound is worst-case tight in the plane, it continues to be so in any higher dimension. The joints problem, in retrospect, was an attempt to remove the triviality from this extension, by forcing the input lines, in a sense, to be truly threedimensional. As follows from the results of Guth and Katz (and even from the weaker previous 3

13 results), one does indeed get improved bounds in truly three-dimensional scenes, when the amount of coplanarity of the input points and lines is kept under control. Back to joints. Elekes, Kaplan and Sharir [39] have extended the study of [55] to consider not just the number of joints but also the number of incidences between the joints of L and the lines of L; that is, each joint is now counted with multiplicity equal to the number of lines incident to it. In fact, the analysis of [39] applies to any set of m points and any set of n lines in R 3, provided that (a) each point is incident to at least three lines, and (b) no plane contains more than some specified number of points. There is some algebraic magic behind the first condition (a); it has to do with the fact that the second fundamental form of the vanishing polynomial f must also vanish at such points (when they are non-singular; see [39, 55], and Section 1.3.5). The second condition, or some alternative one, is needed to avoid situations in which, say, all the given points and lines lie in a common plane, which, in view of the preceding discussion, is crucial in order to bypass the Szemerédi Trotter incidence bound. Under the assumptions in [39], with no plane containing more than n points, the maximum possible number of incidences is Θ(m 1/3 n) when m n, and is equal to the Szemerédi Trotter bound otherwise. Distinct distances. The next development took place in an attempt to apply the new machinery to the planar distinct distances problem of Erdős [43]. In this celebrated problem the goal is to establish a sharp lower bound on the minimum possible number of distinct distances between the elements of a set S of n points in the plane. Erdős noticed that the set of vertices of the n n integer grid generates only O(n/ logn) distinct distances, and conjectured this to be also the lower bound, namely, that any set of n points in the plane determines at least Ω(n/ logn) distinct distances. Again, traditional techniques, becoming progressively more sophisticated during the 65 years since the original problem statement, have been unable to settle the conjecture, and the best lower bound that was achieved, by Katz and Tardos [69], was Ω(n ). Nevertheless, about 15 years ago, Gy. Elekes had come up, in an unpublished note, with an ingenious program to reduce the planar distinct distances problem to an incidence problem between points and curves in three dimensions. To tackle the latter problem, though, he needed a couple of fairly deep conjectures, which neither he nor anybody else knew how to solve at that time. If these conjectures could be established, they would have lead to the almost tight lower bound Ω(n/logn) on the number of distinct distances. In Elekes and Sharir [40], written by Sharir after Elekes s passing away in 2008, Elekes s program has been laid out and developed, and the new algebraic machinery has been applied to it, but this fell short of settling Elekes s conjectures, as the algebraic machinery, available from the 2008 Guth Katz paper and the follow-up ones, was still too weak. This was taken care of in the second breakthrough of Guth and Katz [56], in November 2010, where they introduced new algebraic machinery, based on the polynomial ham sandwich theorem of Stone and Tukey [116] from 1942, which allowed them to establish Elekes s conjectures and 4

14 thereby obtain the aforementioned lower bound Ω(n/ log n) for distinct distances. Specifically, their main result, an extension of the main conjecture of Elekes, is: Given N lines in three dimensions, the number of points that are incident to at least k 3 of these lines is O(N 3/2 /k 2 ), provided that no plane contains more than N 1/2 lines. The case k = 2 is also treated in [56]. There one needs to assume that no plane or regulus (doubly ruled quadric) contains more than N 1/2 lines, and the analysis is based on algebraic properties of ruled surfaces, established by Monge, Salmon and Cayley in the 19th century [81, 90]. It turns out that the question of k 3 can be formulated as a question about incidences between points and lines in three dimensions. Specifically, Guth and Katz showed: 1 Theorem (Guth and Katz [56]). Let P be a set of m distinct points and L a set of n distinct lines in R 3, and let s n be a parameter, such that no plane contains more than s lines of L. Then ( ) I(P,L) = O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n. The application of the polynomial ham sandwich theorem in [56] results in a so-called polynomial partitioning scheme, a new tool that appears to be very powerful in combinatorial and computational geometry, nicely complementing and strengthening the 20-years-old arsenal of geometric partitions based on cuttings [29] and on simplicial partitions [76]. Roughly, it states that, given a set P of m points in R d, and a parameter t < m, one can find a d-variate polynomial f, of degree D = O(t 1/d ), such that each (open) connected component ( cell ) of R d \ Z( f ) contains at most m/t points of P; the number of cells is O(D d ) = O(t). This partitioning of P is not exhaustive, as some (perhaps many, or all) points of P may lie on Z( f ), and they require a special treatment, depending on the specific problem at hand. Handling these points in a systematic manner appears to be a missing fundamental ingredient of the infrastructure of the new paradigm that has not yet been fully resolved (see below for an elaboration of this issue). The power of the new polynomial partitioning technique has quickly been recognized by the community, and has already lead to many interesting new results, and bears a lot of potential for the future to come. It is also one of the major tools used in this thesis. Incidences with curves and surfaces. It is only natural to replace lines by other (simple) geometric objects, like circles or other algebraic curves, or, in three or higher dimensions, planes, spheres and other algebraic surfaces. We give here a brief account of a few such incidence results, before and after the revolution. Points and curves, the planar case. The case of incidences between points and curves has a rich history, starting with the aforementioned case of points and lines in the plane [31, 116, 117], where the worst-case tight bound on the number of incidences is Θ(m 2/3 n 2/3 + m + n), where m is the number of points and n is the number of lines. Still in the plane, Pach and Sharir [83] 1 This bound is not explicitly stated in [56], but it follows directly from the bounds that are established there. 5

15 extended this bound to incidence bounds between points and curves with k degrees of freedom. These are curves with the property that, for each set of k points, there are only µ = O(1) curves that pass through all of them, and each pair of curves intersect in at most µ points; µ is called the multiplicity (of the degrees of freedom). Theorem (Pach and Sharir [83]). Let P be a set of m points in R 2 and let C be a set of n bounded-degree algebraic curves in R 2 with k degrees of freedom and with multiplicity µ. Then ) I(P,C ) = O (m 2k 1 k n 2k 2 2k 1 + m + n, where the constant of proportionality depends on k and µ. Remarks. (1) The result of Pach and Sharir holds for more general families of curves, not necessarily algebraic, but, since algebraicity will be assumed in higher dimensions, we assume it also in the plane. (2) Recently, Sheffer et al. [109] extended the result of Pach and Sharir to the complex plane, by showing that if P is a set of m points in C 2 and C is a set of n bounded-degree algebraic curves in C 2 with k degrees of freedom and with multiplicity µ, then ) I(P,C ) = O (m 2k 1 k +ε n 2k 2 2k 1 + m + n, for any ε > 0, where the constant of proportionality depends on ε,k and µ. Except for the case k = 2 (lines have two degrees of freedom), the bound is not known, and strongly suspected not to be tight in the worst case. Indeed, in a series of papers during the 2000 s [4, 11, 75], an improved bound has been obtained for incidences with circles, parabolas, or other families of curves with certain properties (see [4] for the precise formulation). Specifically, for a set P of m points and a set C of n circles, or parabolas, or similar curves [4], we have ( ) I(P,C ) = O m 2/3 n 2/3 + m 6/11 n 9/11 log 2/11 (m 3 /n) + m + n. (1.1) Some further (slightly) improved bounds, over the bound in Theorem 1.1.2, for more general families of curves in the plane, have been obtained by Chan [25, 26] and by Bien [18]. They are, however, considerably weaker than the bound in (1.1). Recently, Sharir and Zahl [107] have considered families of constant-degree algebraic curves in the plane that belong to an s-dimensional family of curves. This means that each curve in that family can be represented by a constant number of real parameters, so that, in this parametric space, the points representing the curves lie in an s-dimensional algebraic variety F of some constant degree (to which we refer as the complexity of F ). For example, lines or unit circles form 2-dimensional families, and arbitrary circles form a 3-dimensional family. See [107] for more details. Theorem (Sharir and Zahl [107]). Let C be a set of n algebraic plane curves that belong 6

16 to an s-dimensional family F of curves of maximum constant degree E, no two of which share a common irreducible component, and let P be a set of m points in the plane. Then, for any ε > 0, the number I(P,C ) of incidences between the points of P and the curves of C satisfies ) I(P,C ) = O (m 5s 4 2s n 5s 6 5s 4 +ε + m 2/3 n 2/3 + m + n, where the constant of proportionality depends on ε, s, E, and the complexity of the family F. Except for the factor O(n ε ), this is a significant improvement over the bound in Theorem (for s 3), in cases where the assumptions in Theorem imply (as they often do) that C has k = s degrees of freedom. Concretely, when k = s, we obtain an improvement, except for the factor n ε, for the entire meaningful range n 1/s m n 2, in which the bound is superlinear. The factor n ε makes the bound in [107] slightly weaker only when m is close to the lower end n 1/s of that range. Note also that, for s = 3, this bound almost coincides with the one in (1.1). Incidences with curves in three dimensions. The seminal work of Guth and Katz [56], as given in Theorem 1.1.1, has lead to many recent works on incidences between points and lines or other curves in three and higher dimensions; see [24, 57, 98, 102, 103] for a sample of these results. Of particular significance is the recent work of Guth and Zahl [57] on the number of 2-rich points in a collection of curves in R 3, namely, points incident to at least two of the given curves. For the case of lines, Guth and Katz [56] have shown that the number of such points is O(n 3/2 ), when no plane or regulus contains more than n 1/2 lines. Guth and Zahl obtain the same asymptotic bound for general algebraic curves, under analogous (but stricter) restrictive assumptions. For example, by taking circles instead of lines, Guth and Zahl s assumption is that no plane or sphere contains more than O(n 1/2 ) circles. In the general case, the assumption is that no surface that is doubly ruled by curves in a given family of curves contains more than O(n 1/2 ) such curves (we elaborate on this notion below). The study in this thesis requires the extension to three dimensions of the notions of having k degrees of freedom and of being an s-dimensional family of curves. The definitions of these concepts, as given above for the planar case, extend, in a fairly straightforward manner, to three (or higher) dimensions, as will be discussed in more detail later on. We note that these two concepts do not coincide anymore in three or higher dimensions. For example, lines in three dimensions have two degrees of freedom, but they form a 4-dimensional family of curves (this is the number of parameters needed to specify a line in R 3 ). See Section for more details concerning this discrepancy. Points and surfaces. Many of the earlier works on point-surface incidences in three dimensions have only considered special classes of surfaces, most notably planes and spheres (see below). The case of more general surfaces has barely been considered, till the work of Zahl [125], who has studied the general case of incidences between m points and n bounded-degree algebraic surfaces 7

17 in R 3 that have k degrees of freedom. More precisely, in analogy with the case of curves, one needs to assume that for any k points there are at most µ = ) O(1) of the given surfaces that pass through all of them. Zahl s bound is O (m 3k 1 2k n 3k 1 3k 3 + m + n, with the constant of proportionality depending on k, µ, and the maximum degree of the surfaces. By Bézout s theorem, if we require every triple of the given surfaces to have finite intersection, the number of intersection points would be at most E 3, where E is the degree of the surfaces. In particular, E points would then have at most two of the given surfaces passing through all of them. In many instances, though, the actual number of degrees of freedom can be shown to be much smaller. In general, though, one has to allow for the possibility that three (or more) surfaces intersect in a common curve (for example, many planes can intersect in a common line, or many spheres can intersect in a common circle). Handling these general situations ia a major theme studied in Chapter 8 of this thesis. Zahl s bound was later generalized by Basu and Sombra [16] to incidences between points and bounded-degree hypersurfaces in R 4 satisfying certain analogous conditions. Points and planes. Initial partial results on point-plane incidences in three dimensions have been obtained by Edelsbrunner, Guibas and Sharir [36]. More recently, Apfelbaum and Sharir [6] (see also Brass and Knauer [21] and Elekes and Tóth [41]) have shown that if the incidence graph G(P,H), for a set P of m points and a set H of n planes, does not contain a copy of K r,s, for constant parameters r and s, then I(P,H) = O(m 3/4 n 3/4 + m + n). In more generality, Apfelbaum and Sharir [6] have shown that if I = I(P,H) is significantly larger than this bound, then G(P,H) must contain a large complete bipartite subgraph P H, such that P H = Ω(I 2 /(mn)) O(m+n). Moreover, as also shown in [6] (slightly improving a similar result of Brass and Knauer [21]), G(P,H) can be expressed as the union of complete bipartite graphs P i H i so that i ( P i + H i ) = O(m 3/4 n 3/4 + m + n). (This is a specialization to the case d = 3 of a similar result of [6, 21] in any dimension d, and is a special case of the more general analysis of point-surface incidences in three dimensions in this thesis, as will be shortly reviewed.) We note that Fox et al. [47] present a more general framework that includes incidences problems of many kinds, and yields, for incidences between points and planes, almost the same bound, namely O(m 3/4+ε n 3/4+ε + m + n), for any ε > 0, where the constant of proportionality depends on ε. Points and spheres. Earlier works on the special case of point-sphere incidences in three dimensions go back to Chung [30] and to Clarkson et al. [31], and continue with the work of Aronov et al. [9]. Later, Agarwal et al. [1] have bounded the number of non-degenerate spheres with respect to a given point set; this bound was subsequently improved by Apfelbaum and Sharir [7]. 2 2 Given a finite point set P R 3 and a constant 0 < η < 1, a sphere σ R 3 is called η-degenerate (with respect to P), if there exists a circle c σ such that c P η σ P. This definition extends a similar earlier definition for planes (and lines) in Elekes and Tóth [41]. 8

18 The aforementioned recent work of Zahl [125] can be applied in the case of spheres if one assumes that no three, or any larger but constant number, of the spheres intersect in a common circle. In this case the family has k = 3 degrees of freedom any three points determine a unique circle that passes through all of them, and, by assumption, only O(1) spheres contain that circle. Zahl s bound then becomes O(m 3/4 n 3/4 + m + n). In particular, this bound holds for congruent (unit) spheres (where three such spheres can never contain a common circle). The case of incidences with unit spheres have also been studied in Kaplan et al. [65], with the same upper bound; see also [105]. This bound slightly improves an older upper bound in [28]. If many spheres of the family can intersect in a common circle, the bound does no longer hold. The only earlier work that handled this situation is by Apfelbaum and Sharir [6], where it was assumed that the given spheres are non-degenerate. In this case the bound obtained in [6] is O(m 8/11 n 9/11 + m + n). Interestingly, this is also the bound that Zahl s result would have yielded if the spheres had k = 4 degrees of freedom, which however they only almost have : four generic points determine a unique sphere that passes through all of them, but four co-circular points determine an infinity of such spheres. Distinct and repeated distances in three dimensions. The case of spheres is of particular interest, because it arises, in a standard and natural manner, in the analysis of distinct and repeated distances determined by n points in three dimensions. After Guth and Katz s almost complete solution of the number of distinct distances in the plane [56], the three-dimensional case has moved to the research forefront. The prevailing conjecture, due to Erdős, is that the lower bound is Ω(n 2/3 ) (which is the best possible in the worst case, because the n points of the n 1/3 n 1/3 n 1/3 integer grid determine only O(n 2/3 ) distinct distances. However, the current record, obtained by combining the results of Solymosi and Vu [114] with that of Guth and Katz [56], is Ω (n 3/5 ) (a notation hiding polylogarithmic factors), and the problem of closing that gap seems much harder than the two-dimensional problem. A standard reduction to point-sphere incidences goes as follows. Let P denote the set of n points, let t denote the number of distinct distances determined by P, and let δ 1,...,δ t denote the actual distances. Define a set S of nt spheres, whose centers are in P, and whose radii are δ i, for i = 1,...,t. Then I(P,S) = n(n 1), and an upper bound on I(P,S) in terms of n and t immediately implies a lower bound on the number t of distinct distances. Obtaining lower bounds for distinct distances using circles (in the plane) or spheres (in higher dimensions) in the manner just sketched has in general been suboptimal when compared with more effective methods (such as in [56]), but, as we will show in Chapter 8, it can still be used to obtain new lower bounds (larger than Ω(n 2/3 )) in certain favorable special cases, such as the case when the points lie on an algebraic variety of constant degree. The status of the case of unit, or repeated distances is also far from being satisfactory. Here the reduction to point-circle or point-sphere incidences is even more straightforward and natural. That is, let P denote the set of n points and let S denote the set of unit circles (in the plane) or 9

19 unit spheres (in higher dimensions) whose centers lie in P. Then, I(P,S) is equal to twice the number of unit distances. The planar case is stuck with the upper bound O(n 4/3 ) of Spencer et al. [115] from the 1980 s, which is also an immediate consequence of the Pach-Sharir s bound (Theorem 1.1.2) with k = 2 degrees of freedom. The proof has been greatly simplified in [31, 117], but the bound has not been improved (and is actually known to be tight for certain non-euclidean norms [122]). In the plane, the best known lower bound, noted by Erdős, is n 1+c/loglogn. The upper bound O(n 4/3 ) also holds for points on the 2-sphere, and there, surprisingly, it is tight in the worst case (when the repeated distance is 1, say, and the radius of the sphere is 1/ 2) [45], but it is strongly believed that in the plane the correct bound is equal or close to the near-linear lower bound mentioned above. In three dimensions, the aforementioned bound of Zahl [125], also reconstructed in Kaplan et al. [65], with k = 3 degrees of freedom, immediately implies the upper bound O(n 3/2 ) on the number of repeated distances (a slight improvement over the earlier bound of Clarkson et al. [31]), and the best known lower bound is still only Ω(n 4/3 loglogn) [43]. A very recent small improvement of the upper bound has been announced in Zahl [128]. 1.2 Our results The thesis consists of seven studies, presented in four parts and reviewed in the following four respective subsections Incidences between points and lines in R 4 In Chapter 2, we extend the study of Guth and Katz [56], from point-line incidences in three dimensions to point-line incidences in four dimensions, giving worst-case tight or nearly tight bounds on the number of such incidences. This much harder question requires the development of additional tools and techniques from algebraic geometry, most of which are reviewed in Section 1.3 below, and a variety of methods for interfacing them with the problem at hand. Loosely speaking, we study the patterns in which lines can touch one another when they are thrown into four dimensions, where a major subproblem is to understand these patterns when the lines lie in some 3-dimensional algebraic surface. In a preliminary work [99], we proved that, for each ε > 0, there exists an integer c ε, so that the following holds. Let P be a set of m distinct points and L a set of n distinct lines in R 4, and let q,s n be parameters, such that (i) for any polynomial f R[x,y,z,w] of degree c ε, its zero set Z( f ) does not contain more than q lines of L, and (ii) no 2-plane contains more than s lines of L. Then, I(P,L) A ε (m 2/5+ε n 4/5 + m 1/2+ε n 2/3 q 1/12 + m 2/3+ε n 4/9 s 2/9) + A(m + n), (1.2) where A ε depends on ε, and A is an absolute constant. We have subsequently improved and tightened the bound. The improved result, stated next, is presented in Chapter 2, and has appeared in [102]. 10

20 Theorem Let P be a set of m distinct points and L a set of n distinct lines in R 4, and let q,s n be parameters, such that (i) each hyperplane or quadratic hypersurface contains at most q lines of L, and (ii) each 2-flat contains at most s lines of L. Then ( ) ( ) I(P,L) 2 c logm m 2/5 n 4/5 + m + A m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n, (1.3) where A and c are suitable absolute constants. When m n 6/7 or m n 5/3, we have the sharper bound ( ) I(P,L) A m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n. (1.4) In general, except for the factor 2 c logm, the bound is tight in the worst case, for any values of m,n, and for corresponding suitable ranges of q and s. The term m 2/3 n 1/3 s 1/3 comes from the planar Szemerédi Trotter bound [118], and is unavoidable, as it can be attained if we densely pack points and lines into 2-flats, in patterns that attain the Szemerédi Trotter bound. Likewise, the term m 1/2 n 1/2 q 1/4 comes from the bound of Guth and Katz [56] in three dimensions (as in Theorem 1.1.1), and is again unavoidable, as it can be attained if we densely pack points and lines into hyperplanes, in patterns that attain the bound in three dimensions. Our solution employs fairly heavy machinery from algebraic geometry, some going back more than 150 years. For example, an 1846 theorem due to Cayley and Salmon (and, independently obtained by Monge) [81, 90], states that an algebraic surface in three dimensions can fully contain only a bounded number of lines, unless it it is ruled by lines. The study of ruled surfaces, including the way they are embedded in four and higher dimensions, is a central theme in our study. Another related theorem that we use, from 1901, due to Segre and Severi [94, 95], characterizes hypersurfaces in complex 4-space that are infinitely ruled by lines. This machinery is partly presented in Section 1.3, and partly in Chapter 2. Quadrics may increase incidences. In a follow-up work with Ruixiang Zhang [111], presented in Chapter 3, we show that the restrictions made in Theorem are essential, i.e., dropping them would result in a larger number of incidences. Specifically, we show that the condition in assumption (i) of Theorem that quadrics do not contain too many lines, cannot be dropped, in certain cases, by proving the following theorem. Theorem (Solomon and Zhang [111]). For integers m,n, there is a configuration of m points and n lines in R 4, such that all the points (resp., lines) are contained (resp., fully contained) in S := {(x 1,x 2,x 3,x 4 ) R 4 x 1 = x x 2 3 x 2 4}, and (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines in a common hyperplane is O(n/m 1/3 ), and (iii) the number of incidences between the points and lines is 11

21 Ω(m 2/3 n 1/2 + m + n). In particular, when n 9/8 < m < n 3/2, this results in a larger number of incidences than the bound in Theorem Incidences between points and lines in R 3, with applications to Ramsey-type theorems for lines in R 3 In Chapter 4, we give a fairly elementary and simple proof of the Guth-Katz bound in three dimensions, namely, we show that the number of incidences between m points and n lines in R 3, so that no plane contains more than s lines, is ( ) O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between m and n). As already mentioned, the original proof in [56] uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and of providing new tools for tackling similar problems. Such an approach has also recently been undertaken in a follow-up study of Guth [51]. In Chapter 4, we present a different and simpler derivation, with better bounds than those in [51], and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions, and, in a sense, the four-dimensional bound that we present in Chapter 2 is one such application. The results of Chapter 4 have appeared in [100]. Ramsey-type theorems for lines in R 3. In Chapter 5, we prove, jointly with Michael Payne and Jean Cardinal [24], geometric Ramsey-type statements on collections of lines in 3-space. These statements give guarantees on the size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, and reguli in 3-space. Among other things, we prove the following statements. 1. The intersection graph of n lines in R 3 has a clique or an independent set of size Ω(n 1/3 ). 2. Every set of n lines in R 3 has a subset of Ω( n) lines that are all stabbed by one line, or a subset of Ω ( (n/logn) 1/5) lines such that no 6-subset is stabbed by a common line. 3. Every set of n lines in general position in R 3 has a subset of Ω(n 2/3 ) lines that lie on a common regulus, or a subset of Ω(n 1/3 ) lines such that no 4-subset is contained in a regulus. We also refer to a recent work of Pach et al. [84], that studies geometric questions related to the geometry of lines and segments in R 3. 12

22 The results of Chapter 5 have appeared in [24] Improved bounds on incidences of lines with points on a variety When the points lie on some two-dimensional variety in 3-space or three-dimensional variety in 4-space, whose degree is not too large, we show, in Chapter 6 of the thesis, that the number of point-line incidences is substantially smaller than the bounds in Theorem and in Theorem 1.2.1, respectively. In fact, our results do not assume that the two- or three-dimensional variety is embedded in 3-space or 4-space, respectively, and they also hold when the varieties are embedded in higher-dimensional Euclidean spaces. Concretely, our first main result, for this setup, is the following theorem. Theorem (a) The real case: Let P be a set of m distinct points and L a set of n distinct lines in R d, for any d 3, and let 2 s D be two integer parameters, so that all the points and lines lie in a common two-dimensional algebraic variety V of degree D that does not contain any 2-flat, and so that no 2-flat contains more than s lines of L. Then ( ) I(P,L) = O m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + m + n. (1.5) (b) The complex case: Under exactly the same assumptions, when the ambient space is C d, for any d 3, we have ( ) I(P,L) = O m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + D 3 + m + n. (1.6) In both cases, when D and s are constants, we get the linear bound O(m + n). Our next main result for this setup is the following theorem. Theorem (a) The real case: Let P be a set of m distinct points and L a set of n distinct lines in R d, for any d 4, and let s and D be parameters, such that (i) all the points of P lie on a three-dimensional algebraic variety of degree D, without 3-flat or 3-quadric components, and (ii) no 2-flat contains more than s lines of L. Then ( ) I(P,L) = O m 1/2 n 1/2 D + m 2/3 n 1/3 s 1/3 + nd + m. (1.7) (b) The complex case: Under exactly the same assumptions, when the ambient space is C d, for any d 4, we have ( ) I(P,L) = O m 1/2 n 1/2 D + m 2/3 n 1/3 s 1/3 + D 6 + nd + m. (1.8) In both cases, when D and s are constants, we get the linear bound O(m + n). 13

23 A major feature of this analysis is that the bounds also hold over the complex field, as in (1.6) and (1.8) (with a small added term that is negligible when the degree of the ambient variety is small). In contrast, the more general point-line incidences bounds in Theorem and are only known to hold over the reals. The interplay between the real and complex cases will be discussed in detail in Section 1.3. The results of Chapter 6 have appeared in [103] Incidences with curves in three and higher dimensions The results presented so far involve incidences with lines. Our next set of results, presented in Part IV, involve incidences between points and constant-degree algebraic curves in three and higher dimensions. Two different studies address this problem. We first study the general problem, in any dimension. This work, joint with Adam Sheffer, is presented in Chapter 7 and has appeared in [97]. The second study is presented in Chapter 8 and has appeared in [106]. It only studies the threedimensional case, but obtains significantly refined results (over the general results in Chapter 7). Chapter 8 also contains a study of incidences between points and surfaces in R 3, which will be reviewed later in this subsection. Incidences with algebraic curves in R d. In Chapter 7, we prove, jointly with Adam Sheffer, a bound on the number of incidences between points and algebraic curves in R d. Specifically, we prove that the number of incidences between m points and n bounded-degree algebraic curves with k degrees of freedom in R d is O ( m dk d+1 k +ε n dk d+1 dk d d 1 + j=2 m jk k j+1 +ε d( j 1)(k 1) (d j)(k 1) n (d 1)( jk j+1) (d 1)( jk j+1) q j + m + n ), (1.9) for any ε > 0, where the constant of proportionality depends on k,ε and d, provided that no j- dimensional surface of degree c j (k,d,ε), a constant parameter depending on k, d, j, and ε, contains more than q j input curves, and that the q j s satisfy certain mild conditions. The notion of k degrees of freedom, defined in the planar case in the context of Theorem 1.1.2, easily extends, for curves, to any higher dimension. That is, we require that at most µ = O(1) curves of the given family pass through any k specified points, and that any pair of curves intersect in at most µ points. We will return to this notion in the review of Chapter 8 given below. This bound generalizes the planar incidence bound of Pach and Sharir (Theorem 1.1.2) to R d. It also generalizes (with certain restrictions) several other results, including some of the results presented in the thesis. For example, for the case of lines in any dimension d, we have k = 2, and 14

24 the bound in (1.9) is O ( m d+1 2 +ε n d+1 d d 1 + m 2 j=2 j+1 +ε d( j 1) (d j) n (d 1)( j+1) (d 1)( j+1) q j + m + n ). This almost reconstructs the bound of Guth and Katz [56] for d = 3, and the results of Chapter 2 for d = 4. A similar almost reconstruction applies to incidences with circles in three dimensions (where k = 3), as compared with the results of Sharir et al. [98] and those in Chapter 8 (reviewed below). In spite of its general applicability, the bound in (1.9) suffers from several handicaps, also discussed in more detail later on: (i) It has the factors m ε in several of its terms. (ii) The assumptions that it requires, that no j-dimensional surface of degree c j (k,d,ε) contains more than q j curves, is fairly restrictive; these degree bounds, although being constants, can be rather large. (iii) The lower-dimensional bounds in the sum in (1.9) are not sharp when the q j s are small, e.g., when comparing them to the results in [56] and in Chapter 2 for the case of lines. For example, in d = 3 dimensions, the term in (1.9) is O(m 2/3+ε n 1/2 q 1/6 2 ), as opposed to O(m 2/3 n 1/3 q 1/3 2 ) in Theorem 1.1.1, and similar discrepancies occur in d = 4 dimensions. In spite of these handicaps, the result is fairly general and powerful. It partly generalizes a result of Guth [51] in three dimensions (Guth s three-dimensional bound has a better dependency on q 2 ). It also improves a d-dimensional general incidence bound by Fox et al. [47], in the special case of incidences with algebraic curves. This bound is also related to works by Dvir and Gopi [35] and by Hablicsek and Scherr [59] concerning rich lines in high-dimensional spaces. Our bound is a general-purpose bound in R d, and as such, is not known to be tight in most cases. In Chapter 8, we significantly improve this bound, and get rid of the handicaps noted above, in the three-dimensional case. Incidences with curves in three dimensions. We now proceed to review the results in Chapter 8, the final chapter in the thesis. This review is longer than those of the previous chapters, because we also need to discuss several technical assumptions and results, on which our new bounds depend. Preliminaries. This chapter contains two sets of results, one involving incidences with curves in three dimensions, and one involving incidences with surfaces in three dimensions. We begin with the case of curves. In order to state our results, we first define the notions of k degrees of freedom (already mentioned above), of constructibility, and of surfaces infinitely ruled by curves. k degrees of freedom. Let C 0 be an infinite family of irreducible algebraic curves of constant degree E in R 3. Formally, in complete analogy with the planar case, we say that C 0 has k degrees 15

25 of freedom with multiplicity µ, where k and µ are constants, if (i) for every tuple of k points in R 3 there are at most µ curves of C 0 that are incident to all k points, and (ii) every pair of curves of C 0 intersect in at most µ points. As in [83], the bounds that we derive depend more significantly on k than on µ see below. We remark that the notion of k degrees of freedom gets more involved for surfaces, and raises several annoying technical issues. For example, how many points does it take to define, say, a sphere (up to a fixed multiplicity)? Clearly, four generic points do the job (they define a unique sphere passing through all four of them), but four co-circular points do not. While it seems possible to come up with some sort of working definition, we bypass this issue in this thesis, by defining this notion, for a family F of surfaces, only with respect to a given surface V, by saying that F has k degrees of freedom with respect to V if the family of the irreducible components of the curves {σ V σ F }, counted without multiplicity, has k degrees of freedom, in the sense just defined. In the case of spheres, for example, this definition gives four degrees of freedom when V is neither a plane nor a sphere, but only three when V is a plane or a sphere. Constructibility. In the statements of the point-curve incidence theorems, we also assume that C 0 is a constructible family of curves. This notion generalizes the notion of being algebraic, and is discussed in detail in Guth and Zahl [57]. Informally, a set Y C d is constructible if it is a Boolean combination of algebraic sets. The formal definition goes as follows (see, e.g., Harris [61, Lecture 3]). For z C, define v(0) = 0 and v(z) = 1 for z 0. Then Y C d, for some fixed d, is a constructible set if there exist a finite set of polynomials f j : C d C, for j = 1,...,J Y, and a subset B Y {0,1} J Y, so that x Y if and only if (v( f 1 (x)),...,v( f JY (x))) B Y. The constructible sets form a Boolean algebra. This means that finite unions and intersections of constructible sets are constructible, and the complement of a constructible set is constructible. Another fundamental property of constructible sets is that, over C, the projection of a constructible set is constructible; this is known as Chevalley s theorem (see Harris [61, Theorem 3.16] and Guth and Zahl [57, Theorem 2.3]). If Y is a constructible set, we define the complexity of Y to be min(deg f 1 + +deg f JY ), where the minimum is taken over all representations of Y, as described above. As just observed, constructibility of a family C 0 of curves extends the notion of C 0 being s-dimensional. One of the main motivations for using the notion of constructible sets (rather than just s-dimensionality) is the fact, established by Guth and Zahl [57, Proposition 3.3], that the set C 3,E of irreducible curves of degree at most E in complex 3-dimensional space (either affine or projective) is a constructible set of constant complexity that depends only on E. Remark. The definition of constructibility is given over the complex field C. This is in accordance with most of the basic algebraic geometry tools, which have been developed over the complex field; we will elaborate further about it below. 16

26 Surfaces infinitely ruled by curves. Back in three dimensions, a surface V is (singly, doubly, or infinitely) ruled 3 by some family Γ of curves of degree at most E, if each point p V is incident to (at least one, at least two, or infinitely many) curves of Γ that are fully contained in V. As already discussed, the connection between ruled surface theory and incidence geometry goes back to the pioneering work of Guth and Katz [56] and shows up in many subsequent works. A detailed review of ruled surfaces is given in Section 1.3. See also Guth s recent survey [53] and recent book [54], and Kollár [71] for details. For now, we only consider the case of infinitely ruled surfaces. We recall that the only surfaces that are infinitely ruled by lines are planes (see, e.g., Fuchs and Tabachnikov [48, Corollary 16.2]), and that the only surfaces that are infinitely ruled by circles are spheres and planes (see, e.g., Lubbes [74, Theorem 3] and Schicho [93]). It should be noted that, in general, for this definition to make sense, it is important to require that the degree E of the ruling curves be much smaller than deg(v ). Otherwise, every variety V is infinitely ruled by, say, the curves V h, for hyperplanes h, having the same degree as V. A challenging open problem is to characterize all the surfaces that are infinitely ruled by algebraic curves of degree at most E (or by certain special classes thereof). However, the following result of Guth and Zahl provides a useful necessary condition for this property to hold. Theorem (Guth and Zahl [57]). Let V be an irreducible surface, and suppose that it is doubly ruled by curves of degree at most E. Then deg(v ) 100E 2. In particular, an irreducible surface that is infinitely ruled by curves of degree at most E is doubly ruled by these curves, so its degree is at most 100E 2. Therefore, if V is irreducible of degree D larger than this bound, V cannot be infinitely ruled by curves of degree at most E. Incidences with curves in R 3. Theorem (Curves in R 3 ). Let P be a set of m points and C a set of n irreducible algebraic curves of constant degree E, taken from a constructible family C 0, of constant complexity, with k degrees of freedom (and some multiplicity µ) in R 3, such that no surface that is infinitely ruled by curves of C 0 contains more than q curves of C, for a parameter q < n. Then ) I(P,C ) = O (m 3k 2 k n 3k 3 3k 2 + m 2k 1 k n 2k 1 k 1 q 2k 1 k 1 + m + n, (1.10) where the constant of proportionality depends on k, µ, E, and the complexity of the family C 0. Remarks. (1) In certain favorable situations, such as in the cases of lines or circles, discussed above, the surfaces that are infinitely ruled by curves of C 0 have a simple characterization. In such cases the theorem has a stronger flavor, as its assumption on the maximum number of curves on 3 Here we use the simple definition, requiring ruledness et every point of the surface. As noted earlier, this can be done without loss of generality. 17

27 a surface has to be made only for this concrete kind of surfaces. For example, as already noted, for lines (resp., circles) we only need to require that no plane (resp., no plane or sphere) contains more than q of the curves. In general, as mentioned, characterizing infinitely ruled surfaces by a specific family of curves is a difficult task. Nevertheless, we can overcome this issue by replacing the assumption in the theorem by a more restrictive one, requiring that no surface that is infinitely ruled by curves of degree at most E contain more than q curves of C. By Theorem 1.2.5, any infinitely ruled surface of this kind must be of degree at most 100E 2. Hence, an even simpler (albeit weaker) formulation of the theorem is to require that no surface of degree at most 100E 2 contains more than q curves of C. This can indeed be much weaker: In the case of circles, say, instead of making this requirement only for planes and spheres, we now have to make it for every surface of degree at most 400. (2) In several recent works (including the one presented in Chapter 7 of the thesis; see [51, 97, 98]), the assumption in the theorem is replaced by a much more restrictive assumption, that no surface of degree at most c ε contains more than q given curves, where c ε is a constant that depends on another prespecified parameter ε > 0 (where ε affects the resulting incidence bound), and is typically very large (and increases as ε becomes smaller). Getting rid of such an ε-dependent constant (and of the ε in the exponent) is a significant feature of Theorem (3) Theorem generalizes the incidence bound of Guth and Katz [56], obtained for the case of lines. In this case, lines have k = 2 degrees of freedom, they certainly form a constructible (in fact, a 4-dimensional) family of curves, and, as just noted, planes are the only surfaces in R 3 that are infinitely ruled by lines. Thus, in this special case, both the assumptions and the bound in Theorem are identical to those in Guth and Katz [56]. That is, if no plane contains more than q input lines, the number of incidences is O(m 1/2 n 3/4 + m 2/3 n 1/3 q 1/3 + m + n). Improving the bound. The bound in Theorem can be further improved, if we also throw into the analysis the dimensionality s of the family C 0. Actually, as will follow from the proof, the dimensionality that will be used is only that of any subset of C 0 whose members are fully contained in some variety that is infinitely ruled by curves of C 0. As just noted, such a variety must be of constant degree (at most 100E 2, or smaller as in the cases of lines and circles), and the additional constraint that the curves be contained in the variety can typically be expected to reduce the dimensionality of the family. For example, if C 0 is the collection of all circles in R 3, then, since the only surfaces that are infinitely ruled by circles are spheres and planes, the subfamily of all circles that are contained in some sphere or plane is only 3-dimensional (as opposed to the entire C 0, which is 6-dimensional). We capture this setup by saying that C 0 is a family of reduced dimension s if, for each surface V that is infinitely ruled by curves of C 0, the subfamily of the curves of C 0 that are fully contained in V is s-dimensional. In this case we obtain the following variant of Theorem Theorem (Curves in R 3 ). Let P be a set of m points and C a set of n irreducible algebraic 18

28 curves of constant degree E, taken from a constructible family C 0 with k degrees of freedom (and some multiplicity µ) in R 3, such that no surface that is infinitely ruled by curves of C 0 contains more than q of the curves of C, and assume further that C 0 is of reduced dimension s. Then ) ) I(P,C ) = O (m 3k 2 k n 3k 3 3k 2 + O ε (m 2/3 n 1/3 q 1/3 + m 5s 4 2s n 3s 4 5s 4 q 2s 2 5s 4 +ε + m + n, (1.11) for any ε > 0, where the first constant of proportionality depends on k, µ, s, E, and the maximum complexity of any subfamily of C 0 consisting of curves that are fully contained in some surface that is infinitely ruled by curves of C 0, and the second constant also depends on ε. Remarks. (1) Theorem is an improvement of Theorem when s k and m > n 1/k, in cases where q is sufficiently large so as to make the second term in (1.10) dominate the first term; for smaller values of m the bound is always linear. This is true except for the term q ε, which affects the bound only when m is very close to n 1/k (when s = k). When s > k we get a threshold exponent β = ks 4k+2s 5s 4k 2 (which becomes 1/k when s = k), so that the bound in Theorem is stronger (resp., weaker) than the bound in Theorem when m > n β (resp., m < n β ), again, up to the extra factor q ε. (2) The bounds in Theorems and improve, in three dimensions, the result in Chapter 7, in three significant ways: (i) The leading terms in both bounds are essentially the same, but the present bound is sharper, in that it does not include the factor O(n ε ) appearing in Chapter 7. (ii) The assumption here, concerning the number of curves on a low-degree surface, is much weaker than the one made in Chapter 7, where it was required that no surface of some (constant but potentially very large) degree c ε, that depends on ε, contains more than q curves of C (See also Remark (2) following Theorem 1.2.6). (iii) The two variants of the non-leading terms here are significantly smaller than those in Chapter 7, and, in a certain sense (that will be elaborated in Chapter 8) are best possible. Point-circle incidences in R 3. Theorem yields a new bound for the case of incidences between points and circles in R 3, which improves over the previous bound of Sharir, Sheffer, and Zahl [98]. Here, as already discussed, we have k = s = 3 for the case of circles (s = 3 is the dimension of the family of the circles contained in some sphere or plane), so the theorem yields the bound ( ) I(P,C ) = O m 3/7 n 6/7 + m 2/3 n 1/3 q 1/3 + m 6/11 n 5/11 q 4/11+ε + m + n, for any ε > 0, where q is the maximum number of the given circles that are coplanar or cospherical. In fact, the extension of the planar bound (1.1) to higher dimensions, due to Aronov et al. [8], asserts that, for any set C of circles in any dimension, we have ( ) I(P,C ) = O m 2/3 n 2/3 + m 6/11 n 9/11 log 2/11 (m 3 /n) + m + n, (1.12) 19

29 which is slightly better than the general bound of Sharir and Zahl [107] (given in Theorem 1.1.3). If we use this bound, instead of that in Theorem 1.1.3, in the proof of Theorem (specialized for the case of circles; see details in Chapter 8), we get the following slight improvement. Theorem Let P be a set of m points and C a set of n circles in R 3, so that no plane or sphere contains more than q circles of C. Then ( ) I(P,C ) = O m 3/7 n 6/7 + m 2/3 n 1/3 q 1/3 + m 6/11 n 5/11 q 4/11 log 2/11 (m 3 /q) + m + n. Here too we have the three improvements noted in Remark (2) following Theorem In particular, in the sense of part (iii) of that remark, the new bound is best possible with respect to the best known bound (1.12) for the planar or spherical cases; again, see the chapter for details Incidences with surfaces in three dimensions We now review the final set of results in this thesis, involving incidences between points and constant-degree algebraic surfaces in three dimensions. These results are also presented in Chapter 8, and have appeared in [106]. Incidence graph decomposition, for points on a variety and surfaces. In the case of pointsurface incidences, the incidence graph between the points and surfaces can contain large complete bipartite graphs, each involving points on some curve and surfaces containing this curve (unlike earlier studies, reviewed in Section 1.1, we do not have to rule out this possibility, which makes our approach more general). Our bounds estimate the total size of the vertex sets in such a complete bipartite graph decomposition of the incidence graph. In favorable cases, our bounds translate into actual incidence bounds. Overall, here too our results provide a grand generalization of many of the previous studies of (special instances of) this problem. Our first main result on point-surface incidences deals with the special case where the points of P lie on some algebraic variety V of constant degree. Besides being of independent interest, this is a major ingredient of the analysis for the general case of an arbitrary set of points in R 3 and surfaces. In the statements of the following theorems we assume that the set S of the given surfaces is taken from some infinite family F that either has k degrees of freedom with respect to V (with some multiplicity µ), as defined earlier, for suitable constant parameters k (and µ), or is of reduced dimension s with respect to V, for some constant parameter s, meaning that the family Γ := {σ V σ F} is an s-dimensional family of curves (this is reminiscent of the notion of reduced dimension defined above for curves). Theorem Let P be a set of m points on some algebraic surface V of constant degree D in R 3, and let S be a set of n algebraic surfaces in R 3 of maximum constant degree E, taken from some family F of surfaces, which either has k degrees of freedom with respect to V (with some 20

30 multiplicity µ), or is of reduced dimension s with respect to V, for some constant parameters k (and µ) or s. We also assume that the surfaces in S do not share any common irreducible component (which certainly holds when they are irreducible). Then the incidence graph G(P, S) can be decomposed as G(P,S) = (P γ S γ ), (1.13) γ where the union is over all irreducible components of curves γ of the form σ V, for σ S, and, for each such γ, P γ = P γ and S γ is the set of surfaces in S that contain γ. If F has k degrees of freedom then and if F is s-dimensional then we have, for any ε > 0, ) P γ = O (m 2k 1 k n 2k 2 2k 1 + m + n, (1.14) γ ) P γ = O (m 5s 4 2s n 5s 6 5s 4 +ε + m 2/3 n 2/3 + m + n, (1.15) γ where the constants of proportionality depends on D, E, and the complexity of the family F, and either on k and µ in the former case, or on ε and s in the latter case. Moreover, in both cases we have γ S γ = O(n), where the constant of proportionality depends on D and E. Remark. A major feature of this result is that it does not impose any restrictions on the incidence graph, such as requiring it not to contain some fixed complete bipartite graph K r,r, for r a constant, as is done in the preceding studies [16, 65, 125]. We re-iterate that, to allow for the existence of large complete bipartite graphs, the bounds in (1.14) and (1.15), as well as the bound γ S γ = O(n), are not on the number of incidences (that is, on the number of edges in G(P,S), which could be as high as mn) but on the overall size of the vertex sets of the subgraphs in the complete bipartite graph decomposition of G(P,S). This leads to the same asymptotic bound on G(P,S) itself, if one assumes that this graph does not contain K r,r as a subgraph, for a constant r. This kind of compact representation of incidences has already been used in the previous studies (mentioned in Section 1.1) of Brass and Knauer [21], Apfelbaum and Sharir [6], and our recent work [104], albeit only for the special cases of planes or spheres. Another way of bypassing the possible presence of large complete bipartite graphs in G(P,S), used in several earlier works [1, 6, 41], is to assume that the surfaces in S are non-degenerate. These studies only considered the cases of planes and spheres (or of hyperplanes and spheres in higher ) [1, 41]. For spheres, for example, this means that no more than some fixed fraction of the points of P on any given sphere can be cocircular. Although large complete bipartite graphs can exist in G(P,S) in this case, the non-degeneracy assumption allows us to control, in a sharp form, the number of incidences (and shows that the resulting complete bipartite graphs are not so large 21

31 after all). It would be interesting (and, as we believe, doable) to extend our analysis to the case of (suitably defined) more general non-degenerate surfaces. These remarks also apply to the general case (involving points anywhere in R 3 ), given in Theorem below. A mixed incidence bound (for points on most varieties and general surfaces). Our second result on point-surface incidences is an improvement of Theorem 1.2.9, still for the case where the points of P lie on some algebraic variety V of constant degree, where we now also assume that V is not infinitely ruled by the intersection curves of pairs of members of the given family F of surfaces. In this case we obtain an improved, mixed bound, in which G(P,S) can be split into two subgraphs, G 0 (P,S) and G 1 (P,S), where the bound in (1.14) or in (1.15) now holds for G 0 (P,S), i.e., for the actual number of incidences that it represents, and where G 1 (P,S) admits a complete bipartite graph decomposition, as above, for which the sum of the vertex sets is only 4 O(m + n). The actual bound is slightly sharper see below. Specializing the theorem to the case of spheres, as is done below, leads to interesting implications to distinct and repeated distances in three dimensions. Theorem Let P be a set of m points on some irreducible algebraic surface V of constant degree D in R 3, and let S be a set of n algebraic surfaces in R 3 of constant degree E, which do not share any common irreducible component, taken from some infinite constructible family F of surfaces that either has k degrees of freedom with respect to V (with some multiplicity µ) or is s-dimensional with respect to V, for some constant parameters k (and µ) or s. Assume further that V is not infinitely ruled by the family C 0 of the irreducible components of the intersection curves of pairs of surfaces 5 in F. Then the incidence graph G(P,S) can be decomposed as G(P,S) = G 0 (P,S) (P γ S γ ), (1.16) γ where the union is over all irreducible curves γ contained in (one-dimensional) intersections of the form σ σ V, for σ σ S, and, for each such γ, P γ P γ (for some points on some curves, these incident pairs are moved to, and are contained in G 0 (P,S)), and S γ is the set (of size at least two) of surfaces in S that contain γ. Moreover, if F has k degrees of freedom with respect to V (with some multiplicity µ) then ) G 0 (P,S) = O (m 2k 1 k n 2k 2 2k 1 + m + n, (1.17) and if F is s-dimensional with respect to V then, for any ε > 0, ) G 0 (P,S) = O (m 5s 4 2s n 5s 6 5s 4 +ε + m 2/3 n 2/3 + m + n, (1.18) 4 In fact, many bad things must happen for G 1 (P,S) to be nontrivial, and in many situations one would expect G 1 (P,S) to be empty; see below. 5 A stricter assumption is that V is not infinitely ruled by algebraic curves of degree at most E 2, which will hold if we assume that each irreducible component of V has degree larger than 100E 4. 22

32 where the constants of proportionality depends on D, E, and the complexity of the family F, and either on k and µ in the former case, or on ε and s in the latter case. In either case we also have γ P γ = O(m), and S γ = O(n), γ where the constants of proportionality depend on D, E, and the complexity of the family F, and either on k (and µ) in the former case, or on ε and s in the latter case. Remarks. (1) As already alluded to, we note that, typically, one would expect the complete bipartite decomposition part of (1.16) to be empty or trivial. To really be significant, (a) many surfaces of S would have to intersect in a common curve, and, in cases where the multiplicity of these curves is not that large, (b) many curves of this kind would have to be fully contained in V (and also to contain a non-constant number of points of P). Thus, in many cases, in which (a) and (b) do not hold, the bounds in (1.17) or in (1.18) in Theorem are for the overall number of incidences. Note also that both Theorem and Theorem yield a decomposition of (the whole or a portion of) G(P, S) into complete bipartite subgraphs. The major difference is that the bound on the overall vertex set size of these graphs is (relatively) large in Theorem 1.2.9, but it is only linear in m and n (if at all nonzero) in Theorem (2) We note that if V is infinitely ruled by our curves the results break down. For a simple example, take m points and N lines in the plane which form Θ(m 2/3 N 2/3 ) incidences between them. Now pick any surface V in R 3, say the paraboloid z = x 2 + y 2 for specificity, and lift up each of the N lines to a vertical parabola on V. Clearly, V is infinitely ruled by such parabolas, and we get a system of m points and n parabolas with Θ(m 2/3 N 2/3 ) incidences between them. It is also easy to turn this construction into a point-surface incidence structure, in which γ P γ is equal to this bound, which is larger than the lower bound O(m+N) asserted in the theorem. The line y = ax+b in the plane is lifted to the parabola γ a,b = {(x,y,z) R 3 : y = ax + b,z = x 2 + y 2 } contained in the paraboloid V. Define a family S of quadratic surfaces parameterized by a,b,c 0,c 1,c 2 R by S a,b,c0,c 1,c 2 := {(x,y,z) R 3 (z x 2 y 2 )+(y ax b)(c 0 +c 1 x+c 2 y) = 0}. For any c 0,c 1,c 2 R, the quadric S a,b,c0,c 1,c 2 contains the parabola γ a,b, i.e., many surfaces in S intersect in a common parabola. Incidences between points on a variety and spheres. A particular case of interest is when S is a set of spheres. The intersection curves of spheres are circles, and, as already noted, the only surfaces that are infinitely ruled by circles are spheres and planes. Hence, to apply Theorem , we need to assume that the constant-degree surface V that contains the points of P has no planar or spherical components, thereby ensuring that V is not infinitely ruled by circles. Clearly, as already noted, spheres in R 3 have four degrees of freedom, and they form a four-dimensional family of surfaces, with respect to any such variety. We can therefore apply Theorem , with s = 4, and conclude: 23

33 Theorem Let P be a set of m points on some algebraic surface V of constant degree D in R 3, which has no linear or spherical components, and let S be a set of n spheres, of arbitrary radii, in R 3. The incidence graph G(P,S) can be decomposed as G(P,S) = G 0 (P,S) (P γ S γ ), (1.19) where Γ is the set of circles that are contained in V and in at least two spheres of S, and such that, for each γ Γ, P γ = P γ and S γ is the set of all spheres in S that contain γ. We have γ Γ ( ) G 0 (P,S) = O m 1/2 n 7/8+ε + m 2/3 n 2/3 + m + n, (1.20) γ P γ = O(m), and S γ = O(n), γ for any ε > 0, where the constant of proportionality depends on D and ε. Remarks. (1) Since V does not contain a planar or spherical component, the number of circles in Γ is O(D 2 ), as follows by Guth and Zahl [57]. That is, the union in (1.20) is only over a constant number of circles. On the other hand, there might also be incidence edges contained in complete bipartite graphs corresponding to circles that are not contained in V, whose number might be quite large. These incidences are recorded in G 0 (P,S) and their number is bounded in (1.20). (2) Zahl s study [125] yields the bound G(P,S) = O(m 3/4 n 3/4 +m+n), under the assumption that G(P,S) does not contain K r,3, for some (arbitrary) constant r (that is, assuming that every triple of spheres intersect in at most r points of P). Our bound is better for m > n 1/2 (ignoring the n ε factor in our bound). Overall, on one hand, for this rather restrictive assumption, Zahl s result is more general, as it does not require the points to lie on a constant-degree variety, but on the other hand it is more restrictive, due to its assumption on G(P,S), which we do not make. We also note that if we assume that G(P,S) does not contain any K r,r, for r > 3 a constant, the bound in the second part of (1.20) becomes a bound on the number of incidences, so, under this somewhat weaker assumption (than that of Zahl), we improve Zahl s bound for points on a variety and for m > n 1/2. The bound in (1.20) further improves when either (i) the centers of the spheres of S lie on V (or on some other constant-degree variety), or (ii) the spheres of S have the same radius. In both cases, S is only three-dimensional, so the bound improves to ( ) G 0 (P,S) = O m 6/11 n 9/11+ε + m 2/3 n 2/3 + m + n, (1.21) for any ε > 0. When both conditions hold the spheres are congruent and their centers lie on V S is only two-dimensional with respect to V, and the bound improves still further to ( ) G 0 (P,S) = O m 2/3 n 2/3+ε + m + n. 24

34 Using a slightly refined machinery, developed in [105], the latter bound can actually be improved further to G(P,S) = O(m 2/3 n 2/3 + m + n). (1.22) Applications of Theorem and (1.21), (1.22): Distinct distances. As already mentioned, and as will be detailed in the proofs of the following results, the new bounds on point-sphere incidences have immediate applications to the study of distinct and repeated distances determined by a set of n points in R 3, when the points (or a subset thereof see below) lie on some fixed-degree algebraic variety. Specifically, for distinct distances, we have the following results. Theorem (a) Let P be a set of n points on an algebraic surface V of constant degree D in R 3, with no linear or spherical components. Then the number of distinct distances determined by P is Ω(n 7/9 ε ), for any ε > 0, where the constant of proportionality depends on D and ε. (b) Let P 1 be a set of m points on a surface V as in (a), and let P 2 be a set of n arbitrary points in R 3. Then the number of distinct distances determined by pairs of points in P 1 P 2 is ( { }) Ω min m 4/7 ε n 1/7 ε, m 1/2 n 1/2, m, for any ε > 0, where the constant of proportionality depends on D and ε. Remark. In a recent work [105] (not included in this thesis), we have obtained slightly improved bounds, replacing the ε in the exponents by a polylogarithmic factor, using a more refined space decomposition technique. While we believe that the bounds in the theorem are not tight, we note that the bounds in both (a) and (b) (with, say, m = n) are significantly larger than the conjectured best-possible lower bound Ω(n 2/3 ) for arbitrary point sets in R 3. Repeated distances. As another application, we bound the number of unit (or repeated) distances involving points on a surface V, as above. Theorem (a) Let P be a set of n points on some algebraic surface V of constant degree D in R 3, which does not contain any planar or spherical components. Then P determines O(n 4/3 ) unit distances, where the constant of proportionality depends on D. (b) Let P 1 be a set of m points on a surface V as in (a), and let P 2 be a set of n arbitrary points in R 3. Then the number of unit distances determined by pairs of points in P 1 P 2 is ( ) O m 6/11 n 9/11+ε + m 2/3 n 2/3 + m + n, for any ε > 0, where the constant of proportionality depends on D and ε. 25

35 In part (a) we extend, to the case of general constant-degree algebraic surfaces, the known bound O(n 4/3 ), which is worst-case tight when V is a sphere [45]. Part (b) gives (say, for the case m = n) an intermediate bound between O(n 4/3 ) and the best known upper bound O(n 3/2 ) for an arbitrary set of points in R 3 [65, 125]. Another thing to notice is that, for distinct distances, the situation is quite different when V is (or contains) a plane or a sphere, in which case the bound goes up to Ω(n/logn) [56, 119] (see also Sheffer s survey [108] for details). Incidence graph decomposition (for arbitrary points and surfaces). Our final main result on point-surface incidences deals with the general setup involving a set S of algebraic surfaces and an arbitrary set of points in R 3. The analysis in this general setup proceeds by a recursive argument, based on the polynomial partitioning technique of Guth and Katz [56], in which Theorem plays a central role. This result extends a recent result in our preliminary work [105, Theorem 1.4] from spheres to general surfaces, and extends a recent result of Zahl [125], for general algebraic surfaces, to the case where no constraints are imposed on G(P,S). Theorem Let P be a set of m points in R 3, and let S be a set of n surfaces from some s-dimensional family 6 F of surfaces, of constant maximum degree E in R 3. Then the incidence graph G(P,S) can be decomposed as G(P,S) = G 0 (P,S) (P γ S γ ), (1.23) γ where the union is now over all curves γ of intersection of at least two of the surfaces of S, and, for each such γ, P γ = P γ and S γ is the set (of size at least two) of surfaces in S that contain γ. Moreover, we have, for any ε > 0, ( J(P,S) := Pγ + S γ ) ) = O(m 3s 1 2s n 3s 3 3s 1 +ε + m + n, and G 0 (P,S) = O(m + n), γ (1.24) where the constants of proportionality depend on ε, s, D, E, and the complexity of the family F. 1.3 Algebraic Preliminaries In this section we collect and adapt a large part of the machinery from algebraic geometry that we need for our analysis. Some supplementary machinery is developed within the analysis in the subsequent chapters. The proofs are deferred to later chapters in the thesis. In what follows, to facilitate the application of standard techniques in algebraic geometry, it will be more convenient to work over the complex field C, and in complex (affine or projective) spaces. This is in accordance with most of the basic algebraic geometry tools, which have been 6 Here we use the general notion of s-dimensionality, not confined to points on a variety. 26

36 developed over the complex field. Some care has to be exercised when applying them over the reals. Some of the tools that we need to use (See, e.g., Theorem 1.3.2, and the results of Guth and Zahl [57]) apply over the complex field, but not over the reals. On the other hand, when we apply the partitioning method of [56] (as in the proofs of Theorems and ) or when we use Theorem 1.1.3, we (have to) work over the reals. It is a fairly standard practice in algebraic geometry that handles a real algebraic variety V, defined by real polynomials, by considering its complex counterpart V C, namely the set of complex points at which the polynomials defining V vanish. The rich toolbox that complex algebraic geometry has developed allows one to derive various properties of V C, which, with some care, can usually be transported back to the real variety V. This issue arises time and again in this thesis. Roughly speaking, we approach it as follows. We apply the polynomial partitioning technique to the given sets of points and of lines, curves or surfaces, in the original real (affine) space, as we should. Within the cells of the partitioning we then apply some field-independent argument, based either on induction or on some ad-hoc combinatorial argument. Then we need to treat points that lie on the zero set of the partitioning polynomial. We can then switch to the complex field, when it suits our purpose, noting that this step preserves all the real incidences; at worst, it might add additional incidences involving the non-real portions of the variety and of the curves or surfaces. Hence, the bounds that we obtain for this case transport, more or less verbatim, to the real case too Lines on varieties We begin with several basic notions and results in differential and algebraic geometry that we will need (see, e.g., Ivey and Landsberg [63], and Landsberg [73] for more details). For a vector space V (over R or C), let PV denote its projectivization. That is, PV = V \ {0}/, where v w iff w = αv for some non-zero constant α. An algebraic variety is the common zero set of a finite collection of polynomials. We call it affine, if it is defined in the affine space, or projective, if it is defined in the projective space, in terms of homogeneous polynomials. For an (affine) algebraic variety X, and a point p X, let T p X denote the (affine) tangent space of X at the point p. A point p is non-singular if dimt p X = dimx (see Hartshorne [62, Definition I.5 and Theorem I.5.1]). For a point p X, let Σ p denote the set of the complex lines passing through p and contained in X, and let Ξ p denote the union of these lines (here X is implicit in these notations). For p fixed, the lines in Σ p can be represented by their directions, as points in PT p X. In Hartshorne [62, Ex.I.2.10], Ξ p is also called the (affine) cone over Σ p. Clearly, Ξ p T p X. Consider the special case where X is a hypersurface in C 4, i.e., X = Z( f ), for a non-linear polynomial f C[x,y,z,w], which we assume to be irreducible, where Z( f ) = {p C 4 f (p) = 0} 27

37 is the zero set of Z( f ). A line l v = {p + tv t C} passing through p in direction v is said to osculate to Z( f ) to order k at p, if the Taylor expansion of f around p in direction v vanishes to order k, i.e., if f (p) = 0, v f (p) = 0, 2 v f (p) = 0,..., k v f (p) = 0, (1.25) where v f (which for uniformity we also denote as 1 v f ), 2 v f,..., k v f are, respectively, the first, second, and higher order derivatives of f, up to order k, in direction v (where v is regarded as a vector in projective 3-space, and the derivatives are interpreted in a scale-invariant manner we only care whether they vanish or not). That is, v f = f v, 2 v f = v T H f v, where H f is the Hessian matrix of f, and i v f is similarly defined, for i > 2, albeit with more complicated explicit expressions. For simplicity of notation, put F i (p;v) := i v f (p), for i 1. In fact, one can extend the definition of osculation of lines to arbitrary varieties in any dimension (see, e.g., Ivey and Landsberg [63]). For a variety X, a point p X, and an integer k 1, let Σ k p PT p X denote the variety of the lines that pass through p and osculate to X to order k at p; as before, we represent the lines in Σ k p, for p fixed, by their directions, as points in the corresponding projective space. For each k N, there is a natural inclusion Σ p Σ k p. In analogy with the previous notation, we denote by Ξ k p the union of the lines that pass through p with directions in Σ k p. We let F(X) denote the variety of lines (fully) contained in X; this is known as the Fano variety of X, and it is a subvariety of the (2d 2)-dimensional Grassmannian manifold of lines in P d (C); see Harris [61, Lecture 6, page 63] for details, and [61, Example 6.19] for an illustration, and for a proof that this is indeed a variety. We will sometimes denote F(X) also as Σ (or Σ(X)), to conform with the notation involving osculating lines. We also let Σ k denote the variety of the lines osculating to order k at some point of X, and can be thought of as the union of the Σ k p over p X. When representing lines in Σ or Σ k we can no longer use the local representation by directions, and instead represent them, in the customary manner, as points within the Grassmanian manifold. Here too Σ k can be shown to be a variety (within the Grassmannian manifold) and F(X) Σ k for each k. We also have, for any p Z, Σ p F(X) and Σ k p Σ k. Genericity. We recall that a property is said to hold generically (or generally) for polynomials f 1,..., f n, of some prescribed degrees, if there are nonzero polynomials g 1,...,g k in the coefficients of the f i s, such that the property holds for all f 1,..., f n for which none of the polynomials g j is zero (see, e.g., Cox et al. [33, Definition 3.6]). In this case we say that the collection f 1,..., f n is general or generic, with respect to the property in question, namely, with respect to the vanishing of the polynomials g 1,...,g k that define that property Generalized Bézout s theorem An affine (resp. projective) variety X C d (resp. X P d (C)) is called irreducible if, whenever V is written in the form V = V 1 V 2, where V 1 and V 2 are affine (resp., projective) varieties, then 28

38 either V 1 = V or V 2 = V. Theorem (Cox et al. [32, Theorem 4.6.2, Theorem 8.3.6]). Let V be an affine (resp., projective) variety. Then V can be written as a finite union V = V 1 V m, where V i is an irreducible affine (resp., projective) variety, for i = 1,...,m. If one also requires that V i V j for i j, then this decomposition is unique, up to a permutation (see, e.g., [32, Theorem 4.6.4, Theorem 8.3.6]), and is called the minimal decomposition of V into irreducible components. We next state a generalized version of Bézout s theorem, as given in Fulton [49]. It will be a major technical tool in our analysis. Theorem (Fulton [49, Proposition 2.3]). Let V 1,...,V s be subvarieties of P d, and let Z 1,...,Z r be the irreducible components of s i=1 V i. Then r i=1 deg(z i ) s j=1 deg(v j ). A simple application of Theorem yields the following useful result. Lemma A curve C P 4 of degree D can contain at most D lines. This immediately yields the following result, derived in Guth and Katz [55] (see also [39]) in a somewhat different manner. Corollary Let f and g be two trivariate polynomials without a common factor. Z( f,g) := Z( f ) Z(g) contains at most deg( f ) deg(g) lines. Then Generically finite morphisms and the Theorem of the Fibers The following results can be found, e.g., in Harris [61, Chapter 11]. For a map π : X Y of projective varieties, and for y Y, the variety π 1 (y) is called the fiber of π over y. The following result is a slight paraphrasing of Harris [61, Proposition 7.16] and also appears in Sharir and Solomon [103, Theorem 6.3] Theorem (Harris [61, Proposition 7.16]). Let f : X Y be the map induced by the standard projection map π : P d P r (which retains r of the coordinates and discards the rest), where r < d, X P d and Y P r are projective varieties, X is irreducible, and Y is the image of X. Then the 29

39 general fiber 7 of the map f is finite if and only if dim(x) = dim(y ). In this case, the number of points in a general fiber of f is constant. An important technical tool for our analysis is the following so-called Theorem of the Fibers. Theorem (Harris [61, Corollary 11.13]). Let X be a projective variety and π : X P d be a polynomial map (i.e., the coordinate functions x 0 π,...,x d π are homogeneous polynomials); let Y = π(x) denote its image. For any p Y, let λ(p) = dim(π 1 (p)). Then λ(p) is an upper semi-continuous function of p in the Zariski topology 8 on Y ; that is, for any m, the locus of points p Y such that λ(p) m is closed in Y. Moreover, if X 0 X is any irreducible component, Y 0 = π(x 0 ) its image, and λ 0 the minimum value of λ(p) on Y 0, then dim(x 0 ) = dim(y 0 ) + λ Flecnode polynomials and ruled surfaces in three and four dimensions Ruled surfaces in three dimensions. We first review several basic properties of ruled twodimensional surfaces in R 3 or in C 3. Most of these results are considered folklore in the literature, although we have been unable to find concrete rigorous proofs (in the modern jargon of algebraic geometry). For a modern approach to ruled surfaces, there are many references; see, e.g., Hartshorne [62, Section V.2], or Beauville [17, Chapter III]. The theory goes back to the 19th century, as presented in Salmon s monograph [90], and later by Edge [37]. Guth and Katz s paper [56] presents several important properties of ruled surfaces in three dimensions, and more expanded reviews are given in Guth s recent book [54] and survey [53], and in Kollár s paper [71]. We say that a real (resp., complex) surface X is ruled by real (resp., complex) lines if every point p X in a Zariski-open dense set is incident to a real (resp., complex) line that is fully contained in X; see, e.g., [90] or [37] for further details on ruled surfaces. This definition is slightly weaker than the classical definition, where it is required that every point of X be incident to a line contained in X (e.g., as in [90]). It has been used in recent works, see, e.g., [56, 71]. Similarly to the proof of Lemma 3.4 in Guth and Katz [56], a limiting argument implies that the two definitions are equivalent. We note that some care has to be exercised when dealing with ruled surfaces, because ruledness may depend on the underlying field. Specifically, it is possible for a surface defined by real polynomials to be ruled by complex lines, but not by real lines. For example, the sphere defined by x 2 + y 2 + z 2 1 = 0, regarded as a real variety, is certainly not ruled by lines, but as a 7 The meaning of this statement is that the assertion holds for the fiber at any point outside some lower-dimensional exceptional subvariety. 8 The Zariski closure of a set Y is the intersection of all varieties X that contain Y. Y is Zariski closed if it is equal to its closure (and is therefore a variety), and is Zariski open if its complement is Zariski closed. See [62] for further details. 30

40 complex variety it is ruled by (complex) lines. (Indeed, each point (x 0,y 0,z 0 ) on the sphere is incident to the (complex) line (x 0 + αt,y 0 + βt,z 0 + γt), for t C, where α 2 + β 2 + γ 2 = 0 and αx 0 + βy 0 + γz 0 = 0, which is fully contained in the sphere.) In three dimensions, a two-dimensional irreducible ruled surface can be either singly ruled, or doubly ruled (notions that are elaborated below), or a plane (the only infinitely ruled surface; again see below). The discussion so far pertains only to surfaces that are ruled by lines, but in general one can also consider surfaces ruled by other families of curves (e.g., by circles), where the definition of ruledness extends to these cases in a straightforward manner. We will consider ruledness by more general families of curves in Chapter 8. For notational convenience, though, ruled surface, without any extra qualifications refers to a surface ruled by lines. Reguli. A regulus is the surface (in R 3 or C 3 ) spanned by all lines that meet three pairwise skew lines in 3-space. 9 For an elementary proof that a doubly ruled surface over R must be a regulus, we refer the reader to Fuchs and Tabachnikov [48, Theorem 16.4]. As the following lemma shows, the only doubly ruled surfaces are reguli, where a regulus is the union of all lines that meet three pairwise skew lines. There are only two kinds of reguli, both of which are quadrics hyperbolic paraboloids and hyperboloids of one sheet; see, e.g., Fuchs and Tabachnikov [48] for more details. Lemma Let V be an irreducible ruled surface in R 3 or in C 3 which is not a plane, and let C V be an algebraic curve, such that every non-singular point p V \ C is incident to exactly two lines that are fully contained in V. Then V is a regulus. Singly ruled surfaces. Ruled surfaces that are neither planes nor reguli are called singly ruled surfaces (a terminology justified by Theorem 1.3.8, given below). A line l, fully contained in an irreducible singly ruled surface V, such that every point of l is doubly ruled, i.e., every point on l is incident to another line fully contained in V, is called an exceptional line of V. A point p V V that is incident to infinitely many lines fully contained in V is called an exceptional point of V. The following result is another folklore result in the theory of ruled surfaces, used in many studies (such as Guth and Katz [56]). It justifies the terminology singly-ruled surface, by showing that the surface is generated by a one-dimensional family of lines, and that each point on the surface, with the possible exception of points lying on some curve, is incident to exactly one generator (see below). It also shows that there are only finitely many exceptional lines; the property that their number is at most two (see [56]) is presented later. We give (in Chapter 6) a detailed and rigorous proof, to make our presentation as self-contained as possible; we are not aware of any similarly detailed argument in the literature. 9 Technically, in some definitions (cf., e.g., Edge [37, Section I.22]) a regulus is a one-dimensional family of generator lines of the actual surface, i.e., a curve in the Plücker or Grassmannian space of lines, but we use here the alternative notion of the surface spanned by these lines. 31

41 Theorem (a) Let V be an irreducible ruled two-dimensional surface of degree D > 1 in R 3 (or in C 3 ), which is not a regulus. Then, except for finitely many exceptional lines, the lines that are fully contained in V are parameterized by an irreducible algebraic curve Σ 0 (in the parametric Plücker space P 5 that represents lines in 3-space), and thus yield a 1-parameter family of generator lines l(t), for t Σ 0, that depend continuously on the real or complex parameter t. Moreover, if t 1 t 2, and l(t 1 ) l(t 2 ), then there exist sufficiently small and disjoint neighborhoods 1 of t 1 and 2 of t 2, such that all the lines l(t), for t 1 2, are distinct. (b) There exists a one-dimensional curve C V, such that any point p in V \ C is incident to exactly one line fully contained in V. Exceptional lines on a singly ruled surface. In view of Theorem 1.3.8, every point on a singly ruled surface V is incident to at least one generator. Hence an exceptional (non-generator) line is a line l V such that every point on l is incident to a generator (which is different from l). Lemma Let V be an irreducible ruled surface in R 3 or in C 3, which is neither a plane nor a regulus. Then (i) V contains at most two exceptional lines, and (ii) V contains at most one exceptional point. Following Theorem 1.3.8, we refer to irreducible ruled surfaces that are neither planes nor reguli as singly ruled. A line l, fully contained in an irreducible singly ruled surface V, such that every point of l is incident to another line fully contained in V, is called an exceptional line of V (these are the lines mentioned in Theorem 1.3.8(a)). If there exists a point p V V, which is incident to infinitely many lines fully contained in V, then p V is called an exceptional point of V. By Guth and Katz [56], V can contain at most one exceptional point p V (in which case V is a cone with p V as its apex), and (as also asserted in the theorem) at most two exceptional lines. Flecnodes in three dimensions and the Cayley Salmon Monge Theorem. We first recall the classical theorem of Cayley and Salmon, also due to Monge. Consider a polynomial f C[x,y,z] of degree D 3. A flecnode of f is a point p on the zero set Z( f ) of f, for which there exists a line that is incident to p and osculates to Z( f ) at p to order three. That is, if the direction of the line is v then f (p) = 0, and v f (p) = 2 v f (p) = 3 v f (p) = 0, where v f, 2 v f, 3 v f are, respectively, the first, second, and third-order derivatives of f in the direction v. The flecnode polynomial of f, denoted FL f, is the polynomial obtained by eliminating v, via resultants (see, e.g., Cox et al. [33]), from these three homogeneous equations (where p is regarded as a fixed parameter). (See Salmon [90], and the relevant applications thereof in [39, 56], for details concerning flecnode polynomials in three dimensions; see also Ivey and Landsberg [63] for a more modern generalization of this concept.) The Cayley Salmon theorem [90], independently obtained by Monge [81], asserts that an irreducible surface Z( f ) is ruled by lines if and only if FL f vanishes identically on Z( f ). 32

42 Theorem (Cayley and Salmon [90], Monge [81]). Let f C[x,y,z] be an irreducible polynomial of degree D 3. Then Z( f ) is ruled by (complex) lines if and only if Z( f ) Z(FL f ). A simple proof of the Cayley Salmon Monge theorem can be found in Terry Tao s blog [120]. As shown in Salmon [90, Chapter XVII, Section III], the degree of FL f is at most 11D 24. By construction, the flecnode polynomial of f vanishes on all the flecnodes of f, and in particular on all the lines fully contained in Z( f ). We will also be using the following result, established by Guth and Katz [55]; see also [39]. It is in fact an immediate consequence of Corollary Proposition Let f be a trivariate irreducible polynomial of degree D. If Z( f ) fully contains more than 11D 2 24D lines then Z( f ) is ruled by (possibly complex) lines. The flecnode polynomial in four dimensions. The notions of flecnodes and of the flecnode polynomial can be extended to four dimensions. Informally, the four-dimensional flecnode polynomial FL 4 f of a 4-variate polynomial f is defined analogously to the three-dimensional variant FL f, and captures the property that a point on Z( f ) is incident to a line that osculates to Z( f ) up to the fourth order. Specifically, let f C[x,y,z,w] be a polynomial of degree D 4. A flecnode of f is a point p Z( f ) for which there exists a line that passes through p and osculates to Z( f ) to order four at p. Therefore, if the direction of the line is v = (v 0,v 1,v 2,v 3 ), then it osculates to Z( f ) to order four at p if f (p) = 0 and F i (p;v) := i v f (p) = 0, for i = 1,2,3,4, (1.26) where i v f is the ith order derivative of f in the direction v. The four-dimensional flecnode polynomial of f, denoted FL 4 f, is the polynomial obtained by eliminating v from the four equations in the system (1.26). Note that these four polynomials are homogeneous in v (of respective degrees 1, 2, 3, and 4). We thus have a system of four equations in eight variables, which is homogeneous in the four variables v 0,v 1,v 2,v 3. Eliminating those variables results in a single polynomial equation in p = (x,y,z,w). Using standard techniques, as in Cox et al. [33], the resulting polynomial FL 4 f is the multipolynomial resultant Res 4 (F 1,F 2,F 3,F 4 ) of F 1,F 2,F 3,F 4, regarding these as polynomials in v (where the coefficients are polynomials in p). By definition, FL 4 f vanishes at all the flecnodes of f. The following results are immediate consequences of the theory of multipolynomial resultants, presented in Cox et al. [33]. Lemma Given a polynomial f C[x,y,z,w] of degree D 4, its flecnode polynomial FL 4 f has degree O(D). Lemma Given a polynomial f C[x,y,z,w] of degree D 4, every line that is fully contained in Z( f ) is also fully contained in Z(FL 4 f ). Ruled surfaces in four dimensions. Flecnode polynomials are a major tool for characterizing ruled surfaces also in four dimensions. This is manifested in the following theorem of Landsberg 33

43 [73] (which extends Theorem to four dimensions, and is even more general than this), is a crucial tool for our analysis. It is established in [73] as a considerably more general result, but we formulate here a special instance that suffices for our needs. Theorem (Landsberg [73]). Let f C[x,y,z,w] be a polynomial of degree D 4. Then Z( f ) is ruled by (complex) lines if and only if Z( f ) Z(FL 4 f ). When f is of degree 3, we have the following simpler situation, whose easy proof is omitted. Lemma For every polynomial f C[x,y,z,w] of degree 3, Z( f ) is ruled by (possibly complex) lines Flat points and the second fundamental form The three-dimensional case Following the notations in Guth and Katz [55] and in [39] (see also Pressley [85] and Ivey and Landsberg [63] for more basic references), we call a non-singular point p of Z( f ), for f C[x,y,z], linearly flat, if it is incident to at least three distinct lines that are fully contained in Z( f ) (and thus also in the tangent plane T p Z( f )). The condition for a point p to be linearly flat was worked out in [55] (see also [39]), and is as follows. Let p be a non-singular point of Z( f ), and let f (2) denote the second-order Taylor expansion of f at p. That is, we have, for any direction vector v and t C, f (2) (p +tv) = t f (p) v t2 v T H f (p)v. (1.27) If p is linearly flat, there exist three lines l 1, l 2, l 3, contained in the tangent plane T p Z( f ), such that v T H f (p)v = 0, when v is the direction of the lines l 1,l 2 and l 3. (clearly, the first term f (p) v also vanishes for these directions). Using a suitable coordinate frame within T p Z( f ), we can regard v T H f (p)v as a quadratic trivariate homogeneous polynomial, and thus vanishes on the entire lines l 1,l 2 and l 3. Since v T H f (p)v vanishes on three lines inside T p Z( f ), a (generic) line l, fully contained in T p Z( f ) and not passing through p, intersects these lines at three distinct points, at which v T H f v vanishes. Since this is a quadratic polynomial, it must vanish identically on l. Thus, v T H f v is zero for all vectors v T p Z( f ), and thus f (2) vanishes identically on T p Z( f ). In this case, we say that p is a flat point of Z( f ). Therefore, every linearly flat point of Z( f ) is also a flat point of Z( f ) (albeit not necessarily vice versa. 10 ) We next express the set of flat points of Z( f ) as the zero set of three polynomials. In order for v T H f v to vanish identically on T p Z( f ), it is necessary and sufficient that vanish on the three vectors ( f (p) e j ) T H f (p)( f (p) e j ) = 0, for j = 1,2,3, where e j are the standard basis vectors in R 3, 10 For example, for the surface in R 3 defined by the zero set of f = x + y + z + x 3, the point 0 = (0,0,0) Z( f ) is flat (because the second order Taylor expansion of f near 0 is the plane x + y + z = 0), but is not linearly flat, since there is no line incident to 0 and contained in Z( f ). 34

44 for j = 1,2,3 and stand for the vector product in R 3. Therefore, a regular point p Z( f ) is flat if and only if the three polynomials Π j (p) defined by Π j (p) = ( f (p) e j ) T H f (p)( f (p) e j ) vanish at p. The four-dimensional case. We continue with the four-dimensional analog. Extending the above notation to four dimensions, we call a non-singular point p of Z( f ), for f C[x,y,z], linearly flat, if it is incident to at least three distinct 2-flats that are fully contained in Z( f ) (and thus also in the tangent hyperplane T p Z( f )). The condition for a point p to be linearly flat can be worked out as follows, suitably extending the three-dimensional case. Let p be a non-singular point of Z( f ), and let f (2) denote the second-order Taylor expansion of f at p. That is, we have, as in the three-dimensional case, for any direction vector v and t C, f (2) (p +tv) = t f (p) v t2 v T H f (p)v. (1.28) If p is linearly flat, there exist three 2-flats π 1, π 2, π 3, contained in the tangent hyperplane T p Z( f ), such that v T H f (p)v = 0, for all v π 1,π 2,π 3 (clearly, the first term f (p) v also vanishes for any such v). Using a suitable coordinate frame within T p Z( f ), we can regard v T H f (p)v as a quadratic trivariate homogeneous polynomial. Since v T H f (p)v vanishes on three 2-flats inside T p Z( f ), a (generic) line l, fully contained in T p Z( f ) and not passing through p, intersects these 2-flats at three distinct points, at which v T H f v vanishes. Since this is a quadratic polynomial, it must vanish identically on l. Thus, v T H f v is zero for all vectors v T p Z( f ), and thus f (2) vanishes identically on T p Z( f ). In this case, we say that p is a flat point of Z( f ). Therefore, every linearly flat point of Z( f ) is also a flat point of Z( f ) (albeit not necessarily vice versa, as in three dimensions). We next express the set of flat points of Z( f ) as the zero set of a certain collection of polynomials. To do so, we define three canonical 2-flats, on which we test the vanishing of the quadratic form v T H f v. (The preceding analysis shows that, for a linearly flat point, it does not matter which triple of 2-flats is used for testing the linear flatness, as long as they are distinct.) These will be the 2-flats π x p := T p Z( f ) {x = x p }, π y p := T p Z( f ) {y = y p }, and π z p := T p Z( f ) {z = z p }. (1.29) These are indeed distinct 2-flats, unless T p Z( f ) is orthogonal to the x-, y-, or z-axis. Denote by Z( f ) axis the subset of non-singular points p Z( f ), for which T p Z( f ) is orthogonal to one of these axes, and assume in what follows that p Z( f ) \ Z( f ) axis. We can ignore points in Z( f ) axis by assuming that the coordinate frame of the ambient space is generic, to ensure that none of our (finitely many) input points has a tangent hyperplane that is orthogonal to any of the axes. Lemma Let p be a non-singular point of Z( f ) \ Z( f ) axis. Then p is a flat point of Z( f ) if and only if p is a flat point of each of the varieties Z( f {x=xp }),Z( f {y=yp }),Z( f {z=zp }). 35

45 Recall from Elekes et al. [39] that p is flat for f {x=xp } if and only if Π 1 j := Π j( f {x=xp }) vanishes at p, for j = 1,2,3, where Π j (h) = ( h e j ) T H h ( h e j ), and where e 1,e 2,e 3 denote the unit vectors in the respective y-, z-, and w-directions, and the symbol stands for the vector product in {x = x p }, regarded as a copy of C 3. In fact, when x p is also considered as a variable (call it x then), we get that, as in the three-dimensional case, each of Π 1 j, for j = 1,2,3, is a polynomial in x, y, z, w of (total) degree 3D 4. Similarly, the analogously defined polynomials Π 2 j := Π j( f {y=yp }),Π 3 j := Π j( f {z=zp }), for j = 1,2,3, vanish at p if and only if p is a flat point of f {y=yp } and f {z=zp }. By Lemma , we conclude that a non-singular point p Z( f )\Z( f ) axis is flat if and only if Π i j (p) = 0, for 1 i, j 3. We say that a line l Z( f ) is a singular line of Z( f ), if all of its points are singular. We say that a line l Z( f ) is a flat line of Z( f ) if it is not a singular line of Z( f ), and all of its non-singular points are flat. An easy observation is that a flat line can contain at most D 1 singular points of Z( f ) (these are the points on l where all four first-order partial derivatives of f vanish). Similarly, a non-singular line is flat if (and only if) it is incident to at least 3D 3 flat points. The second fundamental form. We use the following notations and results from differential geometry; see Pressley [85] and Ivey and Landsberg [63] for details. For a variety X (in any dimension), the differential dγ of the Gauss mapping γ that maps each point p X to its tangent space T p X, is called the second fundamental form of X. In four dimensions, say, for X = Z( f ), and for any non-singular point p Z( f ), the second fundamental form, locally near p, can be written as (see [63]) a i j du i du j, 1 i, j 3 where x = x(u 1,u 2,u 3 ) is a parametrization of Z( f ), locally near p, and a i j = x ui u j n, where n = n(p) = f (p)/ f (p) is the unit normal to Z( f ) at p. Since the second fundamental form is the differential of the Gauss mapping, it does not depend on the specific local parametrization of f near p. An important property of the second fundamental form is that it vanishes at every non-singular flat point p Z( f ) (see, e.g., Pressley [85] and Ivey and Landsberg [63]). Lemma If a line l Z( f ) is flat, then the tangent space T p Z( f ) is fixed for all the nonsingular points p l Finitely and infinitely ruled surfaces in four dimensions, and u-resultants As already mentioned, in three dimensions the only infinitely ruled surfaces in three are planes (and in fact every triply ruled surface is a plane). The situation is more interesting in four dimensions. Recall again the definition of Ξ p, for a polynomial f and a point p Z( f ), which is the union of all (complex) lines passing through p and fully contained in Z( f ), and that of Σ p, as the set of directions (considered as points in PT p Z( f )) of these lines. Fix a line l Ξ p, and let v = (v 0,v 1,v 2,v 3 ) P 3 represent its direction. Since l Z( f ), the 36

46 four terms F i (p;v) = i v f (p), for i = 1,2,3,4, must vanish at p. These terms, which we denote shortly as F i (v) at the fixed p, are homogeneous polynomials of respective degrees 1,2,3, and 4 in v = (v 0,v 1,v 2,v 3 ). (Note that when D 3, some of these polynomials are identically zero.) The following discussion provides a (partial) algebraic characterization of points p Z( f ) for which Σ p is infinite in four dimensions; that is, points that are incident to infinitely many lines that are fully contained in Z( f ). We refer to this situation by saying that Z( f ) is infinitely ruled at p. To be precise, here we only characterize points that are incident to infinitely many lines that osculate to Z( f ) to order three. The passage from this to the full characterization will be done during the analysis in the Chapter 2. u-resultants. The algebraic tool that we use for this purpose are u-resultants. Specifically, following and specializing Cox et al. [33, Chapter 3.5, page 116], define, for a vector u = (u 0,u 1,u 2,u 3 ) P 3, U(p;u 0,u 1,u 2,u 3 ) = Res 4 ( F 1 (p;v),f 2 (p;v),f 3 (p;v),u 0 v 0 + u 1 v 1 + u 2 v 2 + u 3 v 3 ), where Res 4 ( ) denotes, as earlier, the multipolynomial resultant of the four respective (homogeneous) polynomials, with respect to the variables v 0,v 1,v 2,v 3. For fixed p, this is the so-called u-resultant of F 1 (v),f 2 (v),f 3 (v). Theorem The function U(p;u 0,u 1,u 2,u 3 ) is a homogeneous polynomial of degree 6 in the variables u 0,u 1,u 2,u 3, and is a polynomial of degree O(D) in p = (x,y,z,w). For fixed p Z( f ), U(p;u 0,u 1,u 2,u 3 ) is identically zero as a polynomial in u 0,u 1,u 2,u 3, if and only if there are infinitely many (complex) directions v = (v 0,v 1,v 2,v 3 ), such that the corresponding lines {p +tv t C} osculate to Z( f ) to order three at p. Remark. Theorem shows that the subset of Z( f ) consisting of the points incident to infinitely many lines that osculate to Z( f ) to order three is contained in a subvariety of Z( f ), which is the intersection of Z( f ) with the common zero set of the coefficients of U (considered as polynomials in x, y, z, w). Corollary Fix p Z( f ). The polynomial U(p;u 0,u 1,u 2,u 3 ) is identically zero, as a polynomial in u 0,u 1,u 2,u 3, if and only if there are more than six (complex) lines osculating to Z( f ) to order 3 at p. More to come. We end here the preliminary overview of the algebraic geometry machinery that we use in this thesis. Additional tools, as well as proofs of some of the results reviewed here, will be presented in the relevant subsequent chapters of the thesis. 37

47 38

48 Part I Incidences between points and lines 39

49

50 2 Incidences between points and lines in R 4 41

51 Discrete Comput Geom (2017) 57: DOI /s Incidences Between Points and Lines in R 4 Micha Sharir 1 Noam Solomon 1 Received: 9 January 2016 / Revised: 15 August 2016 / Accepted: 26 August 2016 / Published online: 14 September 2016 Springer Science+Business Media New York 2016 Abstract We show that the number of incidences between m distinct points and n distinct lines in R 4 is O(2 c log m (m 2/5 n 4/5 +m)+m 1/2 n 1/2 q 1/4 +m 2/3 n 1/3 s 1/3 +n), for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor 2 c log m when m n 6/7 or m n 5/3. Except for the factor 2 c log m, the bound is tight in the worst case. Keywords Combinatorial geometry Incidences The polynomial method Algebraic geometry Ruled surfaces 1 Introduction Let P be a set of m distinct points in R 4 and let L be a set of n distinct lines in R 4.Let I (P, L) denote the number of incidences between the points of P and the lines of L; that is, the number of pairs (p,l)with p P,l L, and p l. If all the points of Editor in Charge: János Pach Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. An earlier version of this study appears in Proc. 30th Annu. ACM Sympos. Comput. Geom., 2014, , and the present version is also available in arxiv: v1. Micha Sharir michas@tau.ac.il Noam Solomon noam.solom@gmail.com 1 School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel 123

52 Discrete Comput Geom (2017) 57: P and all the lines of L lie in a common plane, then the classical Szemerédi Trotter theorem [42] yields the worst-case tight bound I (P, L) = O(m 2/3 n 2/3 + m + n). (1) This bound clearly also holds in R 4 (or in any other dimension), by projecting the given lines and points onto some generic plane. Moreover, the bound will continue to be worst-case tight by placing all the points and lines in a common plane, in a configuration that yields the planar lower bound. In the recent groundbreaking paper of Guth and Katz [15], an improved bound has been derived for I (P, L), for a set P of m points and a set L of n lines in R 3, provided that not too many lines of L lie in a common plane. 1 Specifically, they showed: Theorem 1.1 (Guth and Katz [15]) Let P be a set of m distinct points and L a set of n distinct lines in R 3, and let s n be a parameter, such that no plane contains more than s lines of L. Then I (P, L) = O(m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n). (2) This bound is tight in the worst case. In this paper, we establish the following analogous and sharper result in four dimensions. Theorem 1.2 Let P be a set of m distinct points and L a set of n distinct lines in R 4, and let q, s n be parameters, such that (i) each hyperplane or quadric contains at most q lines of L, and (ii) each 2-flat contains at most s lines of L. Then I (P, L) 2 c log m (m 2/5 n 4/5 + m) + A(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n), (3) where A and c are suitable absolute constants. When m n 6/7 or m n 5/3,weget the sharper bound I (P, L) A(m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n). (4) In general, except for the factor 2 c log m, the bound is tight in the worst case, for any values of m, n, and for corresponding suitable ranges of q and s. The proof of Theorem 1.2 will be by induction on m. To facilitate the inductive process, we extend the theorem as follows. We say that a hyperplane or a quadric H in R 4 is q-restricted forasetoflinesl and for an integer parameter q, if there exists a polynomial g H of degree at most O( q), such that each of the lines of L that is contained in H, except for at most q lines, is contained in some irreducible component of H Z(g H ) that is ruled by lines and is not a 2-flat (see below for details). In other 1 The additional requirement in [15], that no regulus contains too many lines, is not needed for the incidence bound given below. 123

53 704 Discrete Comput Geom (2017) 57: words, a q-restricted hyperplane or quadric contains in principle at most q lines of L, but it can also contain an unspecified number of additional lines, all fully contained in ruled (non-planar) components of the zero set of some polynomial of degree O( q). We then have the following more general result. Theorem 1.3 Let P be a set of m distinct points and L a set of n distinct lines in R 4, and let q and s n be parameters, such that (i ) each hyperplane or quadric is q-restricted, and (ii) each 2-flat contains at most s lines of L. Then, I (P, L) 2 c log m (m 2/5 n 4/5 + m) + A(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n), (5) where the parameters A and c are as in Theorem 1.2. As in the preceding theorem, when m n 6/7 or m n 5/3, we get the sharper bound I (P, L) A(m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n). (6) Moreover, except for the factor 2 c log m, the bound is tight in the worst case, as above. The requirement that a hyperplane or quadric H be q-restricted extends (i.e., is a weaker condition than) the simpler requirement that H contains at most q lines of L. Hence, Theorem 1.2 is an immediate corollary of Theorem 1.3. A few remarks are in order. (a) Only the range n m n 2 is of interest; outside this range, regardless of the dimension of the ambient space, we have the well known and trivial upper bound I (P, L) = O(m + n), an immediate consequence of (1). (b) The term m 1/2 n 1/2 q 1/4 comes from the bound of Guth and Katz [15] in three dimensions (as in Theorem 1.1), and is unavoidable, as it can be attained if we densely pack points and lines into hyperplanes, in patterns that realize the bound in three dimensions within each hyperplane; see Sect. 4 for details. (c) Likewise, the term m 2/3 n 1/3 s 1/3 comes from the planar Szemerédi Trotter bound (1), and is too unavoidable, as it can be attained if we densely pack points and lines into 2-planes, in patterns that realize the bound in (1); again, see Sect. 4. (d) Ignoring these terms, and the term n, which is included only to cater for the case m < n, the two terms m 2/5 n 4/5 and m compete for dominance; the former dominates when m = O(n 4/3 ) and the latter when m = (n 4/3 ). Thus the bound in (5) is qualitatively different within these two ranges. (e) The threshold m = n 4/3 also arises in the related problem of joints (points incident to at least four lines not in a common hyperplane) in a set of n lines in 4-space; see [20,29], and a remark below. By a standard argument, the theorem implies the following corollary. Corollary 1.4 Let L be a set of n lines in R 4, satisfying the assumptions (i ) and (ii) in Theorem 1.3, for given parameters q and s. Then, for any k = (2 c log n ),the number m k of points incident to at least k lines of L satisfies ( c log n n 4/3 m k = O k 5/3 + nq1/2 k 2 + ns k 3 + n ). k 123

54 Discrete Comput Geom (2017) 57: Remark (i) It is instructive to compare Corollary 1.4 with the analysis of joints in asetl of n lines. In R d, a joint of L is a point incident to at least d lines of L, not all in a common hyperplane. As shown in [20,29], the maximum number of joints of such a set is O(n d/(d 1) ), and this bound is worst-case tight. In four dimensions, this bound is O(n 4/3 ), which corresponds to the numerator of the first term of the bound in Corollary 1.4. (ii) The other terms cater to configurations involving co-hyperplanar or coplanar lines. For example, when q = n, the second term is O(n 3/2 /k 2 ), in accordance with the bound obtained in Guth and Katz [15] in three dimensions, and when s = n, the third and fourth terms comprise (an equivalent formulation of) the bound (1) of Szemerédi and Trotter [42] for the planar case. (iii) A major interesting and challenging problem is to extend the bound of Corollary 1.4 for any value of k. In particular, is it true that the number of intersection points of the lines (this is the case k = 2) is O(2 4 3 c log n n 4/3 + nq 1/2 + ns)?we conjecture that this is indeed the case. (iv) Another challenging problem is to improve our bound, so as to get rid of, or at least reduce the factor 2 c log m. As stated in the theorems, this can be achieved when m n 6/7 or m n 5/3. Additional remarks and open issues are given in the concluding Sect. 5. Background Incidence problems have been a major topic in combinatorial and computational geometry for the past thirty years, starting with the Szemerédi-Trotter bound [42] back in Several techniques, interesting in their own right, have been developed, or adapted, for the analysis of incidences, including the crossing-lemma technique of Székely [41], and the use of cuttings as a divide-and-conquer mechanism (e.g., see [3]). Connections with range searching and related problems in computational geometry have also been noted, and studies of the Kakeya problem (see, e.g., [43]) indicate the connection between this problem and incidence problems. See Pach and Sharir [27] for a comprehensive (albeit a bit outdated) survey of the topic. The landscape of incidence geometry has dramatically changed in the past seven years, due to the infusion, in two groundbreaking papers by Guth and Katz [14,15] (the first of which was inspired by a similar result of Dvir [6] for finite fields), of new tools and techniques drawn from algebraic geometry. Although their two direct goals have been to obtain a tight upper bound on the number of joints in a set of lines in three dimensions [14], and an almost tight lower bound for the classical distinct distances problem of Erdős [15], the new tools have quickly been recognized as useful for incidence bounds of various sorts. See [10,21,22,38,40,47,48] for a sample of recent works on incidence problems that use the new algebraic machinery. The simplest instances of incidence problems involve points and lines. Szemerédi and Trotter completely solved this special case in the plane [42]. Guth and Katz s second paper [15] provides a worst-case tight bound in three dimensions, under the assumption that no plane contains too many lines; see Theorem 1.1. Under this assumption, the bound in three dimensions is significantly smaller than the planar bound (unless one of m, n is significantly smaller than the other), and the intuition is that this phenomenon should also show up as we move to higher dimensions. Unfortunately, the 123

55 706 Discrete Comput Geom (2017) 57: analysis becomes more involved in higher dimensions, and requires the development or adaptation of progressively more complex tools from algebraic geometry. Most of these tools still appear to be unavailable, and their absence leads either to interesting (new) open problems in the area, or to the need to adapt existing machinery to fit into the new context. The present paper is a first step in this direction, which considers the fourdimensional case. It does indeed derive a sharper, nearly optimal bound, assuming that the configuration of points and lines is truly four-dimensional, in the precise sense spelled out in Theorems 1.2 and 1.3. We also note that studying incidence problems in four (or higher) dimensions has already taken place in several contemporary works, such as in Solymosi and Tao [40], Zahl [48], and Basu and Sombra [1] (and in work in progress by Solymosi and de Zeeuw). These works, though, consider incidences with higher-dimensional varieties, and the study of incidences involving lines, presented in this paper, is new. (There are several ongoing studies, including a companion work joint with Sheffer, that aim to derive weaker but more general bounds involving incidences between points and curves in higher dimensions.) For very recent related studies, see Dvir and Gopi [7] and Hablicsek and Scherr [16]. Our study of point-line incidences in four dimensions has lead us to adapt more advanced tools in algebraic geometry, such as tools involving surfaces that are ruled by lines or by flats, including Severi s 1901 work [34], as well as the more recent works of Landsberg [19,25] on osculating lines and flats to algebraic surfaces in higher dimensions. In a preliminary version of this study [35], we have obtained a weaker and more constrained bound. A discussion of the significant differences between this preliminary work and the present one is given in the overview of the proof, which comes next. Overview of the proof 2 The analysis follows the general approach of Guth and Katz [15], albeit with many significant adaptations and modifications. We use induction on m = P, but we begin the description by ignoring this aspect (for a while). We apply the polynomial partitioning technique of Guth and Katz [15], with some polynomial f R[x, y, z,w] of suitable degree D, and obtain a partition of R 4 into O(D 4 ) cells, each containing at most O(m/D 4 ) points of P. In our first phase, we use D = O(m 2/5 /n 1/5 ), for m = O(n 4/3 ), and D = O(n/m 1/2 ), for m = (n 4/3 ). (7) There are three types of incidences that may arise: an incidence between a point in some cell of the partition and a line crossing that cell, an incidence between a point on the zero set Z( f ) of f and a line not fully contained in Z( f ), and an incidence between a point on Z( f ) and a line fully contained in Z( f ). The above choices of D make it a fairly easy task to bound the number of incidences of the first two types, and 2 In this overview we assume some familiarity of the reader with the new polynomial method of Guth and Katz, and with subsequent applications thereof. Otherwise, the overview can be skipped on first reading. 123

56 Discrete Comput Geom (2017) 57: the hard part is to estimate the number of incidences of the third kind, as we have no control on the number of points and lines contained in Z( f ) in the worst case all the points and lines could be of this kind. At the other end of the spectrum, choosing D to be a constant (as done in our preliminary aforementioned study of this problem [35] and in other recent studies of related problems [13,38,40]) simplifies considerably the handling of incidences on Z( f ), but then the analysis of incidences within the cells of the partition becomes more involved, as the sizes of the subproblems within each cell are too large. In the works just cited (as well as in this paper), this is handled via induction, but the price of a naive inductive approach is three-fold: First, the bound becomes weaker, involving additional factors of the form O(m ε ), for any ε>0 (with a constant of proportionality that depends on ε). Second, the requirement that no hyperplane or quadric contains more than q lines of L has to be replaced by the much more restrictive requirement, that no variety of degree at most c ε contains more than q input lines, where c ε is a (fairly large) constant that depends on ε (and becomes larger as ε gets smaller). Finally, the sharp lower-dimensional terms, such as m 1/2 n 1/2 q 1/4 and m 2/3 n 1/3 s 1/3 in our case (recall that both are worst-case tight), do not pass through the induction successfully, so they have to be replaced by weaker terms; see the preliminary version [35] for such weaker terms, and [38] for a similar phenomenon in a different incidence problem in three dimensions. We note that a recent study by Guth [13] reexamines the point-line incidence problem in R 3 and presents an alternative and simpler analysis (than the original one in [15]), in which he uses a constant-degree partitioning polynomial, and manages to handle successfully the relevant lower-dimensional term m 2/3 n 1/3 s 1/3 through the induction, but the analysis still incurs the extra m ε factors in the bound, and needs the restrictive assumption that no algebraic surface of some large constant maximum degree c ε contains too many lines. In a companion paper [36], we provide yet another simpler derivation (which is somewhat sharper than Guth s) of such an incidence bound in three dimensions. Our approach is to use two different choices of the degree of the partitioning polynomial. We first choose the large value of D specified above, and show that the bound in the right-hand side of (5) accounts for the incidences within the partition cells, for the incidences between points on Z( f ) and lines not fully contained in Z( f ), and for most of the cases involving incidences between points and lines on the zero set Z( f ). We are then left with problematic subsets of points and lines on Z( f ), which are difficult to analyze when the degree is large. (Informally, this happens when the lines lie in certain ruled two-dimensional subvarieties of Z( f ).) To handle them, we retain only these subsets, discard the partitioning, and start afresh with a new partitioning polynomial of a much smaller, albeit still non-constant degree. As the degree is now too small, we need induction to bound the number of incidences within the partition cells. A major feature that makes the induction work well is that the first partitioning step ensures that the surviving set of lines that is passed to the induction is such that each hyperplane or quadric is now O(D 2 )-restricted, with respect to the set of surviving lines, and each 2-flat contains at most O(D) lines of that set (where D is the large degree used in the first partitioning step). As a consequence, the induction works better, and retains the lower-dimensional terms m 1/2 n 1/2 q 1/4 and m 2/3 n 1/3 s 1/3. 123

57 708 Discrete Comput Geom (2017) 57: (In fact, it does not touch them at all, because q and s are not passed to the induction step.) We still pay a small price for this approach, involving the extra factor 2 c log m in the leading terms m 2/5 n 4/5 + m (but not in the lower-dimensional terms); this extra factor is needed to make the induction work, and is a consequence of using a partitioning polynomial of small degree. When m is not too close to n 4/3, as specified in the theorems, induction, and the use of a second partitioning polynomial, are not needed, and a direct analysis yields the sharper bound in (6), without this extra factor. The idea of using a small degree for the partitioning polynomial is not new, and has been applied also in [38,48]. However, the induction process in [38] results in weaker lower-dimensional terms, which we avoid here with the use of two different partitionings. We note that we have recently applied this approach in the aforementioned study of point-line incidences in three dimensions [36], with a simpler analysis (than that in [13,15]) and an improved bound (than the one in [13]). The main (and hard) part of the analysis is still in handling incidences within Z( f ) in the first partitioning step, where the degree of f is large. (Similar issues arise in the second step too, but the bounds there are generally sharper than those obtained in the first step, simply because the degree is smaller.) This is done as follows. We first ignore the singular points on Z( f ). They will be handled separately, as points lying on the zero sets of polynomials of smaller degree (namely, partial derivatives of f ). We also assume that f is irreducible, by considering each irreducible factor of the original f separately (see Sect. 3 for details). This step results in a partition of the points of P and the lines of L among several varieties, each defined by an irreducible factor of f or of some derivative of f, so that it suffices to bound the number of incidences between points and lines assigned to the same variety. The number of cross-variety incidences is shown to be only O(nD), a bound that we are happy to pay. We next define (a four-dimensional variant of) the flecnode polynomial g := FL 4 f of f (see Salmon [32] for the more classical three-dimensional variant, which is used in Guth and Katz [14,15]), which vanishes at those points p Z( f ) that are incident to a line that osculates to Z( f ) (i.e., agrees with Z( f ) near p) up to order four (and in particular to lines that are fully contained in Z( f )); see below for precise definitions. We show that g = FL 4 f is a polynomial of degree O(D). Ifg 0onZ( f ) then Z( f ) is ruled by lines 3 (as follows from Landsberg s work [25], which provides a generalization of the classical Cayley Salmon theorem [15,32]). We handle this case by first reducing it to the case where Z( f ) is infinitely ruled by lines, meaning that most of its points are incident to infinitely many lines that are contained in Z( f ) (otherwise, we can show, using Bézout s theorem, that most points are incident to at most 6 lines, for a total of O(m) incidences), and then by using the aforementioned result of Severi [34] from 1901, which shows that in this case Z( f ) is ruled by 2-flats (each point on Z( f ) is incident to a 2-flat that is fully contained in Z( f )), unless Z( f ) is a hyperplane or a quadric. This allows us to reduce the problem to several planar incidence problems, which are reasonably easier to handle. 3 That is, every point p Z( f ) is incident to a line that is fully contained in Z( f ); see Salmon [8,15,24, 32,37] for definitions. 123

58 Discrete Comput Geom (2017) 57: The other case is where the common zero set Z( f, g) of f and g is two-dimensional. In this case, we decompose Z( f, g) into its irreducible components, and show that the number of incidences between points of P and lines fully contained in irreducible components that are not 2-flats is min{o(md 2 + nd), O(m + nd 4 )}. (8) Both terms are too large for the standard large values of D, but they are non-trivial to establish, and are useful tools for slightly improving the bound and simplifying the analysis considerably when D is not too large see below. The derivation of these bounds is based on a new study of point-line incidences within ruled two-dimensional varieties in 3-space, provided in a companion paper [37]. The irreducible components that are 2-flats are harder to handle, because their number can be O(D 2 ) (as follows from the generalized version of Bézout s theorem [12]), a number that turns out to be too large for the purpose of our incidence bound, when a naive analysis (with a large value of D) is used, so some care is needed in this case. The difficult step in this part is when there are many points, each contained in at least three (and in general many) 2-flats fully contained in Z( f, g) (and thus in Z( f )). Non-singular points of this kind are called linearly flat points of Z( f ), naturally generalizing Guth and Katz s notion of linearly flat points in R 3 [15] (see also Kaplan et al. [10]). Linearly flat points are also flat points, i.e., points where the second fundamental form of Z( f ) vanishes (e.g., see Pressley [28]). Flatness of a point p can be expressed, again by a suitable generalization to four dimensions of the techniques in [10,15], by the vanishing of nine polynomials, each of degree 3D 4, at p, which are constructed from f and from its first and second-order derivatives. The problem can then be reduced to the case where all the points and lines are flat (a line is flat, when not all of its points are singular points of Z( f ), and all of its non-singular points are flat). With a careful (and somewhat intricate) probing into the geometric properties of flat lines, we can bound the number of incidences with flat lines by reducing the problem into several incidence problems in three dimensions (specifically, within hyperplanes tangent to Z( f ) at the flat points), and then using an extension of Guth and Katz s bound (2) for each of these problems, where, in this application, we exploit the fact that each hyperplane contains at most q lines, to obtain a better, q-dependent bound. However, as noted, the terms O(mD 2 ) (when n 6/7 m n 4/3 ) and O(nD 4 ) (when n 4/3 m n 5/3 ) are too large [for the choices of our large values of D in (7)]. We retain and also use them in the second partitioning step, when the degree of the partitioning polynomial is smaller, but finesse them, for the large D, by showing that, after pruning away points and lines whose incidences can be estimated directly [within the bound (6), not using the weaker bounds of (8)], we are left with subsets for which every hyperplane or quadric is O(D 2 )-restricted, and each 2-flat contains at most O(D) lines. However, when m n 6/7 or m n 5/3,thetermsO(mD 2 ), O(nD 4 ) are not too large, and there is no need for this part of the analysis, and a direct application of the bounds in (8) yields the sharper bound in (6) and simplifies the proof considerably. For the remaining range of m and n, we go on to our second partitioning step. We discard f and start afresh with a new partitioning polynomial h of degree E D. 123

59 710 Discrete Comput Geom (2017) 57: As already noted, bounding incidences within the partition cells becomes non-trivial, and we use induction, exploiting the fact that now the parameters q and s are replaced by O(D 2 ) and O(D), respectively. On the flip side of the coin, bounding incidences within Z(h) is now simpler, because E is smaller, and we can use the bounds in (8) (i.e., O(mE 2 +ne) or O(m+nE 4 )) to establish the bound in (5) for the problematic incidences. The reason for using the weaker requirement that each hyperplane and quadric be q-restricted, instead of just requiring that no hyperplane or quadric contain more than q lines of L, is that we do not know how to bound the overall number of lines in a hyperplane or quadric H by O(D 2 ), because of the potential existence of ruled components of Z( f, g) within H, which can accommodate any number of lines. A major difference between this case and the analysis of ruled components in Guth and Katz s study [15] is that here the overall degree of Z( f, g) is O(D 2 ), as opposed to the degree of Z( f ) being only D in [15]. This precludes the application of the techniques of Guth and Katz to our scenario they would lead to bounds that are too large. We also note that our analysis of incidences within Z( f ) is actually carried out (in the projective 4-space) over the complex field, which makes it simpler, and facilitates the application of numerous tools from algebraic geometry that are developed in this setting. The passage from the complex projective setup back to the real affine one is straightforward the former is a generalization of the latter. The real affine setup is needed only for the construction of a polynomial partitioning, which is meaningless over C. Once we are within the variety Z( f ), we can switch to the complex projective setup, and reap the benefits noted above. Note that, in spite of these improvements, Theorem 1.3 still has the peculiar feature, which is not needed in Guth and Katz [15] (for the incidence bound of Theorem 1.1), that also requires that every quadric be q-restricted (or, in the simpler version in Theorem 1.2, contains at most q lines of L). 4 In a recent work in progress, Solomon and Zhang [39] show that this requirement cannot be dropped, by providing a construction of a quadric that contains many points and lines, where the number of incidences between them is significantly larger than the bound in (5) (where q now only bounds the number of lines in a hyperplane). 2 Algebraic Preliminaries In this section we collect and adapt a large part of the machinery from algebraic geometry that we need for our analysis. Some supplementary machinery is developed within the analysis iself. In what follows, to facilitate the application of standard techniques in algebraic geometry, it will be more convenient to work over the complex field C, and in complex projective spaces. We do so even though Theorem 1.3 is stated (and will be proved) only for the real affine case. The passage between the two scenarios, in the proof of the 4 This is not quite the case: Guth and Katz also require that no regulus contains more than s (actually, n) lines, but this is made to bound the number of points incident to just two lines, and is not needed for the incidence bound in Theorem

60 Discrete Comput Geom (2017) 57: theorem, will be straightforward, as discussed in the preceding overview. Concretely, the realness of the underlying field is needed only for the partitioning step itself, which has no (simple) parallel over C. However, after reducing the problem to points and lines contained in Z( f ), it is more convenient to carry out the analysis over C, to allow us to apply the algebraic machinery that we are going to present next. 2.1 Lines on Varieties We begin with several basic notions and results in differential and algebraic geometry that we will need (see, e.g., Ivey and Landsberg [19], and Landsberg [25] formore details). For a vector space V (over R or C), let PV denote its projectivization. That is, PV = V \{0}/, where v w iff w = αv for some non-zero constant α. An algebraic variety is the common zero set of a finite collection of polynomials. We call it affine, if it is defined in the affine space, or projective, if it is defined in the projective space, in terms of homogeneous polynomials. For an (affine) algebraic variety X, and a point p X, lett p X denote the (affine) tangent space of X at the point p. A point p is non-singular if dim T p X = dim X (see Hartshorne [18, Def. I.5 and Thm. I.5.1]). For a point p X, let p denote the set of the complex lines passing through p and contained in X, and let p denote the union of these lines (here X is implicit in these notations). For p fixed, the lines in p can be represented by their directions, as points in PT p X. In Hartshorne [18, Ex. I.2.10], p is also called the (affine) cone over p. Clearly, p T p X. Consider the special case where X is a hypersurface in C 4, i.e., X = Z( f ), fora non-linear polynomial f C[x, y, z,w], which we assume to be irreducible, where Z( f ) ={p C 4 f (p) = 0} is the zero set of Z( f ). A line l v ={p + tv t C} passing through p in direction v is said toosculate to Z( f ) to order k at p, if the Taylor expansion of f around p in direction v vanishes to order k, i.e., if f ( p) = 0, v f (p) = 0, 2 v f (p) = 0,..., k v f (p) = 0, (9) where v f (which for uniformity we also denote as v 1 f ), 2 v f,..., k v f are, respectively, the first, second, and higher order derivatives of f, uptoorderk, in direction v (where v is regarded as a vector in projective 3-space, and the derivatives are interpreted in a scale-invariant manner we only care whether they vanish or not). That is, v f = f v, v 2 f = vt H f v, where H f is the Hessian matrix of f, and v i f is similarly defined, for i > 2, albeit with more complicated explicit expressions. For simplicity of notation, put F i (p; v) := v i f (p), fori 1. In fact, one can extend the definition of osculation of lines to arbitrary varieties in any dimension (see, e.g., Ivey and Landsberg [19]). For a variety X, a point p X, and an integer k 1, let k p PT p X denote the variety of the lines that pass through p and osculate to X to order k at p; as before, we represent the lines in k p,forp fixed, by their directions, as points in the corresponding projective space. For each k N, 123

61 712 Discrete Comput Geom (2017) 57: there is a natural inclusion p k p. In analogy with the previous notation, we denote by k p the union of the lines that pass through p with directions in k p.weletf(x) denote the variety of lines (fully) contained in X; this is known as the Fano variety of X, and it is a subvariety of the (2d 2)-dimensional Grassmannian manifold of lines in P d (C); see Harris [17, Lect. 6, p. 63] for details, and [17, Ex. 6.19] for an illustration, and for a proof that this is indeed a variety. We will sometimes denote F(X) also as (or (X)), to conform with the notation involving osculating lines. We also let k denote the variety of the lines osculating to order k at some point of X, and can be thought of as the union of the k p over p X. When representing lines in or k we can no longer use the local representation by directions, and instead represent them, in the customary manner, as points within the Grassmanian manifold. Here too k can be shown to be a variety (within the Grassmannian manifold) and F(X) k for each k.wealsohave,foranyp Z, p F(X) and k p k. Genericity We recall that a property is said to hold generically (or generally) for polynomials f 1,..., f n, of some prescribed degrees, if there are nonzero polynomials g 1,...,g k in the coefficients of the f i s, such that the property holds for all f 1,..., f n for which none of the polynomials g j is zero (see, e.g., Cox et al. [4, Def. 3.6]). In this case we say that the collection f 1,..., f n is general or generic, with respect to the property in question, namely, with respect to the vanishing of the polynomials g 1,...,g k that define that property. 2.2 Generalized Bézout s Theorem An affine (resp. projective) variety X C d (resp. X P d (C)) is called irreducible if, whenever V is written in the form V = V 1 V 2, where V 1 and V 2 are affine (resp., projective) varieties, then either V 1 = V or V 2 = V. Theorem 2.1 (Cox et al. [5, Thms , 8.3.6]) Let V be an affine (resp., projective) variety. Then V can be written as a finite union V = V 1 V m, where V i is an irreducible affine (resp., projective) variety, for i = 1,...,m. If one also requires that V i V j for i = j, then this decomposition is unique, up to a permutation (see, e.g., [5, Thms , 8.3.6]), and is called the minimal decomposition of V into irreducible components. We next state a generalized version of Bézout s theorem, as given in Fulton [12]. It will be a major technical tool in our analysis. Theorem 2.2 (Fulton [12, Prop. 2.3]) Let V 1,...,V s be subvarieties of P d, and let Z 1,...,Z r be the irreducible components of s i=1 V i. Then r deg(z i ) i=1 s deg(v j ). j=1 123

62 Discrete Comput Geom (2017) 57: A simple application of Theorem 2.2 yields the following useful result. Lemma 2.3 A curve C P 4 of degree D can contain at most D lines. Proof Let t denote the number of these lines, and let C 0 C denote their union. Intersect C 0 with a generic hyperplane H. By Theorem 2.2, the number of intersection points satisfies t deg(c 0 ) deg(h) deg(c) 1 = D, as asserted. This immediately yields the following result, derived in Guth and Katz [14] (see also [10]) in a somewhat different manner. Corollary 2.4 Let f and g be two trivariate polynomials without a common factor. Then Z( f, g) := Z( f ) Z(g) contains at most deg( f ) deg(g) lines. Proof This follows since Z( f, g) is a curve of degree at most deg( f ) deg(g). 2.3 Generically Finite Morphisms and the Theorem of the Fibers The following results can be found, e.g., in Harris [17, Chap. 11]. For a map π : X Y of projective varieties, and for y Y, the variety π 1 (y) is called the fiber of π over y. The following result is a slight paraphrasing of Harris [17, Prop. 7.16] and also appears in Sharir and Solomon [37, Thm.7] Theorem 2.5 (Harris [17, Prop. 7.16]) Let f : X Y be the map induced by the standard projection map π : P d P r (which retains r of the coordinates and discards the rest), where r < d, X P d and Y P r are projective varieties, X is irreducible, and Y is the image of X. Then the general fiber 5 of the map f is finite if and only if dim(x) = dim(y ). In this case, the number of points in a general fiber of f is constant. An important technical tool for our analysis is the following so-called Theorem of the Fibers. Theorem 2.6 (Harris [17, Cor ]) Let X be a projective variety and π : X P d be a polynomial map (i.e., the coordinate functions x 0 π,..., x d π are homogeneous polynomials); let Y = π(x) denote its image. For any p Y, let λ(p) = dim(π 1 (p)). Then λ(p) is an upper semi-continuous function of p in the Zariski topology 6 on Y ; that is, for any m, the locus of points p Y such that λ(p) m 5 The meaning of this statement is that the assertion holds for the fiber at any point outside some lowerdimensional exceptional subvariety. 6 The Zariski closure of a set Y is the intersection of all varieties X that contain Y. Y is Zariski closed if it is equal to its closure (and is therefore a variety), and is Zariski open if its complement is Zariski closed. See [18] for further details. 123

63 714 Discrete Comput Geom (2017) 57: is closed in Y. Moreover, if X 0 X is any irreducible component, Y 0 = π(x 0 ) its image, and λ 0 the minimum value of λ(p) on Y 0, then dim(x 0 ) = dim(y 0 ) + λ Flecnode Polynomials and Ruled Surfaces in Four Dimensions Ruled surfaces in three dimensions We first review several basic properties of ruled two-dimensional surfaces in R 3 or in C 3. Most of these results are considered folklore in the literature, although we have been unable to find concrete rigorous proofs (in the modern jargon of algebraic geometry). For the sake of completeness we provide such proofs in a companion paper [37]. For a modern approach to ruled surfaces, there are many references; see, e.g., Hartshorne [18, Sect. V.2], or Beauville [2, Chapter III]. We say that a real (resp., complex) surface X is ruled by real (resp., complex) lines if every point p X in a Zariski-open dense set is incident to a real (resp., complex) line that is fully contained in X; see, e.g., [32] or[8] for further details on ruled surfaces. This definition is slightly weaker than the classical definition, where it is required that every point of X be incident to a line contained in X (e.g., as in [32]). It has been used in recent works, see, e.g., [15,24]. Similarly to the proof of Lemma 3.4 in Guth and Katz [15], a limiting argument implies that the two definitions are equivalent. We spell out the details in Lemma 6.1 in the appendix (see also Sharir and Solomon [37, Lem. 11]). We note that some care has to be exercised when dealing with ruled surfaces, because ruledness may depend on the underlying field. Specifically, it is possible for a surface defined by real polynomials to be ruled by complex lines, but not by real lines. For example, the sphere defined by x 2 + y 2 + z 2 1 = 0, regarded as a real variety, is certainly not ruled by lines, but as a complex variety it is ruled by (complex) lines. (Indeed, each point (x 0, y 0, z 0 ) on the sphere is incident to the (complex) line (x 0 +αt, y 0 +βt, z 0 +γ t),fort C, where α 2 +β 2 +γ 2 = 0 and αx 0 +βy 0 +γ z 0 = 0, which is fully contained in the sphere.) In three dimensions, a two-dimensional irreducible ruled surface can be either singly ruled,or doubly ruled (notions that are elaborated below), or a plane. As the following lemma shows, the only doubly ruled surfaces are reguli, where a regulus is the union of all lines that meet three pairwise skew lines. There are only two kinds of reguli, both of which are quadrics hyperbolic paraboloids and hyperboloids of one sheet; see, e.g., Fuchs and Tabachnikov [11] for more details. The following (folklore) lemma provides a (somewhat stronger than usual) characterization of doubly ruled surfaces; see [37] for a proof. Lemma 2.7 Let V be an irreducible ruled surface in R 3 or in C 3 which is not a plane. If there exists an algebraic curve C V, such that every non-singular point p V \C is incident to exactly two lines that are fully contained in V, then V is a regulus. When V is an irreducible ruled surface which is neither a plane nor a regulus, it must be singly ruled, in the precise sense spelled out in the following theorem (see also [15]); again, see [37, Thm. 10] for a proof. 123

64 Discrete Comput Geom (2017) 57: Theorem 2.8 (a) Let V be an irreducible ruled two-dimensional surface of degree D > 1 in R 3 (or in C 3 ), which is not a regulus. Then, except for at most two exceptional lines, the lines that are fully contained in V are parametrized by an irreducible algebraic curve 0 in the Plücker space P 5, and thus yield a 1-parameter family of generator lines l(t), fort 0, that depend continuously on the real or complex parameter t. Moreover, if t 1 = t 2, and l(t 1 ) = l(t 2 ), then there exist sufficiently small and disjoint neighborhoods 1 of t 1 and 2 of t 2, such that all the lines l(t), for t 1 2, are distinct. (b) There exists a one-dimensional curve C V, such that any point p in V \ C is incident to exactly one generator line of V. Following this theorem, we refer to irreducible ruled surfaces that are neither planes nor reguli as singly ruled. A line l, fully contained in an irreducible singly ruled surface V, such that every point of l is incident to another line fully contained in V, is called an exceptional line of V (these are the lines mentioned in Theorem 2.8(a)). If there exists a point p V V, which is incident to infinitely many lines fully contained in V, then p V is called an exceptional point of V. By Guth and Katz [15], V can contain at most one exceptional point p V (in which case V is a cone with p V as its apex), and (as also asserted in the theorem) at most two exceptional lines. The flecnode polynomial in four dimensions Let f C[x, y, z,w] be a polynomial of degree D 4. A flecnode of f is a point p Z( f ) for which there exists a line that passes through p and osculates to Z( f ) to order four at p. Therefore, if the direction of the line is v = (v 0,v 1,v 2,v 3 ), then it osculates to Z( f ) to order four at p if f (p) = 0 and F i (p; v) = 0, for i = 1, 2, 3, 4. (10) The four-dimensional flecnode polynomial of f, denoted FL 4 f, is the polynomial obtained by eliminating v from the four equations in the system (10). (See Salmon [32], and the relevant applications thereof in [10,15], for details concerning flecnode polynomials in three dimensions; see also Ivey and Landsberg [19] for a more modern generalization of this concept.) Note that these four polynomials are homogeneous in v (of respective degrees 1, 2, 3, and 4). We thus have a system of four equations in eight variables, which is homogeneous in the four variables v 0,v 1,v 2,v 3. Eliminating those variables results in a single polynomial equation in p = (x, y, z,w). Using standard techniques, as in Cox et al. [4], the resulting polynomial FL 4 f is the multipolynomial resultant Res 4 (F 1, F 2, F 3, F 4 ) of F 1, F 2, F 3, F 4, regarding these as polynomials in v (where the coefficients are polynomials in p). By definition, FL 4 f vanishes at all the flecnodes of f. The following results are immediate consequences of the theory of multipolynomial resultants, presented in Cox et al. [4]. Lemma 2.9 Given a polynomial f C[x, y, z,w] of degree D 4, its flecnode polynomial FL 4 f has degree O(D). Proof The polynomial F i,fori = 1,...,4, is a homogeneous polynomial in v of degree d i = i over C[x, y, z,w]. By[4, Thm. 4.9], putting d := ( 4 i=1 d i ) 3 = 7, 123

65 716 Discrete Comput Geom (2017) 57: the multipolynomial resultant FL 4 f = Res 4 (F 1, F 2, F 3, F 4 ) is equal to D 3 D 3, where D 3 is a polynomial of degree ( d+3) ( 3 = 10 ) 3 = 120 in the coefficients of the polynomials Fi, and D 3 is a polynomial of degree d 1d 2 d 3 + d 1 d 2 d 4 + d 1 d 3 d 4 + d 2 d 3 d 4 = = 50 in these coefficients (see Cox et al. [4, Chap. 3.4, exercises 1,3,6,12,19]). Since each coefficient of any of the polynomials F i is of degree at most D 1, we deduce that FL 4 f is of degree at most O(D). Lemma 2.10 Given a polynomial f C[x, y, z,w] of degree D 4, every line that is fully contained in Z( f ) is also fully contained in Z(FL 4 f ). Proof Every point on any such line is a flecnode of f,sofl 4 f vanishes identically on the line. Ruled Surfaces in four dimensions Flecnode polynomials are a major tool for characterizing ruled surfaces. This is manifested in the following theorem of Landsberg [25], which is a crucial tool for our analysis. It is established in [25] as a considerably more general result, but we formulate here a special instance that suffices for our needs. Theorem 2.11 (Landsberg [25]) Let f C[x, y, z,w] be a polynomial of degree D 4. Then Z( f ) is ruled by (complex) lines if and only if Z( f ) Z(FL 4 f ). We note that Theorem 2.11 extends the classical Cayley Salmon theorem in three dimensions (see Salmon [32]). A quick review of this result is given below. We also note that we will use a refined version of this theorem, also due to Landsberg, given as Theorem 3.8 in Sect. 3. When f is of degree 3, we have the following simpler situation. Lemma 2.12 For every polynomial f C[x, y, z,w] of degree 3, Z( f ) is ruled by (possibly complex) lines. Proof Let v = (v 0,v 1,v 2,v 3 ) C 4 be a direction. First notice that for a point p C 4, the line through p in direction v is contained in Z( f ) if and only if the first three equations in (10) are satisfied, because all the other terms in the Taylor expansion of f (p + tv) always vanish for a polynomial f of degree 3. This is a system of three homogeneous polynomials in v 0,v 1,v 2,v 3, of degrees 1, 2, 3, respectively. By Bézout s theorem, as stated in Theorem 2.2 below, the number of solutions (complex projective, counted with multiplicities) of this system is either six or infinite, so there is at least one (possibly complex) line that passes through p and is contained in Z( f ). Back to three dimensions In three dimensions the analysis is somewhat simpler, and goes back to the 19th century, in Salmon s work [32] ond others. The flecnode polynomial FL f of f, defined in an analogous manner, is of degree 11 deg( f ) 24 [32]. Theorem 2.11 is replaced by the Cayley Salmon theorem [32], with the analogous assertion that Z( f ) is ruled by lines if and only if Z( f ) Z(FL f ). A simple proof of the Cayley Salmon theorem can be found in Terry Tao s blog [44]. We will be using the following result, established by Guth and Katz [14]; see also [10]. 123

66 Discrete Comput Geom (2017) 57: Proposition 2.13 Let f be a trivariate irreducible polynomial of degree D. If Z( f ) fully contains more than 11D 2 24D lines then Z( f ) is ruled by (possibly complex) lines. Proof Apply Corollary 2.4 to FL f and f, to conclude that FL f and f must have a common factor. Since f is irreducible, this factor must be f itself, and then the Cayley Salmon theorem implies that Z( f ) is ruled. 2.5 Flat Points and the Second Fundamental Form We continue with the four-dimensional setup. Extending the notation in Guth and Katz [14] (see also [10], and also Pressley [28] and Ivey and Landsberg [19] for more basic references), we call a non-singular point p of Z( f ) linearly flat, if it is incident to at least three distinct 2-flats that are fully contained in Z( f ) (and thus also in the tangent hyperplane T p Z( f )). (The original definition, in [10,15], for the three-dimensional case, is that a non-singular point p Z( f ) is linearly flat if it is incident to three distinct lines that are fully contained in Z( f )) The condition for a point p to be linearly flat can be worked out as follows, suitably extending the technique used in three dimensions in [10,14]. Although this extension is fairly routine, we are not aware of any previous concrete reference, so we spell out the details for the sake of completeness. Let p be a non-singular point of Z( f ), and let f (2) denote the second-order Taylor expansion of f at p. That is, we have, for any direction vector v and t C, f (2) (p + tv) = t f (p) v t2 v T H f (p)v. (11) If p is linearly flat, there exist three 2-flats π 1,π 2,π 3, contained in the tangent hyperplane T p Z( f ), such that v T H f (p)v = 0, for all v π 1,π 2,π 3 (clearly, the first term f (p) v also vanishes for any such v). Using a suitable coordinate frame within T p Z( f ), we can regard v T H f (p)v as a quadratic trivariate homogeneous polynomial. Since v T H f (p)v vanishes on three 2-flats inside T p Z( f ), a (generic) line l, fully contained in T p Z( f ) and not passing through p, intersects these 2-flats at three distinct points, at which v T H f v vanishes. Since this is a quadratic polynomial, it must vanish identically on l. Thus, v T H f v is zero for all vectors v T p Z( f ), and thus f (2) vanishes identically on T p Z( f ). In this case, we say that p is a flat point of Z( f ). Therefore, every linearly flat point of Z( f ) is also a flat point of Z( f ) (albeit not necessarily vice versa). 7 The same definition applies in three dimensions too. We next express the set of linearly flat points of Z( f ) as the zero set of a certain collection of polynomials. To do so, we define three canonical 2-flats, on which we test the vanishing of the quadratic form v T H f v. (The preceding analysis shows that, for a linearly flat point, it does not matter which triple of 2-flats is used for testing the linear flatness, as long as they are distinct.) These will be the 2-flats 7 For example, for the surface in R 3 defined by the zero set of f = x + y + z + x 3, the point 0 = (0, 0, 0) Z( f ) is flat (because the second order Taylor expansion of f near 0 is the plane x + y + z = 0), but is not linearly flat, since there is no line incident to 0 and contained in Z( f ). 123

67 718 Discrete Comput Geom (2017) 57: πp x :=T p Z( f ) {x = x p }, πp y := T p Z( f ) {y = y p }, and πp z := T p Z( f ) {z = z p }. (12) These are indeed distinct 2-flats, unless T p Z( f ) is orthogonal to the x-, y-, or z-axis. Denote by Z( f ) axis the subset of non-singular points p Z( f ),forwhicht p Z( f ) is orthogonal to one of these axes, and assume in what follows that p Z( f )\ Z( f ) axis. We can ignore points in Z( f ) axis by assuming that the coordinate frame of the ambient space is generic, to ensure that none of our (finitely many) input points has a tangent hyperplane that is orthogonal to any of the axes. Lemma 2.14 Let p be a non-singular point of Z( f ) \ Z( f ) axis. Then p is a flat point of Z( f ) if and only if p is a flat point of each of the varieties Z( f {x=x p }), Z( f {y=yp }), Z( f {z=z p }). Proof Note that the three varieties in the lemma are two-dimensional varieties within the corresponding three-dimensional cross-sections x = x p, y = y p, and z = z p,of 4-space. If p is a flat point of Z( f ) \ Z( f ) axis, then the second-order Taylor expansion f (2) vanishes identically on T p Z( f ). Bytheassumptiononp, wehave T p Z( f {x=x p }) = T p Z( f ) {x = x p }, T p Z( f {y=yp }) = T p Z( f ) {y = y p }, and T p Z( f {z=z p }) = T p Z( f ) {z = z p }, and these are three distinct 2-flats. Therefore, f (2) {x=x p } vanishes identically on T p Z( f {x=x p }), implying that p is a flat point of Z( f {x=x p }); similarly p is a flat point of Z( f {y=yp }) and of Z( f {z=z p }). For the other direction, notice that if p satisfies the assumptions in the lemma, and is a flat point of each of Z( f {x=x p }), Z( f {y=yp }), and Z( f {z=z p }), then f (2) vanishes on three distinct 2-flats contained in T p Z( f ) (namely, the intersection of T p Z( f ) with {x = x p }, {y = y p } and {z = z p }), which are distinct since p / Z( f ) axis. Since f (2) is quadratic, the argument given above implies that it is identically 0 on T p Z( f ). Recall from Elekes et al. [10] that p is flat for f {x=x p } if and only if 1 j := j ( f {x=x p }) vanishes at p, for j = 1, 2, 3, where j (h) = ( h e j ) T H h ( h e j ), and where e 1, e 2, e 3 denote the unit vectors in the respective y-, z-, and w-directions, and the symbol stands for the vector product in {x = x p }, regarded as a copy of C 3. In fact, when x p is also considered as a variable (call it x then), we get that, as in the three-dimensional case, each of 1 j,for j = 1, 2, 3, is a polynomial in x, y, z,wof (total) degree 3D 4. Similarly, the analogously defined polynomials 2 j := j ( f {y=yp }), 3 j := j ( f {z=z p }), for j = 1, 2, 3, vanish at p if and only if p is a flat point of f {y=yp } and f {z=z p }. By Lemma 2.14, we conclude that a non-singular point p Z( f ) \ Z( f ) axis is flat if and only if i j (p) = 0, for 1 i, j 3. We say that a line l Z( f ) is a singular line of Z( f ), if all of its points are singular. We say that a line l Z( f ) is a flat line of Z( f ) if it is not a singular line of 123

68 Discrete Comput Geom (2017) 57: Z( f ), and all of its non-singular points are flat. An easy observation is that a flat line can contain at most D 1 singular points of Z( f ) (these are the points on l where all four first-order partial derivatives of f vanish). Similarly, a non-singular line is flat if (and only if) it is incident to at least 3D 3 flat points. The second fundamental form We use the following notations and results from differential geometry; see Pressley [28] and Ivey and Landsberg [19] for details. For a variety X, the differential dγ of the Gauss mapping γ that maps each point p X to its tangent space T p X, is called the second fundamental form of X. In four dimensions, for X = Z( f ), and for any non-singular point p Z( f ), the second fundamental form, locally near p, can be written as (see [19]) a ij du i du j, 1 i, j 3 where x = x(u 1, u 2, u 3 ) is a parametrization of Z( f ), locally near p, and a ij = x ui u j n, where n = n(p) = f (p)/ f (p) is the unit normal to Z( f ) at p. Since the second fundamental form is the differential of the Gauss mapping, it does not depend on the specific local parametrization of f near p. An important property of the second fundamental form is that it vanishes at every non-singular flat point p Z( f ) (see, e.g., Pressley [28] and Ivey and Landsberg [19]). Lemma 2.15 If a line l Z( f ) is flat, then the tangent space T p Z( f ) is fixed for all the non-singular points p l. Proof The proof applies a fairly standard argument in differential geometry (see, e.g., Pressley [28]); see also a proof of a similar claim for the three-dimensional case in [10, Appendix]. Fix a non-singular point p l, and assume that x = x(u 1, u 2, u 3 ) is a parametrization of Z( f ), locally near p. We assume, as we may, that the relevant neighborhood N p of p consists only of non-singular points. For any point (a, b, c) in the corresponding parameter domain, x u1, x u2, x u3 span the tangent space to Z( f ) at x(a, b, c). Indeed, since x(u 1, u 2, u 3 ) is a local parametrization, its differential (dx) (a,b,c) : T (a,b,c) C 3 T x(a,b,c) Z( f ) is an isomorphism. Hence, the image of this latter map is spanned by x u1, x u2, x u3 at x(a, b, c). In particular, we have x ui n = 0, i = 1, 2, 3, over N p. We now differentiate these equations with respect to u j,for j = 1, 2, 3, and obtain x ui u j n + x ui n u j 0 onl N p, for 1 i, j 3. The first term vanishes because l is flat, so, as noted above, the second fundamental form vanishes at each non-singular point of l. We therefore have x ui n u j 0 onl N p, for i, j = 1, 2,

69 720 Discrete Comput Geom (2017) 57: Since x u1, x u2, x u3 span the tangent space T q Z( f ), for each q N p,itfollowsthat n u j (q) is orthogonal to T q Z( f ) for each q l N p, and thus must be parallel to n(q) in this neighborhood. However, since n is of unit length, we have n n 1, and differentiating this equation yields n u j n 0 on l N p, for j = 1, 2, 3. Since n u j (q) is both parallel and orthogonal to n(q), it must be identically zero on l N p,for j = 1, 2, 3. Write l = p + tv, t C, and define h(t) := n(p + tv), fort C. Then, in a suitable tensor notation, h (t) = (n u1 (p + tv), n u2 (p + tv), n u3 (p + tv)) v 0, locally near t = 0. Thus, n(p + tv) is constant locally near t = 0, implying that n is constant along l, locally near p. It still remains to show that n is constant on the set of all the non-singular points of Z( f ) contained in l. Set Z s (l) := {t C p + tv is a singular point of Z( f )}. As l is not singular, Z s (l) D 1 (as already observed). The map t n(p + tv) is constant in a neighborhood of every point t of Z ns (l) := C \ Z s (l). Since Z ns (l) is a connected set, 8 n has a fixed value at all the non-singular points on l, as asserted. Since the tangent hyperplanes T p Z( f ) along l all contain the line l itself, and all have the same normal, we deduce that T p Z( f ) is fixed for all non-singular points p l. 2.6 Finitely and Infinitely Ruled Surfaces in Four Dimensions, and u-resultants Recall again the definition of p, for a polynomial f C[x, y, z,w] and a point p Z( f ), which is the union of all (complex) lines passing through p and fully contained in Z( f ), and that of p, as the set of directions (considered as points in PT p Z( f )) of these lines. Fix a line l p, and let v = (v 0,v 1,v 2,v 3 ) P 3 represent its direction. Since l Z( f ), the four terms F i (p; v) = v i f (p), fori = 1, 2, 3, 4, must vanish at p. These terms, which we denote shortly as F i (v) at the fixed p, are homogeneous polynomials of respective degrees 1, 2, 3, and 4 in v = (v 0,v 1,v 2,v 3 ). (Note that when D 3, some of these polynomials are identically zero.) In this subsection we provide a (partial) algebraic characterization of points p Z( f ) for which p is infinite; that is, points that are incident to infinitely many lines that are fully contained in Z( f ). We refer to this situation by saying that Z( f ) is infinitely ruled at p. To be precise, here we only characterize points that are 8 This property holds for C but not for R. 123

70 Discrete Comput Geom (2017) 57: incident to infinitely many lines that osculate to Z( f ) to order three. The passage from this to the full characterization will be done during the analysis in the next section. u-resultants The algebraic tool that we use for this purpose are u-resultants. Specifically, following and specializing Cox et al. [4, Chap. 3.5, p. 116], define, for a vector u = (u 0, u 1, u 2, u 3 ) P 3, U(p; u 0, u 1, u 2, u 3 ) = Res 4 (F 1 (p; v), F 2 (p; v), F 3 (p; v), u 0 v 0 + u 1 v 1 + u 2 v 2 + u 3 v 3 ), where Res 4 ( ) denotes, as earlier, the multipolynomial resultant of the four respective (homogeneous) polynomials, with respect to the variables v 0,v 1,v 2,v 3. For fixed p, this is the so-called u-resultant of F 1 (v), F 2 (v), F 3 (v). Theorem 2.16 The function U(p; u 0, u 1, u 2, u 3 ) is a homogeneous polynomial of degree six in the variables u 0, u 1, u 2, u 3, and is a polynomial of degree O(D) in p = (x, y, z,w). For fixed p Z( f ), U(p; u 0, u 1, u 2, u 3 ) is identically zero as a polynomial in u 0, u 1, u 2, u 3, if and only if there are infinitely many (complex) directions v = (v 0,v 1,v 2,v 3 ), such that the corresponding lines {p + tv t C} osculate to Z( f ) to order three at p. Proof By definition, the osculation property in the theorem, for given p and v, is equivalent to F 1 (p; v) = F 2 (p; v) = F 3 (p; v) = 0. Regarding F 1, F 2, F 3 as homogeneous polynomials in v, the degree of U in u 0, u 1, u 2, u 3 is deg(f 1 ) deg(f 2 ) deg(f 3 ) = 3!=6 (see Cox et al. [4, Exe b]). Put d = deg(f 1 ) + deg(f 2 ) + deg(f 3 ) + 1 = 7. Then the total degree of U in the coefficients of F i, each being a polynomial in p of degree at most D, isatmost ( d ( 3) = 7 3) = 35 (see also the proof of Lemma 2.9 and Cox et al. [4, Exes c, ]), and thus the degree of U as a polynomial in p is O(D). Put H(u,v) = u 0 v 0 + u 1 v 1 + u 2 v 2 + u 3 v 3, and, for any v C 4, denote by H v the hyperplane H(u,v) = 0. Fix p Z( f ), and regard F 1, F 2, F 3, H(u, ) as polynomials in v. If the osculation property holds at p (for infinitely many lines) then Z(F 1, F 2, F 3 ) is infinite, so it is at least 1-dimensional. Thus, for any u = (u 0, u 1, u 2, u 3 ) C 4, the variety Z(F 1, F 2, F 3, H(u,v)) is non-empty, so the multipolynomial resultant of these four polynomials (in v) vanishes at u. Since this holds for all u C 4, It follows from Cox et al. [4, Prop ] that U 0. Suppose then that the osculation property does not hold (for infinitely many lines) at p, soz(f 1, F 2, F 3 ) is finite. Pick any u 0 / v Z(F 1,F 2,F 3 ) H v. Then, for every v Z(F 1, F 2, F 3 ),wehaveh(u 0,v) = 0, implying that Z(F 1, F 2, F 3, H(u 0, )) ={v Z(F 1, F 2, F 3 ) H(u 0,v)= 0} =. Therefore, by the properties of multipolynomial resultants, U(u 0 ) = 0, and U is not identically zero. Remark Theorem 2.16 shows that the subset of Z( f ) consisting of the points incident to infinitely many lines that osculate to Z( f ) to order three is contained in a subvariety 123

71 722 Discrete Comput Geom (2017) 57: of Z( f ), which is the intersection of Z( f ) with the common zero set of the coefficients of U (considered as polynomials in x, y, z,w). Corollary 2.17 Fix p Z( f ). The polynomial U(p; u 0, u 1, u 2, u 3 ) is identically zero, as a polynomial in u 0, u 1, u 2, u 3, if and only if there are more than six (complex) lines osculating to Z( f ) to order 3 at p. Proof The polynomial F i is either 0 or of degree i (in v, for a fixed value of p), for i = 1, 2, 3. By Theorem 2.2, the number of their common zeros v = (v 0,v 1,v 2,v 3 ) is either six (counting complex projective solutions with multiplicity; see also the proof of Theorem 2.16) or infinite. The result then follows from Theorem Proof of Theorem 1.3 Let P, L, m, n, q, and s be as in the theorem. The proof proceeds by induction on m, where we establish the inequality I (P, L) 2 c log m (m 2/5 n 4/5 + m)+ A(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n), (13) where c and A are constants that will be fixed later, and where the base cases of the induction are the ranges m n and m M 0, for a sufficiently large constant M 0. In both cases we have I (P, L) A(m + n), for a suitable choice of A. 9 Assume then that the bound holds for all m < m, and consider an instance involving sets P, L, with P =m > L = n, and m > M 0. As already discussed, the bound in (5) is qualitatively different in the two ranges m = O(n 4/3 ) and m = (n 4/3 ), and the analysis will occasionally have to bifurcate accordingly. Nevertheless, the bifurcation is mainly in the choice of various parameters, and in manipulating them. Most of the technical details that deal with the algebraic structure of the problem are identical. We will therefore present the analysis jointly for both cases, and bifurcate only locally, when the induction itself, or tools that prepare for the induction, get into action, and require different treatments in the two cases. As promised in the overview, we will use two different partitioning schemes, one with a polynomial of large degree, and one with a polynomial of small degree. We start naturally with the first scheme. An important issue to bear in mind is that, unlike most of the material in the preceding section, where the underlying field was C, the analysis in this section is over the reals. Nevertheless, this is essentially needed only for constructing a polynomial partitioning, which is meaningless over C. Once this is done, the analysis of incidences between points and lines on the zero set of the partitioning polynomial can be carried out over the complex field just as well as over R, and then the machinery reviewed and developed in the previous section can be brought to bear. 9 When m n (or when n m), an immediate application of the Szemerédi Trotter theorem yields the linear bound O(m + n). 123

72 Discrete Comput Geom (2017) 57: First partitioning scheme Fix a parameter r, given by r = { cm 8/5 /n 4/5 if m an 4/3, cn 4 /m 2 if m an 4/3, where a and c are suitable constants. Note that, in both cases, 1 r m, for a suitable choice of the constants of proportionality, unless either m = (n 2 ) or n = (m 2 ), extreme cases that have already been handled. We refer to the cases n 1/2 m an 4/3 and an 4/3 < m n 2 as the cases of small m and of large m, respectively. We now apply the polynomial partitioning theorem of Guth and Katz (see [15] and [22, Thm. 2.6]), to obtain an r-partitioning 4-variate (real) polynomial f of degree D = O(r 1/4 ) { c 0 m 2/5 /n 1/5 if m an 4/3, c 0 n/m 1/2 if m an 4/3, (14) for another suitable constant c 0. That is, every connected component of R 4 \ Z( f ) contains at most m/r points of P, where, as above, Z( f ) denotes the zero set of f. By Warren s theorem [46](seealso[22]), the number of components of R 4 \ Z( f ) is O(D 4 ) = O(r). Set P 0 := P Z( f ) and P := P \ P 0. We recall that, although the points of P are more or less evenly partitioned among the cells of the partition, no nontrivial bound can be provided for the size of P 0 ; in the worst case, all the points of P could lie in Z( f ). Each line l L is either fully contained in Z( f ) or intersects it in at most D points (since the restriction of f to l is a univariate polynomial of degree at most D). Let L 0 denote the subset of lines of L that are fully contained in Z( f ) and put L = L \ L 0. We have I (P, L) = I (P 0, L 0 ) + I (P 0, L ) + I (P, L ). (15) As can be expected (and noted earlier), the harder part of the analysis is the estimation of I (P 0, L 0 ). Indeed, it might happen that Z( f ) is a hyperplane, and then the best (and worst-case tight) bound we can offer is the bound specified by Theorem 1.1. It might also happen that Z( f ) contains some 2-flat, in which case we are back in the planar scenario, for which the best (and worst-case tight) bound we can offer is the Szemerédi Trotter bound (1). Of course, the assumptions of the theorem come to the rescue, and we will see below how exactly they are used. We first bound the second and third terms of (15). We have I (P 0, L ) L D nd, (16) because, as just noted, a line not fully contained in Z( f ) can intersect this set in at most D points. To estimate I (P, L ), we put, for each cell τ of the partition, P τ = P τ, and let L τ denote the set of the lines of L that cross τ; put m τ = P τ m/r, and 123

73 724 Discrete Comput Geom (2017) 57: n τ = L τ. Since every line l L crosses at most 1 + D components of R 4 \ Z( f ) (because it has to pass through Z( f ) in between cells), we have n τ L (1 + D) n(1 + D). (17) τ Clearly, we have I (P, L ) = τ I (P τ, L τ ). We now bifurcate depending on the value of m. Estimating I (P, L ): The case of small m. Here we use the easy upper bound (which holds for any pair of sets P τ, L τ ) I (P τ, L τ ) = O( P τ 2 + L τ ) = O((m/r) 2 + n τ ). Summing these bounds over the cells, using (17), and recalling the value of r (and of D), we get I (P, L ) = τ I (P τ, L τ ) = O(m 2 /r + nr 1/4 ) = O(m 2/5 n 4/5 ). Estimating I (P, L ): The case of large m. Here we use the dual (generally applicable) upper bound I (P τ, L τ ) = O( L τ 2 + P τ ), which, by splitting L τ into L τ / P τ 1/2 subsets of size at most P τ 1/2, becomes I (P τ, L τ ) = O( P τ 1/2 L τ + P τ ) = O((m/r) 1/2 n τ + m τ ). Summing these bounds over the cells, using (17), and recalling the value of r, we get I (P, L )= τ I (P τ, L τ ) = O((m/r) 1/2 nr 1/4 + m) = O(m 1/2 n/d + m) = O(m). Combining both bounds, we have: I (P 0, L ) + I (P, L ) = O(m 2/5 n 4/5 + m). (18) Note that in this part of the analysis we do not need the assumptions involving q and s the large degree trivializes the analysis within the cells of the partition. Estimating I (P 0, L 0 ). We next bound the number of incidences between points and lines that are contained in Z( f ). To simplify the notation, write P for P 0 and L for L 0, and denote their respective cardinalities as m and n. (The reader should keep this convention in mind, as we will undo it towards the end of the analysis.) To be precise, we will not be able to account explicitly for all types of these incidences (for the present choices of D). Our strategy is to obtain an explicit bound for a subset of the incidences, which is subsumed by the bound in (5), and then prune away those 123

74 Discrete Comput Geom (2017) 57: lines and points that participate in these incidences. We will be left with problematic subsets of points and lines, and we will then handle them in a second, new, inductionbased partitioning step. A major goal for the first stage is to show that, for the set of surviving lines, the parameters q and s can be replaced by the respective parameters O(D 2 ) and O(D) that pass well through the induction; see below for details. 10 By the nature of its construction, f is in general reducible (see [15]). However, to apply successfully certain steps of the forthcoming analysis, we will need to assume that f is irreducible, so we will apply the analysis separately to each irreducible factor of f, and then sum up the resulting bounds. (The actual problem decomposition is subtler see below.) Write the irreducible factors of f, in an arbitrary order, as f 1,..., f k,forsome k D. The points of P are partitioned among the zero sets of these factors, by assigning each point p P to the first factor in this order whose zero set contains p. A line l L is similarly assigned to the first factor whose zero set fully contains l (there always exists such a factor). Then I (P, L) is the sum, over i = 1,...,k,ofthe number of incidences between the points and the lines that are assigned to the (same) ith factor, plus the number of incidences between points and lines assigned to different factors. The latter kind of incidences is easier to handle. Indeed, if (p,l)is an incident pair in P L, so that p is assigned to f i and l is assigned to f j,fori = j (necessarily i < j), then the incidence occurs at an intersection of l with Z( f i ). By construction, l is not fully contained in Z( f i ),soitintersectsitinatmostdeg( f i ) points, so the overall number of incidences on l of this kind is at most i = j deg f i < D, and the overall number of such incidences is therefore at most nd. For the former kind of incidences, we assume in what follows that we have a single irreducible polynomial f, and denote by P and m, for short, the set of points assigned to f and its cardinality, and by L and n the set of lines assigned to f (and thus fully contained in Z( f )) and its cardinality. We continue to denote the degree of f as D. (Again, we will undo these conventions towards the end of the analysis.) This is not yet the end of the reduction, because, in most of the analysis about to unfold, we need to assume that the points of P are non-singular points of Z( f ). To reduce the setup to this situation we proceed as follows. We construct a sequence of partial derivatives of f that are not identically zero on Z( f ). For this we assume, as we may, that f, and each of its derivatives, are square-free; whenever this fails, we replace the corresponding derivative by its square-free counterpart before continuing to differentiate. Without loss of generality, assume that this sequence is f, f x, f xx, and so on. Denote the j-th element in this sequence as f j,for j = 0, 1,... (so f 0 = f, f 1 = f x, and so on). Assign each point p P to the first polynomial f j in the sequence for which p is non-singular; more precisely, we assign p to the first f j for which f j (p) = 0but f j+1 (p) = 0 (recall that f 0 (p) is always 0 by assumption. Similarly, assign each line l to the first polynomial f j in the sequence for which l is fully contained in Z( f j ) but not fully contained in Z( f j+1 ) (again, by assumption, there always exists such f j ). If l is assigned to f j then it can only contain points p 10 Note that in general the bounds O(D 2 ) and O(D) are not necessarily smaller than their respective original counterparts q and s. Nevertheless, they uniformly depend on m and n in a way that makes them fit the induction process, whereas the parameters q and s, over which we have no control, do not. 123

75 726 Discrete Comput Geom (2017) 57: that were assigned to some f k with k j. Indeed, if l contained a point p assigned to f k with k < j then f k+1 (p) = 0butl is fully contained in Z( f k+1 ), since k + 1 j; this is a contradiction that establishes the claim. Fix a line l L, which is assigned to some f j. An incidence between l and a point p P, assigned to some f k,fork > j, can be charged to the intersection of l with Z( f j+1 ) at p (by construction, p belongs to Z( f j+1 )). The number of such intersections is at most D j 1, so the overall number of incidences of this sort, over all lines l L, iso(nd). It therefore suffices to consider only incidences between points and lines that are assigned to the same zero set Z( f i ). The reductions so far have produced a finite collection of up to O(D) polynomials, each of degree at most D, so that the points of P are partitioned among the polynomials and so are the lines of L, and we only need to bound the number of incidences between points and lines assigned to the same polynomial. This is not the end yet, because the various partial derivatives might be reducible, which we want to avoid. Thus, in a final decomposition step, we split each derivative polynomial f j into its irreducible factors, and reassign the points and lines that were assigned to Z( f j ) to the various factors, by the same first come first served rule used above. The overall number of incidences that are lost in this process is again O(nD). The overall number of polynomials is O(D 2 ), as can easily be checked. Note also that the last decomposition step preserves non-singularity of the points in the special sense defined above; that is, as is easily verified, a point p Z( f j ) with f j+1 (p) = 0, continues to be a non-singular point of the irreducible component it is reassigned to. We now fix one such final polynomial, still call it f, denote its degree by D (which is upper bounded by the original degree D), and denote by P and L the subsets of the original sets of points and lines that are assigned to f, and by m and n their respective cardinalities. (Again, this simplifying convention will be undone towards the end of the analysis.) We now may assume that P consists exclusively of non-singular points of the irreducible variety Z( f ). If D 3, then, by Lemma 2.12, Z( f ) is ruled by lines. Hypersurfaces ruled by lines will be handled in the later part of the analysis. (Note that the cases D = 1or D = 2 can be controlled by assumption (i ) of the theorem (see below), whereas the case D = 3 requires a different treatment.) Suppose then that D 4. The flecnode polynomial FL 4 f of f (see Sect. 2.4) vanishes identically on every line of L (and thus also on P, assuming that each point of P is incident to at least one line of L). If FL 4 f does not vanish identically on Z( f ), then Z( f, FL 4 f ) := Z( f ) Z(FL4 f ) is a twodimensional variety (see, e.g., Hartshorne [18, Exercise I.1.8]). It contains P and all the lines of L (by Lemma 2.10), and is of degree O(D 2 ) (by Theorem 2.2). The other possibility is that FL 4 f vanishes identically on Z( f ), and then Theorem 2.11 implies that Z( f ) is ruled by lines. This latter case, which requires several more refined tools from algebraic geometry, will be analyzed later. First Case: Z( f, FL 4 f ) is Two-Dimensional Put g = FL 4 f. In the analysis below, we only use the facts that deg(g) = O(D), and that Z( f, g) is two-dimensional, so the analysis applies for any such g; this comment 123

76 Discrete Comput Geom (2017) 57: will be useful in later steps of the analysis. Recall that in this part of the analysis f is assumed to be an irreducible polynomial of degree 4. We have a set P of m points and a set L of n lines in C 4, so that P is contained in the two-dimensional algebraic variety Z( f, g) C 4. By pruning away all the lines containing at most max(d, deg(g)) points of P, weloseo(nd) incidences, and all the surviving lines are contained in Z( f, g), as is easily checked. For simplicity of notation, we continue to denote by L the set of surviving lines. Let Z( f, g) = s i=1 V i be the decomposition of Z( f, g) into its irreducible components, as described in Section 2.2. By Theorem 2.2, wehave s i=1 deg(v i ) deg( f ) deg(g) = O(D 2 ). Incidences Within Non-planar Components of Z( f, g). Our next step is to analyze the number of incidences between points and lines within the components of Z( f, g) that are not 2-flats. For this we first need the following bound on point-line incidences within a two-dimensional surface in three dimensions. This part of the analysis is taken from our paper [37]. We also refer to Sect. 2.4 for properties of ruled surfaces. For a point p on an irreducible singly ruled surface V, which is not the exceptional point of V,welet V (p) denote the number of generator lines passing through p and fully contained in V (so if p is incident to an exceptional line, we do not count that line in V (p)). We also put V (p) := max{0, V (p) 1}. Finally, if V is a cone and p V is its exceptional point (that is, apex), we put V (p V ) = V (p V ) := 0. We also consider a variant of this notation, where we are also given a finite set L of lines (where not all lines of L are necessarily contained in V ), which does not contain any of the (at most two) exceptional lines of V. For a point p V,weletλ V (p; L) denote the number of lines in L that pass through p and are fully contained in V, with the same provisions as above, namely that we do not count incidences with exceptional lines, nor do we count incidences with an exceptional point, and put λ V (p; L) := max{0,λ V (p; L) 1}.If V is a cone with apex p V, we put λ V (p V ; L) = λ V (p V ; L) = 0. We clearly have λ V (p; L) V (p) and λ V (p; L) V (p), for each point p. Lemma 3.1 Let V be an irreducible singly ruled two-dimensional surface of degree D > 1 in R 3 or in C 3. Then, for any line l, except for the (at most) two exceptional lines of V, we have V (p) D ifl is not fully contained in V, p l V p l V V (p) D ifl is fully contained in V. The following lemma provides the needed infrastructure for our analysis, and is taken from Sharir and Solomon [37, Thm. 15]. Lemma 3.2 Let V be a possibly reducible two-dimensional algebraic surface of degree D > 1 in R 3 or in C 3, with no linear components. Let P be a set of m distinct points on V and let L be a set of n distinct lines fully contained in V. Then there exists a subset L 0 LofatmostO(D 2 ) lines, such that the number of incidences between P and L \ L 0 satisfies 123

77 728 Discrete Comput Geom (2017) 57: I (P, L \ L 0 ) = O(m 1/2 n 1/2 D 1/2 + m + n). (19) Sketch of Proof. We provide the following sketch of the proof; the full details are given in the companion paper [37]. Consider the irreducible components W 1,...,W u of V. We first argue that the number of lines that are either contained in the union of the non-ruled components, or those contained in more than one ruled component of V is O(D 2 ), and we place all these lines, as well as the exceptional lines of any singly ruled component, in the exceptional set L 0. We may thus assume that each surviving line in L 1 := L \ L 0 is contained in a unique ruled component of V, and is a generator of that component. The strategy of the proof is to consider each line l of L 1, and to estimate the number of its incidences with the points of P in an indirect manner, via Lemma 3.1, applied to l and to each of the ruled components W j of V. Specifically, we fix some threshold parameter ξ, and dispose of points that are incident to at most ξ lines of L 1, losing at most mξ incidences. Let P 1 denote the set of surviving points. Now if a line l L 1 is incident to a point p P 1, it meets at least ξ other lines of L 1 at p. It follows from Lemma 3.1 that the overall number of such lines, over all points in P 1 l, is roughly D, so the number of such points on l is at most roughly D/ξ, for a total of nd/ξ incidences of this kind. Choosing ξ = (nd/m) 1/2 yields the bound O(m 1/2 n 1/2 D 1/2 ), and the lemma follows. We can now proceed, by deriving two upper bounds for certain types of incidences between P and L. The first bound is relevant for the range m = O(n 4/3 ), and the second bound is relevant for the range m = (n 4/3 ). Nevertheless, both bounds apply to the entire range of m and n. Proposition 3.3 The number of incidences involving non-singular points of Z( f ) that are contained in components of Z( f, g) that are not 2-flats is min{o(md 2 + nd), O(m + nd 4 )}. (20) Proof We first establish the bound O(mD 2 + nd). Letp Z( f ) be a non-singular point. The irreducible decomposition of S p := Z( f, g) T p Z( f ) is the union of oneand two-dimensional components. Clearly, S p contains all the lines that are incident to p and are fully contained in Z( f, g); it is a variety, embedded in 3-space (namely, in T p Z( f )), of degree O(D 2 ). The union of the one-dimensional components is a curve of degree O(D 2 ), so, by Lemma 2.3, it can contain at most O(D 2 ) lines; when summing over all p P, the total number of incidences with those lines is O(mD 2 ). It remains to bound incidences involving the two-dimensional components of S p that are not 2-flats. By Sharir and Solomon [36, Lem. 5], the number of lines incident to p inside these two-dimensional components of S p is at most O(D 2 ), except possibly for lines that lie in a component that is a cone and has p as its apex. Summing over all p P, we get a total of O(mD 2 ) incidences for this case too, ignoring lines that lie only in conic (or flat) components. Note that each two-dimensional component of S p is necessarily also a twodimensional irreducible component of Z( f, g). Hence the analysis performed so far 123

78 Discrete Comput Geom (2017) 57: takes care of all incidences except for those that occur on conic two-dimensional components of Z( f, g) (and on 2-flats, which we totally ignore in this proposition). Let V be a conic component of Z( f, g) with apex p V, which is not a 2-flat. We note that V cannot fully contain a line that is not incident to p V. Indeed, suppose to the contrary that V contained such a line l. Since V is a cone with apex p V, for each point a l, the line connecting a to p V is fully contained in V, and therefore the 2-flat containing p V and l is fully contained in V.AsV is irreducible and is not a 2-flat, we obtain a contradiction, showing that no such line exists. We conclude that any point on V, except for p V, is incident to at most one line that is fully contained in V (a generator through p V ), for a total of O(m) incidences. Since Z( f, g) is of degree O(D 2 ), the number of conic components of Z( f, g) is O(D 2 ), so, summing this bound over all components V, we get again the bound O(mD 2 ) on the number of relevant non-apex incidences. Therefore, it remains to bound the number of incidences between the points of P c := {p V p V is an apex of an irreducible conic component V of Z( f, g)} and the lines of L. Since there are at most O(D 2 ) irreducible components of Z( f, g), we have P c cd 2, for some suitable constant c. WenextletL c denote the set of lines in L containing fewer than cd points of P c, and claim that any point p P c is incident to fewer than D lines of L\L c. Indeed, otherwise, we would get at least D lines incident to p, each containing at least cd+1 points of P c, i.e., at least cd points other than p. As these points are all distinct, we would get that P c 1 + D cd > cd 2, a contradiction. On the other hand, by definition of L c,wehave I (P c, L c ) = O(nD). We have thus shown that the number of incidences involving points of P c is I (P c, L) = I (P c, L c ) + I (P c, L \ L c ) = O(nD) + O(mD) = O(nD + md), well within the bound that we seek to establish. The second bound We next establish the second bound O(m + nd 4 ).LetV be an irreducible two-dimensional component of Z( f, g). IfV is not ruled, then by Proposition 2.13, it contains at most 11 deg(v ) 2 24 deg(v )<11 deg(v ) 2 lines. Summing over all irreducible components of Z( f, g) that are not ruled, we get at most 11 V deg(v )2 = O(D 4 ) lines. Let l be one of those lines, and let p l P. For any other line λ L that passes through p, we charge its incidence with p to its intersection with l. This yields a total of O(nD 4 ) incidences, to which we add O(m) for incidences with those points that lie on only one line of L, for a total of O(m + nd 4 ) incidences. We next analyze the irreducible components of Z( f, g) that are ruled but are not 2-flats. Let V 1,...,V k denote these components, for some k = O(D 2 ). Project all these components onto some generic hyperplane, and regard them as a single (reducible) ruled surface in 3-space, whose degree is k i=1 deg(v i ) = O(D 2 ). Lemma 3.2 then yields a subset L 0 of L of size O(D 4 ), and shows that 123

79 730 Discrete Comput Geom (2017) 57: I (P, L \ L 0 ) = O(m 1/2 n 1/2 D + m + n). The lines of L 0 are simply added to the set of O(D 4 ) lines not belonging to ruled components. This does not affect the asymptotic bound O(nD 4 ) derived above. In total we get incidences. Since O(m 1/2 n 1/2 D + m + nd 4 ) m 1/2 n 1/2 D 1 2 (m + nd2 ), we obtain the second bound asserted in the proposition. Remark The term O(nD 4 ) appears to be too weak, and can probably be improved, using ideas similar to those in the proof of Lemma 3.2. Since such an improvement does not have a significant effect on our analysis, we leave it as an interesting problem for further research. Restrictedness of hyperplanes and quadrics, and lines on 2-flats The bounds in Proposition 3.3 might be too large, for the current choices of D, because of the respective terms O(mD 2 ) and O(nD 4 ). (Technically, the m and n in the definition of D are not necessarily the same as the m and n that denote the size of the current subsets of the original P and L, but let us assume that they are the same for the present discussion.) For example, when m = O(n 4/3 ) and D = (m 2/5 /n 1/5 ) (recall that this is the large value of D for this range), we have md 2 = (m 9/5 /n 2/5 ), and this is m 2/5 n 4/5 when m n 6/7. Similarly, when m = (n 4/3 ) and D = (n/m 1/2 ) (which is the value chosen for this range), we have nd 4 = (n 5 /m 2 ), and this is m when m n 5/3. These bounds will be used in the second partitioning step, where we use a smaller-degree partitioning polynomial, and for m outside the problematic ranges, i.e., for m n 6/7 or m n 5/3 ; see below for details. Otherwise, for the current D, these bounds need to be finessed and replaced by the following alternative analysis. 11 In the first step of this analysis, we estimate the number of lines contained in a hyperplane or a quadric (when Z( f, g) is two-dimensional), and establish the following properties. Lemma 3.4 Each hyperplane or quadric H is O(D 2 )-restricted for the lines of L that are contained in non-planar components of Z( f, g). Proof Fix a hyperplane or quadric H. Recall that all the lines in the current set L are contained in Z( f, g). LetV be an irreducible component of Z( f, g), which is not a 2-flat. If V H is a curve, then (recalling Theorem 2.2) its degree is at most 11 As the calculations worked out above indicate, the bounds in Proposition 3.3 will be within the bound (5)whenm is sufficiently small (below n 4/3 ) or sufficiently large (above n 5/3 ). For such values of m we can bypass the induction process, and obtain the desired bounds directly, in a single step. See a more detailed description towards the end of this section. 123

80 Discrete Comput Geom (2017) 57: deg(v ) (when H is a hyperplane) or 2 deg(v ) (when H is a quadric), and can therefore contain at most 2 deg(v ) lines, by Lemma 2.3. Therefore, the union of all the irreducible components V of Z( f, g) which intersect H in a curve, contains at most 2 V deg(v ) = O(D2 ) lines. Assume then that V H is two-dimensional. Since V is irreducible, we must have V H = V,soV is fully contained in H. Moreover, V is an irreducible two-dimensional surface contained in Z( f ) H, and therefore must be an irreducible component of Z( f ) H, which is a two-dimensional surface of degree D. By Theorem 2.2, V H deg(v ) deg(z( f ) H) deg( f ) D. IfV is not ruled by lines (and, by assumption, is not a 2-flat), then by Proposition 2.13, it contains at most 11 deg(v ) 2 lines, and summing over all such components V within H, we get a total of at most V 11 deg(v )2 = O(D 2 ) lines. The remaining irreducible (two-dimensional) components V of Z( f, g) that meet H (if such components exist) are fully contained in H, and are ruled by lines. As already observed, these components are also irreducible components of Z( f ) H, and so, with the exception of O(D 2 ) lines (those contained in the components already analyzed), all the lines of L that lie in H are contained in components of Z( f ) H that are ruled by lines. Since f restricted to H is a polynomial of degree D, and since we are interested in lines of L that are not contained in planar components of Z( f ) H we conclude that H is O(D 2 )-restricted, with respect to the subset of L mentioned in the lemma. We next analyze the number of lines contained in a 2-flat. Lemma 3.5 Let π be a 2-flat that is not fully contained in Z( f, g). Then the number of lines fully contained in Z( f ) π is O(D). Proof The intersection Z( f ) π is either π itself, or a curve of degree D. The latter case implies (using Lemma 2.3) that π contains at most D lines that are fully contained in Z( f ). In the former case π Z( f ). By assumption, π is not contained in Z( f, g), implying that g intersects π in a curve of degree O(D) (since π Z( f, g) = π Z(g)), and can therefore contain at most O(D) lines that are fully contained in Z( f ). Recap Summing up what has been done so far, we can classify the incidences in I (P, L) into the following types. Recall that the analysis is confined to a single irreducible factor f of the original polynomial or of some higher partial derivative of such a factor. (a) We treat the cases where f is linear or quadratic separately, using a variant of Theorem 1.1, which takes into account the restrictedness of hyperplanes and quadrics; see Proposition 3.6 below. (b) We treat the case where Z( f ) is ruled by lines separately (this is the second case in the analysis, when Z( f, FL (4) f ) is three-dimensional). If f is not ruled by lines and is of degree 4 (recall that each surface of degree at most 3 is ruled by lines see Lemma 2.12), then there are two kinds of incidences that need to be considered. (c) Incidences between points and lines that are contained in irreducible components of Z( f, FL (4) f ) (or, more generally, of Z( f, g), for other suitable polynomials g) 123

81 732 Discrete Comput Geom (2017) 57: that are not 2-flats. We have bounded the number of these incidences in Proposition 3.3 in two different ways, but we also ignored these incidences, passing them to the induction in the second partitioning step, to be presented later, where we now know that each hyperplane and quadric is O(D 2 )-restricted, and each 2-flat contains at most O(D) lines of L. For both properties to hold, we first have to get rid of all the lines of L that are contained in 2-flats within Z( f, g), and we will perform this pruning after bounding the number of incidences involving lines that are contained in such 2-flats. This will make the O(D 2 )-restrictedness in Lemma 3.4 hold with respect to the entire (pruned) set L, and will make Lemma 3.5 hold for each 2-flat. (d) Incidences between points and lines that are contained in some irreducible component of Z( f, g) that is a 2-flat. These incidences will be analyzed explicitly below, using the properties of flat points and lines, as presented in Sect This classification of incidences, especially those of types (c) and (d), holds in general, for any polynomial g satisfying the properties assumed in this treatment (that it has degree O(D) and that Z( f, g) is two-dimensional), and their treatment also applies to these more general scenarios. Incidences within hyperplanes and quadrics We next derive a bound that we will use several times later on, in cases where we can partition P and L (or, more precisely, subsets thereof) among some finite collection of hyperplanes and quadrics, so that all the relevant incidences occur between points and lines that are assigned to the same surface. Recall that we have already applied a similar partitioning among the factors of f and of its derivatives. The prime application of this bound will be to incidences of type (a) above, but it will also be used in the analysis of type (d) incidences, and in ) is three-dimensional, i.e., when ) = Z( f ). In particular, we emphasize that the following proposition does ) be two-dimensional. the analysis of the second case (b), where Z( f, FL (4) f Z( f, FL (4) f not require that Z( f, FL (4) f Proposition 3.6 Let H 1,...,H t be a finite collection of hyperplanes and quadrics. Assume that the points of P and the lines of L are partitioned among H 1,...,H t,so that each point p P (resp., each line l L) is assigned to a unique hyperplane or quadric that contains p ( resp., fully contains l), and assume further that each H i is q-restricted and that each 2-flat contains at most s lines of L. Then the overall number of incidences between points and lines that are assigned to the same surface (i.e., hyperplane or quadric) is O(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + n). (21) Proof For i = 1,...,t, letl i (resp., P i ) denote the set of lines of L (resp., points of P), that are assigned to H i, and put n i := L i, m i := P i.wehave i m i = m and i n i = n. For each i, since H i is q-restricted, there exists a polynomial g i = g Hi, of degree O( q), such that all the lines of L i, with the exception of at most q of them, are fully contained in ruled components of H i Z(g i ) that are not 2-flats. Write L i = Li nr L r i, where Lr i is the subset of those lines that are fully contained in ruled 123

82 Discrete Comput Geom (2017) 57: components of H i Z(g i ) that are not 2-flats, and Li nr is the complementary subset, of size at most q. The lines in L r i are contained in the union W r of the ruled components of H i Z(g i ) that are not 2-flats. We also remove from L r i the subset L r i0 of O(q) lines, as provided by Lemma 3.2 (including all the lines that are fully contained in more than one component W i ), and put them in Li nr ; we continue to use the same notations for these modified sets. To apply Lemma 3.2 to the case where H i is a quadric, we first project the configuration onto some generic 3-space, and note that by Sharir and Solomon [35, Lem. 2.1], the projection of W r does not contain any 2-flat. Since the size of Li nr is still O(q), we have, by Theorem 1.1, I (P i, Li nr ) = O(m 1/2 i Li nr 3/4 + m 2/3 i Li nr 1/3 s 1/3 + m i + Li nr ) = O(m 1/2 i n 1/2 i q 1/4 + m 2/3 i n 1/3 i s 1/3 + m i + n i ). (Note that Theorem 1.1 is directly applicable when H i is a hyperplane, and that it can also be applied when H i is a quadric, by projecting the configuration onto some generic hyperplane, similar to what we have just noted for the application of Lemma 3.2.) We next bound I (P i, L r i ), using Lemma 3.2 (when H i is a quadric, we apply it to the generic projection of W r to three dimensions, as above). Since deg(g i ) = O( q), W r is of degree O( q), and thus also its projection to three dimensions (see, e.g., [17]), in case H i is a quadric. We have already removed from L r i the subset L r i0 provided by the lemma, and so the lemma yields the bound That is, we have: I (P i, L r i ) = O(m1/2 i n 1/2 i q 1/4 + m i + n i ). I (P i, L i ) = O(m 1/2 i n 1/2 i q 1/4 + m 2/3 i n 1/3 i s 1/3 + m i + n i ). Summing these bounds for i = 1,...,t, and using Hölder s inequality (twice), we get the bound asserted in (21). The case where f is linear or quadratic (These are the cases D = 1, 2) Let us apply Proposition 3.6 right away to bound the number of incidences when our (irreducible) f is linear or quadratic, that is, when Z( f ) is a hyperplane or a quadric. Proposition 3.6 (together with assumption (i) of the theorem) then implies the following bound. I (P, L) = O(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + n), (22) which is subsumed by the main bound (5). Incidences within 2-flats fully contained in Z( f, g) Assuming generic directions of the coordinate axes, we may assume that, for every non-singular point p P, T p Z( f ) is not orthogonal to any of the axes. This allows us to use the flatness criterion developed in Sect. 2.5 to each point of P. As in previous steps of the analysis, we simplify the notation by denoting the subsets of the points and lines that lie in the 2-flat components of Z( f, g) as P and L, and their respective sizes as m and n. Each point p P (resp., each line 123

83 734 Discrete Comput Geom (2017) 57: l L) under consideration is contained (resp., fully contained) in at least one of the 2-flats π 1,...,π k that are fully contained in Z( f, g) (these are the linear irreducible components of Z( f, g), and we have k = O(D 2 )). Let P (2) (resp., P (3) ) denote the set of points p P that lie in at most two (resp., at least three) of these 2-flats. Assign each point p P (2) to the (at most) two 2-flats containing it. Note that if p P (2), then every line l that is incident to p can be contained in at most two of the 2-flats π i, and we assign l to those 2-flats. Let L (2) denote the set of lines l L such that l is incident to at least one point in P (2) (and is thus contained in at most two 2-flats π i ), and put L (3) = L \L (2).Fori = 1,...,k,letL (2) i (resp., P (2) i ) denote the set of lines of L (2) (resp., points of P (2) ), that are contained in π i, and put n i := L (2) i, m i = P (2) i. (Note that we ignore here lines that are not fully contained in one of these 2-flats; these lines are fully contained in other components of Z( f, g) and their contribution to the incidence count has already been taken care of.) By construction, k m i 2m, i=1 and k n i 2n. i=1 Moreover, a point p P (2) can be incident only to lines of L that are contained in one of the (at most) two 2-flats that contain p, sowehavei (P (2), L (2) ) ki=1 I (P (2) i, L (2) i ). The Szemerédi Trotter bound (1) yields I (P (2) i, L (2) i ) = O(m 2/3 i n 2/3 i + m i + n i ), i = 1,...,k. (23) By assumption (ii) of the theorem, n i s for each i = 1,...,k, so, summing over i = 1,...,k and using Hölder s inequality, we obtain I (P (2), L (2) ) k ( k I (P (2) i, L (2) i ) = O i=1 i=1 (( k = O ( m 2/3 i n 2/3 i + m i + n i ) ) m 2/3 i n 1/3 i i=1 (( k ) 2/3 ( k = O m i i=1 s 1/3) ) + m + n i=1 ) 1/3s ) n 1/3 i + m + n = O(m 2/3 n 1/3 s 1/3 + m + n). (24) Consider next the points of P (3), each contained in at least three 2-flats that are fully contained in Z( f ). All the points of P (3) are linearly flat (see Sect. 2.5 for details), and are therefore flat. Notice that each such point can be incident to lines of L (2) and to lines of L (3). We prune away each line l L that contains fewer than 3D points of P (3), losing at most 3nD incidences in the process. Each of the surviving lines contains at least 3D 3 flat points, and is therefore flat, because the degrees of the nine polynomials whose vanishing at p captures the flatness of p, areallatmost3d 4. In other words, we are left with the task of bounding 123

84 Discrete Comput Geom (2017) 57: the number of incidences between flat points and flat lines. To simplify this part of the presentation, we again rename the sets of these points and lines as P and L, and denote their sizes by m and n, respectively. Incidences between flat points and lines By Lemma 2.15, all the (non-singular) points of a flat line have the same tangent hyperplane. We assign each flat point p P (resp., flat line l L) tot p Z( f ) (resp., to T p Z( f ) for some (any) non-singular point p P l; again we only consider lines incident to at least one such point). We have therefore partitioned P and L among distinct hyperplanes H 1,...,H t, and we only need to count incidences between points and lines assigned to the same hyperplane. By assumptions (i) and (ii) of the theorem, the conditions of Proposition 3.6 hold, implying that the number of these incidences is O(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + n). (25) As promised, after having bounded the number of incidences within the 2-flats that are fully contained in Z( f, g), we remove from L the lines that are contained in such 2-flats, and continue the analysis with the remaining subset. In summary, combining the bounds in (24) and (25), Proposition 3.3, and Lemmas 3.4 and 3.5, the overall outcome of the analysis for the first case is summarized in the following proposition. (In the proposition, f is one of the irreducible factors of the original polynomial or of one of its derivatives, and P and L refer to the subsets assigned to that factor.) Proposition 3.7 Let g be any polynomial of degree O(D) such that Z( f, g) is twodimensional, let P be a set of m points contained in Z( f, g), and let L be a set of n lines contained in Z( f, g). Then I (P, L) = I (P, L ) + O(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + nd), (26) where P and L are subsets of P and L, respectively, so that each hyperplane or quadric is O(D 2 )-restricted with respect to L, and each 2-flat contains at most O(D) lines of L. We also have the explicit estimate I (P, L ) = min{o(md 2 + nd), O(m + nd 4 )}. (27) Remark (1) As already noted, lines that are contained in 2-flats that are fully contained in Z( f, g) have already been taken care of, and thus do not belong to L,sothe application of Lemma 3.5 shows that every 2-flat contains only O(D) lines of L, and the application of Lemma 3.4 shows that every hyperplane or quadric is O(D 2 )-restricted. (2) When m and n are such that the bound on I (P, L ) in (27) is dominated by O(m 2/5 n 4/5 + m), we use these bounds explicitly, and get the induction-free refined bound in (6). This remark will be expanded and highlighted later, as we spell out the details of the induction process. 123

85 736 Discrete Comput Geom (2017) 57: Second Case: Z( f ) is Ruled by Lines We next consider the case where the four-dimensional flecnode polynomial FL 4 f vanishes identically on Z( f ). By Theorem 2.11, this implies that Z( f ) is ruled by (possibly complex) lines. In what follows we assume that D 3 (the cases D = 1, 2 have already been treated earlier, using Proposition 3.6). We prune away points p P, with p 6 (the number of incidences involving these points is at most 6m = O(m)). For simplicity of notation, we still denote the set of surviving points by P. Thus we now have p > 6, for every p P. Recalling the properties of the u-resultant of f (that is, the u-resultant associated with F 1 (p; v), F 2 (p; v), F 3 (p; v)), as reviewed in Sect. 2.6, we have, by Corollary 2.17, that U(p; u 0, u 1, u 2, u 3 ) 0 (as a polynomial in u 0,...,u 3 ) for every p P. We will use the following theorem of Landsberg, which generalizes Theorem It is stated here in a specialized and slightly revised form, but still for an arbitrary hypersurface in any dimension, and for any choice of the parameter k. Recall that k is the union of k p over all p X, namely, it is the set of all lines that osculate to Z( f ) to order three at some point on Z( f ). The actual application of the theorem will be for X = Z( f ) (and d = 4, k = 3). We refer the reader to Sect. 2.1 for notations and further details. Theorem 3.8 (Landsberg [19, Thm ]) Let X P d (C) be a hypersurface, and let k 2 be an integer, such that there is an irreducible component 0 k k satisfying, for every point p in a Zariski open set O Z( f ), dim 0,p k > d k 1, where k 0,p is the set of lines in 0 k incident to p. Then, for each point p O, all lines in k 0,p are contained in X. To appreciate the theorem, we note that, informally, lines through a fixed point p have d 1 degrees of freedom, and the constraint that such a line osculates to X to order k removes k degrees of freedom, leaving d k 1 degrees. The theorem asserts that if the dimension of this set of lines is larger, for most points on X, then these lines are fully contained in X. Note also that this is a local-to-global theorem the large dimensionality condition has to hold at every point of some Zariski open subset of Z( f ), for the conclusion to hold. IfU(p; u 0, u 1, u 2, u 3 ) does not vanish identically (as a polynomial in u 0, u 1, u 2, u 3 ) at every point p Z( f ), then at least one of its coefficients, call it c U, does not vanish identically on Z( f ). In this case, as U vanishes identically at every point of P (as a polynomial in u 0, u 1, u 2, u 3 ), it follows that P is contained in the two-dimensional variety Z( f, c U ). Since c U has degree O(D) in x, y, z,w(by Theorem 2.16), we can proceed exactly as we did in the case where Z( f, FL 4 f ) was 2-dimensional. That is, we obtain the bound (26) in Proposition 3.7, namely, I (P, L) = I (P, L ) + O(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + nd), (28) where P and L are subsets of P and L, respectively, so that each hyperplane or quadric is O(D 2 )-restricted with respect to L, and each 2-flat contains at most O(D) lines of L. We also have the explicit estimate 123

86 Discrete Comput Geom (2017) 57: I (P, L ) = min{o(md 2 + nd), O(m + nd 4 )}. Therefore, since this case does not require the following analysis, it suffices to consider the complementary situation, where we assume that U(p; u 0, u 1, u 2, u 3 ) 0 at every point p Z( f ) (as a polynomial in u 0, u 1, u 2, u 3 ). By Theorem 2.16, 3 p is infinite, so its dimension is positive, for each such p. Informally, the analysis proceeds as follows. Since 3 p is (at least) one-dimensional for every point p Z( f ),theset 3, which is the union of 3 p, over all p Z( f ), has (at least) three degrees of freedom three for specifying p, at least one for specifying the line in 3 p, and one removed because the same line may arise at each of its points (if it is fully contained in Z( f )). In what follows we show that we can find a single irreducible component 0 3 of 3, which is three-dimensional, and such that for any point p Z( f ), the variety 0,p 3 is at least one-dimensional. This will facilitate the application of Theorem 3.8 in our context. Theorem 3.9 There exists an irreducible component 0 3 of 3 of dimension at least three, such that for each non-singular p Z( f ), the variety 0,p 3 is at least onedimensional. Proof The proof makes use of the Theorem of the Fibers and related results, as reviewed in Sect Put W := {(p,l) p l, l 3 p } Z( f ) 3. Note that W is naturally embedded in P 3 P 5, where the second component contains the Plücker hypersurface of lines in 3-space. W can formally be defined as the zero set of homogeneous polynomials; one polynomial defines the Plücker quadric, other polynomials express the condition p l, and other polynomials are those defining the projective variety 3 p, whose elements are now represented by their Plücker coordinates in the appropriate projective space (see Sect. 2.1 for details). Therefore, W is a projective variety. Let 1 : W Z( f ), 2 : W 3 be the (restrictions to W of the) projections to the first and second factors of the product. For an irreducible component 3 0 of 3 (which is also a projective variety), put W 0 := 1 2 ( 3 0 ) ={(p,l) W l 3 0,p }. Since W and 3 0 are projective varieties, so is W 0. (Indeed, if W = Z({ f i (p,l)}), and 3 0 = Z({g j(l)}), for suitable sets of homogeneous polynomials { f i }, {g j }, then W 0 = Z({ f i (p,l),g j (l)}).) Let W 0 denote some irreducible component of W 0, and put Y := 1 ( W 0 ) Z( f ). By the projective extension theorem (see, e.g., Cox et al. [5, Thm. 8.6]), Y is also a projective variety. 123

87 738 Discrete Comput Geom (2017) 57: For a point p Y, the fiber of the map 1 W 0 : W 0 Y over p is contained in {p} 0,p 3 ={(p,l) l 3 0,p } (this is the fiber of 1 W0 over p, which clearly contains the fiber of 1 W 0 over p, as W 0 W 0 ). We will show that there exists some component 0 3, and some irreducible component W 0 of W 0 = 2 1 ( 3 0 ), such that (i) Y = 1( W 0 ) is equal to Z( f ), and (ii) for every point p Z( f ), the fiber of 1 W 0 : W 0 Y over p is (at least) one-dimensional; in this case we say that 0 3 and W 0 form a one-dimensional line cover of Z( f ). Suppose that we have found such a pair 0 3, W 0. As noted above, the fiber of 1 W 0 over p is contained in (or equal to) {p} 0,p 3, and dim({p} 0,p 3 ) = dim( 3 0,p ). Therefore, since Y = Z( f ), this would imply that, for every p Z( f ), wehavedim( 0,p 3 ) 1, which is what we want to prove. We pick some component 0 3, and some irreducible component W 0 of W 0 = 2 1 ( 3 0 ), and analyze when do 3 0 and W 0 form a one-dimensional line cover of Z( f ). Put, as above, Y = 1 ( W 0 ). For a point p Y, put λ(p) = dim( 1 1 ({p})), W 0 and let λ = min p Y λ(p). As noted above, λ(p) dim( 0,p 3 ). By the Theorem of the Fibers (Theorem 2.6), applied to the map 1 W 0 : W 0 Y Z( f ),wehave dim( W 0 ) = dim(y ) + λ. (29) Observe that λ 1. Indeed, if λ = 2, then there exists some non-singular point p Y, such that 0,p 3 is (at least) two-dimensional, implying that Z( f ) is a threedimensional cone; since p is non-singular, Z( f ) is thus a hyperplane, contrary to our assumptions. Assume first that Y = 1 ( W 0 ) is equal to Z( f ) (this is part (i) of the definition of a one-dimensional line cover). If λ = 1, part (ii) of this property also holds, and we are done. Assume then that λ = 0. By the first part of the Theorem of the Fibers (Theorem 2.6), the subset Y 1 ={p Y λ(p) 1} is Zariski closed in Y,soitis a subvariety of Z( f ), of dimension at most 2. Hence, for each p in the Zariski open complement Y \ Y 1, the fiber 1 1 ({p}) is finite. W 0 The remaining case is when Y = 1 ( W 0 ) is properly contained in Z( f ). Since Z( f ) is irreducible, Y is of dimension at most two. To recap, we have proved that for each component 0 3 of 3, and each component W 0 of W 0, if the associated Y is properly contained in Z( f ), then the image of W 0 under 1 (that is, Y ) is at most two-dimensional; we refer to this situation as being of the first kind. If Y = Z( f ) but λ = 0 (these are refered to as situations of the second kind), then, except for a two-dimensional subvariety Y 1 of Z( f ), the fibers of the map 1 W 0 are finite. However, in the case under consideration, we have argued that, for any non-singular point p Z( f ), the fiber 1 1 (p) ={p} 3 p is (at least) one-dimensional. We apply this analysis to all the irreducible components 0 3 of 3, and to all the irreducible components of the corresponding W 0 = 2 1 ( 3 0 ).LetY denote the union of all the images Y of the first kind, and of all the excluded subvarieties Y 1 of the second kind. Being a finite union of two-dimensional varieties, Y is two-dimensional. 123

88 Discrete Comput Geom (2017) 57: The union, over the irreducible components 0 3 of 3, of all the corresponding components W 0, covers W, and therefore, for any non-singular point p Z( f ), the union over all the components W 0 of the fibers of 1 W 0 over p is equal to the fiber of 1 over p, which is one-dimensional (and thus infinite). We claim that there must exist some irreducible component 0 3 of 3, and a corresponding irreducible component W 0 of W 0, such that Y = 1 ( W 0 ) is equal to Z( f ), and the corresponding λ is equal to 1. Indeed, if this were not the case, take any nonsingular point p in Z( f ) \ Y. Since p is not in the image 1 ( W 0 ), for any W 0 of the first kind, the fiber of 1 W 0 at p is empty. Similarly, since p is not in the excluded set Y 1 for any W 0 of the second kind, the fiber of 1 W 0 at p is finite. But then the fiber of 1 at p, being a finite union of (empty or) finite sets, must be finite, a contradiction that establishes the claim. Since for every p Y = Z( f ), λ λ(p) dim( 0,p 3 ), it follows that all the fibers 0,p 3 are (at least) one-dimensional, completing the proof. Remark One interesting corollary of the Theorem of the Fibers is that if we know that for any point p in a Zariski open subset O of Z( f ), the fiber of 1 over p (which is equal to {p} 3 p ) is one-dimensional, then this is true for the entire Z( f ). Indeed, by the Theorem of the Fibers (Theorem 3.9), the set {p Z( f ) dim( 1 1 ({p})) 1} is Zariski closed, and, since it contains the Zariski open set O, it must be equal to Z( f ). By the preceding remark, Theorem 3.8 (with d = 4, k = 3, O = Z( f ), and 0 3 as specified by Theorem 3.9) then implies that Z( f ) is infinitely ruled by lines, in the sense defined in Sect. 2.6; that is, each point p Z( f ) is incident to infinitely many lines that are fully contained in Z( f ), and, moreover, 0,p 3 = 0,p (which is the set of lines in 0 incident to p). That is, we have shown that 0 3 = 0. In other words, for each p Z( f ), 0,p is of dimension at least 1, or, equivalently, the cone 0,p (which is the union of the lines in 0,p ) is at least two-dimensional. If, for some nonsingular p Z( f ), the cone 0,p were three-dimensional, then, as already noted, Z( f ) would be a hyperplane, contrary to assumption. Thus, for each non-singular point p Z( f ), the cone 0,p is two-dimensional, and 0,p is one-dimensional. We also have dim( 0 ) = dim( 0 3 ) 3. We thus have Corollary 3.10 The union of lines in 3 0 = 0 is equal to Z( f ), and dim( 0 ) = dim( 3 0 ) 3. Severi s theorem The following theorem is a major ingredient in the present part of our analysis. It has been obtained by Severi [34] in 1901, and a variant of it is also attributed to Segre [33]; it is mentioned in a recent work of Rogora [31], in another work of Mezzetti and Portelli [26], and also appears in the unpublished thesis of Richelson [30]. Severi s paper is not easily accessible (and is written in Italian). As a small service to the community, we sketch in Appendix A a proof of this theorem (or rather of a special case of the theorem that arises in our context), suggested to us by A.J. de Jong. 123

89 740 Discrete Comput Geom (2017) 57: Theorem 3.11 (Severi s Theorem [34]) Let X P d (C) be a k-dimensional irreducible variety, and let 0 be an irreducible component of maximal dimension of F(X), such that the lines of 0 cover X. Then the following holds. 1. If dim( 0 ) = 2k 2, then X is a copy of P k (C)(that is, a complex projective k-flat). 2. If dim( 0 ) = 2k 3, then either X is a quadric, or X is ruled by copies of P k 1 (C), i.e., every point 12 p X is incident to a copy of P k 1 (C) that is fully contained in X. As is easily checked, the maximum dimension of 0 is 2k 2. Note also that the cases where dim 0 < 2k 3 are not treated by the theorem (although they might occur); see Rogora [31] for a (partial) treatment of these cases. We apply Severi s theorem to Z( f ) and to the component 0 obtained in Theorem 3.9 and Corollary 3.10, with k = 3 and with dim( 0 ) = 3 = 2k 3. We thus conclude that either Z( f ) is a quadric, a case ruled out in the present part of the analysis (which assumed that deg( f ) 3), or it is ruled by 2-flats. The case where Z( f ) is ruled by 2-flats. In the remaining case, every point p Z( f ) (see the footnote in Theorem 3.11) is incident to at least one 2-flat τ p Z( f ). Let D p denote the set of 2-flats that pass through p and are contained in Z( f ). For a non-singular point p Z( f ),if D p > 2, then p is a (linearly flat and thus) flat point of Z( f ). Recall that we have bounded the number of incidences involving flat points (and lines) by partitioning them among a finite number of hyperplanes, and by bounding the incidences within each hyperplane. (Recall that lines incident to fewer than 3D 3 points of P have been pruned away, losing only O(nD) incidences, and that the remaining lines are all flat.) Repeating this argument here, we obtain the bound O(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + n). In what follows we therefore assume that all points of P are non-singular and non-flat (call these points ordinary for short), and therefore D p =1 or 2, for each such p. Put H 1 (p) (resp., H 1 (p), H 2 (p)) for the 2-flat (resp., two 2-flats) in D p, when D p =1 (resp., D p =2). Clearly, each line in L, containing at least one ordinary point p Z( f ), is fully contained in at most two 2-flats fully contained in Z( f ) (namely, the 2-flats of D p ). Assign each ordinary point p P to each of the at most two 2-flats in D p, and assign each line l L that is incident to at least one ordinary point to the at most two 2-flats that fully contain l and are fully contained in Z( f ) (it is possible that l is not assigned to any 2-flat see below). Changing the notation, enumerate these 2-flats, over all ordinary points p P, asu 1,...,U k, and, for each i = 1,...,k, letp i and L i denote the respective subsets of points and lines assigned to U i, and let m i and n i denote their cardinalities. We then have i m i 2m and i n i 2n, and the total 12 Similar to the definition in Sect. 2.4 for the case of lines, it suffices to require this property for every point in some Zariski-open subset of X. Here too one can show that the two definitions are equivalent. See also the companion paper [37, Lem.11]. 123

90 Discrete Comput Geom (2017) 57: number of incidences within the 2-flats U i (excluding lines not assigned to any 2-flat) is at most k i=1 I (P i, L i ). This incidence count can be obtained exactly as in the first case of the analysis, using the bound in (24). That is, we have k I (P i, L i ) = O(m 2/3 n 1/3 s 1/3 + m + n). i=1 As noted, this bound does not take into account incidences involving lines which are not contained in any of the 2-flats U i (and are therefore not assigned to any such 2-flat). It suffices to consider only lines of this sort that are non-singular and non-flat, since singular or flat lines are only incident to singular or flat points, and we assumed above that all the points of P are ordinary points. If l is a non-singular and non-flat line, and is not fully contained in any of the U i, we call it a piercing line of Z( f ). Lemma 3.12 If l is a piercing line of Z( f ), then the union of lines fully contained in Z( f ) and intersecting l is equal to Z( f ). Proof Let V denote this union. By a suitable extension to four dimensions of a similar result of Sharir and Solomon [36, Lem.5],V is a variety in the complex projective setting, which we assume throughout this part of the analysis. Clearly V Z( f ).IfV is strictly contained in Z( f ), then, since Z( f ) is irreducible, V must be a finite union of irreducible components V 1,...,V k, each of dimension at most two. Let p l be an ordinary point of Z( f ) (since l is non-singular and non-flat, such a point exists), and let H 1 (p) be one of the at most two 2-flats in D p. Note that H 1 (p) is contained in V (because it is a union of lines fully contained in Z( f ) and intersecting l at p). We claim that there exists some V j such that H 1 (p) V j. Indeed, otherwise, the intersection H 1 (p) V j would be (at most) one-dimensional for each j = 1,...,k (a variety strictly contained in a 2-flat is of dimension at most one), and therefore ( k ) V H 1 (p) = V j H 1 (p) = j=1 k (V j H 1 (p)) j=1 is a finite union of varieties of dimension at most one, contradicting the fact that H 1 (p) is contained in V (and is of dimension two). This contradiction establishes the claim. Since H 1 (p) and V j are two-dimensional irreducible varieties and H 1 (p) V j,it follows that H 1 (p) = V j. In other words, for each ordinary point p l there exists a 2-flat H 1 (p) D p which is equal to some component V j. Consider only the components V j that coincide with such a 2-flat. Since there are only finitely many components V j of this kind, one of them, call it V j0, has to intersect l in infinitely many points, and therefore l V j0. That is, l is contained in the 2-flat V j0 that is fully contained in Z( f ). Now pick any ordinary point p P l. By definition, since p V j0, V j0 must be one of the (at most) two 2-flats in D p. But then l is fully contained in that 2-flat, which is one of the U i s, and therefore l is not a piercing line. This contradiction completes the proof. 123

91 742 Discrete Comput Geom (2017) 57: Remark The last step of the proof shows that if a non-singular and non-flat line l contains a point of P then it is piercing (if and) only if it is not contained in any 2-flat fully contained in Z( f ). Lemma 3.13 Let p Z( f ) be an ordinary point. Then p is incident to at most one piercing line. Proof Assume to the contrary that p is incident to two piercing lines l 1,l 2 L. We claim that the 2-flat π 12 that is spanned by l 1 and l 2 is fully contained in Z( f ) (and thus, by the preceding remark, l 1 and l 2 are not piercing lines). Indeed, for any point q l 1, Lemma 3.12 implies that there exists some line l q = l 1, incident to q, that intersects l 2 and is fully contained in Z( f ). When q varies along the nonsingular points of l 1, we get an infinite collection of lines, fully contained in both Z( f ) and π 12, i.e., in their intersection Z( f ) π 12.Ifπ 12 is not contained in Z( f ) then Z( f ) π 12 = π 12 is a degree-d plane curve, so by Lemma 2.3, it contains at most D lines, and therefore cannot contain the infinite union of lines p l p. Therefore, each ordinary point p P is incident to at most one piercing line, and the total contribution of incidences involving ordinary points and piercing lines is at most m. In summary, combining the bounds that we have obtained for the various subcases of the second case, we get the following proposition. As in the first case, here f refers to a single irreducible factor (of the original polynomial or one of its derivatives), D to its degree, and P and L refer to the subsets of the original respective sets of points and lines, that are assigned to f. Proposition 3.14 Let P be a set of m points contained in Z( f ), and let L be a set of n lines contained in Z( f ), and assume that Z( f ) is ruled by lines and that f is of degree 3. Then I (P, L) = I (P, L ) + O ( m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + nd ), (30) where P and L are subsets of P and L, respectively, so that each hyperplane or quadric is O(D 2 )-restricted with respect to L, and each 2-flat contains at most O(D) lines of L. We also have the explicit estimate I (P, L ) = min{o(md 2 + nd), O(m + nd 4 )}. (31) The Induction In summary, after having exhausted all possible cases, we are in the following situation; we finally undo the shorthand notations that we have used, and re-express the various bounds in terms of the original parameters. The first partitioning step has resulted in a collection of irreducible polynomials, which we write as f 1,..., f k, with respective degrees D 1,...,D k, all upper bounded 123

92 Discrete Comput Geom (2017) 57: by the degree D chosen in (27) for the original values of m and n. The points of P have been partitioned among the zero sets Z( f 1 ),...,Z( f k ), into respective pairwise disjoint subsets P 1,...,P k, including a leftover subset P of points outside all the zero sets, and the lines of L have been partitioned among the zero sets, into respective pairwise disjoint subsets L 1,...,L k, so that the zero set to which a line is assigned fully contains it, and including a leftover subset L of lines not fully contained in any zero set. Put m i = P i, n i = L i,fori = 1,...,k, and m = P, n = L. Then m + k i=1 m i = m, and n + k i=1 n i = n. Then I (P, L) is I (P, L )+ k i=1 I (P i, L i ) plus the number of incidences between points assigned to some Z( f i ) and lines not fully contained in Z( f i ). (Note that I (P \ P, L ) also counts incidences of this kind.) As we have argued, the total number of these additional incidences is O(nD). That is, we have, for any choice of the degree D, k I (P, L) I (P, L ) + O(nD) + I (P i, L i ). (32) For each i, the preceding analysis culminates in the following bound. I (P i, L i ) = I (Pi, L i ) + O(m1/2 i n 1/2 i q 1/4 + m 2/3 i n 1/3 i s 1/3 + m i + n i D), (33) where, for each i, Pi and Li are respective subsets of P i and L i, so that each hyperplane or quadric is O(D 2 )-restricted with respect to Li, and each 2-flat contains at most O(D) lines of Li. We also have the explicit estimate I (P i, L i ) = min{o(m i D 2 + n i D), O(m i + n i D 4 )}, for each i. (34) In addition, for the large values of D in (14), we have i=1 I (P, L ) = O(m 2/5 n 4/5 + m). (35) Induction-free derivation of the bound To proceed with the analysis, for general values of m and n, we bound the various quantities I (Pi, L i ) using induction. However, as asserted in the theorems, the cases where m n 6/7 or m n 5/3 admit an inductionfree argument that yields the improved bound in (6), and we first dispose of these cases. (Recall that these are the original values of m and n, the respective sizes of the entire input sets P and L.) Assume first that m n 6/7. We substitute (33), the first bounds in (34), and (35)into (32). Using the Cauchy-Schwarz and Hölder s inequalities, we have i m1/2 i n 1/2 i m 1/2 n 1/2 and i m2/3 i n 1/3 i m 2/3 n 1/3.Wealsohave i m i m and i n i n.in total we thus get I (P, L) = O(m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + md 2 + nd) = O(m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n), where we have used the fact that md 2 + nd = O(m 2/5 n 4/5 + n) for the choice D = O(m 2/5 /n 1/5 ) in (27). This establishes (6) for this case. The case m n 5/3 is 123

93 744 Discrete Comput Geom (2017) 57: handled in the same manner, using the second bounds O(m i + n i D 4 ) in (34) instead, and the fact that the sum of these bounds is O(m) when m n 5/3. The induction via a new partitioning We now proceed with the general case, where induction is needed. To simplify the notation, we (again, but only temporarily) drop the indices, and consider one of many (possibly a nonconstant number of) subproblems, involving a set P (=Pi )ofm ( m i) points and a set L (= Li )ofn ( n i) lines, so that each hyperplane or quadric is O(D 2 )-restricted for L, and each 2-flat contains at most O(D) lines of L; here D (=D i ) is the degree of the corresponding factor f (= f i ), which is upper bounded by the value in (14). In what follows we will use this latter bound for (an upper bound on) the D i s. To make the induction work, we choose a degree E, typically much smaller than D (see below for the actual value), and construct a new partitioning polynomial h of degree E for P. (Although P Z( f ) and each line of L is fully contained in Z( f ), we ignore here f completely, possibly losing some structural properties of P and L, and consider only the partitioning induced by h.) With an appropriate value of r = (E 4 ), we obtain O(r) cells, each containing at most m/r points of P, and each line of L either crosses at most E + 1 cells, or is fully contained in Z(h). Set P 0 := P Z(h) and P := P \ P 0. Similarly, denote by L 0 the set of lines of L that are fully contained in Z(h), and put L := L \ L 0. We repeat the whole analysis done so far, but with h and its degree E instead of f and D, for the points of P and the lines of L. That is, we apply, to our P and L, the bounds given in (32), (33), and (34) [but not the one in (35)], with E instead of D. Moreover, in this application we exploit the property that each hyperplane or quadric is O(D 2 )-restricted with respect to L, and each 2-flat contains at most O(D) lines of L. We thus get the following recurrence (where the parameters k, P i, L i, etc., are new and depend on h, butwe recycle the notation in the interest of simplicity). I (P, L) I (P, L ) + O(nE) + = I (P, L ) + O(nE) + + k i=1 k I (P i, L i ) i=1 k i=1 O(m 1/2 i n 1/2 i D 1/2 + m 2/3 I (P i, L i ) i n 1/3 i D 1/3 + m i + n i E). Concretely, P is the subset of the points of P contained in the cells of the h-partition, L is the subset of lines of L not fully contained in Z(h), P i and L i are the subsets of the points and lines assigned to the various irreducible factors h i of h and of its derivatives, and Pi, L i are the excluded subsets, as provided in Propositions 3.7 and Using the Cauchy-Schwarz and Hölder s inequalities in the second sum, we get, for a suitable absolute constant a, 123

94 Discrete Comput Geom (2017) 57: We have I (P, L) I (P, L ) + a(m 1/2 n 1/2 D 1/2 + m 2/3 n 1/3 D 1/3 + m + ne) k + I (Pi, L i ). i=1 k k I (Pi, L i ) a ( i=1 i=1 ) min{m i E 2 + n i E, m i + n i E 4 } min{a (me 2 + ne), a (m + ne 4 )}, for a suitable absolute constant a. That is, slightly increasing the coefficient a, we have I (P, L) I (P, L ) + a(m 1/2 n 1/2 D 1/2 + m 2/3 n 1/3 D 1/3 + m + ne) + min{ame 2, ane 4 }. (36) We next turn to bound I (P, L ). For each cell τ of R 4 \ Z(h), put P τ := P τ, and let L τ denote the set of the lines of L that cross τ; put m τ = P τ m/r (where r = (E 4 )), and n τ = L τ. Since every line l L crosses at most E +1 components of R 4 \ Z(h),wehave τ n τ n(1 + E). To simplify the application of the induction hypothesis within the cells of the partition, we want to make the subproblems be of uniform size, so that m τ = m/e 4 and n τ = n/e 3 for each τ (the latter quantity, up to some constant, is the average number of lines crossing a cell). This is easy to enforce: To achieve m τ = m/e 4,we simply partition P τ into m τ /(m/e 4 ) =O(1) subsets, each consisting of at most m/e 4 points, and analyze each subset separately. Similarly, if τ is crossed by ξn/e 3 lines, for ξ>1, we treat τ as if it occurs ξ times, where each incarnation involves all the points of (each of the constantly many corresponding subsets of) P τ, and at most n/e 3 lines of L τ. As is easily verified, the number of subproblems remains O(E 4 ), with a larger constant of proportionality. We apply the induction hypothesis [i.e., the inequality (13)] for each cell τ. Itis here that the extra factor 2 c log m in the bound in the theorem comes into play; as noted earlier, its role is to make the induction step work. We obtain I (P τ, L τ ) 2 c log m τ (m 2/5 τ n 4/5 τ + m τ )+β A(m 1/2 τ n 1/2 τ D 1/2 + m 2/3 τ n 1/3 τ D 1/3 + n τ ) = 2 c log(m/e 4) ((m/e 4 ) 2/5 (n/e 3 ) 4/5 + m/e 4 ) + β A((m/E 4 ) 1/2 (n/e 3 ) 1/2 D 1/2 +(m/e 4 ) 2/3 (n/e 3 ) 1/3 D 1/3 + n/e 3 ), for a suitable absolute constant β. Summing this bound over all cells τ, that is, multiplying it by O(E 4 ), we get, for a suitable absolute constant b, 123

95 746 Discrete Comput Geom (2017) 57: I (P τ, L τ ) b 2 c log(m/e 4) (m 2/5 n 4/5 + m) τ + ba(m 1/2 n 1/2 D 1/2 E 1/2 + m 2/3 n 1/3 D 1/3 E 1/3 + ne). (37) We have 2 c ( log(m/e 4) = 2 c log m 4logE = 2 c log m ( ) < 2 c log m 1 2logE log m = 1 4logE log m ) 1/2 2 c log m 2 2c log E/ log m. We choose E to ensure that 2 2c log E/ log m > 2b, or 2c log E log m > log(2b), or log E > log(2b) log m. 2c That is, we choose E > 2 c log m, for c = log(2b) 2c < c/3, (38) where the last constraint can be enforced if c is chosen sufficiently large. With this constraint on the choice of E, (37) becomes I (P τ, L τ ) 2 1 2c log m (m 2/5 n 4/5 + m) τ + ba(m 1/2 n 1/2 D 1/2 E 1/2 + m 2/3 n 1/3 D 1/3 E 1/3 + ne). (39) Adding this bound to the one in (36), we get I (P, L) 2 1 2c log m (m 2/5 n 4/5 + m) + (ba+ a)(m 1/2 n 1/2 D 1/2 E 1/2 + m 2/3 n 1/3 D 1/3 E 1/3 + ne) + am + min{ame 2, ane 4 }. (40) Returning to the original notations, we have just bounded I (P i = 1,...,k. Concretely, we have shown that, for each i, i, L i ), for any I (Pi, L i ) 2 1 2c log m i (m 2/5 i n 4/5 i + m i ) + (ba+ a)(m 1/2 i n 1/2 i D 1/2 E 1/2 i + m 2/3 i n 1/3 i D 1/3 E 1/3 i + n i E i ) + am i + min{am i Ei 2, an i Ei 4 }, (41) where E i is the degree of the new partitioning polynomial that is constructed for P i and L i. 123

96 Discrete Comput Geom (2017) 57: We now add up these bounds, using (32), (33), and (35), and replacing the E i s by a common upper bound E that we will choose shortly. We thus get the following bound, where now P and L stand, respectively, for the original, entire input sets of points and lines. I (P, L) γ(m 2/5 n 4/5 + m) + γ nd k + γ (m 1/2 i n 1/2 i q 1/4 + m 2/3 i n 1/3 i s 1/3 + m i + n i D) γ + i=1 k i=1 2 c log m i (m 2/5 i n 4/5 i + m i ) k (m 1/2 i n 1/2 i D 1/2 E 1/2 + m 2/3 i n 1/3 i D 1/3 E 1/3 + n i E + m i ) (42) i=1 k min{am i E 2, an i E 4 }, (43) i=1 for a suitable absolute constant γ. With several applications of the Cauchy Schwarz and Hölder s inequalities we get I (P, L) (γ c log m )(m 2/5 n 4/5 + m) + γ(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + nd) + γ(m 1/2 n 1/2 D 1/2 E 1/2 + m 2/3 n 1/3 D 1/3 E 1/3 + ne + m) + min{ame 2, ane 4 }. (44) We now bifurcate depending on the relation between m and n, where now, as in the recurrence just derived, m and n refer to the original values of these parameters. The case m = O(n 4/3 ). Recall that here we take D = O(m 2/5 /n 1/5 ).Itiseasily checked that, for this choice of D, each of the terms m 1/2 n 1/2 D 1/2, m 2/3 n 1/3 D 1/3, m, and nd n, iso(m 2/5 n 4/5 ), because n 1/2 m = O(n 4/3 ). We choose 13 E = 2 c log m. This turns (44) into the bound I (P, L) (γ c log m + μ2 2c log m )(m 2/5 n 4/5 + m) γ(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 ), for suitable absolute constants μ and γ. The choice of c, and the assumption that m M 0 and that M 0 is sufficiently large, ensure that γ + μ2 2c log m < 1 2 2c log m, 13 This rather minuscule value of E is only needed when m n 4/3 ; for smaller values of m, much larger values of E can be chosen. 123

97 748 Discrete Comput Geom (2017) 57: and thus we get I (P, L) 2 c log m (m 2/5 n 4/5 + m) + γ(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 ), which is the bound asserted in (5). The case m = (n 4/3 ).HerewetakeD = O(n/m 1/2 ). It is easily checked that, for this choice of D, each of the terms m 1/2 n 1/2 D 1/2, m 2/3 n 1/3 D 1/3, m 2/5 n 4/5, and nd n,iso(m), because m = (n 4/3 ). We choose, as before, E = 2 c log m (or a larger value when applicable), and note that, for m M 0 sufficiently large, the term ne 4 is also O(m). This turns (44) into the bound I (P, L) (γ c log m + μ2 2c log m )(m 2/5 n 4/5 + m) + γ(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 ), for suitable absolute constants μ and γ. As above, the choice of c, and the assumption that m M 0 and that M 0 is sufficiently large, ensure that and thus we get γ + μ2 2c log m < 1 2 2c log m, I (P, L) 2 c log m (m 2/5 n 4/5 + m) + γ(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 ), again establishing the bound in (5). Therefore, in both cases, we completed, at last, the induction step and thus establishing the general upper bound (5) in the theorem. The improved bound in (6), for m n 6/7 or for m n 5/3, has already been established. With the lower bound construction, given in the following section, the proof of the theorem is completed. 4 The Lower Bound In this section we present a construction that shows that the bound asserted in the theorem is worst-case tight (except for the factor 2 c log m ), for each m and n, and for q and s in suitable corresponding ranges, made precise below. The construction is a generalization to four dimensions of a construction due to Elekes; see [9]. (A three-dimensional generalization has been used in Guth and Katz [15] for their lower bound construction.) We have already remarked that the lower order terms m 1/2 n 1/2 q 1/4 and m 2/3 n 1/3 s 1/3 are both worst-case tight, as they can be attained by a suitable packing of points and lines into hyperplanes (for the first term) or planes (for the second term). Specifically, assume that s q, and create n/q parallel hyperplanes, and place on each of them q lines and mq/n points in a configuration that attains the 123

98 Discrete Comput Geom (2017) 57: three-dimensional lower bound as in Guth and Katz [15]. Note that in this construction no plane contains more than q s lines, as desired. Overall, we get (n/q) ((mq/n) 1/2 q 3/4 ) = (m 1/2 n 1/2 q 1/4 ) incidences. A similar (and simpler) construction can be carried out for the second term m 2/3 n 1/3 s 1/3. We therefore focus on the term m 2/5 n 4/5 (the remaining terms m and n are trivial to attain). We fix two integer parameters k and l, with concrete values that will be set later, and take P to be the set of vertices of the integer grid {(x, y, z,w) 1 x k, 1 y, z,w 2kl}. We have P =8k 4 l 3. We then take L to be the set of all lines of the form y = ax + b, z = cx + d, w = ex + f, (45) where 1 a, c, e l and 1 b, d, f kl.wehave L =k 3 l 6. Note that each line in L has k incidences with the points of P, one for each x = 1, 2,...,k, so I (P, L) = k 4 l 6 = ( P 2/5 L 4/5 ), as is easily checked. Note that L 1/2 P 8 L 4/3, which is (asymptotically) the range of interest for this bound to be significant: when P < L 1/2 we have the trivial bound I (P, L) = O( L ), and when P > L 4/3, the leading term in the bound changes qualitatively to O(m), which is trivial for a lower bound. Moreover, for any pair of integers m, n, with n 1/2 m n 4/3, we can find k and l for which P = (m) and L = (n). Specifically, choose k = (m 2/5 /n 1/5 ) and l = (n 4/5 /m 3/5 ); both are 1 for the range of m and n under consideration. To complete the construction, we show that no hyperplane or quadric contains more than q 0 := O ( L 6/5 / P 2/5) = O(k 2 l 6 ) lines of L, and no plane contains more than s 0 := O ( L 7/5 / P 4/5) = O(kl 6 ) lines of L. As an easy calculation shows, these threshold values of q and s are such that, for q > q 0 or s > s 0,the corresponding lower-dimensional term m 1/2 n 1/2 q 1/4 or m 2/3 n 1/3 s 1/3 dominates the leading term m 2/5 n 4/5 (for the former domination to arise, we need to assume, as above, that q s), making the above construction pointless (see below for more details). The actual values of q and s that we will now derive are actually much smaller. To estimate our q and s, leth be an arbitrary hyperplane. If h is orthogonal to the x-axis then it does not contain any line of L, as is easily checked, so we may assume that h intersects any hyperplane of the form x = i in a 2-plane π i. The intersection of P with x = i is a 2kl 2kl 2kl lattice, that we denote as Q i. Every line λ L in h meets π i at a single point (as noted, it cannot be fully contained in π i ), which is necessarily a point in Q i (every line of L contains a point of every Q i ).Thesizeof π i Q i is easily seen to be O((kl) 2 ), and each point is incident to at most l 2 lines that 123

99 750 Discrete Comput Geom (2017) 57: lie in h. To see this latter property, substitute the equations (45) ofalineofl into the linear equation defining h,sayax + By + Cz+ Dw 1 = 0 (where B, C and D are not all 0). This yields a linear equation in x, whose x-coefficient has to vanish. This in turn yields a linear equation in a, c, and e, which can have at most l 2 solutions over [1,...,l] 3 (it is easily checked that the x-coefficient cannot be identically zero for all choices of a, c, e). The number of lines of point-line incidences of P and L within h is thus O(l 2 (kl) 2 ) = O(k 2 l 4 ). Since each line is incident to k points, necessarily all lying in h, it follows that the number of lines of L in h is O(k 2 l 4 /k) = O(kl 4 ), which is always smaller than q 0. This analysis easily extends to show that no quadric contains more than O(kl 4 ) lines of L; we omit the routine details. Finally, let π be a 2-plane, where again we may assume that π is not orthogonal to the x-axis. Then π meets a hyperplane x = i in a line μ, and μ Q i contains at most kl points. Every line λ in π meets μ at one of these points and, arguing as above, each such point can be incident to at most l lines that lie in π (now instead of one linear equation in a, c, e, we get two). Hence, π contains at most kl 2 /k = l 2 lines of L, which is always smaller than s 0. We have thus shown that the bound in Theorem 1.3 is (almost) tight in the worst case. The bound will be tight when P L 6/7, which occurs when k l 3/2,asan easy calculation shows. Remark As the analysis shows, the various constructions impose certain constraints on the values of q and s, and are therefore not as general (in terms of these parameters) as one might hope. It would be interesting to extend the constructions so that they apply to more general values of q and s. 5 Conclusion The results of this paper (almost) settle the problem of point-line incidences in four dimensions, but they raise several interesting and challenging open problems. Among them are: (a) Get rid of the factor 2 c log m in the bound. We have achieved this improvement when m is not too close to n 4/3, so to speak, allowing us to use the weak but noninductive bounds and complete the analysis in one step. We believe that the ranges of m where this can be done can be enlarged, e.g. by improving the weak bounds. A concrete step in this direction would be to improve the term O(nD 4 ) in the second bound in Proposition 3.3, which, as already remarked, appears to be too weak. It would also be interesting to improve the bound using the strategy in [36,38], which generates a sequence of ranges of m, converging to m = (n 4/3 ), where in each range the improved bound (6) holds, with a different constant of proportionality A. (For readers familiar with the approaches in [35,38], we note that the reason this technique does not appear to apply here is the multitude of subproblems, each with its own m i, n i. The induction in [35,38] generates subproblems in which the relation between m and n falls into a range already handled. Here though we do not know how to enforce this property, as we have little control over the values of m, n in the resulting subproblems. 123

100 Discrete Comput Geom (2017) 57: (b) Extend (and sharpen) the bound of Corollary 1.4 for any value of k. In particular, is it true that the number of intersection points of the lines (this is the case k = 2; the intersection points are also known as 2-rich points)iso(n 4/3 + nq 1/2 + ns)? We conjecture that this is indeed the case. (In this conjecture we assume that we have already managed to get rid of the factor 2 c log m, as in (a) above.) A deeper question, extending a similar open problem in three dimensions that has been posed by Guth and others (see, e.g., Katz s expository note [23]), is whether the above conjectured bound can be improved when q = o(n 2/3 ) and s = o(n 1/3 ), that is, when the second and third terms in the conjectured bound become much smaller than the term n 4/3. We also note that if we could establish such a bound for the number of k-rich points, for any constant k (when q and s are not too large), then the case of large m (that is, m = (n 4/3 )) would become vacuous, as only O(n 4/3 ) points could be incident to more than k lines. (c) Extend the study to five and higher dimensions. In a preliminary ongoing study, joint with Adam Sheffer, we can do it using a constant-degree partitioning polynomial, with the disadvantages discussed above (slightly weaker bounds, significantly more restrictive assumptions, and inferior lower-dimensional terms). The leading terms in the resulting bounds, for points and curves in R d,are O(m 2/(d+1)+ε n d/(d+1) + m 1+ε ), for any ε > 0. See also Dvir and Gopi [7] and Hablicsek and Scherr [16] for recent related studies. Obtaining sharper results in such general settings, like the ones obtained in this paper, is quite challenging algebraically, although some of the tools developed in this work seem promising for higher dimensions too. (d) If we are given in advance that the points and lines lie in some algebraic surface of a given degree D > 2, can we improve the bound and/or simplify the analysis? In our companion work [37] we achieve these goals for the three-dimensional case, improving the bound of Guth and Katz [15] in such special cases. (e) Elaborating on item (a) above, we note that the culprit Proposition 3.3, which produces the weak bounds that force us to go into the induction, is only used in the case where Z( f, g) is two-dimensional, and the difficulty there lies in bounding the number of incidences within a two-dimensional ruled surface (be it either one irreducible ruled surface of large degree, or the union of many irreducible ruled surfaces of small degree). The analysis of the three-dimensional analogous situation (addressed in Guth and Katz [15]), cannot be applied here, since the degree of the underlying surface in four dimensions is O(D 2 ) instead of D in [15]. In a recent study of Szermerédi-Trotter type theorems in three dimensions [24], Kollár uses the arithmetic genus of curves to prove effective bounds on the number of point-line incidences in three dimensions. In four dimensions, the situation is more involved, but we hope that the arithmetic genus of the surface Z( f, g) may yield effective bounds for the number of incidences within this surface. Acknowledgements Work on this paper by Noam Solomon and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S. Israel Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. We would like to thank several people whose advice, comments and guidance have helped us a lot in our work on the paper. They are János Kollár, Martin Sombra, Aise J. de Jong, and Saugata Basu. 123

101 752 Discrete Comput Geom (2017) 57: We also thank the anonymous referees for their helpful comments on the paper. In addition, as noted, part of the work on the paper was carried out during the special semester on Algebraic Techniques for Combinatorial and Computational Geometry, held at the Institute for Pure and Applied Mathematics at UCLA, in the Spring of We are grateful for the pleasant working environment provided by IPAM, and for the helpful interaction with additional colleagues, including Larry Guth, Nets Hawk Katz, Terry Tao, Jordan Ellenberg, and many others. Appendix: Severi s Theorem In this appendix we sketch a proof of Severi s theorem (Theorem 3.11). First, recall from Sect. 2.1 that a real (resp., complex) surface X is ruledbyreal (resp., complex) lines if every point p X in a Zariski open dense set is incident to a real (resp., complex) line that is fully contained in X. This definition has been used in several recent works (see, e.g., [15]); this is a slightly weaker condition than the classical condition that requires that every point of X be incident to a line contained in X. Nevertheless, as we show next, the two are equivalent. Lemma 6.1 Let f R[x, y, z] (resp., f R[x, y, z,w]) be an irreducible polynomial such that there exists a Zariski open dense set U Z( f ), so that each point in the set is incident to a line, fully contained in Z( f ). Then FL f (resp., FL 4 f ) vanishes identically on Z( f ), and Z( f ) is ruled by lines. Proof By assumption and definition, FL f (resp., FL 4 f ) vanishes on U. If it vanishes on Z( f ), Theorem 2.11 implies that Z( f ) is ruled. Otherwise, Z( f, FL f ) (resp., Z( f, FL 4 f )) is properly contained in Z( f ) and contains U. Since Z( f ) is irreducible, this latter variety must be of dimension at most 1 (resp., 2). On the other hand, Z( f, FL f ) (resp., Z( f, FL 4 f )) is Zariski closed set (by definition of the Zariski topology) and therefore contains its Zariski closure. As U is Zariski dense, its Zariski closure is Z( f ). Remark In Sharir and Solomon [37], we have proved the same statement without using the Flecnode polynomial. This phenomenon generalizes to k-flats instead of lines (and the proof translates verbatim). Lemma 6.2 Let V be an irreducible variety for which there exists a Zariski open subset U V with the property that each point p U is incident to a k-flat that is fully contained in V. Then this property holds for every point of V. We now proceed to sketch a proof of Severi s theorem. For convenience, we repeat its statement. Theorem 3.11 (Severi s Theorem [34]) Let X P d (C) be a k-dimensional irreducible variety, and let 0 be a component of maximal dimension of F(X), such that the lines of 0 cover X. Then the following holds. 1. If dim( 0 ) = 2k 2, then X is a copy of P k (C)(that is, a complex projective k-flat). 123

102 Discrete Comput Geom (2017) 57: If dim( 0 ) = 2k 3, then either X is a quadric, or X is ruled by copies of P k 1 (C), i.e., every point p X is incident to a copy of P k 1 (C) that is fully contained in X. We sketch a proof in the case k = 3, d = 4, under the simplifying assumption that for any non-singular x X, 0,x is infinite; this assumption holds in our application of the theorem (by the informal dimensionality argument mentioned in the paper, it holds on average in general for these parameters). Our proof is based on a sketch provided by A. J. de Jong, via private communication, and we are very grateful for his assistance. Sketch of Proof For x X, we recall that 0,x denotes the cone of lines (i.e., union of lines) of 0,x The proof consists of the following steps. (1) Assume first that dim( 0 ) = 2k 2 = 4. Then there exists some non-singular point x 0 X with dim( 0,x0 ) = 2. Indeed, if, for all non-singular points x X, dim( 0,x ) 1, then dim( 0 )<4 (see the analysis in Theorem 3.9, and the preceding analysis), contradicting the assumption in this case. By an argument that has already been sketched earlier, this implies that dim( 0,x0 ) = 3, i.e., the cone of lines in 0,x0 through x 0 is three-dimensional, and therefore X = 0,x0. As x 0 is non-singular, it follows that X must be a hyperplane, as claimed. (2) Consider next the case where dim( 0 ) = 2k 3 = 3, and for any non-singular point x X, 0,x is 1-dimensional (as just argued, if 0,x is two-dimensional for some non-singular x X, then X is a hyperplane). In other words, 0,x, parameterized by the direction of its lines, is a curve in PT x X = P 2 (C); put e x for its degree. If e x = 1, then 0,x contains a 2-flat. We next define a plane-flecnode polynomial system associated with X, that expresses, for a point x X, the existence of a 2-flat H, such that H osculates to X to order 3 at x. Since X is a hypersurface, we can write X = Z( f ), for a suitable 4-variate polynomial f (see Sect. 2), and assume that f is irreducible (as X is irreducible). We represent a 2-flat through the origin in C 4 (ignoring the lower-dimensional family of 2-flats that cannot be represented in this manner) as H v0,v 1,v 2,v 3 := {(x, y, z,w) z = v 0 x + v 1 y,w= v 2 x + v 3 y}, (46) for v 0,v 1,v 2,v 3 C. The 2-flat H v0,v 1,v 2,v 3 is said to osculate to X = Z( f ) to order k at p, if the Taylor expansion of f at p along H satisfies f (p + (x, y,v 0 x + v 1 y,v 2 x + v 3 y)) = O(x k+1 + y k+1 ). (47) This translates into a system of homogeneous polynomial equations in v 0,v 1,v 2,v 3, involving the partial derivatives of f up to order k. Specializing to the case k = 3, the plane-flecnode polynomial system, PFL f, associated with f, is obtained by eliminating v 0,v 1,v 2,v 3 from these equations (for osculation up to order 3). This is the multipolynomial resultant system of the polynomials defining these equations 123

103 754 Discrete Comput Geom (2017) 57: up to order 3, with respect to v 0,v 1,v 2,v 3 (see Van der Waerden [45, Chap. XI] for details). Another theorem of Landsberg [25, Thm. 1] states that, if, for every g PFL f, g vanishes identically on X, then X is ruled by 2-flats, which finishes the proof in this case. Therefore, we may assume that X Z(PFL f ) is a Zariski closed proper subset of X. By definition of PFL f, it follows that for every non-singular point x X \ Z(PFL f ) (namely, outside the Zariski closed set Z(PFL f )), we have e x > 1. Indeed, if e x = 1, then, as observed above, there is a 2-flat incident to x, and fully contained in X, implying that for every g PFL f, g(x) = 0, contradicting the assumption that x X \ Z(PFL f ). For a generic hyperplane H in P 4 (C), which is not contained in Z(PFL f ), put S H := X H. As observed above, X Z(PFL f ) is properly contained in X, which in turn implies that, for a generic hyperplane H in P 4 (C), S H is not fully contained in Z(PFL f ). Indeed, let g be a polynomial in PFL f that does not vanish identically on X. Then X Z(g) = Z( f, g) is strictly contained in X = Z( f ), and since Z( f ) is irreducible, it follows that Z( f, g) is two-dimensional. Therefore, for a generic hyperplane H, X Z(PFL f ) H is contained in the one-dimensional variety Z( f, g) H, and thus cannot contain the two-dimensional variety S H. Let x X be a non-singular point, and let H be a hyperplane in P 4 (C), which is incident to X and not contained in Z(PFL f ). We claim that for a generic H, there are e x distinct lines that are incident to x and fully contained in S H. Indeed, the intersection of the hyperplane H with T x X is a 2-flat in T x X containing x. Taking its projectivization (where the point x is regarded as 0), namely, PT x X = P 2,the (generic) 2-flat T x X H becomes a (generic) line. The degree of 0,x PT x X is e x. Therefore, the intersection of x with a line in PT x X = P 2 consists of e x points, which are distinct since the line is generic. Therefore, its intersection with 0,x consists of e x distinct points. These e x distinct (projective) points represent e x distinct lines, incident to x and fully contained in X H = S H, as claimed. We say that a pair (x, H), where H is a hyperplane in P 4 (C) and x S H,isadequate if there are e x distinct lines incident to x that are fully contained in S H. Since a generic point x is non-singular, the previous paragraph implies that a generic pair (x, H) is adequate. Therefore, by changing the order of quantifiers, fixing a generic hyperplane H, a generic point x S H is such that the pair (x, H) is adequate. By Bertini s Theorem (see, e.g., Harris [17, Thm ]), the irreducibility of X implies that for a generic hyperplane H, the surface S H is an irreducible surface in H = P 3 (C). For a generic point x S H, that is, outside an algebraic curve C H in S H, the pair (x, H) is adequate. Therefore, there are e x distinct lines that are incident to x and fully contained in S H, which, by Lemma 6.1, implies that S H is a ruled surface. Moreover, for any x S H \ Z(PFL f ),wehavee x > 1. As observed above, PFL f does not vanish identically on S H, implying that Z(PFL f ) S H is a Zariski closed proper subset of S H, i.e., an algebraic curve contained in S H. Adding this curve to C H, it follows that outside this algebraic curve, each point of S H is incident to at least two lines fully contained in S H. By Sharir and Solomon [37, Lem. 9], this implies that S H is either a 2-flat or a regulus. If X is of degree greater than two, then, for a generic hyperplane H, S H is a (two-dimensional) surface of degree greater than two. 123

104 Discrete Comput Geom (2017) 57: Therefore, X must be of degree at most two, namely, X is either a hyperplane or a quadric. If X is a hyperplane, then 0 is four-dimensional, contrary to the present assumption, so finally, we deduce that X is a quadric, and the proof is complete. References 1. Basu, S., Sombra, M.: Polynomial partitioning on varieties of codimension two and pointhypersurface incidences in four dimensions. Discrete Comput. Geom. 55(1), (2016). Also in arxiv: Beauville, A.: Complex Algebraic Surfaces, vol. 34. Cambridge University Press, Cambridge (1996) 3. Clarkson, K., Edelsbrunner, H., Guibas, L., Sharir, M., Welzl, E.: Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom. 5, (1990) 4. Cox, D., Little, J., O Shea, D.: Using Algebraic Geometry. Springer, Heidelberg (2005) 5. Cox, D., Little, J., O Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, Heidelberg (2007) 6. Dvir, Z.: On the size of Kakeya sets in finite fields. J. Am. Math. Soc. 22, (2009) 7. Dvir, Z., Gopi, S.: On the number of rich lines in truly high dimensional sets. In: Proceedings of 30th Annual ACM Symposium on Computational Geometry, pp (2015) 8. Edge, W.L.: The Theory of Ruled Surfaces. Cambridge University Press, Cambridge (2011) 9. Elekes, G.: Sums versus products in number theory, algebra and Erdős geometry a survey. Paul Erdős and His Mathematics II. Bolyai Mathematical Society Studies, vol. 11, pp Bolyai Mathematical Society, Budapest (2002) 10. Elekes, G., Kaplan, H., Sharir, M.: On lines, joints, and incidences in three dimensions. J. Comb. Theory, Ser. A 118, (2011). Also in arxiv: Fuchs, D., Tabachnikov, S.: Mathematical Omnibus: Thirty Lectures on Classic Mathematics. American Mathematical Society, Providence, RI (2007) 12. Fulton, W.: Introduction to Intersection Theory in Algebraic Geometry, Expository Lectures from the CBMS Regional Conference Held at George Mason University, June 27 July 1, 1983, Vol. 54. AMS Bookstore (1984) 13. Guth, L.: Distinct distance estimates and low-degree polynomial partitioning. Discrete Comput. Geom. 48, 1 17 (2014). Also in arxiv: Guth, L., Katz, N.H.: Algebraic methods in discrete analogs of the Kakeya problem. Adv. Math. 225, (2010). Also in arxiv: v1 15. Guth, L., Katz, N.H.: On the Erdős distinct distances problem in the plane. Ann. Math. 181, (2015). Also in arxiv: Hablicsek, M., Scherr, Z.: On the number of rich lines in high dimensional real vector spaces. Discrete Comput. Geom. 55(1), 1 8 (2014). Also in arxiv: Harris, J.: Algebraic Geometry: A First Course, vol Springer, New York (1992) 18. Hartshorne, R.: Algebraic Geometry. Springer, New York (1983) 19. Ivey, T.A., Landsberg, J.M.: Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics, vol. 61. American Mathematical Society, Providence, RI (2003) 20. Kaplan, H., Sharir, M., Shustin, E.: On lines and joints. Discrete Comput. Geom. 44, (2010) 21. Kaplan, H., Matoušek, J., Safernová, Z., Sharir, M.: Unit distances in three dimensions. Comb. Prob. Comput. 21, (2012). Also in arxiv: Kaplan, H., Matoušek, J., Sharir, M.: Simple proofs of classical theorems in discrete geometry via the Guth Katz polynomial partitioning technique. Discrete Comput. Geom. 48, (2012). Also in arxiv: Katz, N.H.: The flecnode polynomial: a central object in incidence geometry, in arxiv: Kollár, J.: Szemerédi Trotter-type theorems in dimension 3. Adv. Math. 271, (2015). Also in arxiv: Landsberg, J.M.: Is a linear space contained in a submanifold? On the number of derivatives needed to tell. J. Reine Angew. Math. 508, (1999) 26. Mezzetti, E., Portelli, D.: On Threefolds Covered by Lines. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 70, no. 1. Springer, Heidelberg (2000) 123

105 756 Discrete Comput Geom (2017) 57: Pach, J., Sharir, M.: Geometric incidences. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp American Mathematical Society, Providence, RI (2004) 28. Pressley, A.: Elementary Differential Geometry. Springer Undergraduate Mathematics Series. Springer, London (2001) 29. Quilodrán, R.: The joints problem in R n. SIAM J. Discrete Math. 23(4), (2010) 30. Richelson, S.: Classifying Varieties with Many Lines, a senior thesis, Harvard University (2008) Rogora, E.: Varieties with many lines. Manuscr. Math. 82(1), (1994) 32. Salmon, G.: A Treatise on the Analytic Geometry of Three Dimensions, vol. 2, 5th edn. Hodges, Figgis and co. Ltd., Dublin (1915) 33. Segre, B.: Sulle V n contenenti più di n k S k, Nota I e II. Rend. Accad. Naz. Lincei 5( ), (1948) 34. Severi, F.: Intorno ai punti doppi impropri etc. Rend. Cir. Math. Palermo 15(10), (1901) 35. Sharir, M., Solomon, N.: Incidences between points and lines in R 4. In: Proceedings of 30th Annual Symposium on Computational Geometry, pp (2014) 36. Sharir, M., Solomon, N.: Incidences between points and lines in three dimensions, Intuitive Geometry. Also In: Pach, J. (ed.) Proceedings of 31st Annual Symposium on Computational Geometry. Also in arxiv: (2015) 37. Sharir, M., Solomon, N.: Incidences between points and lines on a two-dimensional variety, in arxiv: Sharir, M., Sheffer, A., Zahl, J.: Improved bounds for incidences between points and circles. Comb. Prob. Comput. 24(03), (2015). Also in arxiv: Solomon, N., Zhang, R., Solomon,: Highly incidental patterns on a quadratic hypersurface in R 4,in arxiv: Solymosi, J., Tao, T.: An incidence theorem in higher dimensions. Discrete Comput. Geom. 48, (2012) 41. Székely, L.: Crossing numbers and hard Erdős problems in discrete geometry. Combt. Prob. Comput. 6, (1997) 42. Szemerédi, E., Trotter, W.T.: Extremal problems in discrete geometry. Combinatorica 3, (1983) 43. Tao, T.: From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE. Notices Am. Math. Soc. 48(3), (2001) 44. Tao, T.: The Cayley Salmon theorem via classical differential. 03/28/the-cayley-salmon-theorem-via-classical-differential-geometry/ 45. van der Waerden, B.L., Artin, E., Noether, E.: Modern Algebra, vol. 2. Springer, Heidelberg (1966) 46. Warren, H.E.: Lower bound for approximation by nonlinear manifolds. Trans. Am. Math. Soc. 133, (1968) 47. Zahl, J.: An improved bound on the number of point-surface incidences in three dimensions. Contrib. Discrete Math. 8(1), (2013). Also in arxiv: Zahl, J.: A Szemerédi Trotter type theorem in R 4. Discrete Comput. Geom. 54(3), (2015). Also in arxiv:

106 3 Highly incidental patterns on a quadratic hypersurface in R 4 97

107 Discrete Mathematics 340 (2017) Contents lists available at ScienceDirect Discrete Mathematics journal homepage: Note Highly incidental patterns on a quadratic hypersurface in R 4 Noam Solomon a, *, Ruixiang Zhang b a School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel b Department of Mathematics, Princeton University, Princeton, NJ 08540, United States a r t i c l e i n f o a b s t r a c t Article history: Received 14 January 2016 Received in revised form 29 November 2016 Accepted 8 December 2016 Available online 3 January 2017 Keywords: Combinatorial geometry Incidences In Sharir and Solomon (2015), Sharir and Solomon showed that the number of incidences between m distinct points and n distinct lines in R 4 is O ( m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + n ), (1) provided that no 2-flat contains more than s lines, and no hyperplane or quadric contains more than q lines, where the O hides a multiplicative factor of 2 c log m for some absolute constant c. In this paper we prove that, for integers m, n satisfying n 9/8 < m < n 3/2, there exist m points and n lines on the quadratic hypersurface in R 4 {(x 1, x 2, x 3, x 4 ) R 4 x 1 = x x2 3 x2 4 }, such that (i) at most s = O(1) lines lie on any 2-flat, (ii) at most q = O(n/m 1/3 ) lines lie on any hyperplane, and (iii) the number of incidences between the points and the lines is Θ(m 2/3 n 1/2 ), which is asymptotically larger than the upper bound in (1), when n 9/8 < m < n 3/2. This shows that the assumption that no quadric contains more than q lines (in the above mentioned theorem of Sharir and Solomon (2015)) is necessary in this regime of m and n. By a suitable projection from this quadratic hypersurface onto R 3, we obtain m points and n lines in R 3, with at most s = O(1) lines on a common plane, such that the number of incidences between the m points and the n lines is Θ(m 2/3 n 1/2 ). It remains an interesting question to determine if this bound is also tight in general Elsevier B.V. All rights reserved. 1. Introduction Let P be a set of m distinct points in R 2 and let L be a set of n distinct lines in R 2. Let I(P, L) denote the number of incidences between the points of P and the lines of L; that is, the number of pairs (p, l), such that p P, l L and p l. The classical Szemerédi Trotter theorem [12] yields the worst-case tight bound I(P, L) = O ( m 2/3 n 2/3 + m + n ). (2) Work on this paper by Noam Solomon was supported by Grant 892/13 from the Israel Science Foundation. Ruixiang Zhang was supported by Princeton University. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. * Corresponding author. addresses: noam.solom@gmail.com (N. Solomon), ruixiang@math.princeton.edu (R. Zhang) X/ 2016 Elsevier B.V. All rights reserved.

108 586 N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) This bound clearly also holds in three, four, or any higher dimensions which can be easily proved by projecting the given lines and points onto some generic plane. Moreover, the bound will continue to be worst-case tight by placing all the points and lines in a common plane, in a configuration that yields the planar lower bound. In the groundbreaking paper of Guth and Katz [4], an improved bound has been derived for I(P, L), for a set P of m points and a set L of n lines in R 3, provided that not too many lines of L lie in a common plane. 1 Specifically, they showed: Theorem 1.1 (Guth and Katz [4]). Let P be a set of m distinct points and L a set of n distinct lines in R 3, and let s n be a parameter, such that no plane contains more than s lines of L. Then I(P, L) = O ( m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n ). Remark. When s = Θ( n), this bound is known to be tight, by a generalization to three dimensions of Elekes planar construction of points and lines on an integer grid (see Guth and Katz [4] for the details). For smaller values of s, it is an open problem to give lower bounds or improve the upper bound, and the case s = O(1) is of particular interest. In Theorem 1.5 we give an improved upper bound, and it remains a question (see Question 4.1) whether it is tight. In a recent paper of Sharir and Solomon [7], the following analogous and sharper result in four dimensions was established. Theorem 1.2. Let P be a set of m distinct points and L a set of n distinct lines in R 4, and let q, s n be parameters, such that (i) each hyperplane or quadric contains at most q lines of L, and (ii) each 2-flat contains at most s lines of L. Then I(P, L) 2 c log m ( m 2/5 n 4/5 + m ) + A ( m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n ), (3) where A and c are suitable absolute constants. When m n 6/7 or m n 5/3, there is the sharper bound I(P, L) A ( m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n ). (4) In general, except for the factor 2 c log m, the bound is tight in the worst case, for any values of m, n, and for corresponding suitable ranges of q and s. The term m 2/3 n 1/3 s 1/3 comes from the planar Szemerédi Trotter bound (2), and is unavoidable, as it can be attained if we densely pack points and lines into 2-flats, in patterns that realize the bound in (2). Likewise, the term m 1/2 n 1/2 q 1/4 comes from the bound of Guth and Katz [4] in three dimensions (as in Theorem 1.1), and is again unavoidable, as it can be attained if we densely pack points and lines into hyperplanes, in patterns that realize the bound in three dimensions. In this paper we show that the condition in assumption (i) of Theorem 1.2 that quadrics also do not contain too many lines, cannot be dropped, by proving the following theorem. Theorem 1.3. For each positive integer k and each α > 0, there exists m = Θ(k 3+3α ) points and n = Θ(k 2+4α ) lines on the quadratic hypersurface S := {(x 1, x 2, x 3, x 4 ) R 4 x 1 = x x2 3 x2 4 } in R 4, such that there are at most O(1) lines lying on any 2-flat and O(k 1+3α ) lines lying on any hyperplane, and I(P, L) = Θ(k 3+4α ). Given integers m and n, there are k, α such that m = Θ(k 3+3α ) points and n = Θ(k 2+4α ). Substituting these values in Theorem 1.3, we obtain the following corollary. Corollary 1.4. For integers m, n, there is a configuration of m points and n lines in R 4, such that all the points (resp., lines) are contained (resp., fully contained) in S, and (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines in a common hyperplane is O(n/m 1/3 ), and (iii) the number of incidences between the points and lines is Ω(m 2/3 n 1/2 + m + n). Remarks. (1) For integers m, n, satisfying n 9/8 < m < n 3/2, the number incidences Ω(m 2/3 n 1/2 ) in Corollary 1.4 is asymptotically larger than the bound of Eq. (3) for the number of incidences O(m 2/5 n 4/5 +m 1/2 n 1/2 q 1/4 +m 2/3 n 1/3 +m+n) = O(m 2/5 n 4/5 + m 5/12 n 3/4 + m + n) (as q = O(n/m 1/3 )). This implies that the condition in assumption (i) of Theorem 1.2 cannot be dropped, in this regime of m and n. (2) We note that the number of 2-rich points determined by n lines in R 4 is O(n 3/2 ), provided that at most O( n) of the lines lie on a common plane or regulus. 2 To see this, project the lines onto some (generic) hyperplane H, such that no two lines are projected onto the same line, and similarly, no two 2-rich points are projected onto the same 2-rich point, and such that at most O( n) lines lie on a common plane or regulus. Then, the number of 2-rich points in the configuration of n lines in R 4 is 1 The additional requirement in [4], that no regulus contains too many lines, is not needed for the bound given below. 2 A regulus is a quadratic surface that is doubly ruled by lines. For more details about reguli, see e.g., Sharir and Solomon [6].

109 N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) equal to the number of 2-rich points in the configuration of the projected lines onto H. By Guth and Katz [4], the number of 2-rich points determined by the projected lines is O(n 3/2 ), and therefore the same holds for the number of 2-rich points in the original configuration of lines in R 4. We also notice that in a configuration of m points and n lines in R 4, the 1-rich points (i.e., points that are incident to exactly one line) contribute at most m incidences. Therefore, in Corollary 1.4, as s = O(1), the assumption that m n 3/2 causes no loss of generality. Proof techniques. It is a common practice to take geometric objects to be integer points on certain hypersurfaces (especially quadratic ones) and varieties passing through a lot of such points, in order to obtain lower bounds for their incidences. For some most recent applications of this method, see [9,14,15]. In this paper we obtain our incidence lower incidence bound by taking integer points and low height lines on the above hypersurface S. Projection to R 3. As remarked above, Guth and Katz [4] proved that the number of incidences between m points and n lines in R 3 is I(P, L) = O ( m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n ), provided that no plane contains more than s lines of L. When s = Θ( n), this bound is tight, by a generalization to three dimensions of Elekes construction of points and lines on an integer grid in the plane (see Guth and Katz [4] for the details). For smaller values of s, it is an open problem to give lower bounds or improve the upper bound, where the case s = O(1), is of particular interest. By choosing a generic projection from R 4 to R 3, we show that Corollary 1.4 directly implies the following Theorem. Theorem 1.5. For integers m, n, there is a configuration of m points and n lines in R 3, such that (i) the number of lines in any common plane is s = O(1), and (ii) the number of incidences between the points and lines is Ω(m 2/3 n 1/2 + m + n). Remark. When n 3/4 m n 3/2, the term m 2/3 n 1/2 dominates over m and n, showing that in this regime of m and n, the construction in Theorem 1.5, of m points and n lines with O(1) lines in a common plane, yields a super-linear number of incidences. As observed above, the bound of Guth and Katz [4] implies that the number of 2-rich points determined by the n lines is O(n 3/2 ), so the assumption that m n 3/2 causes no loss of generality. Background. Incidence problems have been a major topic in combinatorial and computational geometry for the past thirty years, starting with the Szemerédi Trotter bound [12] back in Several techniques, interesting in their own right, have been developed, or adapted, for the analysis of incidences, including the crossing-lemma technique of Székely [11], and the use of cuttings as a divide-and-conquer mechanism (e.g., see [2]). Connections with range searching and related problems in computational geometry have also been noted, and studies of the Kakeya problem (see, e.g., [13]) indicate the connection between this problem and incidence problems. See Pach and Sharir [5] for a comprehensive survey of the topic. The simplest instances of incidence problems involve points and lines. Szemerédi and Trotter solved completely this special case in the plane [12]. Guth and Katz s second paper [4] provides a worst-case tight bound in three dimensions, under the assumption that no plane contains too many lines; see Theorem 1.1. Under this assumption, the bound in three dimensions is significantly smaller than the planar bound (unless one of m, n is significantly smaller than the other), and the intuition is that this phenomenon should also show up as we move to higher dimensions. The first attempt in higher dimensions was made by Sharir and Solomon in [8]. In a recent work, Sharir and Solomon [7] gave a tight bound in fourdimensions provided that the number of lines fully contained in a common hyperplane or quadric is bounded by a parameter q, and the number of lines fully contained in a common 2-flat is bounded by a parameter s. Whereas the condition that no common hyperplane contains more than a bounded number of lines was known to be necessary, it remained an open question whether the condition that the number of lines in a common quadric is bounded is necessary. In this paper, we show that when n 9/8 < m < n 3/2, this condition is indeed necessary, by describing an explicit quadratic hypersurface in R 4 containing more incidences than the bound prescribed by the main theorem of [7]. This is the content of Theorem 1.3, and Corollary 1.4. We remark that in [14], another example of points on a quadratic hypersurface in F 4 with highly incidental pattern was noticed. There F is a finite field. Our current quadratic hypersurface and our counting techniques in R 4 are slightly different. The reader may find it interesting to compare the results here to the results in [14]. Another interesting remark is that in three dimensions, there are certain quadratic surfaces, called reguli, such that if one allows too many lines to lie on such a regulus, the number of 2-rich points determined by them can be larger than the Guth Katz bound [4] of O(n 3/2 ). The quadratic hypersurface in R 4 presented in this paper can be thought of as a higher degree analogs of regulus. However, If one only cares about incidences between points and lines (instead of the number of 2-rich points determined by the lines), the existence of many lines on a regulus (or any quadratic surface in R 3 ) do not yield more than a linear number of incidences. 2. Proof of Theorem 1.3 Proof. We start by recalling the quadric S = {(x 1, x 2, x 3, x 4 ) R 4 x 1 = x x2 3 x2 4 }, (5) on which the construction takes place, and define the set of points by P = {(x 1, x 2, x 3, x 4 ) S x i Z, i = 1,..., 4, x 1 200k 2+2α, x 2, x 3, x 4 100k 1+α }, (6)

110 588 N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) and the set of lines { L = {x + tv t R} S x = (x 1,..., x 4 ), v = (v 1,..., v 4 ), x i, v i Z, i = 1,..., 4, x 1 k 2+2α, x 2, x 3, x 4 k 1+α, k 1+2α v 1 8k 1+2α, v 2, v 3 k α, v = v2 + 2 v2, 3 v 1 = 2x 2 v 2 + 2x 3 v 3 2x 4 v 4, } gcd(v 2, v 3, v 4 ) = 1, and v 4 kα, 2 for any positive integer k and any α > 0. Since a point on S is uniquely determined by its last three coordinates, we have P = {(x 2, x 3, x 4 ) Z 3 x 2, x 3, x 4 100k 1+α } = Θ(k 3+3α ). The analysis of (an asymptotically tight bound on) the number of lines of L is a bit more involved. A line {x + tv t R} in L (assuming x S, x 1 k 2+2α, x 2, x 3, x 4 k 1+α ) is fully contained in S if and only if v 1 = 2x 2 v 2 + 2x 3 v 3 2x 4 v 4 and v 2 4 = v2 2 + v2 3. It follows by Benito and Varona [1] that the number of primitive integer triples (v 2, v 3, v 4 ) (i.e., without a common divisor) satisfying v 2 4 = v2 2 + v2 3, v 2, v 3 k α, and v 4 kα 2 is Θ(kα ). For each such (v 2, v 3, v 4 ), we claim that there are Ω(k 3+3α ) (and trivially also O( P ) = O(k 3+3α )) points x P, such that v 1 = 2x 2 v 2 + 2x 3 v 3 2x 4 v 4 satisfying k1+2α v 4 1 8k 1+2α. Indeed, note that v 2, v 3 v 4. Choosing x 2, x 3 x 4, k 1+α x k 1+α (there are at least k3+3α choices of such 32 triples (x 2, x 3, x 4 )) implies that 2x 2 v 2 + 2x 3 v 3 2 x 2 v x 3 v 3 2 x 4 4 ( v 2 + v 3 ) x 4 v 4. Here v 1 x 4 v 4 k1+2α. The inequality v 4 1 8k 1+2α is immediate. Moreover, each line l satisfying the above conditions is incident to O(k) different points of P (and can thus be expressed in O(k) different ways as {x + tv t R} S, for x 1 k 2+2α, x 2, x 3, x 4 k 1+α ). Indeed, parameterize l as {x + tv t R} S, where x, v satisfy v 1 k1+2α 4, x 1 k 2+2α, and v = (v 1, v 2, v 3, v 4 ) is primitive (i.e., its coordinates do not have a common factor). Notice that if t > 8k, then the first coordinate of x + tv has absolute value greater than k 2+2α, and that if t Z, then x + tv Z 4 (since v is primitive and x Z 4 ). In either case, x + tv P. This implies that l P {x + tv t Z, t 8k}, and thus l P 16k = O(k) as claimed. Therefore, the total number of lines is Ω( k3+3α k α ) = Ω(k 2+4α ). k It is easy to see that each line in L is incident to Ω(k) points in P. It follows that L = O(k 2+4α ). Hence L = Θ(k 2+4α ). Since each line has Θ(k) integer points in P on it, we have I(P, L) = Θ(k 3+4α ). We now bound the number of lines fully contained in any 2-flat, and then bound the number of lines on any hyperplane. The bounds will be uniform (i.e., independent of the specific 2-flat or hyperplane). Let π denote any 2-flat, and we analyze the number of lines that are fully contained in π S. We claim that S contains no planes, so π S. Assume the contrary, then we parameterize π = {(u 1 s + r 1 t + w 1, u 2 s + r 2 t + w 2, u 3 s + r 3 t + w 3, u 4 s + r 4 t + w 4 ) s, t R}, for constants u i, r i, w i R, i = 1, 2, 3, 4 where (u 1, u 2, u 3, u 4 ) and (r 1, r 2, r 3, r 4 ) are both nonzero and not proportional to each other. Comparing the coefficients of quadratic terms in the identity u 1 s + r 1 t + w 1 (u 2 s + r 2 t + w 2 ) 2 + (u 3 s + r 3 t + w 3 ) 2 (u 4 s + r 4 t + w 4 ) 2, we deduce (u 2, u 3, u 4 ) and (r 2, r 3, r 4 ) are proportional to each other. Hence we may assume u 2 = u 3 = u 4 = 0. But this forces u 1 = 0, a contradiction. Therefore π is not contained in S. Thus the intersection π S is a curve of degree at most two, so there are at most two lines fully contained in π S. Next, we take any hyperplane H, and analyze the number of lines fully contained in S H. The surface S H is a quadratic 2-surface contained in H. We will use the classification of (real) quadratic surfaces in R 3 (see, e.g., Sylvester s original paper [10]), and distinguish between two cases.

111 N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) If the equation of H can be expressed as x 1 = ϕ(x 2, x 3, x 4 ), where ϕ is a linear form, then each point x H S satisfies the equations { x x2 3 x2 = 4 ϕ(x 2, x 3, x 4 ), (7) x H. This is either a cone, i.e., is linearly equivalent to x x2 3 x2 4 = 0, or a hyperboloid of one or two sheets, i.e., is linearly equivalent to x x2 3 x2 = 4 1 or x2 + 2 x2 3 x2 4 = 1, respectively. It is easy to verify (and well known) that there are no lines on the hyperboloid of two sheets. We therefore assume that S H is either a cone or a hyperboloid of one sheet. In these cases, there are at most two lines of L with any given direction that are fully contained in S H. Note that if a line {x + tv t R} L is fully contained in S H, then v 1 = ϕ(v 2, v 3, v 4 ) (where we let ϕ denote the linear homogeneous part of ϕ), and v 2 = 4 v2 + 2 v2 3 (being the homogeneous part of degree two in t), v 2, v 3 k α and v 4 k α. As observed above, there are O(k α ) such triples (v 2, v 3, v 4 ). Therefore, the number of lines in L that lie in S H is O(k α ). In the remaining case, the equation of H is of the form ϕ(x 2, x 3, x 4 ) = 0, where ϕ is a linear form. We can assume, without loss of generality, that the equation of H is x 2 = ψ(x 3, x 4 ), where ψ is a linear form (the remaining case x 4 = 0 is simpler to handle). In this case, for every point x S H, we have { x 1 = ψ(x 3, x 4 ) 2 + x 2 3 x2, 4 x H. The classification of (real) quadratic surfaces implies that this can be an elliptic paraboloid, a parabolic cylinder or a hyperbolic paraboloid. An elliptic paraboloid contains no lines and the corresponding case is trivial. If S H is a parabolic cylinder, then all lines on it are parallel. It is straightforward that there are O(k 2+2α ) points in P that lie on it (by counting possible pairs (x 3, x 4 )). Hence there are O(k 1+2α ) lines in L that are fully contained in S H. In the rest of the discussion we assume S H is a hyperbolic paraboloid. In this case, similarly to the case of the one-sheeted hyperboloid, there are at most two lines with the same direction. Moreover, the direction (v 1, v 2, v 3, v 4 ) of any line on S H satisfies v 2 = ψ(v 3, v 4 ) and v 2 4 = v2 2 + v2 3 (where we let ψ denote the linear homogeneous part of ψ). Thus once we fix v 1 and v 3 or v 4 (depending on ψ), we have limited the possible direction (v 1, v 2, v 3, v 4 ) in a set with 2 elements. Hence there are O(k 1+3α ) lines that are fully contained in S H. Finally, we show that for α < 1 2, the number of incidences is (asymptotically) larger than Θ(m2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 + m + n), which is the bound of Eq. (3), with m = Θ(k 3+3α ), n = Θ(k 2+4α ), q = O(k 1+3α ), and s = O(1). We have m 2/5 n 4/5 = O(k 6+6α+8+16α 5 ) = O(k 14+22α 5 ), and the exponent is smaller than 3 + 4α, as α < 1 2. Similarly, m 1/2 n 1/2 q 1/4 = O(k 6+6α+4+8α+1+3α 4 ) = O(k 11+17α 4 ), and the exponent is smaller than 3 + 4α, as α < 1 < 1. Similarly, 2 m 2/3 n 1/3 = O(k 6+6α+2+4α 3 ) = O(k 8+10α 3 ), and the exponent is smaller than 3 + 4α for every α. Since both m and n are O(k 3+4α ), the claim is proved. 3. Proof of Theorem 1.5 The proof of Theorem 1.5 follows easily by Corollary 1.4, together with the following lemma. Lemma 3.1. Let L be a set of n lines in R 4 such that at most s lines lie on a common 2-flat. There exists a projection from R 4 onto a hyperplane H R 4, such that at most s lines lie on any common plane in H. Proof of Lemma 3.1. Let π 1,..., π k denote the set of 2-flats containing at least two lines in L, then k ( n 2). For a generic hyperplane H R 4, the projection p : R 4 H maps π i onto a plane π i contained in H. We pick, as we may, a hyperplane H, so that p is bijective on π 1,..., π k. Denote by L the set of projected lines in R 3. It is easy to verify that the set of planes in H containing at least two lines in L consists precisely of π,..., π 1 k. Moreover, the number of lines in L that are contained in π i is equal to the number of lines in L that are contained in π i, thus completing the proof. 4. Discussion and open questions In Corollary 1.4, we show a concrete irreducible quadratic hypersurface S in R 4, together with a set of m points and n lines that lie on S, for n 9/8 < m < n 3/2, such that (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines in any common hyperplane is O(n/m 1/3 ), and (iii) the number of incidences between the points and lines is Ω(m 2/3 n 1/2 ), which is asymptotically larger than Θ(m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 + m + n) in this regime of m and n. A natural question is

112 590 N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) to extend this result to other regimes by a similar construction. The condition (i) is natural and should not be hard to achieve, since if a plane is not contained in a quadratic hypersurface, then by the generalized version of Bézout s theorem [3] it can contain at most two lines. Here are a few natural questions that arise 1. Can we generalize our construction, such that in (ii) we are allowed to have a more general q, not necessarily n/m 1/3, s.t. the number of lines in any common hyperplane is O(q), and we still get a lower bound of incidences asymptotically larger than Θ(m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 + m + n)? 2. Can we find a similar construction when m < n 9/8? 3. How powerful is the natural generalization of this construction for R d, when d > 4? Notice that for d > 4, finding the precise bound for the number of incidences between a set P of m points and a set L of n lines in R d is already an interesting open question. It is probably too early for us to answer this question before we find the correct bound. 4. In three dimensions, it remains a question to determine if Theorem 1.5 is tight. Question 4.1. Let P be a set of m distinct points and L a set of n distinct lines in R 3, and assume that no plane contains more than s = O(1) lines of L. Then what is a good or tight upper bound of I(P, L)? Would O(m n 2 + m + n) suffice? We do not know the answer to this question yet. It seems to require new techniques. Acknowledgments We thank Micha Sharir for his invaluable advice, and the anonymous referees for their helpful comments. References [1] M. Benito, J.L. Varona, Pythagorian triples with legs less than n, J. Comput. Appl. Math. 143 (1) (2002) [2] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, E. Welzl, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990) [3] W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, in: Expository Lectures from the CBMS Regional Conference Held at George Mason University, June 27 July 1, 1983, vol. 54, AMS Bookstore, [4] L. Guth, N.H. Katz, On the Erdős distinct distances problem in the plane, Ann. of Math. (2) 181 (2015) Also in arxiv: [5] J. Pach, M. Sharir, Geometric incidences, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, in: Contemporary Mathematics, vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp [6] M. Sharir, N. Solomon, Incidences between points and lines on a two- and three-dimensional varieties, in arxiv: [7] M. Sharir, N. Solomon, Incidences between points and lines in four dimensions, Discrete Comput. Geom. (2016) in press. Also in Proc. 56th IEEE Symp. on Foundations of Computer Science (2015), , and in arxiv: [8] M. Sharir, N. Solomon, Incidences between points and lines in R 4, in: Proc. 30th Annu. ACM Sympos. Comput. Geom., 2014, pp [9] A. Sheffer, Lower bounds for incidences with hypersurfaces, Discrete Anal. (2016) in press. Also in arxiv: [10] J.J. Sylvester XIX, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares, Lond. Edinb. Dublin Phil. Mag. J. Sci. 4 (23) (1852) [11] L. Székely, Crossing numbers and hard Erdős problems in discrete geometry, Combin. Probab. Comput. 6 (1997) [12] E. Szemerédi, W.T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983) [13] T. Tao, From rotating needles to stability of waves: Emerging connections between combinatorics, analysis, and PDE, Notices Amer. Math. Soc. 48 (3) (2001) [14] T. Tao, A new bound for finite field Besicovitch sets in four dimensions, Pacific J. Math. 222 (2) (2005) [15] R. Zhang, Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem, Selecta Math. (N.S.) (2014) 1 18.

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116 4 Incidences between points and lines in R 3 107

117 Incidences between points and lines in three dimensions Micha Sharir Noam Solomon arxiv: v1 [math.co] 12 Jan 2015 January 13, 2015 Abstract We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R 3, so that no plane contains more than s lines, is ( ) O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between m and n). This bound, originally obtained by Guth and Katz [9] as a major step in their solution of Erdős s distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth [7]. The present paper presents a different and simpler derivation, with better bounds than those in [7], and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions. 1 Introduction Let P be a set of m distinct points in R 3 and let L be a set of n distinct lines in R 3. Let I(P, L) denote the number of incidences between the points of P and the lines of L; that is, the number of pairs (p, l) with p P, l L, and p l. If all the points of P and all the lines of L lie in a common plane, then the classical Szemerédi Trotter theorem [26] yields the worst-case tight bound ( ) I(P, L) = O m 2/3 n 2/3 + m + n. (1) This bound clearly also holds in three dimensions, by projecting the given lines and points onto some generic plane. Moreover, the bound will continue to be worst-case tight by Work on this paper by Noam Solomon and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S. Israel Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. michas@post.tau.ac.il School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. noam.solom@gmail.com 1

118 placing all the points and lines in a common plane, in a configuration that yields the planar lower bound. In the 2010 groundbreaking paper of Guth and Katz [9], an improved bound has been derived for I(P, L), for a set P of m points and a set L of n lines in R 3, provided that not too many lines of L lie in a common plane. Specifically, they showed: 1 Theorem 1 (Guth and Katz [9]). Let P be a set of m distinct points and L a set of n distinct lines in R 3, and let s n be a parameter, such that no plane contains more than s lines of L. Then ( ) I(P, L) = O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n. This bound was a major step in the derivation of the main result of [9], which was to prove an almost-linear lower bound on the number of distinct distances determined by any finite set of points in the plane, a classical problem posed by Erdős in 1946 [6]. Their proof uses several nontrivial tools from algebraic and differential geometry, most notably the Cayley Salmon theorem on osculating lines to algebraic surfaces in R 3, and additional properties of ruled surfaces. All this machinery comes on top of the main innovation of Guth and Katz, the introduction of the polynomial partitioning technique; see below. In this paper, we provide a simple derivation of this bound, which bypasses most of the techniques from algebraic geometry that are used in the original proof. A recent related study by Guth [7] provides another simpler derivation of a similar bound, but (a) the bound obtained in [7] is slightly worse, involving extra factors of the form m ε, for any ε > 0, and (b) the assumptions there are stronger, namely that no algebraic surface of degree at most c ε, a (potentially large) constant that depends on ε, contains more than s lines of L (in fact, Guth considers in [7] only the case s = n). It should be noted, though, that Guth also manages to derive a (slightly weaker but still) near-linear lower bound on the number of distinct distances. As in the classical work of Guth and Katz [9], and in the follow-up study of Guth [7], here too we use the polynomial partitioning method, as pioneered in [9]. The main difference between our approach and those of [7, 9] is the choice of the degree of the partitioning polynomial. Whereas Guth and Katz [9] choose a large degree, and Guth [7] chooses a constant degree, we choose an intermediate degree. This reaps many benefits from both the high-degree and the constant-degree approaches, and pays a small price in the bound (albeit much better than in [7]). Specifically, our main result is a simple and fairly elementary derivation of the following result. Theorem 2. Let P be a set of m distinct points and L a set of n distinct lines in R 3, and let s n be a parameter, such that no plane contains more than s lines of L. Then ) ( ) I(P, L) A m,n (m 1/2 n 3/4 + m + B m 2/3 n 1/3 s 1/3 + n, (2) where B is an absolute constant, and, for another suitable absolute constant b > 1, ( A m,n = O b log(m 2 n) ) ( log(n 3 /m 2 ), for m n 3/2, and O b log(m 3 /n 4 ) ) log(m 2 /n 3 ), for m n 3/2. (3) 1 We skip over certain subtleties in their bound: They also assume that no regulus contains more than s input lines, but then they are able also to bound the number of intersection points of the lines. Moreover, if one also assumes that each point is incident to at least three lines then the term m in the bound can be dropped. 2

119 Remarks. (1) Only the range n m n 2 is of interest; outside this range, regardless of the dimension of the ambient space, we have the well known and trivial upper bound O(m + n). (2) The term m 2/3 n 1/3 s 1/3 comes from the planar Szemerédi Trotter bound (1), and is unavoidable, as it can be attained if we densely pack points and lines into planes, in patterns that realize the bound in (1). (3) Ignoring this term, the two terms m 1/2 n 3/4 and m compete for dominance; the former dominates when m n 3/2 and the latter when m n 3/2. Thus the bound in (2) is qualitatively different within these two ranges. (4) The threshold m = n 3/2 also arises in the related problem of joints (points incident to at least three non-coplanar lines) in a set of n lines in 3-space; see [8]. A concise rephrasing of the bound in (2) and (3) is as follows. We partition each of the ranges m n 3/2, m > n 3/2 into a sequence of subranges n α j 1 < m n α j, j = 0, 1,... (for m n 3/2 ), or n α j 1 > m n α j, j = 0, 1,... (for m n 3/2 ), so that within each range the bound asserted in the theorem holds for some fixed constant of proportionality (denoted as A m,n in the bound), where these constants vary with j, and grow, exponentially in j, as prescribed in (3), as m approaches n 3/2 (from either side). Informally, if we keep m sufficiently away from n 3/2, the bound in (2) holds with a fixed constant of proportionality. Handling the border range m n 3/2 is also fairly straightforward, although, to bypass the exponential growth of the constant of proportionality, it results in a slightly different bound; see below for details. Our proof is elementary to the extent that, among other things, it avoids any explicit handling of singular and flat points on the zero set of the partitioning polynomial. While these notions are relatively easy to handle in three dimensions (see, e.g., [5, 8]), they become more complex notions in higher dimensions (as witnessed, for example, in our companion work on the four-dimensional setting [22]), making proofs based on them harder to extend. Additional merits and features of our analysis are discussed in detail in the concluding section. In a nutshell, the main merits are: (i) We use two separate partitioning polynomials. The first one is of high degree, and is used to prune away some points and lines, and to establish useful properties of the surviving points and lines. The second partitioning step, using a polynomial of low degree, is then applied, from scratch, to the surviving input, exploiting the properties established in the first step. This idea seems to have a potential for further applications. (ii) Because of the way we use the polynomial partitioning technique, we need induction to handle incidences within the cells of the second partition. One of the nontrivial achievements of our technique is the ability to retain The planar term O(m 2/3 n 1/3 s 1/3 ) in the bound in (2) through the inductive process. Without such care, this term does not pass well through the induction, which has been a sore issue in several recent works on related problems (see [19, 20, 21]). This is one of the main reasons for using two separate partitioning steps. Background. Incidence problems have been a major topic in combinatorial and computational geometry for the past thirty years, starting with the aforementioned Szemerédi- Trotter bound [26] back in Several techniques, interesting in their own right, have 3

120 been developed, or adapted, for the analysis of incidences, including the crossing-lemma technique of Székely [25], and the use of cuttings as a divide-and-conquer mechanism (e.g., see [3]). Connections with range searching and related algorithmic problems in computational geometry have also been noted, and studies of the Kakeya problem (see, e.g., [27]) indicate the connection between this problem and incidence problems. See Pach and Sharir [16] for a comprehensive (albeit a bit outdated) survey of the topic. The landscape of incidence geometry has dramatically changed in the past six years, due to the infusion, in two groundbreaking papers by Guth and Katz [8, 9], of new tools and techniques drawn from algebraic geometry. Although their two direct goals have been to obtain a tight upper bound on the number of joints in a set of lines in three dimensions [8], and a near-linear lower bound for the classical distinct distances problem of Erdős [9], the new tools have quickly been recognized as useful for incidence bounds. See [5, 12, 13, 20, 24, 30, 31] for a sample of recent works on incidence problems that use the new algebraic machinery. The simplest instances of incidence problems involve points and lines, tackled by Szemerédi and Trotter in the plane [26], and by Guth and Katz in three dimensions [9]. Other recent studies on incidence problems include incidences between points and lines in four dimensions (Sharir and Solomon [21, 22]), and incidences between points and circles in three dimensions (Sharir, Sheffer and Zahl [20]), not to mention incidences with higherdimensional surfaces, such as in [1, 12, 24, 30, 31]. In a companion paper (with Sheffer) [19], we study the general case of incidences between points and curves in any dimension, and derive reasonably sharp bounds (albeit weaker in several respects than the one derived here). That tools from algebraic geometry form the major key for successful solution of difficult problems in combinatorial geometry, came as a big surprise to the community. It has lead to intensive research of the new tools, aiming to extend them and to find new applications. A major purpose of this study, as well as of Guth [7], is to show that one can still tackle successfully the problems using less heavy algebraic machinery. This offers a new, simplified, and more elementary approach, which we expect to prove potent for other applications too, such as those just mentioned. Looking for simpler, yet effective techniques that would be easier to extend to more involved contexts (such as incidences in higher dimensions) has been our main motivation for this study. A more detailed supplementary discussion (which would be premature at this point) of the merits and other issues related to our technique is given in a concluding section. 2 Proof of Theorem 2 The proof proceeds by induction on m. As already mentioned, the bound in (2) is qualitatively different in the two ranges m n 3/2 and m n 3/2. The analysis bifurcates accordingly. While the general flow is fairly similar in both cases, there are many differences too. The case m < n 3/2. We partition this range into a sequence of ranges m n α 0, n α 0 < m n α 1,..., where α 0 = 1/2 and the sequence {α j } j 0 is increasing and converges to 4

121 3/2. More precisely, as our analysis will show, we can take α j = j+2, for j 0. The induction is actually on the index j of the range n α j 1 < m n α j, and establishes (2) for m in this range, with a coefficient A j (written in (2, 3) as A m,n ) that increases with j. This paradigm has already been used in Sharir et al. [20] and in Zahl [31], for related incidence problems, albeit in a somewhat less effective manner; see the discussion at the end of the paper. The base range of the induction is m n, where the trivial general upper bound on point-line incidences, in any dimension, yields I = O(m 2 + n) = O(n), so (2) holds for a sufficiently large choice of the initial constant A 0. Assume then that (2) holds for all m n α j 1 for some j 1, and consider an instance of the problem with n α j 1 < m n 3/2 (the analysis will force us to constrain this upper bound in order to complete the induction step, thereby obtaining the next exponent α j ). Fix a parameter r, whose precise value will be chosen later (in fact, and this is a major novelty of our approach, there will be two different choices for r see below), and apply the polynomial partitioning theorem of Guth and Katz (see [9] and [13, Theorem 2.6]), to obtain an r-partitioning trivariate (real) polynomial f of degree D = O(r 1/3 ). That is, every connected component of R 3 \ Z(f) contains at most m/r points of P, where Z(f) denotes the zero set of f. By Warren s theorem [29] (see also [13]), the number of components of R 3 \ Z(f) is O(D 3 ) = O(r). Set P 1 := P Z(f) and P 1 := P \ P 1. A major recurring theme in this approach is that, although the points of P 1 are more or less evenly partitioned among the cells of the partition, no nontrivial bound can be provided for the size of P 1 ; in the worst case, all the points of P could lie in Z(f). Each line l L is either fully contained in Z(f) or intersects it in at most D points (since the restriction of f to l is a univariate polynomial of degree at most D). Let L 1 denote the subset of lines of L that are fully contained in Z(f) and put L 1 = L \ L 1. We then have I(P, L) = I(P 1, L 1 ) + I(P 1, L 1 ) + I(P 1, L 1 ). We first bound I(P 1, L 1 ) and I(P 1, L 1 ). As already observed, we have I(P 1, L 1) L 1 D nd. We estimate I(P 1, L 1 ) as follows. For each (open) cell τ of R3 \ Z(f), put P τ = P τ (that is, P 1 τ), and let L τ denote the set of the lines of L 1 that cross τ; put m τ = P τ m/r, and n τ = L τ. Since every line l L 1 crosses at most 1 + D components of R3 \ Z(f), we have n τ n(1 + D), and I(P 1, L 1 ) = I(P τ, L τ ). τ τ For each τ we use the trivial bound I(P τ, L τ ) = O(m 2 τ + n τ ). Summing over the cells, we get ( I(P 1, L 1 ) = τ I(P τ, L τ ) = O r (m/r) 2 + τ n τ ) = O ( m 2 /r + nd ) = O(m 2 /D 3 +nd). For the initial value of D, we take D = m 1/2 /n 1/4 (which we get from a suitable value of r = Θ(D 3 )), and get the bound I(P 1, L 1 ) + I(P 1, L 1 ) = O(m1/2 n 3/4 ). 5

122 This choice of D is the one made in [9]. It is sufficiently large to control the situation in the cells, by the bound just obtained, but requires heavy-duty machinery from algebraic geometry to handle the situation on Z(f). We now turn to Z(f), where we need to estimate I(P 1, L 1 ). Since all the incidences involving any point in P 1 and/or any line in L 1 have already been accounted for, we discard these sets, and remain with P 1 and L 1 only. We forget the preceding polynomial partitioning step, and start afresh, applying a new polynomial partitioning to P 1 with a polynomial g of degree E, which will typically be much smaller than D, but still non-constant. Before doing this, we note that the set of lines L 1 has a special structure, because all its lines lie on the algebraic surface Z(f), which has degree D. We exploit this to derive the following lemmas. We emphasize, since this will be important later on in the analysis, that Lemmas 3 7 hold for any choice of (r and) D. We note that in general the partitioning polynomial f may be reducible, and apply some of the following arguments to each irreducible factor separately. Clearly, there are at most D such factors. Lemma 3. Let π be a plane which is not a component of Z(f). Then π contains at most D lines of L 1. Proof. Suppose to the contrary that π contains at least D + 1 lines of L. Every generic line λ in π intersects these lines in at least D + 1 distinct points, all belonging to Z(f). Hence f must vanish identically on λ, and it follows that f 0 on π, so π is a component of Z(f), contrary to assumption. Lemma 4. The number of incidences between the points of P 1 that lie in the planar components of Z(f) and the lines of L 1, is O(m 2/3 n 1/3 s 1/3 + nd). Proof. Clearly, f can have at most D linear factors, and thus Z(f) can contain at most D planar components. Enumerate them as π 1,..., π k, where k D. Let P 1 denote the subset of the points of P 1 that lie in these planar components. Assign each point of P 1 to the first plane π i, in this order, that contains it, and assign each line of L 1 to the first plane that fully contains it; some lines might not be assigned at all in this manner. For i = 1,..., k, let P i denote the set of points assigned to π i, and let L i denote the set of lines assigned to π i. Put m i = P i and n i = L i. Then i m i m and i n i n; by assumption, we also have n i s for each i. Then I( P i, L i ) = O(m 2/3 i n 2/3 i + m i + n i ) = O(m 2/3 i n 1/3 i s 1/3 + m i + n i ). Summing over the k planes, we get, using Hölder s inequality, I( P i, L i ) = O(m 2/3 i n 1/3 i s 1/3 + m i + n i ) i i ( ) 2/3 ( ) 1/3 ( ) = O m i n i s 1/3 + m + n = O m 2/3 n 1/3 s 1/3 + m + n. i i We also need to include incidences between points p P 1 and lines l L 1 not assigned to the same plane as p (or not assigned to any plane at all). Any such incidence (p, l) can 6

123 be charged (uniquely) to the intersection point of l with the plane π i to which p has been assigned. The number of such intersections is O(nD), and the lemma follows. Lemma 5. Each point p Z(f) is incident to at most D 2 lines of L 1, unless Z(f) has an irreducible component that is either a plane containing p or a cone with apex p. Proof. Fix any line l that passes through p, and write its parametric equation as {p + tv t R}, where v is the direction of l. Consider the Taylor expansion of f at p along l f(p + tv) = D i=1 1 i! F i(p; v)t i, where F i (p; v) is the i-th order derivative of f at p in direction v; it is a homogeneous polynomial in v (p is considered fixed) of degree i, for i = 1,..., D. For each line l L 1 that passes through p, f vanishes identically on l, so we have F i (p; v) = 0 for each i. Assuming that p is incident to more than D 2 lines of L 1, we conclude that the homogeneous system F 1 (p; v) = F 2 (p; v) = = F D (p; v) = 0 (4) has more than D 2 (projectively distinct) roots. The classical Bézout s theorem, applied in the projective plane where the directions v are represented (e.g., see [4]), asserts that, since all these polynomials are of degree at most D, each pair of polynomials F i (p; v), F j (p; v) must have a common factor. The following slightly more involved inductive argument shows that in fact all these polynomials must have a common factor. 2 Lemma 6. Let f 1,..., f n C[x, y, z] be n homogeneous polynomials of degree at most D. If Z(f 1,..., f n ) > D 2, then all the f i s have a nontrivial common factor. Proof. The proof is via induction on n. The case n = 2 is precisely the classical Bézout s theorem in the projective plane. Assume that the inductive claim holds for n 1 polynomials. By assumption, Z(f 1,..., f n 1 ) Z(f 1,..., f n ) > D 2, so the induction hypothesis implies that there is a polynomial g that divides f i, for i = 1,..., n 1; assume, as we may, that g = GCD(f 1,..., f n 1 ). If there are more than deg(g)deg(f n ) points in Z(g, f n ), then again, by the classical Bézout s theorem in the projective plane, g and f n have a nontrivial common factor, which is then also a common factor of f i, for i = 1,..., n, completing the proof. Otherwise, put f i = f i /g, for i = 1,..., n 1. Notice that Z(f 1,..., f n 1 ) = Z( f 1,..., f n 1 ) Z(g), implying that each point of Z(f 1,..., f n ) belongs either to Z(g) Z(f n ) or to Z( f 1,..., f n 1 ) Z(f n ). As Z(f 1,..., f n ) > D 2 and Z(g, f n ) deg(g)deg(f n ) deg(g)d, it follows that Z( f 1,..., f n 1 ) Z( f 1,..., f n 1, f n ) (D deg(g))d > (D deg(g)) 2. Hence, applying the induction hypothesis to the polynomials f 1,..., f n 1 (all of degree at most D deg(g)), we conclude that they have a nontrivial common factor, contradicting the fact that g is the greatest common divisor of f 1,..., f n 1. Continuing with the proof of Lemma 5, there is an infinity of directions v that satisfy (4), so there is an infinity of lines passing through v and contained in Z(f). The union of 2 See also [17] for a similar observation. 7

124 these lines can be shown to be a two-dimensional algebraic variety, 3 contained in Z(f), so Z(f) has an irreducible component that is either a plane through p or a cone with apex p, as claimed. Lemma 7. The number of incidences between the points of P 1 that lie in the (non-planar) conic components of Z(f), and the lines of L 1, is O(m + nd). Proof. Let σ be such an (irreducible) conic component of Z(f) and let p be its apex. We observe that σ cannot contain any line that is not incident to p, because such a line would span with p a plane contained in σ, contradicting the assumption that σ is irreducible and non-planar. It follows that the number of incidences between P σ := P 1 σ and L σ, consisting of the lines of L 1 contained in σ, is thus O( P σ + L σ ) (p contributes L σ incidences, and every other point at most one incidence). Applying a similar first-comefirst-serve assignment of points and lines to the conic components of Z(f), as we did for the planar components in the proof of lemma 4, and adding the bound O(nD) on the number of incidences between points and lines not assigned to the same component, we obtain the bound asserted in the lemma. Remark. Note that in both Lemma 4 and Lemma 7, we bound the number of incidences between points on planar or conic components of Z(f) and all the lines of L 1. Pruning. To continue, we remove all the points of P 1 that lie in some planar or conic component of Z(f), and all the lines of L 1 that are fully contained in such components. With the choice of D = m 1/2 /n 1/4, we lose in the process O(m 2/3 n 1/3 s 1/3 + m + nd) = O(m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 ) incidences (recall that the term m is subsumed by the term m 1/2 n 3/4 for m < n 3/2 ). Continue, for simplicity of notation, to denote the sets of remaining points and lines as P 1 and L 1, respectively, and their sizes as m and n. Now each point is incident to at most D 2 lines (a fact that we will not use for this value of D), and no plane contains more than D lines of L 1, a crucial property for the next steps of the analysis. That is, this allows us to replace the input parameter s, bounding the maximum number of coplanar lines, by D; this is a key step that makes the induction work. A new polynomial partitioning. We now return to the promised step of constructing a new polynomial partitioning. We adapt the preceding notation, with a few modifications. We choose a degree E, typically much smaller than D, and construct a partitioning polynomial g of degree E for P 1. With an appropriate value of r = Θ(E 3 ), we obtain O(r) open cells, each containing at most m/r points of P 1, and each line of L 1 either crosses at most E + 1 cells, or is fully contained in Z(g). Set P 2 := P 1 Z(g) and P 2 := P 1 \ P 2. Similarly, denote by L 2 the set of lines of L 1 that are fully contained in Z(g), and put L 2 := L 1 \ L 2. We first dispose of incidences involving 3 It is simply the variety given by the equations (4), rewritten as F 1(p; x p) = F 2(p; x p) = = F D(p; x p) = 0. It is two-dimensional because it is contained in Z(f), hence at most two-dimensional, and it cannot be one-dimensional since it would then consist of only finitely many lines (see, e.g., [22, Lemma 2.3]). 8

125 the lines of L 2. (That is, now we first focus on incidences within Z(g), and only then turn to look at the cells.) By Lemma 4 and Lemma 7, the number of incidences involving points P 2 that lie in some planar or conic component of Z(g), and all the lines of L 2, is O(m 2/3 n 1/3 s 1/3 + m + ne) = O(m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + n). (For E D, this might be a gross overestimation, but we do not care.) We remove these points from P 2, and remove all the lines of L 2 that are contained in such components; continue to denote the sets of remaining points and lines as P 2 and L 2. Now each point is incident to at most E 2 lines of L 2 (Lemma 5), so the number of remaining incidences involving points of P 2 is O(mE 2 ); for E suitably small, this bound will be subsumed by O(m 1/2 n 3/4 ). Unlike the case of a large D, namely, D = m 1/2 /n 1/4, here the difficult part is to treat incidences within the cells of the partition. Since E D, we cannot use the naive bound O(n 2 + m) within each cell, because that would make the overall bound too large. Therefore, to control the incidence bound within the cells, we proceed in the following inductive manner. For each cell τ of R 3 \ Z(g), put P τ := P 2 τ, and let L τ denote the set of the lines of L 2 that cross τ; put m τ = P τ m/r, and n τ = L τ. Since every line l L 1 (that is, of L 2 ) crosses at most 1 + E components of R3 \ Z(g), we have τ n τ n(1 + E). It is important to note that at this point of the analysis the sizes of P 1 and of L 1 might be smaller than the original respective values m and n. In particular, we may no longer assume that P 1 > L 1 α j 1, as we did assume for m and n. Nevertheless, in what follows m and n will denote the original values, which serve as upper bounds for the respective actual sizes of P 1 and L 1, and the induction will work correctly with these values; see below for details. In order to apply the induction hypothesis within the cells of the partition, we want to assume that m τ n α j 1 τ for each τ. To ensure that, we require that the number of lines of L 2 that cross a cell be at most n/e2. Cells τ that are crossed by κn/e 2 lines, for κ > 1, are treated as if they occur κ times, where each incarnation involves all the points of P τ, and at most n/e 2 lines of L τ. The number of subproblems remains O(E 3 ). Arguing similarly, we may also assume that m τ m/e 3 for each cell τ (by duplicating each cell into a constant number of subproblems, if needed). We therefore require that m E 3 ( n E 2 ) αj 1. (Note that, as already commented above, these are only upper bounds on the actual sizes of these subsets, but this will have no real effect on the induction process.) That is, we require ( m ) 1/(3 2αj 1 ) E. (5) n α j 1 With these preparations, we apply the induction hypothesis within each cell τ, recalling that no plane contains more than D lines 4 of L 2 L 1, and get ( ) ( ) I(P τ, L τ ) A j 1 mτ 1/2 nτ 3/4 + m τ + B m 2/3 τ n 1/3 τ D 1/3 + n τ A j 1 ((m/e 3 ) 1/2 (n/e 2 ) 3/4 + m/e 3) ( + B +(m/e 3 ) 2/3 (n/e 2 ) 1/3 D 1/3 + n/e 2). 4 This was the main reason for carrying out the first partitioning step, as already noted. 9

126 Summing these bounds over the cells τ, that is, multiplying them by O(E 3 ), we get, for a suitable absolute constant b, I(P 2, L 2 ) = ) ( ) I(P τ, L τ ) ba j 1 (m 1/2 n 3/4 + m + B m 2/3 n 1/3 E 1/3 D 1/3 + ne. τ We now require that E = O(D). Then the last term satisfies ne = O(nD) = O(m 1/2 n 3/4 ), and, as already remarked, the preceding term m is also subsumed by the first term. The second term, after substituting D = O(m 1/2 /n 1/4 ), becomes O(m 5/6 n 1/4 E 1/3 ). Hence, with a slightly larger b, we have I(P 2, L 2 ) ba j 1m 1/2 n 3/4 + bbm 5/6 n 1/4 E 1/3. Adding up all the bounds, including those for the portions of P and L that were discarded during the first partitioning step, we obtain, for a suitable constant c, ( I(P, L) c m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + n + me 2) + ba j 1 m 1/2 n 3/4 + bbm 5/6 n 1/4 E 1/3. We choose E to ensure that the two E-dependent terms are dominated by the term m 1/2 n 3/4. That is, m 5/6 n 1/4 E 1/3 m 1/2 n 3/4, or E n 3/2 /m, and me 2 m 1/2 n 3/4, or E n 3/8 /m 1/4. Since n 3/2 /m = ( n 3/8 /m 1/4) 4, and both sides are 1, the latter condition is stricter, and we ignore the former. As already noted, we also require that E = O(D); specifically, we require that E m 1/2 /n 1/4. In conclusion, recalling (5), the two constraints on the choice of E are { } ( m ) 1/(3 2αj 1 ) n 3/8 m1/2 n α j 1 E min, m1/4 n 1/4, (6) and, for these constraints to be compatible, we require that and that ( m n α j 1 ) 1/(3 2αj 1 ) n 3/8 ( m n α j 1 2(7 2α j 1 ), m 1/4, or m n 9+2αj 1 ) 1/(3 2αj 1 ) m 1/2 n 1/4, which fortunately always holds, as is easil;y checked, since m n 3/2 and α j 1 1/2. Note that we have not explicitly stated any concrete choice of E; any value satisfying (6) will do. We put α j := 9 + 2α j 1 2(7 2α j 1 ), and conclude that if m n α j then the bound asserted in the theorem holds, with A j = ba j 1 + c and B = c. This completes the induction step. Note that the recurrence A j = ba j 1 + c solves to A j = O(b j ). 10

127 It remains to argue that the induction covers the entire range m = O(n 3/2 ). Using the above recurrence for the α j s, with α 0 = 1/2, it easily follows that α j = j + 2, for each j 0, showing that α j converges to 3/2, implying that the entire range m = O(n 3/2 ) is covered by the induction. To calibrate the dependence of the constant of proportionality on m and n, we note that, for n α j 1 m < n α j, the constant is O(b j ). We have j + 1 = α j 1 log m log n, or j log m log n 3 2 log m log n = log(m2 n) log(n 3 /m 2 ). This establishes the expression for A m,n given in the statement of the theorem. Handling the middle ground m n 3/2. Some care is needed when m approaches n 3/2, because of the potentially unbounded growth of the constant A j. To handle this situation, we simply fix a value j, in the manner detailed below, write m = kn α j, solve k separate problems, each involving m/k = n α j points of P and all the n lines of L, and sum up the resulting incidence bounds. We then get ( ) I(P, L) akb j (m/k) 1/2 n 3/4 + (m/k) ( ) + kb (m/k) 2/3 n 1/3 s 1/3 + n = ak 1/2 b j m 1/2 n 3/4 + ab j m + k 1/3 Bm 2/3 n 1/3 s 1/3 + kbn, for a suitable absolute constant a. Recalling that α j = j+2, we have k m/n α j n 3/2 /n α j = n 2/(j+2). Hence the coefficient of the leading term in the above bound is bounded by an 1/(j+2) b j, and we (asymptotically) minimize this expression by choosing j = j 0 := log n/ log b. With this choice all the other coefficients are also dominated by the leading coefficient, and we obtain ( I(P, L) = O 2 2 log b ( )) log n m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n. (7) In other words, the bound in (2) and (3) holds for any m n 3/2, but, for m n α j 0 one should use instead the bound in (7), which controls the exponential growth of the constants of proportionality within this range. The case m > n 3/2. The analysis of this case is, in a sense, a mirror image of the preceding analysis, except for a new key lemma (Lemma 8). For the sake of completeness, we repeat a sizeable portion of the analysis, providing many of the relevant (often differing) details. We partition this range into a sequence of ranges m n α 0, n α 1 m < n α 0,..., where α 0 = 2 and the sequence {α j } j 0 is decreasing and converges to 3/2. The induction is on 11

128 the index j of the range n α j m < n α j 1, and establishes (2) for m in this range, with a coefficient A j (written in (2,3) as A m,n ) that increases with j. The base range of the induction is m n 2, where the trivial general upper bound on point-line incidences in any dimension, dual to the one used in the previous case, yields I = O(n 2 + m) = O(m), so (2) holds for a sufficiently large choice of the initial constant A 0. Assume then that (2) holds for all m n α j 1 for some j 1, and consider an instance of the problem with n 3/2 m < n α j 1 (again, the lower bound will increase, to n α j, to facilitate the induction step). For a parameter r, to be specified later, apply the polynomial partition theorem to obtain an r-partitioning trivariate (real) polynomial f of degree D = O(r 1/3 ). That is, every connected component of R 3 \ Z(f) contains at most m/r points of P, and the number of components of R 3 \ Z(f) is O(D 3 ) = O(r). Set P 1 := P Z(f) and P 1 := P \ P 1. Each line l L is either fully contained in Z(f) or intersects it in at most D points. Let L 1 denote the subset of lines of L that are fully contained in Z(f) and put L 1 = L \ L 1. As before, we have We have I(P, L) = I(P 1, L 1 ) + I(P 1, L 1 ) + I(P 1, L 1 ). I(P 1, L 1) L 1 D nd, and we estimate I(P 1, L 1 ) as follows. For each cell τ of R3 \ Z(f), put P τ = P τ (that is, P 1 τ), and let L τ denote the set of the lines of L 1 that cross τ; put m τ = P τ m/r, and n τ = L τ. As before, we have τ n τ n(1 + D), so the average number of lines that cross a cell is O(n/D 2 ). Arguing as above, we may assume, by possibly increasing the number of cells by a constant factor, that each n τ is at most n/d 2. Clearly, we have I(P 1, L 1) = τ I(P τ, L τ ). For each τ we use the trivial dual bound, mentioned above, I(P τ, L τ ) = O(n 2 τ + m τ ). Summing over the cells, we get I(P 1, L 1) = τ I(P τ, L τ ) = O ( D 3 (n/d 2 ) 2 + m ) = O ( n 2 /D + m ). For the initial value of D, we take D = n 2 /m, noting that 1 D 3 m because n 3/2 m n 2, and get the bound I(P 1, L 1) + I(P 1, L 1) = O(n 2 /D + m + nd) = O(m + n 3 /m) = O(m), where the latter bound follows since m n 3/2. It remains to estimate I(P 1, L 1 ). Since all the incidences involving any point in P 1 and/or any line in L 1 have been accounted for, we discard these sets, and remain with P 1 and L 1 only. As before, we forget the preceding polynomial partitioning step, and start afresh, applying a new polynomial partitioning to P 1 with a polynomial g of degree E, which will typically be much smaller than D, but still non-constant. 12

129 For this case we need the following lemma, which can be regarded, in some sense, as a dual (albeit somewhat more involved) version of Lemma 5. Unlike the rest of the analysis, the best way to prove this lemma is by switching to the complex projective setting. This is needed for one key step in the proof, where we need the property that the projection of a complex projective variety is a variety. Once this is done, we can switch back to the real affine case, and complete the proof. Here is a very quick review of the transition to the complex projective setup. A real affine algebraic variety X, defined by a collection of real polynomials, can also be regarded as a complex projective variety. (Technically, one needs to take the projective closure of the complexification of X; details about these standard operations can be found, e.g., in Bochnak et al. [2, Proposition ] and in Cox et al. [4, Definition 8.4.6].) If f is an irreducible polynomial over R, it might still be reducible over C, but then it must have the form f = gḡ, where g is an irreducible complex polynomial and ḡ is its complex conjugate. (Indeed, if h is any irreducible factor of f, then h is also an irreducible factor of f, and therefore h h is a real polynomial dividing f. As f is irreducible over R, the claim follows.) In the following lemma, adapting a notation used in earlier works, we say that a point p P 1 is 1-poor (resp., 2-rich) if it is incident to at most one line (resp., to at least two lines) of L 1. Recall also that a regulus is a doubly-ruled surface in R 3 or in C 3. It is the union of all lines that pass through three fixed pairwise skew lines; it is a quadric, which is either a hyperbolic paraboloid or a one-sheeted hyperboloid. Lemma 8. Let f be an irreducible polynomial in C[x, y, z], such that Z(f) is not a complex plane nor a complex regulus, and let L 1 be a finite set of lines fully contained in Z(f). Then, with the possible exception of at most two lines, each line l L 1 is incident to at most O(D 3 ) 2-rich points. Proof. The strategy of the proof is to charge each incidence of l with some 2-rich point p to an intersection of l with another line of L 1 that passes through p, and to argue that, in general, there can be only O(D 3 ) such other lines. This in turn will be shown by arguing that the union of all the lines that are fully contained in Z(f) and pass through l is a onedimensional variety, of degree O(D 3 ), from which the claim will follow. As we will show, this will indeed be the case except when l is one of at most two exceptional lines on Z(f). Fix a line l as in the lemma, assume for simplicity that it passes through the origin, and write it as {tv 0 t C}; since l is a real line, v 0 can be assumed to be real. Consider the union V (l) of all the lines that are fully contained in Z(f) and are incident to l; that is, V (l) is the union of l with the set of all points p Z(f) \ l for which there exists t C such that the line connecting p to tv 0 l is fully contained in Z(f). In other words, for such a t and for each s C, we have f((1 s)p + stv 0 ) = 0. Regarding the left-hand side as D a polynomial in s, we can write it as G i (p; t)s i 0, for suitable (complex) polynomials i=0 G i (p; t) in p and t, each of total degree at most D. In other words, p and t have to satisfy the system G 0 (p; t) = G 1 (p; t) = = G D (p; t) = 0, (8) which defines an algebraic variety σ(l) in P 4 (C). Note that, substituting s = 0, we have G 0 (p; t) f(p), and that the limit points (tv 0, t) (corresponding to points on l) also satisfy 13

130 this system, since in this case f((1 s)tv 0 + stv 0 ) = f(tv 0 ) = 0 for all s. In other words, V (l) is the projection of σ(l) into P 3 (C), given by (p, t) p. For each p Z(f) \ l this system has only finitely many solutions in t, for otherwise the plane spanned by p and l 0 would be fully contained in Z(f), contrary to our assumption. By the projective extension theorem (see, e.g., [4, Theorem 8.6]), the projection of σ(l) into P 3 (C), in which t is discarded, is an algebraic variety τ(l). We observe that τ(l) is contained in Z(f), and is therefore of dimension at most two. Assume first that τ(l) is two-dimensional. As f is irreducible over C, we must have τ(l) = Z(f). This implies that each point p Z(f) \ l is incident to a (complex) line that is fully contained in Z(f) and is incident to l. In particular, Z(f) is ruled by complex lines. By assumption, Z(f) is neither a complex plane nor a complex regulus. We may also assume that Z(f) is not a complex cone, for then each line in L 1 is incident to at most one 2-rich point (namely, the apex of Z(f)), making the assertion of the lemma trivial. It then follows that Z(f) is an irreducible singly ruled (complex) surface. As argued in Guth and Katz [9] (see also our companion paper [23] for an independent analysis of this situation, which caters more explicitly to the complex setting too), Z(f) can contain at most two lines l with this property. Excluding these (at most) two exceptional lines l, we may thus assume that τ(l) is (at most) a one-dimensional curve. Clearly, by definition, each point (p, t) σ(l), except for p l, defines a line λ, in the original 3-space, that connects p to tv 0, and each point q λ satisfies (q, t) σ(l). Hence, the line {(q, t) q λ} is fully contained in σ(l), and therefore the line λ is fully contained in τ(l). Since τ(l) is one-dimensional, this in turn implies (see, e.g., [22, Lemma 2.3]) that τ(l) is a finite union of (complex) lines, whose number is at most deg(τ(l)). This also implies that σ(l) is the union of the same number of lines, and in particular σ(l) is also one-dimensional, and the number of lines that it contains is at most deg(σ(l)). We claim that this latter degree is at most O(D 3 ). This follows from a well-known result in algebra (see, e.g., Schmid [18, Lemma 2.2]), that asserts that, since σ(l) is a onedimensional curve in P 4 (C), and is the common zero set of polynomials, each of degree O(D), its degree is O(D 3 ). This completes the proof of the lemma. (The passage from the complex projective setting back to the real affine one is trivial for this property.) Corollary 9. Let f be a real or complex trivariate polynomial of degree D, such that (the complexification of) Z(f) does not contain any complex plane nor any complex regulus. Let L 1 be a set of n lines fully contained in Z(f), and let P 1 be a set of m points contained in Z(f). Then I(P 1, L 1 ) = O(m + nd 3 ). Proof. Write f = s i=1 f i for its decomposition into irreducible factors, for s D. We apply Lemma 8 to each complex factor f i of the f. By the observation preceding Lemma 8,some of these factors might be complex (non-real) polynomials, even when f is real. That is, regardless of whether the original f is real or not, we carry out the analysis in the complex projective space P 3 (C), and regard Z(f i ) as a variety in that space. Note also that, by focussing on the single irreducible component Z(f i ) of Z(f), we consider only points and lines that are fully contained in Z(f i ). We thus shrink P 1 and 14

131 L 1 accordingly, and note that the notions of being 2-rich or 1-poor are now redefined with respect to the reduced sets. All of this will be rectified at the end of the proof. Assign each line l L 1 to the first component Z(f i ), in the above order, that fully contains l, and assign each point p P 1 to the first component that contains it. If a point p and a line l are incident, then either they are both assigned to the same component Z(f i ), or p is assigned to some component Z(f i ) and l, which is assigned to a later component, is not contained in Z(f i ). Each incidence of the latter kind can be charged to a crossing between l and Z(f i ), and the total number of these crossings is O(nD). It therefore suffices to consider incidences between points and lines assigned to the same component. Moreover, if a point p is 2-rich with respect to the entire collection L 1 but is 1-poor with respect to the lines assigned to its component, then all of its incidences except one are accounted by the preceding term O(nD), which thus takes care also of the single incidence within Z(f i ). By Lemma 8, for each f i, excluding at most two exceptional lines, the number of incidences between a line assigned to (and contained in) Z(f i ) and the points assigned to Z(f i ) that are still 2-rich within Z(f i ), is O(deg(f i ) 3 ) = O(D 3 ). Summing over all relevant lines, we get the bound O(nD 3 ). Finally, each irreducible component Z(f i ) can contain at most two exceptional lines, for a total of at most 2D such lines. The number of 2-rich points on each such line l is at most n, since each such point is incident to another line, so the total number of corresponding incidences is at most O(nD), which is subsumed by the preceding bound O(nD 3 ). The number of incidences with 1-poor points is, trivially, at most m. This completes the proof of the corollary. Pruning. In the preceding lemma and corollary, we have excluded planar and reguli components of Z(f). Arguing as in the case of small m, the number of incidences involving points that lie on planar components of Z(f) is O(m 2/3 n 1/3 s 1/3 + m) (see Lemma 4), and the number of incidences involving points that lie on conic components of Z(f) is O(m + nd) = O(m) (see Lemma 7). A similar bound holds for points on the reguli components. Specifically, we assign each point and line to a regulus that contain them, if one exists, in the same first-come first-serve manner used above. Any point p can be incident to at most two lines that are fully contained in the regulus to which it is assigned, and any other incidence of p with a line l can be uniquely charged to the intersection of l with that regulus, for a total (over all lines and reguli) of O(nD) incidences. We remove all points that lie in any such component and all lines that are fully contained in any such component. With the choice of D = n 2 /m, we lose in the process O(m 2/3 n 1/3 s 1/3 + m + nd) = O(m + m 2/3 n 1/3 s 1/3 ) incidences (recall that nd m for m n 3/2 ). For the remainder sets, which we continue to denote as P 1 and L 1, respectively, no plane contains more than O(D) lines of L 1, as argued in Lemma 3. A new polynomial partitioning. We adapt the notation used in the preceding case, with a few modifications. We choose a degree E, typically much smaller than D, and construct a partitioning polynomial g of degree E for P 1. With an appropriate value of 15

132 r = Θ(E 3 ), we obtain O(r) cells, each containing at most m/r points of P 1, and each line of L 1 either crosses at most E + 1 cells, or is fully contained in Z(g). Set P 2 := P 1 Z(g) and P 2 := P 1 \ P 2. Similarly, denote by L 2 the set of lines of L 1 that are fully contained in Z(g), and put L 2 := L 1 \ L 2. We first dispose of incidences involving the lines of L 2. By Lemma 4 and the preceding arguments, the number of incidences involving points of P 2 that lie in some planar, conic, or regulus component of Z(g), and all the lines of L 2, is O(m 2/3 n 1/3 s 1/3 + m + ne). We remove these points from P 2, and remove all the lines of L 2 that are contained in such components. Continue to denote the sets of remaining points and lines as P 2 and L 2. By Corollary 9, the number of incidences between P 2 and L 2 is O(m + ne 3 ). To complete the estimation, we need to bound the number of incidences in the cells of the partition, which we do inductively, as before. Specifically, for each cell τ of R 3 \Z(g), put P τ := P 2 τ, and let L τ denote the set of the lines of L 2 that cross τ; put m τ = P τ m/r, and n τ = L τ. Since every line l L 0 crosses at most 1 + E components of R 3 \ Z(g), we have τ n τ n(1 + E), and, arguing as above, we may assume that each n τ is at most n/e 2, and each m τ is at most m/e 3. To apply the induction hypothesis in each cell, we therefore require that m E 3 ( n E 2 ) αj 1. (As before, the actual sizes of P1 and L 1 might be smaller than the respective original values m and n. We use here the original values, and note, similar to the preceding case, that the fact that these are only upper bounds on the actual sizes is harmless for the induction process.) That is, we require ( n α j 1 E m ) 1/(2αj 1 3). (9) With these preparations, we apply the induction hypothesis within each cell τ, recalling that no plane contains more than D lines of L 2 L 1, and get ( ) ( I(P τ, L τ ) A j 1 mτ 1/2 nτ 3/4 + m τ + B m 2/3 τ n 1/3 A j 1 ((m/e 3 ) 1/2 (n/e 2 ) 3/4 + m/e 3) + B τ D 1/3 + n τ ) ( (m/e 3 ) 2/3 (n/e 2 ) 1/3 D 1/3 + n/e 2). Summing these bounds over the cells τ, that is, multiplying them by O(E 3 ), we get, for a suitable absolute constant b, I(P 2, L 2) = ) ( ) I(P τ, L τ ) ba j 1 (m 1/2 n 3/4 + m + bb m 2/3 n 1/3 E 1/3 D 1/3 + ne. τ Requiring that E m/n, the last term satisfies ne m, and the first term is also at most O(m) (because m n 3/2 ). The second term, after substituting D = O(n 2 /m), becomes O(m 1/3 ne 1/3 ). Hence, with a slightly larger b, we have I(P 2, L 2 ) ba j 1m + bbm 1/3 ne 1/3. Collecting all partial bounds obtained so far, we obtain ( I(P, L) c m 2/3 n 1/3 s 1/3 + m + ne 3) + ba j 1 m + bbm 1/3 ne 1/3, 16

133 for a suitable constant c. dominated by m. That is, We choose E to ensure that the two E-dependent terms are m 1/3 ne 1/3 m, or E m 2 /n 3, and ne 3 m, or E m 1/3 /n 1/3. In addition, we also require that E m/n, but, as is easily seen, both of the above constraints imply that E m/n, so we get this latter constraint for free, and ignore it in what follows. As is easily checked, the second constraint E m 1/3 /n 1/3 is stricter than the first constraint E m 2 /n 3 for m n 8/5, and the situation is reversed when m n 8/5. So in our inductive descent of m, we first consider the second constraint, and then switch to the first constraint. Hence, in the first part of this analysis, the two constraints on the choice of E are ( n α j 1 m ) 1/(2αj 1 3) E m1/3 n 1/3, and, for these constraints to be compatible, we require that ( n α j 1 m ) 1/(2αj 1 3) m1/3 n 1/3, or m n 5αj 1 3 2α j 1. We start the process with α 0 = 2, and take α 1 := 5α 0 3 = 7/4. As this is still larger than 2α 0 8/5, we perform two additional rounds of the induction, using the same constraints, leading to the exponents α 2 = 5α 1 3 2α 1 = 23 14, and α 3 = 5α 2 3 2α 2 = < 8 5. To play it safe, we reset α 3 := 8/5, and establish the induction step for m n 8/5. We can then proceed to the second part, where the two constraints on the choice of E are ( n α j 1 m ) 1/(2αj 1 3) E m2 n 3, and, for these constraints to be compatible, we require that ( n α j 1 m ) 1/(2αj 1 3) m2 n 3, or m n 7αj 1 9 4α j 1 5. We define, for j 4, α j = 7α j 1 9 4α j 1 5. Substituting α 3 = 8/5 we get α 4 = 11/7, and in general a simple calculation shows that α j = j 2, for j 3. This sequence does indeed converge to 3/2 as j, implying that the entire range m = Ω(n 3/2 ) is covered by the induction. 17

134 In both parts, we conclude that if m n α j then the bound asserted in the theorem holds with A j = ba j 1 + c,, and B = c. This completes the induction step. Finally, we calibrate the dependence of the constant of proportionality on m and n, by noting that, for n α j m < n α j 1, the constant is O(b j ). We have j 6 = α j 1 log m log n, or j 3 log m log n 4 2 log m log n 3 = log ( m 3 /n 4) log (m 2 /n 3 ). (Technically, this only handles the range j 3, but, for an asymptotic bound, we can extend it to j = 1, 2 too.) This establishes the explicit expression for A m,n for this range, as stated in the theorem, and completes its proof. Again, as in the case of a small m, we need to be careful when m approaches n 3/2. Here we can fix a j, assume that n 3/2 m < n α j, and set k := m/n α j, where α j = 3/2 2/(j +2) is the j-th index in the hierarchy for m n 3/2. That is, k n α j α j = 1 4j j + 2. As before, we now solve k separate subproblems, each with m/k points of P and all the lines of L, and sum up the resulting incidence bounds. The analysis is similar to the one used above, and we omit its details. It yields almost the same bound as in (7), where the slightly larger upper bound on k leads to the slightly larger bound ( ( )) I(P, L) = O log b log n m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n, with a slightly different absolute constant b. 3 Discussion In this paper we derived an asymptotically tight bound for the number of incidences between a set P of points and a set L of lines in R 3. This bound has already been established by Guth and Katz [9], where the main tool was the use of partitioning polynomials. As already mentioned, the main novelty here is to use two separate partitioning polynomials of different degrees; the one with the higher degree is used as a pruning mechanism, after which the maximum number of coplanar lines of L can be better controlled (by the degree D of the polynomial), which is a key ingredient in making the inductive argument work. The second main tool of Guth and Katz was the Cayley Salmon theorem. This theorem says that a surface in R 3 of degree D cannot contain more than 11D 2 24D lines, unless it is ruled by lines. This is an ancient theorem, from the 19th century, combining algebraic and differential geometry, and its re-emergenece in recent years has kindled the interest of the combinatorial geometry community in classical (and modern) algebraic geometry. New proofs of the theorem were obtained (see, e.g., Terry Tao s blog [28]), and generalizations to higher dimensions have also been developed (see Landsberg [15]). However, the theorem only holds over the complex field, and using it over the reals requires some care. There is also an alternative way to bound the number of point-line incidences using flat and singular points. However, as already remarked, these two, as well as the Cayley Salmon machinery, are non-trivial constructs, especially in higher dimensions, and their 18

135 generalization to other problems in combinatorial geometry (even incidence problems with curves other than lines or incidences with lines in higher dimensions) seem quite difficult (and are mostly open). It is therefore of considerable interest to develop alternative, more elementary interfaces between algebraic and combinatorial geometry, which is a primary goal of the present paper (as well as of Guth s recent work [7]). In this regard, one could perhaps view Lemma 5 and Corollary 9 as certain weaker analogs of the Cayley Salmon theorem, which are nevertheless easier to derive, without having to use differential geometry. Some of the tools in Guth s paper [7] might also be interpreted as such weaker variants of the Cayley Salmon theory. It would be interesting to see suitable extensions of these tools to higher dimensions. Besides the intrinsic interest in simplifying the Guth Katz analysis, the present work has been motivated by our study of incidences between points and lines in four dimensions. This has begun in a year-old companion paper [21], where we have used the the polynomial partitioning method, with a polynomial of constant degree. This, similarly to Guth s work in three dimensions [7], has resulted in a slightly weaker bound and considerably stricter assumptions concerning the input set of lines. In a more involved follow-up study [22], we have managed to improve the bound, and to get rid of the restrictive assumptions, using two partitioning steps, with polynomials of non-constant degrees, as in the present paper. However, the analysis in [22] is not as simple as in the present paper, because, even though there are generalizations of the Cayley Salmon theorem to higher dimensions (due to Landsberg, as mentioned above), it turns out that a thorough investigation of the variety of lines fully contained in a given hypersurface of non-constant degree, is a fairly intricate and challenging problem, raising many deep questions in algebraic geometry, some of which are still unresolved. One potential application of the techniques used in this paper, mainly the interplay between partitioning polynomials of different degrees, is to the problem, recently studied by Sharir, Sheffer and Zahl [20], of bounding the number of incidences between points and circles in R 3. That paper uses a partitioning polynomial of constant degree, and, as a result, the term that caters to incidences within lower-dimensional spaces (such as our term m 2/3 n 1/3 s 1/3 ) does not go well through the induction mechanism, and consequently the bound derived in [20] was weaker. We believe that our technique can improve the bound of [20] in terms of this lower-dimensional term. A substantial part of the present paper (half of the proof of the theorem) was devoted to the treatment of the case m > n 3/2. However, under the appropriate assumptions, the number of points incident to at least two lines was shown by Guth and Katz [9] to be bounded by O(n 3/2 ). A recent note by Kollár [14] gives a simplified proof, including an explicit multiplicative constant. In his work, Kollár does not use partitioning polynomials, but employs more advanced algebraic geometric tools, like the arithmetic genus of a curve, which serves as an upper bound for the number of singular points. If we accept (pedagogically) the upper bound O(n 3/2 ) for the number of 2-rich points as a black box, the regime in which m > n 3/2 becomes irrelevant, and can be discarded from the analysis, thus greatly simplifying the paper. A challenging problem is thus to find an elementary proof that the number of points incident to at least two lines is O(n 3/2 ) (e.g., without the use of the Cayley Salmon theorem or the tools used by Kollár). Another challenging (and probably harder) problem is to 19

136 improve the bound of Guth and Katz when the bound s on the maximum number of mutually coplanar lines is n 1/2 : In their original derivation, Guth and Katz [9] consider mainly the case s = n 1/2, and the lower bound constrcution in [9] also has s = n 1/2. Another natural further research direction is to find further applications of partitioning polynomials of intermediate degrees. References [1] S. Basu and M. Sombra, Polynomial partitioning on varieties and point-hypersurface incidences in four dimensions, in arxiv: [2] J. Bochnak, M. Coste and M. F. Roy, Real Algebraic Geometry, Springer Verlag, Heidelberg, [3] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), [4] D. Cox, J. Little and D. O Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer Verlag, Heidelberg, [5] G. Elekes, H. Kaplan and M. Sharir, On lines, joints, and incidences in three dimensions, J. Combinat. Theory, Ser. A 118 (2011), Also in arxiv: [6] P. Erdős, On sets of distances of n points, Amer. Math. Monthly 53 (1946), [7] L. Guth, Distinct distance estimates and low-degree polynomial partitioning, in arxiv: [8] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem, Advances Math. 225 (2010), Also in arxiv: v1. [9] L. Guth and N. H. Katz, On the Erdős distinct distances problem in the plane, Annals Math. 181 (2015), Also in arxiv: [10] J. Harris, Algebraic Geometry: A First Course, Vol Springer-Verlag, New York, [11] R. Harshorne, Algebraic Geometry, Springer-Verlag, New York [12] H. Kaplan, J. Matoušek, Z. Safernová and M. Sharir, Unit distances in three dimensions, Combinat. Probab. Comput. 21 (2012), Also in arxiv: [13] H. Kaplan, J. Matoušek and M. Sharir, Simple proofs of classical theorems in discrete geometry via the Guth Katz polynomial partitioning technique, Discrete Comput. Geom. 48 (2012), Also in arxiv: [14] J. Kollár, Szemerédi Trotter-type theorems in dimension 3, in arxiv: [15] J. M. Landsberg, is a linear space contained in a submanifold? On the number of derivatives needed to tell, J. Reine Angew. Math. 508 (1999),

137 [16] J. Pach and M. Sharir, Geometric incidences, in Towards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics, Vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp [17] O. Raz, M. Sharir, and F. De Zeeuw, Polynomials vanishing on Cartesian products: The Elekes Szabó Theorem revisited, manuscript, [18] J. Schmid, On the affine Bézout inequality, Manuscripta Mathematica 88(1) (1995), [19] M. Sharir, A. Sheffer, and N. Solomon, Incidences with curves in R d, manuscript, [20] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between points and circles, Combinat. Probab. Comput., in press. Also in Proc. 29th ACM Symp. on Computational Geometry (2013), , and in arxiv: [21] M. Sharir and N. Solomon, Incidences between points and lines in R 4, Proc. 30th Annu. ACM Sympos. Comput. Geom., 2014, [22] M. Sharir and N. Solomon, Incidences between points and lines in four dimensions, in arxiv: [23] M. Sharir and N. Solomon, Incidences between points and lines on a two-dimensional variety, manuscript, [24] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput. Geom. 48 (2012), [25] L. Székely, Crossing numbers and hard Erdős problems in discrete geometry, Combinat. Probab. Comput. 6 (1997), [26] E. Szemerédi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), [27] T. Tao, From rotating needles to stability of waves: Emerging connections between combinatorics, analysis, and PDE, Notices AMS 48(3) (2001), [28] T. Tao, The Cayley Salmon theorem via classical differential geometry, March [29] H. E. Warren, Lower bound for approximation by nonlinear manifolds, Trans. Amer. Math. Soc. 133 (1968), [30] J. Zahl, An improved bound on the number of point-surface incidences in three dimensions, Contrib. Discrete Math. 8(1) (2013). Also in arxiv: [31] J. Zahl, A Szemerédi-Trotter type theorem in R 4, in arxiv:

138 5 Ramsey-type theorems for lines in 3-space 129

139 Discrete Mathematics and Theoretical Computer Science DMTCS vol. 18:3, 2016, #14 Ramsey-Type Theorems for Lines in 3-space arxiv: v2 [math.co] 15 Sep 2016 Jean Cardinal 1 Michael S. Payne 2 Noam Solomon 3 1 Université libre de Bruxelles (ULB), Belgium 2 Monash University, Melbourne, Australia 3 Tel-Aviv University, Israel received 26 th Jan. 2016, revised 20 th Aug. 2016, accepted 21 st Aug We prove geometric Ramsey-type statements on collections of lines in 3-space. These statements give guarantees on the size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, and reguli in 3-space. Among other things, we prove the following: The intersection graph of n lines in R 3 has a clique or independent set of size Ω(n 1/3 ). Every set of n lines in R 3 has a subset of ( n lines that are all stabbed by one line, or a subset of Ω (n/ log n) 1/5) such that no 6-subset is stabbed by one line. Every set of n lines in general position in R 3 has a subset of Ω(n 2/3 ) lines that all lie on a regulus, or a subset of Ω(n 1/3 ) lines such that no 4-subset is contained in a regulus. The proofs of these statements all follow from geometric incidence bounds such as the Guth-Katz bound on pointline incidences in R 3 combined with Turán-type results on independent sets in sparse graphs and hypergraphs. As an intermediate step towards the third result, we also show that for a fixed family of plane algebraic curves with s degrees of freedom, every set of n points in the plane has a subset of Ω(n 1 1/s ) points incident to a single curve, or a subset of Ω(n 1/s ) points such that at most s of them lie on a curve. Although similar Ramsey-type statements can be proved using existing generic algebraic frameworks, the lower bounds we get are much larger than what can be obtained with these methods. The proofs directly yield polynomial-time algorithms for finding subsets of the claimed size. Keywords: Geometric Ramsey theory, Erdős-Hajnal property, incidence bounds 1 Introduction Ramsey theory studies the conditions under which particular discrete structures must contain certain substructures. Ramsey s theorem says that for every n, every sufficiently large graph has either a clique or an independent set of size n. Early geometric Ramsey-type statements include the Happy Ending Problem on convex quadrilaterals in planar point sets, and the Erdős-Szekeres Theorem on subsets in convex position [9]. We prove a number of Ramsey-type statements involving lines in R 3. The combinatorics of lines in space is a widely studied topic which arises in many applications such as computer graphics, motion ISSN c 2016 by the author(s) Distributed under a Creative Commons Attribution 4.0 International License

140 2 Jean Cardinal, Michael S. Payne, Noam Solomon planning, and solid modeling [4]. Our proofs combine two main ingredients: geometric information in the form of bounds on the number of incidences among the objects, and a Turán-type theorem that converts this information into a Ramsey-type statement. We establish a general lemma that allows us to streamline the proofs. Ramsey s Theorem for graphs and hypergraphs only guarantees the existence of rather small cliques or independent sets. However, as discussed below, for the geometric relations we study the bounds are known to be much larger. Therefore we are interested in finding the correct asymptotics. In particular, we are interested in the Erdős-Hajnal property. A class of graphs has this property if each member with n vertices has either a clique or an independent set of size n δ for some constant δ > 0. This comes from the Erdős-Hajnal conjecture which states that, for each graph H, the family of graphs excluding H as an induced subgraph has this property. Our results yield new Erdős-Hajnal exponents for each of the classes of (hyper)graphs studied. The results presented here make use of important recent advances in combinatorial geometry. The key example is the bound on the number of incidences between points and lines in R 3 given by Guth and Katz [12] in their recent solution of the Erdős distinct distances problem. Such results have sparked a lot of interest in the field, and it can be expected that further progress will yield further Ramsey-type results. 1.1 A general framework In general we consider two classes of geometric objects P and Q in R d and a binary incidence relation contained in P Q. For a finite set P P and a fixed integer t 2, we say that a t-subset S ( ) P t is degenerate whenever there exists q Q such that every p S is incident to q. Hence the incidence relation together with the integer t induces a t-uniform hypergraph H = (P, E), where E ( ) P t is the set of all degenerate t-subsets of P. A clique in this hypergraph is a subset S P such that ( S t) E. Similarly, an independent set is a subset S P such that ( S t) E =. In what follows, the families P and Q will mostly consist of lines or points in 3-space. We are interested in Ramsey-type statements stating that the t-uniform hypergraph H induced by a set P P of size n has either a clique of size ω(n) or an independent set of size α(n). 1.2 Previous results We first briefly survey some known results that fit into this framework. In many cases, either P or Q is a set of points. When P is a set of points, finding a large independent set amounts to finding a large subset of points in some kind of general position defined with respect to Q. When Q is the set of points, we are dealing with intersections between the objects in P. In particular, the case t = 2 corresponds to the study of geometric intersection graphs. General position subset problems A set in R d is usually said to be in general position whenever no d + 1 points lie on a hyperplane. For points and lines in the plane, Payne and Wood proved that the Erdős-Hajnal property essentially holds with exponent 1/2 [19]. Cardinal et al. proved an analogous result in R d [3]. Theorem 1.1 ([19, 3]). Fix d 2. Every set of n points in R d contains n cohyperplanar points or Ω((n/ log n) 1/d ) points in general position.

141 Ramsey-Type Theorems for Lines in 3-space 3 In both cases, the proofs rely on incidence bounds, in particular the Szemerédi-Trotter Theorem [24] in the plane, and the point-hyperplane incidence bounds due to Elekes and Tóth [8] in R d. In this paper we formalise the technique used in those proofs in order to easily apply it to other incidence relations. Erdős-Hajnal properties for geometric intersection graphs A survey of Erdős-Hajnal properties for geometric intersection graphs was produced by Fox and Pach [10]. A general Ramsey-type statement for the case where P is the set of plane convex sets has been known for a long time. In what follows, a vertically convex set is a set whose intersection with any vertical line is a line segment. Theorem 1.2 (Larman et al. [16]). Any family of n compact, connected and vertically convex sets in the plane contains at least n 1/5 members that are either pairwise disjoint or pairwise intersecting. Larman et al. also showed that there exist arrangements of k line segments with at most k pairwise crossing and at most k pairwise disjoint segments. This lower bound was improved successively by Károlyi et al. [14], and Kyncl [15]. More recently Fox and Pach studied intersection graphs of a large variety of other geometric objects [11]. For example they proved the following about families of s-intersecting curves in the plane families such that no two curves cross more than s times. Theorem 1.3 (Fox-Pach [11]). For each ǫ > 0 and positive integer s, there is δ = δ(ǫ, s) > 0 such that if G is an intersection graph of a s-intersecting family of n curves in the plane, then G has a clique of size at least n δ or an independent set of size at least n 1 ǫ. Erdős-Hajnal properties for hypergraphs have been proved by Conlon, Fox, and Sudakov [6]. Semi-algebraic sets and relations A very general version of the problem for the case t = 2 has been studied by Alon et al. [1]. Here Ramseytype results are provided for intersection relations between semialgebraic sets of constant description complexity in R d. It was shown that intersection graphs of such objects always have the Erdős-Hajnal property. The proof combines a linearisation technique with a space decomposition theorem due to Yao and Yao [27]. The following general statement can be extracted from their proof. Theorem 1.4. Consider a relation R on elements of a family F of semi-algebraic sets of constant description complexity. Suppose that each element f F can be parameterized by a point f R d, and that the relation R can be mapped into a semi-algebraic set R in R 2d. For each g F, let Σ g = {f R d : (f, g ) R }. Let Q be the smallest dimension of a space R Q in which the description of Σ g becomes linear, and let k be the number of bilinear inequalities in the definition of R in R Q. Then the graph of the relation R satisfies the Erdős-Hajnal property with exponent 1/(2k(Q + 1)). A similar result is given for the so-called strong version of the Erdős-Hajnal property: for every such intersection relation, there exists a constant ǫ and a pair of subfamilies F 1, F 2 F, each of size at least ǫ F, such that either every element of F 1 intersects every element of F 2, or no element of F 1 intersects any element of F 2. The exponent for the usual Erdős-Hajnal statement is a function of this ǫ. As an example, Alon et al. applied their machinery to prove the following result on arrangement of lines in R 3. Theorem 1.5 (Alon et al. [1]). Every family of n pairwise skew lines in R 3 contains at least k n 1/6 elements l 1, l 2,..., l k such that l i passes above l j for all i < j.

142 4 Jean Cardinal, Michael S. Payne, Noam Solomon For the problems we consider, however, the exponents we obtain are significantly larger than what can be obtained from Theorem 1.4. A general version of this problem in which degenerate t-tuples are defined by a finite number of polynomial equations and inequalities of bounded description complexity has recently been studied by Conlon et al. [5]. They show that the Ramsey numbers in this general setting grow like towers of height t 1, and that this is asymptotically tight. Such a setting is relevant here, since we also consider Erdős-Hajnal statements for some geometric hypergraphs. 1.3 Summary of our results In Section 2 we give a simple lemma that allows to convert geometric incidence bounds into bounds on the number of degenerate subsets, hence on the number of hyperedges of the hypergraphs of interest. We also recall the statements of the Turán bound for hypergraphs due to Spencer. Section 3 deals with the case where P and Q are lines and points in R 3. A natural object to consider is the intersection graph of lines in R 3, for which we prove the Erdős-Hajnal property with exponent 1/3. Theorem 3.7. The intersection graph of n lines in R 3 has a clique or independent set of size Ω(n 1/3 ). This makes use of the Guth-Katz incidence bound between points and lines in R 3 [13]. We further show that this exponent can be raised to 1/2 if we consider lines in the projective 3-space. We also show how to obtain bounds on the size of independent sets for t = 3, in which a subset of lines in general position is defined as a set of lines with no three intersecting in the same point. Section 4 deals with the setting where both P and Q are lines in R 3. We prove the following theorem. Theorem 4.1. Let L be a set of n lines in R 3. Then either there is a subset of ( n lines of L that are all stabbed by one line, or there is a subset of Ω (n/ log n) 1/5) lines of L such that no 6-subset is stabbed by one line. The proof involves lifting the set of lines to a set of points and hyperplanes in R 5, and applying the Ramsey-type result on points and hyperplanes due to Cardinal et al. [3]. The latter in turn relies on a point-hyperplane incidence bound due to Elekes and Tóth [8]. Finally, in Section 5 we introduce the notion of a subset of lines in general position in R 3 with respect to reguli, defined as loci of lines intersecting three pairwise skew lines. We use the Pach-Sharir bound on incidences between points and curves in the plane [18] to obtain the following result. Theorem 5.5. Let L be a set of n pairwise skew lines in R 3. Then there are Ω(n 2/3 ) lines on a regulus, or Ω(n 1/3 ) lines, no 4-subset of which lie on a regulus. We also explain how to use a line-regulus incidence bound due to Aronov et al. [2] for an alternative proof of this result. The large subsets whose existence our results guarantee can be found in polynomial time. In each case, a degenerate t-subset is incident to only one element of Q (for example, three collinear points lie on only one line). Furthermore, the cliques given by our results are of a particular type: all the elements intersect a single element of Q (for example, a collinear set of points). Thus the largest such clique in the hypergraph H can be found in polynomial time by checking all the elements of Q that determine a degenerate t- subset (for example, all lines determined by the point set). If the clique size is small, Turán-type theorems yield an independent set of a guaranteed minimum size. These theorems are constructive, hence the large independent set can be found efficiently.

143 Ramsey-Type Theorems for Lines in 3-space 5 2 Preliminaries In order to prove the existence of large independent sets in hypergraphs with no large clique, we proceed in two steps. First, we use incidence bounds to get upper bounds on the density of the (hyper)graph. Then we apply Turán s Theorem or its hypergraph analogue to find a lower bound on the size of the independent set. This is an extension of the method used to prove Theorem 1.1 in [19, 3]. The use of incidence bounds is also reminiscent from the technique used by Pach and Sharir for the repeated angle problem [17]. The following lemma will allow us to quickly convert incidence bounds into density conditions. Recall that we consider two families P and Q with an incidence relation in P Q, and that a t-subset S of P is said to be degenerate whenever there exists q Q such that every p S is incident to q. Lemma 2.1. Let P be a subset of P with P = n, such that no element of Q is incident to more than l elements of P. Let us denote by P k the number of elements of Q incident to at least k elements of P, and suppose P k g(n)/k a for some function g and some real number a. Then the number of degenerate t-subsets induced by P is at most g(n) if t < a, m g(n) log l if t = a, g(n)l t a if t > a. Proof: Let P j be the number of elements of Q incident to exactly j elements of P. Then l ( ) ) j l l j l l m = P j < P j j t < P j (t k t 1 k t 1 t = j=t j=1 l k t 1 P k g(n) k=1 j=1 l k t 1 a, k=1 where we use that j k=1 kt 1 = j t /t + O(j t 1 ), and t = O(1). The final sum simplifies differently depending on the relative values of t and a. We recall the statement of Turán s Theorem. Theorem 2.2 (Turán [25]). Let G be a graph with n vertices and m edges. Then α(g) m < n/2 then α(g) > n/2. Otherwise α(g) n 2 /4m. k=1 The hypergraph version of this result was proved by Spencer. k=1 j=k P j n 2m n +1. Thus if Theorem 2.3 (Spencer [23]). Let H be a t-uniform hypergraph with n vertices and m edges. If m < n/t then α(h) > n/2. Otherwise α(h) t 1 n t t/(t 1) (m/n). 1/(t 1) 3 Points and lines in R 3 The recent resolution of Erdős distinct distance problem by Guth and Katz involves new bounds on the number of incidences between points and lines in R 3 [12]. This breakthrough has fostered research on point-line incidence bounds in space. In this section and the next, we exploit those recent results to obtain various new Ramsey-type statements on point-line incidence relations in space.

144 6 Jean Cardinal, Michael S. Payne, Noam Solomon 3.1 General position with respect to lines Theorem 1.1 for d = 2 states that in a set P of n points in the plane there exist either n collinear points, or Ω( n/ log n) points with no three collinear. Payne and Wood [19] conjectured that the true size should be Ω( n), but this small improvement has proven elusive. Here we consider the same question but with P = R 3, Q defined as the set of lines in R 3, and t = 3. Hence we consider that a set P R 3 is in general position when no three points are collinear. So far this is the same question as in the planar case, since a point set in higher dimensional space can always be projected to the plane in a way that maintains the collinearity relation. However, under a small extra assumption, namely that among the n points in R 3, at most n/ log n are coplanar, we are able to remove the log n factor in the independent set. This sheds some light on the nature of potential counterexamples to the conjecture of Payne and Wood. We will use the following result of Dvir and Gopi [7], which is deduced from Guth and Katz [13]. Theorem 3.1. Given a set P of n points in R 3, such that at most s points are contained in a plane, the number P k of lines containing at least k points is P k n2 k 4 + ns k 3 + n k. Theorem 3.2. Any set of n points in R 3 such that at most n/ log n of the points lie in a plane contains either n collinear points or Ω( n) with no three collinear. Proof: We apply Lemma 2.1 on each term of the bound in Theorem 3.1. We obtain that the number of degenerate 3-subsets of points is m n 2 + ns log l + nl 2, where l = n and s = n/ log n. Hence the dominating term is n 2. Applying Theorem 2.3 yields an independent set of size Ω( n). In fact, this theorem holds in R d for d > 3. To see this, we take a generic projection of R d onto R 3. The condition that at most n/ log n lines are coplanar remains true under a generic projection. 3.2 Line intersection graphs in R 3 We now consider the setting in which the family P is the set of lines in R 3 and Q = R 3. The first subcase we consider is t = 2, or in other words, intersection graphs of lines. Note that in an intersection graph of lines in R 3, every clique of size k 2 corresponds either to a subset of k lines having a common intersection point, or to a subset of k lines lying in a plane. However, k lines lying in a plane do not form a clique if some of them are parallel. We consider a set L of n lines in R 3, such that no more than l lines intersect in a point, and at most s lines lie in a common plane or a regulus. We recall that a regulus is a degree two algebraic surface, which is the union of all the lines in R 3 that intersect three pairwise-skew lines in R 3. It is a doubly-ruled surface; each point on a regulus is incident to precisely two lines fully contained in the regulus. Moreover, there are two rulings for the regulus; every line from one ruling intersects every line from the other ruling, and does not intersect any line from the same ruling. We first recall two important theorems of Guth and Katz [13]. In what follows, P k denotes the number of points incident to at least k lines in L.

145 Ramsey-Type Theorems for Lines in 3-space 7 Theorem 3.3 ([13, Theorem 4.5]). If L is a set of n lines, so that no plane contains more than s lines, then for k 3 we have P k n3/2 k 2 + ns k 3 + n k. Theorem 3.4 ([13, Theorem 2.11],[21]). If L is a set of n lines, so that no plane or regulus contains more than s lines, then P 2 n 3/2 + ns. Note the difference between the two statements: the assumption that no regulus contains more than s lines is required for the case k = 2 only. Applying Lemma 2.1 to the bounds in Theorems 3.3 and 3.4 yields the following. Proposition 3.5. Given a set L of n lines, so that no plane or regulus contains more than s lines, and no point is incident to more than l lines of L, the number of line-line incidences is O(n 3/2 log l + ns + nl). Lemma 3.6. Consider a set L of n lines in R 3, such that no plane contains more than s lines, and no point is incident to more than l lines of L. Let G be the intersection graph L. If s, l n 1/2, then α(g) n/ log l. Moreover, if r := max{s, l} n 1 2 +ǫ for some ǫ > 0, then α(g) n/r. Proof: If there is some regulus containing at least n 1/2 lines, we divide the lines into the two rulings of the regulus. One ruling contains at least half the lines, and as the lines in one ruling do not intersect one another, it follows that α(g) n 1/2. We may therefore assume that the number of lines contained in a common regulus is at most n 1/2. If s, l n 1/2, the first term in the bound in Proposition 3.5 dominates, and applying Theorem 2.2 gives α(g) n/ log l. If r n 1 2 +ǫ, one of the latter terms dominates, and we apply Theorem 2.2 to get α(g) n/r. Theorem 3.7. The intersection graph of n lines in R 3 has a clique or independent set of size Ω(n 1/3 ). Proof: Suppose that such a graph G has α(g) n 1/3. Then by Lemma 3.6, max{s, l} n 2/3. If l n 2/3 we are done, so s n 2/3. Therefore, we may assume that there is a plane containing n 2/3 lines. Divide these lines into classes of pairwise parallel lines. If some class contains at least n 1/3 lines, we have α(g) n 1/3. Otherwise, there are at least n 1/3 different parallel classes. Choosing one line from each class yields a clique of size n 1/3. Note that the Erdős-Hajnal property for intersection graphs of lines in R 3 can be directly established from Theorem 1.4 by Alon et al. [1], but with a much smaller exponent. In their setting, we can represent the intersection relation between lines using Plücker coordinates in R 5, and using two inequalities. This yields k = 2 and Q = 5, and an Erdős-Hajnal exponent of 1/24. Although it is likely that it can be improved by shortcutting steps in the general proof, any exponent we would get would still be far from 1/3. We now make a connection with intersection graphs of lines in space and line graphs. Recall that the line graph of a graph G has the set of edges E(G) as vertex set, and an edge between two edges of G whenever they are incident to the same vertex of G. Observe that for every graph G, the line graph of G can be represented as the intersection graph of lines in R 3 by drawing G on a vertex set in general enough position in R 3, and extending the edges of the drawing to lines. By applying Vizing s Theorem, which says that the edge chromatic number of every graph is at most + 1, we may see that the class

146 8 Jean Cardinal, Michael S. Payne, Noam Solomon of line graphs has the Erdös Hajnal property with exponent 1/2. The question of the exact Erdös Hajnal exponent for intersection graphs of lines in R 3 remains open it lies somewhere between 1/3 and 1/2. Finally we note that for sets of lines in projective space, coplanar sets of lines always form a clique. The following stronger result can be directly obtained. Theorem 3.8. For every intersection graph G of n lines in P 3, either ω(g) n or α(g) = Ω( n/ log n). Hence intersection graphs of lines in the projective plane satisfy the Erdős-Hajnal property with exponent roughly 1/ Independent Sets of Lines for t = 3 We now consider the case in which P is the set of lines in R 3, Q = R 3 and t = 3. This can be seen as a kind of three-dimensional version of the dual of the result of Payne and Wood (Theorem 1.1 with d = 2). Theorem 3.9. Consider a collection L of n lines in R 3, such that at most s lie in a plane, with s n/ log n. Then there exists a point incident to n lines, or a subset of Ω( n) lines such that at most two intersect in one point. Proof: We let l be the largest number of lines intersecting in one point, and suppose l < n. Applying Lemma 2.1 and Theorem 3.3, we get that the number of triples sharing a point is at most m ln 3/2 + ns log l + nl 2 n 2. Then by Theorem 2.3 we have an independent set of size Ω( n). If the above theorem is stated with dependence on l, we get Ω(n 3/4 / l). If s is allowed to be as large as n, we are back in the dual of general position subset selection, and we get Ω( n/ log n), the same as Theorem Stabbing lines in R 3 Three lines in R 3 are typically intersected by a fourth line, except in certain degenerate cases. Thus it makes sense to study configurations of lines in R 3, and to consider a set of 4 or more lines degenerate if all its elements are intersected by another line. Here we provide a result for 6-tuples of lines. We define a 6-tuple of lines to be degenerate if all six lines are intersected (or stabbed ) by a single line in R 3. We call this line a stabbing line for the 6-tuple of lines. So in our framework this is the setting in which both P and Q are the set of lines in R 3, and t = 6. We make use of the Plücker coordinates and coefficients for lines in R 3, which are a common tool for dealing with incidences between lines, see e.g. Sharir [20]. Let a = (a 0 : a 1 : a 2 : a 3 ), b = (b 0 : b 1 : b 2 : b 3 ) be two points on a line l, given in projective coordinates. By definition, the Plücker coordinates of l are given by (π 01 : π 02 : π 12 : π 03 : π 13 : π 23 ) P 5, where π ij = a i b j a j b i for 0 i < j 3. Similarly, the Plücker coefficients of l are given by (π 23 : π 13 : π 03 : π 12 : π 02 : π 01 ) P 5,

147 Ramsey-Type Theorems for Lines in 3-space 9 i.e., these are the Plücker coordinates written in reverse order with two signs flipped. The important property is that two lines l 1 and l 2 are incident if and only if the Plücker coordinates of l 1 lie on the hyperplane defined by the Plücker coefficients of l 2 and vice versa. Therefore, we define P, and Q to be the points in P 5 defined by the Plücker coordinates of the lines in L, and the hyperplanes defined by the Plücker coefficients of the lines in R 3, respectively. The incidence relation between points in P and hyperplanes in Q is the standard incidence relation between points and hyperplanes. The integer t is set to 6, and a 6-tuple of points in P is degenerate whenever there is a hyperplane in Q which is incident to all six points in the 6-tuple. We prove the following Ramsey-type result for stabbing lines in R 3. Theorem 4.1. Let L be a set of n lines in R 3. Then either there is a subset of ( n lines of L that are all stabbed by one line, or there is a subset of Ω (n/ log n) 1/5) lines of L such that no 6-subset is stabbed by one line. Theorem 4.1 is an immediate consequence of the following generalisation of Theorem 1.1. The difference is that the set of hyperplanes H is arbitrary instead of being the set of all hyperplanes in R d. Theorem 4.2. Let H be a set of hyperplanes in R d. Then, every set of( n points in R d with at most l points on any hyperplane in H, where l = O(n 1/2 ), contains a subset of Ω (n/ log l) 1/d) points so that every hyperplane in H contains at most d of these points. Theorem 4.2, with d = 5, applied to the points and hyperplanes given by the Plücker coordinates and coefficients, implies Theorem 4.1. Theorem 4.2 follows from the following generalized version of Lemma 4.5 of Cardinal et al. [3]. Lemma 4.3. Fix d 2 and a set H of hyperplanes in R d. Let P be a set of n points in R d with no more than l points in a hyperplane in H, for some l = O(n 1/2 ). Then, the number of (d + 1)-tuples in P that lie in a hyperplane in H is O(n d log l). The difference between this lemma and the original version in [3] is that the set of hyperplanes H is arbitrary, rather than being the set of all hyperplanes. The proof is similar to that of Cardinal et al., and is given in Appendix A. The following result provides a simple upper bound. Theorem 4.4. For every constant integer t 4, there exists an arrangement L of n lines in R 3 such that there is no subset of more than O( n) lines that are all stabbed by one line, nor any subset of more than O( n) lines with no t stabbed by one line. Proof: Construct L as follows: pick n parallel planes, each containing n lines, with no three intersecting and no two parallel. Consider a subset stabbed by one line. Either it has three coplanar lines; then it must be fully contained in one of the planes and contains at most n lines; or it has no three coplanar lines, hence contains at most two lines from each plane, and has at most 2 n lines. Now consider a subset such that no t lines are stabbed by one. Then it contains at most t 1 lines from each plane, and has at most (t 1) n lines.

148 10 Jean Cardinal, Michael S. Payne, Noam Solomon 5 Lines and reguli in R 3 Consider the case in which P is the class of lines in R 3, Q is the class of reguli, and t = 4. Let P be a set of n lines, and assume that the lines in P are pairwise skew. Every triple of lines in P therefore determines a single regulus, and we may consider the set of all reguli determined by P. We consider the containment relation rather than intersection we are interested in 4-tuples that all lie in the same regulus. In order to prove our result, we first reformulate previously known incidence bounds between points and curves in the plane. 5.1 General position with respect to algebraic curves We first consider the case where P = R 2 and Q is a family of algebraic curves of bounded degree. We define the number of degrees of freedom of a family of algebraic curves C to be the minimum value s such that for any s points in R 2 there are at most c curves passing through all of them, for some constant c. Moreover, C has multiplicity type r if any two curves in C intersect in at most r points. We consider a set of points to be in general position with respect to C when no s + 1 points lie on a curve in C. It is possible to extract Ramsey-type statements for this situation directly from Theorem 1.1 via linearisation. For example, let us consider the special case of circles, where s = 3. Given a set of points in the plane, we can lift it onto a paraboloid in R 3 in such a way that a subset of the original set lies on a circle (possibly degenerated into a line) if and only if the corresponding lifted points lie on a hyperplane in R 3. By applying Theorem 1.1 on the lifted set, we can show that there exists a subset of n points incident to a circle, or a subset of Ω((n/ log n) 1/3 ) points such that at most three of them lie on a circle. We show how we can improve on this. In order to apply our technique, we need Szemerédi-Trotter-type bounds on the number of incidences between points and curves. This has been studied by Pach and Sharir [18]. Theorem 5.1 ([18]). Let P be a set of n points in the plane and let C be a set of m bounded degree plane algebraic curves with s degrees of freedom and multiplicity type r. Then the number of point-curve incidences is at most ( ) I(P, C) C(r, s) n s/(2s 1) m (2s 2)/(2s 1) + n + m where C(r, s) is a constant depending only on r and s. Pach and Sharir proved Theorem 5.1 for simple curves with s degrees of freedom and multiplicity type r. It is well known that one may replace simple curves with bounded degree algebraic curves, since such curves may be cut into a constant number of simple pieces. Note that a set of bounded degree algebraic curves has constant multiplicity type if no two curves share a common component. Wang et al. [26] recently proved another result for incidences between points and algebraic curves, though for our purposes Theorem 5.1 is stronger. Theorem 5.2. Consider a family C of bounded degree algebraic curves in R 2 with constant multiplicity type and s degrees of freedom, for some s > 2. Then in any set of n points in R 2, there exists a subset of Ω(n 1 1/s ) points incident to a single curve of C, or a subset of Ω(n 1/s ) points such that at most s of them lie on a curve of C. Proof: Set t = s + 1 and count the number of degenerate t-subsets. We denote by P k the number of curves of C containing at least k points of P. A direct corollary of Theorem 5.1 is that, for values of k

149 Ramsey-Type Theorems for Lines in 3-space 11 larger than some constant, P k ns k 2s 1 + n k. On the other hand, for smaller values of k, the trivial bound P k n s holds since for any s points, there are at most a constant number of curves passing through all of them. Suppose now that no curve contains more than l n 1 1/s points of P. Since s > 2, it follows that t < 2s 1. Using Lemma 2.1, we deduce that the number of degenerate t-subsets is m n s + nl s n s. Thus by Theorem 2.3 there exists an independent set of size at least t 1 n t t/(t 1) (m/n) = 1/(t 1) Ω(n1/s ). As an example, we can instantiate the result as follows for circles in the plane. Corollary 5.3. In any set of n points in R 2, there exists a subset of Ω(n 2/3 ) points incident to a circle, or a subset of Ω(n 1/3 ) points such that no four of them lie on a circle. Using the standard point-line duality, Theorem 1.1 states that for every arrangement of n lines in R 2, either there exists a point contained in n lines, or there exists a set of Ω((n/ log n) 1/2 ) lines inducing a simple arrangement, that is, such that no point belongs to more than two lines. We provide a similar dual version of Theorem 5.2. This corresponds to the case where P is a family of algebraic curves with s degrees of freedom, Q = R 2, and t = 3. As mentioned before, the case t = 2, or intersection graphs, has been studied previously [10, 11]. The proof is very similar to that of Theorem 5.2 and omitted. Theorem 5.4. Consider a family C of bounded degree algebraic curves in R 2 with constant multiplicity type and s degrees of freedom, for some s > 2. Then in any arrangement C of m such curves, there exists a subset of Ω(m 1 1/s ) curves intersecting in one point, or a subset of Ω(m 1/s ) curves inducing a simple subarrangement, that is, such that at most two intersect in one point. 5.2 Ramsey-type results for lines and reguli in R 3 We now come back to our original problem in which P is the class of lines in R 3, Q is the class of reguli, and t = 4. Here we restrict the finite arrangement P P to be pairwise skew, that is, pairwise nonintersecting and nonparallel. We also consider the containment relation, that is, l P is incident to R Q if it is fully contained in it. Recall that a regulus can be defined as a quadratic ruled surface which is the locus of all lines that are incident to three lines in general position. This surface is a doubly ruled surface, that is, every point on a regulus is incident to precisely two lines fully contained in it. There are only two kinds of reguli, both of which are quadrics hyperbolic paraboloids and hyperboloids of one sheet; see for instance Sharir and Solomon [22] for more details. Theorem 5.5. Let L be a set of n pairwise skew lines in R 3. Then there are Ω(n 2/3 ) lines on a regulus, or Ω(n 1/3 ) lines, no 4-subset of which lie on a regulus.

150 12 Jean Cardinal, Michael S. Payne, Noam Solomon Proof: We map the lines in L to a set P of points in R 4. This can be done for instance by associating with each line the x- and y-coordinates of the two points of intersection with the planes z = 0 and z = 1. (We may assume no line is parallel to these planes). Under this mapping, a ruling of a regulus corresponds to an algebraic curve in R 4. Let C be the finite set of all curves corresponding to a ruling of a regulus determined by three lines in L. Note that any triple of points in R 4 is contained in at most one such curve, because three lines in R 3 lie in at most one ruling of one regulus. (A pair of parallel or intersecting lines are not contained in a ruling of any regulus, even though they are contained in many reguli). Apply a generic projection π from R 4 to R 2, and consider the arrangement of points P = π(p ) together with the set of projected curves C = π(c). Such a projection preserves the incidences between points and curves in R 4, and only creates new intersections between pairs of curves (i.e. simple crossings). Three or more curves in C intersect in a point if and only if their preimages in C intersect in a point. The set of curves C has three degrees of freedom, since for any three points in R 2 there are at most two curves passing through all of them. Otherwise, if three curves pass through three points, the corresponding curves in C also intersect in three points in R 4, a contradiction. Moreover, the curves in C are algebraic of bounded degree, do not share common components, and thus have constant multiplicity type. Applying Theorem 5.2 with s = 3, we obtain that there are Ω(n 2/3 ) points of π(p ) on one curve, or Ω(n 1/3 ) points of π(p ), no four of which lie on a curve. The result follows. The bounds can be shown to be tight in the following sense. Theorem 5.6. There exists a set P of n pairwise skew lines in R 3 such that there is no subset of more than O(n 2/3 ) lines on a regulus, and no more than O(n 1/3 ) lines such that no 4-subset lie on a regulus. Proof: The set P is constructed as follows: take a set of n 1/3 distinct reguli, and for each regulus take n 2/3 lines in one of its rulings, giving n pairwise skew lines. Consider a subset of P contained in a regulus. Either it is one of the chosen reguli, and it contains at most n 2/3 lines, or it contains at most two lines from each regulus, and has size at most 2n 1/3. On the other hand, consider a subset of lines with no four on a regulus. It can contain at most three lines from each chosen regulus, and therefore has size at most 3n 1/3. Alternative proof. Aronov et al. [2] proved the following bound on the number of incidences between lines and reguli in 3-space. Theorem 5.7 (Aronov et al.[2]). Let L be a set of n lines in R 3, and let R be a set of m reguli in R 3. Then the number of incidences between the lines of L and the reguli of R is O(n 4/7 m 17/21 +n 2/3 m 2/3 +m+n). From this bound, one may derive an alternative proof of Theorem 5.5, of which we now give a brief sketch. First bound P k, defined as the number of reguli containing at least k lines. From the above Theorem, we get P k n 3 /k 21/4 + n 2 /k 3 + n/k. Then from Lemma 2.1 we know that if no regulus contains more than l lines, then the number of degenerate 4-tuples of lines is m n 3 + n 2 l + nl 3. Hence either l is larger than n 2/3, or m n 3 and from Theorem 2.3 there exists an independent set of lines of size Ω(n 1/3 ).

151 Ramsey-Type Theorems for Lines in 3-space 13 Acknowledgments The authors wish to thank the reviewers for their comments, including those on earlier, preliminary versions of this paper. References [1] Noga Alon, János Pach, Rom Pinchasi, Rados Radoicic, and Micha Sharir. Crossing patterns of semi-algebraic sets. J. Comb. Theory, Ser. A, 111(2): , [2] Boris Aronov, Vladlen Koltun, and Micha Sharir. Incidences between points and circles in three and higher dimensions. Discrete & Computational Geometry, 33(2): , [3] Jean Cardinal, Csaba D. Tóth, and David R. Wood. General Position Subsets and Independent Hyperplanes in d-space. ArXiv e-prints, 2014, To appear in Journal of Geometry. [4] Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, and Jorge Stolfi. Lines in space: Combinatorics and algorithms. Algorithmica, 15(5): , [5] David Conlon, Jacob Fox, János Pach, Benny Sudakov, and Andrew Suk. Ramsey-type results for semi-algebraic relations. In Proc. Symposium on Computational Geometry (SoCG), pages , [6] David Conlon, Jacob Fox, and Benny Sudakov. Erdős-Hajnal-type theorems in hypergraphs. J. Comb. Theory, Ser. B, 102(5): , [7] Zeev Dvir and Sivakanth Gopi. On the number of rich lines in truly high dimensional sets. In Proc. 31st International Symposium on Computational Geometry (SoCG), [8] György Elekes and Csaba D. Tóth. Incidences of not-too-degenerate hyperplanes. In Proceedings of the 21st ACM Symposium on Computational Geometry (SoCG), pages 16 21, [9] Paul Erdős and George Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2: , [10] Jacob Fox and János Pach. Erdős-Hajnal-type results on intersection patterns of geometric objects. Bolyai Society Mathematical Studies Horizon of Combinatorics, 17:79 103, [11] Jacob Fox and János Pach. Coloring K k -free intersection graphs of geometric objects in the plane. Eur. J. Comb., 33(5): , [12] Larry Guth and Nets H. Katz. Algebraic methods in discrete analogs of the Kakeya problem. Advances Math., 225: , [13] Larry Guth and Nets H. Katz. On the Erdős distinct distances problem in the plane. Annals Math., 181: , [14] Gyula Károlyi, János Pach, and Géza Tóth. Ramsey-type results for geometric graphs, I. Discrete & Computational Geometry, 18(3): , 1997.

152 14 Jean Cardinal, Michael S. Payne, Noam Solomon [15] Jan Kyncl. Ramsey-type constructions for arrangements of segments. Eur. J. Comb., 33(3): , [16] David Larman, Jiri Matoušek, János Pach, and Jeno Torocsik. A Ramsey-type result for convex sets. Bull. London Math. Soc., 26(2): , [17] János Pach and Micha Sharir. Repeated angles in the plane and related problems. J. Comb. Theory, Ser. A, 59(1):12 22, [18] János Pach and Micha Sharir. On the number of incidences between points and curves. Combinatorics, Probability & Computing, 7(1): , [19] Michael S. Payne and David R. Wood. On the general position subset selection problem. SIAM J. Discrete Math., 27(4): , [20] Micha Sharir. On joints in arrangements of lines in space and related problems. J. Comb. Theory, Ser. A, 67.1:89 99, [21] Micha Sharir and Noam Solomon. Incidences between points and lines in three dimensions. ArXiv e-prints, 2015, [22] Micha Sharir and Noam Solomon. Incidences between points and lines on a two-dimensional variety. ArXiv e-prints, 2015, [23] Joel Spencer. Turán s theorem for k-graphs. Discrete Mathematics, 2(2): , [24] Endre Szemerédi and William T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3(3): , [25] Paul Turán. On an extremal problem in graph theory. Mat. Fiz. Lapok, 48: , [26] Hong Wang, Ben Yang, and Ruixiang Zhang. Bounds of incidences between points and algebraic curves. ArXiv e-prints, 2013, [27] Andrew Chi-Chih Yao and F. Frances Yao. A general approach to d-dimensional geometric queries (extended abstract). In Proceedings of the 17th Annual ACM Symposium on Theory of Computing (STOC), pages , A Proof of Lemma 4.3 For the proof we need the following observation regarding generic projection maps. Lemma A.1. Let P be a finite set of points in R d, and let A be a finite set of (d 2)-flats in R d. Let π be a generic projection from R d to a hyperplane. Then a point p P lies on a (d 2)-flat A A if and only if π(p) π(a). Proof: The forward implication is clear. For the other direction, suppose p / A. Then the affine span of {p} A is a hyperplane, that is, it is (d 1)-dimensional. By the genericity of π, the image π(span({p} A)) must also be (d 1)-dimensional, so π(p) / π(a). We also need the following result of Elekes and Tóth [8]. Given a point set P, a hyperplane h is said to be γ-degenerate if at most γ P h points of P h lie on a (d 2)-flat.

153 Ramsey-Type Theorems for Lines in 3-space 15 Theorem A.2. For every d 3 there exist constants C d > 0 and γ d > 0 such that for every set of n points in R d, the number h k of γ d -degenerate hyperplanes containing at least k points of P is at most For convenience we restate Lemma 4.3. ( ) n d nd 1 C d + kd+1 k d 1. Lemma 4.3. Fix d 2 and a set H of hyperplanes in R d. Let P be a set of n points in R d with no more than l points in a hyperplane in H, for some l = O(n 1/2 ). Then, the number of (d + 1)-tuples in P that lie in a hyperplane in H is O(n d log l). Proof: The proof is an adaptation of the proof of Lemma 4.5 in Cardinal et al. [3]. It proceeds by induction on d 2. The base case is d = 2. We wish to bound the number of triples of points of P, lying on a line in H. Let h k (resp., h k ) denote the number of lines of H containing exactly (resp., at least) k points of P. The number of triples of points lying on a line of H is l k=3 h ( k ) l k 3 k=3 k2 h k l ( k=3 k2 n 2 k + n 3 k ) n 2 log l + l 2 n n 2 log l, (1) where h k n2 k + n 3 k follows by the Szemerédi-Trotter Theorem [24]. We now consider the general case d 3. Let P be a set of n points in R d, with no more than l points in a hyperplane in H, where H is a given set of hyperplanes in R d, and l = O(n 1/2 ). Let γ := γ d > 0 be the constant specified in Theorem A.2. We distinguish between the following three types of (d + 1)-tuples: Type 1: (d + 1)-tuples of P contained in a (d 2)-flat in a hyperplane in H. Let F be the set of (d 2)-flats that are contained in some hyperplane in H and spanned by the points P. Let s k denote the number of flats in F that contain exactly k points of P. We project P onto a (d 1)-flat K via a generic projection π to obtain a set of points P := π(p ) in R d 1. Let H be the set of hyperplanes π(γ) for each Γ F. By Lemma A.1, P Γ = P π(γ) for each Γ F. Thus s k is also the number of hyperplanes in H containing k points of P. Moreover, the hyperplanes in H contain at most l points of P. Applying the induction hypothesis on P with respect to H we deduce that the number of d-tuples in P that lie in a hyperplane in H is l ( ) k s k n d 1 log l. d Therefore, the number of (d + 1)-tuples of P lying on a (d 2)-flat in F is l k=d+1 k=d ( ) k s k d + 1 l k=d+1 ( ) k ks k ln d 1 log l n d log l. d

154 16 Jean Cardinal, Michael S. Payne, Noam Solomon Type 2: (d + 1)-tuples of P that span a γ-degenerate hyperplane in H. Let h k denote the number of γ-degenerate hyperplanes in H containing exactly k points of P. Using Theorem A.2, we get l k=d+1 h k( k d+1) l k=d+1 kd h k l ( k=d+1 kd n d k d+1 + nd 1 k d 1 ) n d log l + l 2 n d 1 n d log l. Type 3: (d + 1)-tuples of P that span a hyperplane in H that is not γ-degenerate. Recall that if a hyperplane H spanned by P is not γ-degenerate, then more than a γ fraction of its points lie in some (d 2)-flat. Consider a (d 2)-flat L containing exactly k points of P. A point in P \ L can be on at most one hyperplane containing L. Let n r denote the number of hyperplanes in H containing L and exactly r points of P \ L. Then r n rr n, and by assumption on the hyperplanes in H, we have r l. We will assign each tuple of Type 3 to a (d 2)-flat that causes it to be Type 3. Fix a (d 2)-flat L with k points and consider a hyperplane H H that is not γ-degenerate because it contains L. That is, suppose H contains r + k points, and k > γ(r + k), so r < O(k). All tuples that span H contain at least one point not in L. Hence the number of tuples that span H is O(rk d ). Assign these tuples to L. The total number of tuples of Type 3 that will be assigned to L in this way is therefore at most ( ) O n r rk d nk d. r Let F be the set of (d 2)-flats that have at least one Type 3 tuple assigned to them. Thus F is a finite set. Let s k denote the number of flats in F that contain exactly k points of P. We project P onto a (d 1)-flat K via a generic projection π to obtain a set of points P := π(p ) in R d 1. Let H be the set of hyperplanes π(γ) for each Γ F. By Lemma A.1, P Γ = P π(γ) for each Γ F. Thus s k is also the number of hyperplanes in H containing k points of P. Moreover, the hyperplanes in H contain at most l points of P. Applying the induction hypothesis on P with respect to H we deduce that the number of d-tuples in P that lie in a hyperplane in H is l ( ) k s k n d 1 log l. d k=d Moreover, d 1 k=1 s kk d n d 1. Therefore, the number of (d + 1)-tuples of Type 3 is at most l s k nk d n k=1 l s k k d n d log l. k=1 Summing over all three cases, the proof is complete. (2)

155 146

156 Part III Incidences between points and lines on varieties 147

157

158 6 Incidences between points and lines on two- and three-dimensional varieties 149

159 Incidences between points and lines on two- and three-dimensional varieties Micha Sharir Noam Solomon August 31, 2017 Abstract Let P be a set of m points and L a set of n lines in R 4, such that the points of P lie on an algebraic three-dimensional variety of degree D that does not contain hyperplane or quadric 1 components, and no 2-flat contains more than s lines of L. We show that the number of incidences between P and L is ( ) I(P, L) = O m 1/2 n 1/2 D + m 2/3 n 1/3 s 1/3 + nd + m, for some absolute constant of proportionality. This significantly improves the bound of the authors [37], for arbitrary sets of points and lines in R 4, when D is not too large. Moreover, when D and s are constant, we get a linear bound. The same bound holds when the three-dimensional surface is embedded in any higher-dimensional space. The bound extends (with a slight deterioration, when D is large) to the complex field too. For a complex three-dimensional variety, of degree D, embedded in C 4 (or in any higher-dimensional C d ), under the same assumptions as above, we have ( ) I(P, L) = O m 1/2 n 1/2 D + m 2/3 n 1/3 s 1/3 + D 6 + nd + m. For the proof of these bounds, we revisit certain parts of [37], combined with the following new incidence bound, for which we present a direct and fairly simple proof. Going back to the real case, let P be a set of m points and L a set of n lines in R d, for d 3, which lie in a common two-dimensional algebraic surface of degree D that does not contain any 2-flat, so that no 2-flat contains more than s lines of L (here we require that the lines of L also be contained in the surface). Then the number of incidences between P and L is ( ) I(P, L) = O m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + m + n. Work on this paper by Noam Solomon and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S. Israel Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), by the Blavatnik Computer Science Research Fund at Tel Aviv University, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. An earlier version of the paper, which only contains some of the results and in a weaker form, is: M. Sharir and N. Solomon, Incidences between points and lines on a two-dimensional variety, in arxiv: School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. michas@post.tau.ac.il School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. noam.solom@gmail.com 1 A quadric is an algebraic variety of degree two. 1

160 When d = 3, this improves the bound of Guth and Katz [15] for this special case, when D n 1/2. Moreover, the bound does not involve the term O(nD). This term arises in most standard approaches, and its removal is a significant aspect of our result. Again, the bound is linear when D = O(1). This bound too extends (with a slight deterioration, when D is large) to the complex field. For a complex two-dimensional variety, of degree D, when the ambient space is C 3 (or any higher-dimensional C d ), under the same assumptions as above, we have ( ) I(P, L) = O m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + D 3 + m + n. These new incidence bounds are among the very few bounds, known so far, that hold over the complex field. The bound for two-dimensional (resp., three-dimensional) varieties coincides with the bound in the real case when D = O(m 1/3 ) (resp., D = O(m 1/6 )). 1 Introduction Let P be a set of m points and L a set of n lines in R d or in C d. Let I(P, L) denote the number of incidences between the points of P and the lines of L; that is, the number of pairs (p, l) with p P, l L, and p l. If all the points of P and all the lines of L lie in a common 2-flat, then, in the real case, the classical Szemerédi Trotter theorem [44] yields the worst-case tight bound ( ) I(P, L) = O m 2/3 n 2/3 + m + n. (1) The same bound also holds in the complex plane, as shown later by Tóth [45] and Zahl [47]. This bound clearly also holds in R d and in C d, for any d, by projecting the given lines and points onto some generic 2-flat. Moreover, the bound will continue to be worst-case tight by placing all the points and lines in a common 2-flat, in a configuration that yields the planar lower bound. In the 2010 groundbreaking paper of Guth and Katz [15], an improved bound for the real case has been derived for I(P, L), for a set P of m points and a set L of n lines in R 3, provided that not too many lines of L lie in a common plane. Specifically, they showed: 2 Theorem 1.1 (Guth and Katz [15]). Let P be a set of m points and L a set of n lines in R 3, and let s n be a parameter, such that no plane contains more than s lines of L. Then ( ) I(P, L) = O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n. This bound (or, rather, an alternative formulation thereof) was a major step in the derivation of the main result of [15], which was an almost-linear lower bound on the number of distinct distances determined by any finite set of points in the plane, a classical problem posed by Erdős in 1946 [8]. Guth and Katz s proof uses several nontrivial tools from algebraic and differential geometry, most notably the Cayley Salmon Monge theorem on osculating lines to algebraic surfaces in R 3, and various properties of ruled surfaces. All this machinery comes on top of the major innovation of Guth and Katz, the introduction of the polynomial partitioning technique. 2 Actually, Theorem 1.1 is not stated explicitly in [15], but it follows immediately from the bounds that they derive. 2

161 For the purpose of the analysis in this paper, it is important to recall, right away, that the polynomial partitioning technique holds only over the reals. This will be the major stumbling block that we will face as we handle the complex case. We overcome (or rather bypass) it by exploiting the assumption that in this case all the lines are also contained in the given variety; see Section 3 for details. In four dimensions, and for the real case, the authors established in [37] a sharper and (almost) optimal bound. More precisely, they have shown: Theorem 1.2. Let P be a set of m points and L a set of n lines in R 4, and let s q n be parameters, such that (i) no hyperplane or quadric contains more than q lines of L, and (ii) no 2-flat contains more than s lines of L. Then, ( ) ( ) I(P, L) 2 c log m m 2/5 n 4/5 + m + A m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n, (2) for suitable (absolute) constant parameters A and c. Moreover, except for the factor 2 c log m, the bound is tight in the worst case, for all m and n and suitable ranges of q and s. For certain ranges of m and n the bound holds without that factor. Our results. In this work we contest the leading terms O ( m 1/2 n 3/4) (for d = 3) and 2 c log m m 2/5 n 4/5 (for d = 4), and present situations in which they can be significantly improved. A major feature of this work is that, in the setups considered here, the analysis can also be carried over to the complex domain, except for a small penalty that we pay for bypassing the polynomial partitioning technique, which, as noted, only holds over the reals. Concretely, we assume that the points of P lie on some algebraic variety, and derive significantly improved bounds when the degree of the variety is not too large. In the former case we assume that the points and the lines lie on a two-dimensional variety, which is allowed to be embedded in any R d, for d 3. In the latter case we assume that the points (but not necessarily the lines) lie on a three-dimensional variety, embedded in any R d, for d 4. In the former (resp., latter) case we also assume that the variety contains no plane (resp., no hyerplane or quadric). Thus, in addition to improving the respective bounds in Theorems 1.1 and 1.2, for the special cases under consideration, and extending them to the complex domain, we obtain an extra bonus by extending the results to two-dimensional and three-dimensional varieties embedded in any higher dimension. Points on a two-dimensional variety. We derive two closely related results, one that holds over the real field and one that holds also over the complex field. It is simplest to think of the variety as embedded in R 3 or in C 3. The real case is a special case of the setup of Guth and Katz [15], where there is no need to use the polynomial partitioning method, because we assume that the points and lines all lie in a common surface (the zero set of a polynomial) of degree 3 D. This very assumption is also the one that lets us derive the (slightly weaker) version that holds over C, thereby constituting a significant progress over the existing theory of incidences in three (and higher) dimensions. To be more precise, over the reals we do apply the polynomial partitioning technique (as a step in the application of the Guth-Katz bound), but only to a small subset of the lines. Concretely, our first main result, for this setup, is the following theorem. 3 See later, in Section 2, for a discussion of the notions of degree and dimension over the reals. 3

162 Theorem 1.3. (a) The real case: Let P be a set of m points and L a set of n lines in R d, for any d 3, and let 2 s D be two integer parameters, so that all the points and lines lie in a common two-dimensional algebraic variety V of degree D that does not contain any 2-flat, and so that no 2-flat contains more than s lines of L. Then ( ) I(P, L) = O m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + m + n. (3) (b) The complex case: Under exactly the same assumptions, when the ambient space is C d, for any d 3, we have ( ) I(P, L) = O m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + D 3 + m + n. (4) The assumption that s is at most D can be dropped, because, for any 2-flat π, the intersection π V is a plane algebraic curve of degree at most D in π (this holds since V does not contain any 2-flat), and can therefore contain at most D lines. We also have the following easy and interesting corollary. Corollary 1.4. Let P be a set of m points and L a set of n lines in R d or in C d, for any d 3, such that all the points lie in a common two-dimensional algebraic variety of constant degree that does not contain any 2-flat. Then I(P, L) = O (m + n), where the constant of proportionality depends on the degree of the surface. For d = 3, the corollary can also be derived, for the real case, from the analysis in Guth and Katz [15], using a somewhat different approach. Moreover, although not explicitly stated there, it seems that the argument in [15] also works over C. As a matter of fact, the corollary can also be extended (with a different bound though) to the case where the containing surface may have planar components. See a remark to that effect in Corollary 5.1 in the concluding section. We remark that the fact that V does not have planar components (and all the lines of L are contained in V ) is what enables us to get rid of the term O(nD) in the bound. This term is unavoidable, so to speak, in a general setting. For example, if we only assume that the points lie on V but the lines are arbitrary, we will incur the term O(nD) that we are trying to avoid. In the case of three-dimensional varieties, though, discussed later in the introduction (see Remark (2) following Theorem 1.5), we cannot avoid this term even when the lines lie on the variety, which is why we do not impose this property in that context. We also exploit the proof technique of Theorem 1.3 to derive an upper bound of O(nD) on the number of 2-rich points determined by a set of n lines contained in a variety, as above, in both the real and complex cases. See Section 5 for details. The significance of Theorem 1.3 is fourfold: (a) First and foremost, the theorem yields a new incidence result for points and lines on a two-dimensional variety over the complex field, in three and higher dimensions. Incidence results over the complex domain are rather rare. They include (as already mentioned) Tóth s extension of the Szemerédi-Trotter bound to the complex plane [45], which was the only result of that kind that predated the introduction of the algebraic machinery by Guth 4

163 and Katz, and several more recent works [39, 41, 42, 47] (where the latter work [47] provides an alternative algebraic derivation of Tóth s bound). (b) In the real three-dimensional case, the bound improves the Guth Katz bound when D n 1/2, for points and lines in two-dimensional varieties V that do not contain planes. Note that the threshold n 1/2 is a natural one because, as is well known and easy to show, any set of n lines in R 3 admits a polynomial of degree O(n 1/2 ) whose zero set contains all the lines; a simple modification of the construction applies in higher dimensions too. Of course the comparison is far from perfect, because this polynomial may have many linear components, in which case our bound does not apply. Still, it offers some basis for evaluating the quality of our bound. In three dimensions, this threshold is in fact larger than the standard degree O(m 1/2 /n 1/4 ) used in the analysis of Guth and Katz [15], when m < n 3/2. (c) Another significant feature of our bound is that it does not contain the term nd, which arises naturally in [15] and other works, and seems to be unavoidable when P is an arbitrary set of points. When D is not a constant, this becomes a crucial feature of the new bound, which has already been exploited in the analysis in [37], and is also used in the second main result of this paper, Theorem 1.5 below. 4 See an additional discussion of this feature at the end of the paper. (d) Our result offers a sharper point-line incidence bound in arbitrary dimensions, for the special case assumed in the theorem (which again holds over the complex field too). Theorem 1.3(a) has been used, as one of the key tools, in the analysis in our paper [37] on incidences between points and lines in four dimensions. In this application, the absence of the term nd is a crucial feature of our result, which was required in the scenario considered in [37]. The proof of Theorem 1.3 makes extensive use of several properties of ruled surfaces in R 3 or in C 3. While these results exist as folklore in the literature, and short proofs are provided for some of them, e..g., in [15], we include here detailed and rigorous proofs thereof, making them more accessible to the combinatorial geometry community. Other recent expositions include Guth s recent survey [12] and book [13], and a survey by Kollár [20]. 5 Points on a three-dimensional variety. Our second main result deals with the case where the points lie on a three-dimensional variety, embedded in R 4 or in C 4, or in any higher dimension. Similar to the case of two-dimensional varieties discussed above, we have to be careful here too, because hyperplanes and 3-quadrics (in R 4, and, a posteriori, in C 4 too) admit too many incidences in the worst case. That is, by a generalization of Elekes s construction [6], there exists a configuration of m points and n lines in a 3-flat with Θ(m 1/2 n 3/4 ) incidences. More recently, Solomon and Zhang [40] established an analogous statement for three-dimensional quadrics, when n 9/8 < m < n 3/2. Concretely, for such values of m and n, they have constructed a quadric S R 4, a set P of m points on S, and a set L of n lines contained in S, so that (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines in any hyperplane is O(n/m 1/3 ), and (iii) the number of 4 Although the bound in Theorem 1.5 does contain the term nd, it still crucially relies on the absence of this term in the bound for two-dimensional varieties. 5 While there is (naturally) some overlap between these surveys and our exposition, the main technical properties that we present do not seem to be rigorously covered in the other works. 5

164 incidences between the points and lines is Ω(m 2/3 n 1/2 ), which is asymptotically larger than the corresponding bound in (2) (for s = O(1)), when n 9/8 m n 3/2. In other words, when studying incidences with points on a variety in R 4 or in C 4, the cases where the variety is (or contains) a hyperplane or a quadric are special and do not yield the sharper bounds that we derive below. In the real case, the case of a hyperplane puts us back in R 3, where the best bound is Guth and Katz s in Theorem 1.1, and the case of a quadric reduces to the same setup via a suitable generic projection onto R 3. In the second main result of this paper, we show that if all the points lie on a three-dimensional algebraic variety of degree D without 3-flat or 3-quadric components, and if no 2-flat contains more than s lines, then, if D and s are not too large, the bound becomes significantly smaller. Moreover, here too we get a real version and a complex version of the theorem which nearly coincide. Specifically, we show: Theorem 1.5. (a) The real case: Let P be a set of m points and L a set of n lines in R d, for any d 4, and let s and D be parameters, such that (i) all the points of P lie on a three-dimensional algebraic variety of degree D, without any linear or quadratic three-dimensional components, and (ii) no 2-flat contains more than s lines of L. Then ( ) I(P, L) = O m 1/2 n 1/2 D + m 2/3 n 1/3 s 1/3 + nd + m. (5) When D and s are constants, we get the linear bound O(m + n). (b) The complex case: Under exactly the same assumptions, when the ambient space is C d, for any d 4, we have ( ) I(P, L) = O m 1/2 n 1/2 D + m 2/3 n 1/3 s 1/3 + D 6 + nd + m. (6) Remarks. (1) Note that for D < min{m 1/2 /n 1/4, n 1/4 }, our bound for the real case is sharper than the bound of Guth and Katz [15] (note that m 1/2 /n 1/4 is the degree of the partitioning polynomial used in the analysis of [15] for m n 3/2 ). On the other hand, when D > min{m 1/2 /n 1/4, n 1/4 }, our bound is not the best possible. Indeed, in this case we can project P and L onto some generic 3-flat, and apply instead the bound of [15] to the projected points and lines (we also show that a generic choice of the image 3-flat ensures that no 2-flat contains more than s of the projected lines), which is sharper than ours for these values of D. In the complex case, we also need to assume that D is small enough so that the term D 6 does not dominate the other terms, so as to make the bound look like the bound in the real case. (2) As already noted earlier, here we do not have to insist that the lines of L be contained in the variety. A line not contained in a variety of degree D can intersect it in at most D points, so the number of incidences with such lines is at most nd. The actual argument that yields this term is more involved, because we apply the argument to a variety of larger degree; see Section 4 for details. For two-dimensional varieties, since we want to avoid this term, we require the lines to be contained in the variety. Here, since we allow this term, the lines can be arbitrary. (3) The assumption that the points of P lie on a variety is not as restrictive as it might sound, because, in four dimensions, one can always construct a polynomial f of degree 6

165 O(m 1/4 ) whose zero set contains all the given points, or alternatively, a polynomial f of degree O(n 1/3 ) whose zero set contains all the given lines. The assumptions become restrictive, and the bound becomes more interesting, when D is significantly smaller. In addition, the constructed polynomial f could be one for which Z(f) does contain 3-flats or 3-quadrics, so another restrictive aspect of our assumptions is that they exclude these situations. (4) The assumption that Z(f) does not contain any linear or quadratic component is also significant, because having such components would make the problem behave like the problem of point-line incidences in three dimensions, as studied in Guth and Katz [15]. The bound would then deteriorate to their bound, and hold only over the reals. Additional discussion is provided in Section 5. Similar to the case of a two-dimensional variety, we also have here the following easy and interesting corollary (it does not hold when V contains 3-flats or 3-quadrics, in light of the lower bound constructions in [37, 40]). Corollary 1.6. Let P be a set of m points and L a set of n lines in R d or in C d, for any d 4, such that all the points lie in a common three-dimensional algebraic variety V of constant degree that does not contain any 3-flats or 3-quadrics, and no 2-flat contains more than O(1) lines of L. Then I(P, L) = O (m + n), where the constant of proportionality depends on the degree of V. Theorem 1.3 is a key technical ingredient in the proof of Theorem 1.5. The proofs of both theorems are somewhat technical, and use a battery of sophisticated tools from algebraic geometry. Some of these tools are borrowed and adapted from our previous work [37]. Other tools involve properties of ruled surfaces, which, as already said, are established here rigorously, for the sake of completeness. Since most of the presentation and derivation of these results is within the scope of algebraic and differential geometry, we give the necessary background in the following section, to make the presentation in this paper easier to follow for non-experts. Reader might consider skipping Section 2 on first reading, using it as a reference for the various algebraic tools that are used later in the paper. The proof of Theorem 1.3 is then presented in Section 3, and the proof of Theorem 1.5 is presented in Section 4. The concluding Section 5 discusses our results, establishes a few consequences thereof, and raises several related open problems. 2 Algebraic tools and ruled surfaces In this section we review the preliminary algebraic (and differential) geometry infrastructure needed for our analysis, and then go on to establish the properties of ruled surfaces that we will use. These properties are considered folklore in the literature; having failed to find rigorous proofs of them (except for several short proofs or proof sketches for some of them), we provide here such proofs for the sake of completeness. Some of the notions covered in this section are also discussed in our study [37] on point-line incidences in four dimensions. Degree and dimension of real variaties. In principle, all the varieties considered in the paper are regarded as complex algebraic varieties, for which the notions of degree and 7

166 dimension are classical well known concepts (discussed, e.g., in Harris [16]). When we talk about a real variety V, we mean a variety defined as the zero set of real polynomials, making the degree and dimension of V still well defined. When we consider such a variety in an incidence problem over the reals, though, we replace V by the subset V R of its real points. Singularity. The notion of singularities is a major concept, treated in full generality in algebraic geometry (see, e.g., Kunz [21, Theorem VI.1.15] and Cox et al. [3]). Here we only recall some of their properties, and only for a few special cases that are relevant to our analysis. Let V be a two-dimensional variety in R 3 or C 3 of degree D, given as the zero set Z(f) of some trivariate polynomial f. Assuming f to be square-free, a point p Z(f) is singular if f(p) = 0. For any point p Z(f), let f(p + x) = f µ (x) + f µ+1 (x) +... be the Taylor expansion of f near p, where f j is the j-th order term in the expansion (which is a homogeneous polynomial of x of degree j), and where we assume that there are no terms of order (i.e., degree) smaller than µ. The terms f j also depend on p, which we regard as fixed in the present discussion. In general, we have f 1 (x) = f(p) x, f 2 (x) = 1 2 xt H f (p)x, where H f is the Hessian matrix of f, and the higher-order terms are similarly defined, albeit with more involved expressions. If p is singular, we have µ 2. In this case, we say that p is a singular point of V = Z(f) of multiplicity µ = µ V (p). For any point p Z(f), we call the hypersurface Z(f µ ) the tangent cone of Z(f) at p, and denote it by C p Z(f). If µ = 1, then p is nonsingular and the tangent cone coincides with the (well-defined) tangent plane T p Z(f) to Z(f) at p. We denote by V sing the locus of singular points of V. This is a subvariety of dimension at most 1; see, e.g., Solymosi and Tao [41, Proposition 4.4]. We say that a line l is a singular line for V if all of its points are singular points of V. Similarly, let γ be a one-dimensional algebraic curve in R 2 or in C 2, specified as Z(f), for some bivariate square-free polynomial f. Then p Z(f) is singular if f(p) = 0. The multiplicity µ of a point p γ is defined as in the three-dimensional case, and we denote it as µ γ (p); the multiplicity is at least 2 when p is singular. The singular locus γ sing of γ is now a discrete set. Indeed, the fact that f is square-free guarantees that f has no common factor with any of its first-order derivatives, and Bézout s Theorem (see, e.g., [3, Theorem 8.7.7]) then implies that the common zero set of f, f x, f y, and f z is a (finite) discrete set. Still in two dimensions, a line l, not contained in the curve γ, can intersect it in at most D points, counted with multiplicity. To define this concept formally, as in, e.g., Beltrametti [2, Section 3.4], let l be a line and let p l γ, such that l is not contained in the tangent cone of γ at p. The intersection multiplicity of γ and l at p is the smallest order of a nonzero term of the Taylor expansion of f at p in the direction of l. As it happens, the intersection multiplicity is also equal to µ γ (p) (informally, this is the number of branches of γ that l crosses at p, counted with multiplicity; see [3, Section 8.7] for a treatment on the intersection multiplicity in the plane). The intersection between a line l and a curve γ (not containing l) consists of at most deg(γ) points, counted with their intersection multiplicities. This standard property is a crucial ingredient of one of the key lemmas (Lemma 3.1) in the proof of Theorem 1.3 in Section 3. 8

167 Assume that V is irreducible. By Guth and Katz [14] (see also Elekes et al. [7, Corollary 2]), the number of singular lines contained in V is at most D(D 1). Flatness. We say that a non-singular point x V is flat if the second-order Taylor expansion of f at x vanishes on the tangent plane T x V, or alternatively, if the second fundamental form of V vanishes at x (see, e.g., Pressley [26]). As argued, e.g., in Elekes et al. [7], if x is a non-singular point of V and there exist three lines incident to p that are contained in V (this property is captured by calling p a linearly flat point) then x is a flat point. Following Guth and Katz [14], Elekes et al. [7, Proposition 6] proved that a non-singular point x V is flat if and only if certain three polynomials, each of degree at most 3D 4, vanish at p. A non-singular line l is said to be flat if all of its non-singular points are flat. By Guth and Katz [14] (see also Elekes et al. [7, Proposition 7]), the number of flat lines contained in V is at most D(3D 4), unless V is a plane. As in the proof of Theorem 1.5, the notions of linear flatness and flatness can be extended to any higher dimension. For example, for a three-dimensional surface V in R 4 or in C 4, which is the zero set of some (square-free) polynomial f of degree D, a non-singular point x V is said to be linearly flat, if it is incident to at least three 2-flats that are contained in V = Z(f) (and thus also in the tangent hyperplane T p Z(f)). Linearly flat points can then be shown to be flat, meaning that the second fundamental form of f vanishes at them. This property, at a point p, can be expressed by several polynomials of degree at most 3D 4 vanishing at p (see [37, Section 2.5]). As in the three-dimensional case, the second fundamental form vanishes identically on Z(f) if and only if Z(f) is a hyperplane. This property holds in any dimension; see, e.g. [18, Exercise ]). As in three dimensions, we call a line contained in V flat if all its non-singular points are flat. Ruled surfaces. For a modern approach to ruled surfaces, there are many references; see, e.g., Hartshorne [17, Section V.2], or Beauville [1, Chapter III]; see also Salmon [30] and Edge [5] for earlier treatments of ruled surfaces. Three relevant very recent additions are the survey [12] and book [13] of Guth, as well as a survey in Kollár [20], where this topic is addressed in detail. We say that a real (resp., complex) surface V is ruled by real (resp., complex) lines if every point p in a Zariski-open 6 dense subset of V is incident to a real (complex) line that is contained in V. This definition has been used in several recent works, see, e.g., [15, 20]; it is a slightly weaker condition than the classical condition where it is required that every point of V be incident to a line contained in V (e.g., as in [30]). Nevertheless, similarly to the proof of Lemma 3.4 in Guth and Katz [15], a limit argument implies that the two definitions are in fact equivalent. We give, in Lemma 2.2 below, a short algebraic proof of this fact, for the sake of completeness. Flecnodes in three dimensions and the Cayley-Salmon-Monge Theorem. We first recall the classical theorem of Cayley and Salmon, also due to Monge. Consider a polynomial f C[x, y, z] of degree D 3. A flecnode of f is a point p Z(f) for which there exists a line that is incident to p and osculates to Z(f) at p to order three. That is, if the direction of the line is v then f(p) = 0, and v f(p) = 2 vf(p) = 3 vf(p) = 0, 6 See Cox et al. [3, Section 4.2] for details concerning the Zariski topology. 9

168 where v f, 2 vf, 3 vf are, respectively, the first, second, and third-order derivatives of f in the direction v (compare with the definition of singular points, as reviewed earlier, for the explicit forms of v f and 2 vf). The flecnode polynomial of f, denoted FL f, is the polynomial obtained by eliminating v from these three homogeneous equations (where p is regarded as a fixed parameter). We thus have a system of three equations in six variables, which is homogeneous in the three variables defining v. Eliminating those variables results in a single polynomial equation in p = (x, y, z). Using standard techniques, as in Cox et al. [4], the resulting polynomial FL f is the multipolynomial resultant 7 Res 3 (F 1, F 2, F 3 ) of F 1, F 2, F 3, regarding these as polynomials in v (where the coefficients are polynomials in p). As shown in Salmon [30, Chapter XVII, Section III], the degree of FL f is at most 11D 24. By construction, the flecnode polynomial of f vanishes on all the flecnodes of f, and in particular on all the lines contained in Z(f). Theorem 2.1 (Cayley and Salmon [30], Monge [24]). Let f C[x, y, z] be a polynomial of degree D 3. Then Z(f) is ruled by (complex) lines if and only if Z(f) Z(FL f ). Note that the correct formulation of Theorem 2.1 is over C; earlier applications, over R, as the one in Guth and Katz [15], require some additional arguments to establish their validity; see Katz [19] for a discussion of this issue. Lemma 2.2. Let f C[x, y, z] be an irreducible polynomial such that there exists a nonempty Zariski open dense set in Z(f) so that each point in the set is incident to a line that is contained in Z(f). Then FL f vanishes identically on Z(f), and Z(f) is ruled by lines. Proof. Let U Z(f) be the set assumed in the lemma. By assumption and definition, FL f vanishes on U, so U, and its Zariski closure, are contained in Z(f, FL f ). Since U is open, it must be two-dimensional. Indeed, otherwise its complement would be a (nonempty) two-dimensional subvariety of Z(f) (a Zariski closed set is a variety). In this case, the complement must be equal to Z(f), since f is irreducible, which is impossible since U is nonempty. Hence Z(f, FL f ) is also two-dimensional, and thus, by the same argument just used, must be equal to Z(f). Theorem 2.1 then implies that Z(f) is ruled by (complex) lines, as claimed. 8 The notions of flecnodes and of the flecnode polynomial can be extended to four dimensions, as done in [37]. Informally, the four-dimensional flecnode polynomial FL 4 f of f is defined analogously to the three-dimensional variant FL f, and captures the property that a point on Z(f) is incident to a line that osculates to Z(f) up to the fourth order. It is obtained by eliminating the direction v of the osculating line from the four homogeneous equations given by the vanishing of the first four terms of the Taylor expansion of f(p + tv) near p. Clearly, FL 4 f vanishes identically on every line that is contained in Z(f). As in the three-dimensional case, its degree can be shown to be O(D). Landsberg [22] derives an analog of Theorem 2.1 that holds for three-dimensional surfaces (see [37, Theorem 2.11]). Specifically, Landsberg s theorem asserts that if FL 4 f vanishes 7 The resultant method, or Macaulay s resultant method dates back to the 19th century. It is thoroughly covered in Van der Waerden [46, Chapter XI] and in Cox et al. [4]. 8 An alternative proof, based on a compactness argument, has been suggested to us by Martin Sombra (personal communication). 10

169 identically on Z(f), then Z(f) is ruled by (possibly complex) lines. We will discuss this in more detail in Section 4. See also another result of Landsberg, given as Theorem 4.2 in Section 4, that we will also require. These theorems, in three and four dimensions, play an important role in the proofs of the main theorems. Theorem of the fibers and related tools. The main technical tool for the analysis is the following so-called Theorem of the Fibers. Both Theorem 2.3 and Theorem 2.4 hold (only) for the complex field C. Theorem 2.3 (Harris [16, Corollary 11.13]). Let X be a projective variety and π : X P d be a homogeneous polynomial map (i.e., the coordinate functions x 0 π,..., x d π are homogeneous polynomials); let Y = π(x) denote the image of X. For any p Y, let 9 λ(p) = dim(π 1 ({p})). Then λ(p) is an upper semi-continuous function of p in the Zariski topology on Y ; that is, for any m, the locus of points p Y such that λ(p) m is Zariski closed in Y. Moreover, if X 0 X is any irreducible component, Y 0 = π(x 0 ) its image, and λ 0 the minimum value of λ(p) on Y 0, then dim(x 0 ) = dim(y 0 ) + λ 0. We also need the following theorem and lemma from Harris [16]. Theorem 2.4 (Harris [16, Proposition 7.16]). Let f : X Y be the map induced by the standard projection map π : P d P r (which retains the first r coordinates and discards the rest), where r < d, where X P d and Y P r are projective varieties, X is irreducible, and Y is the image of X (which is also irreducible). Then the general fiber 10 of the map f is finite if and only if dim(x) = dim(y ). In this case, the number of points in a general fiber of f is constant (which depends on the degree of X). In particular, when Y is two-dimensional (and d > r 2 are arbitrary), there exist an integer c f and an algebraic curve C f Y, such that for any y Y \C f, we have f 1 (y) = c f. With the notations of Theorem 2.4, the set of points y Y, such that the fiber of f over y is not equal to c f, is a Zariski closed proper subvariety of Y. For more details, we refer the reader to Shafarevich [34, Theorem II.6.4], and to Hartshorne [17, Exercise II.3.7]. Lemma 2.5 (Harris [16, Theorem 11.14]). Let π : X Y be a polynomial map between two projective varieties X, Y, with Y = f(x) irreducible. Suppose that all the fibers π 1 ({p}) of π, for p Y, are irreducible and of the same dimension. Then X is also irreducible. Reguli. We rederive here the following (folklore) characterization of doubly ruled surfaces in R 3 or C 3, namely, irreducible algebraic surfaces, each of whose points is incident to at least two lines that are contained in the surface. Recall that a regulus is the surface spanned by all lines that meet three pairwise skew lines in 3-space. 11 For an elementary proof that 9 The dimension of an algebraic variety is defined in any textbook in algebraic geometry (see, e.g., Cox et al. [3, Definitions 7, 10]). 10 The meaning of this statement is that the assertion holds for the fiber at any point outside some lowerdimensional exceptional subvariety. 11 Technically, in some definitions (cf., e.g., Edge [5, Section I.22]) a regulus is a one-dimensional family of generator lines of the actual surface, i.e., a curve in the Plücker or Grassmannian space of lines in the Klein quadric, but we use here the alternative notion of the surface spanned by these lines. See below, under Lines in a variety, for a comment to that effect. 11

170 a doubly ruled surface must be a regulus, we refer the reader to Fuchs and Tabachnikov [9, Theorem 16.4]. Their proof however is analytic and works only over the reals. Lemma 2.6. Let V be an irreducible ruled surface in R 3 or in C 3 which is not a plane, and let C V be an algebraic curve, such that every non-singular point p V \ C is incident to at least two lines that are contained in V. Then V is a regulus. 12 Proof. As mentioned above (see also [14]), the number of singular lines in V is finite (it is smaller than deg(v ) 2 ). For any non-singular line l, contained in V, but not in C, the union of lines U l intersecting l and contained in V is a subvariety of V (see, e.g., Sharir and Solomon [35, Lemma 8] for the easy proof). By assumption, each non-singular point in l \ C is incident to another line (other than l) contained in V, and thus U l is the union of infinitely many lines, and is therefore two-dimensional. Since V is irreducible, it follows that U l = V. Next, pick any triple of non-singular and non-concurrent lines l 1, l 2, l 3 that are contained in V and intersect l at distinct non-singular points of l \ C. There has to exist such a triple, for otherwise we would have an infinite family of concurrent (or parallel) lines incident to l and contained in V (where the point of concurrency lies outside l), and the plane that they span would then have to be contained in (the irreducible) V, contrary to assumption. See Figure 1 for an illustration. The argument given for l applies equally well to l 1, l 2, and l 3 (by construction, neither of them is contained in C), and implies that U l1 = U l2 = U l3 = V. Assume that there exists some line l V intersecting l 1 at some non-singular point p l 1 \C, and that l l 2 =. We treat lines here as projective varieties, so this assumption means that l and l 2 are skew to one another; parallel lines are considered to be intersecting. Since p l 1 V = U l2, there exists some line ˆl intersecting l 2, such that ˆl l 1 = {p}. Hence there exist three lines, namely l 1, l and ˆl, that are incident to p and contained in V. Since p is non-singular, it must be a flat point (as mentioned above; see [7]). Repeating this argument for 3deg(V ) non-singular points p l 1, it follows that l 1 contains at least 3deg(V ) flat points, and is therefore, by the properties of flat points noted earlier, a flat line. As is easily checked, l 1 can be taken to be an arbitrary non-singular line among those incident to l, so it follows that every non-singular point on V is flat, and therefore, as shown in [7, 14], V is a plane, contrary to assumption. Therefore, every non-singular line that intersects l 1 at a non-singular point also intersects l 2, and, similarly, it also intersects l 3. This implies that the intersection of V and the surface R generated by the lines intersecting l 1, l 2, and l 3 is two-dimensional, and is therefore equal to V, since V is irreducible. Since l 1, l 2 and l 3 are pairwise skew, R = V is a regulus, as asserted. Most of the basic algebraic geometry tools have been developed over the complex field C, and some care has to be exercised when applying them over the reals. A major part of the theory developed in this section is of this nature. For example, both Theorems 2.3 and 2.4 hold only over the complex field. As another important example, one of the main tools at our disposal is the Cayley Salmon Monge theorem (Theorem 2.1), whose original formulation also applies only over C. Expanding on a previously made comment, we note 12 Over R, a regulus is either a hyperbolic paraboloid or a one-sheeted hyperboloid. Over C, balls (equivalent to hyperboloids) and paraboloids (equivalent to hyperbolic paraboloids) are also reguli, and are indeed doubly ruled by complex lines. Not all complex quadrics are reguli, though: for example, the cylinder y = x 2 is not a regulus. 12

171 l p ˆl l l 1 l 2 l 3 Figure 1: The structure of U l in the proof of Lemma 2.6. that even when V is a variety defined as the zero set of a real polynomial f, the vanishing of the flecnode polynomial FL f only guarantees that the set of complex (and real) points of V is ruled by complex lines. A very simple example that illustrates this issue is the unit sphere σ, given by x 2 + y 2 + z 2 = 1, which is certainly not ruled by real lines, but the flecnode polynomial of f(x, y, z) = x 2 + y 2 + z 2 1 vanishes on σ (since the equation 3 vf(p) = 0 that participates in its construction is identically zero for any quadratic polynomial f). This is the condition in the Cayley Salmon Monge theorem that guarantees that σ is ruled by (complex) lines, and indeed it does, as is easily checked; in fact, for the same reason, every quadric is ruled by complex lines. This issue has not been directly addressed in Guth and Katz [15], although their theory can be adjusted to hold for the real case too, as noted later in Katz [19]. This is just one example of many similar issues that one must watch out for. It is a fairly standard practice in algebraic geometry that handles a real algebraic variety V, defined by real polynomials, by considering its complex counterpart V C, namely the set of complex points at which the polynomials defining V vanish. The rich toolbox that complex algebraic geometry has developed allows one to derive various properties of V C, but some care might be needed when transporting these properties back to the real variety V, as the preceding note concerning the Cayley Salmon Monge theorem illustrates. Fortunately, though, passing to the complex domain (and sometimes also to the projective setting) does not pose any difficulties for deriving upper bounds in incidence problems every real incidence will be preserved, and at worst we will be counting additional incidences, on the non-real portion of the extended varieties. With this understanding, and with the appropriate caution, we will move freely between the real and complex domains, as convenient. We note that most of the results developed in Section 3 of this paper also apply over C, except for one crucial step (where we resort to the application of the result of Guth and Katz [15], which holds only over the reals), due to which we do not know how to extend Theorem 1.3(a) to the complex domain. Nevertheless, we can derive the weaker variant of it, Theorem 1.3(b), for the complex case see a remark to that effect in Section 5. Lines on a variety. In preparation for the key technical Theorem 2.7, given below, we make the following comments. Lines in three dimensions are parameterized by their Plücker coordinates, as follows (see, e.g., Griffiths and Harris [11, Section 1.5]). For two 13

172 distinct points x, y P 3, given in projective coordinates as x = (x 0, x 1, x 2, x 3 ) and y = (y 0, y 1, y 2, y 3 ), let l x,y denote the (unique) line in P 3 incident to both x and y. The Plücker coordinates of l x,y are given in projective coordinates in P 5 as (π 0,1, π 0,2, π 0,3, π 2,3, π 3,1, π 1,2 ), where π i,j = x i y j x j y i. Under this parameterization, the set of lines in P 3 corresponds bijectively to the set of points in P 5 lying on the Klein quadric (also known as the Grassmannian manifold of lines in P 3 ) given by the quadratic equation π 0,1 π 2,3 +π 0,2 π 3,1 +π 0,3 π 1,2 = 0 (which is indeed always satisfied by the Plücker coordinates of a line). Given a surface V in P 3, the set of lines contained in V, represented by their Plücker coordinates in P 5, is a subvariety of the Klein quadric (or the Grassmaniann of lines in P 4 ), which is denoted by F (V ), and is called the Fano variety of V ; see Harris [16, Lecture 6, page 63] for details, and [16, Example 6.19] for an illustration, and for a proof that F (V ) is indeed a variety. The Plücker coordinates are continuous, in the sense that if one takes two points l, l on the Klein quadric that are near each other, the lines in P 3 that they correspond to are also near to one another, in an obvious sense whose precise details are omitted here. Remark. This representation of lines, which can be extended to higher dimensions too, is also useful in this study of ruled surfaces, as it offers an alternative definition of a ruled surface in terms of its Fano variety. That is, in the context of the Plücker coordinates, a ruled surface can be defined as a one-dimensional family of lines, that is, a curve on the Klein quadric. With this point of view, the union of lines in the projective three-space, which are the Klein pre-images of points on this curve is referred to as the point set of the ruled surface. We refer the reader to Pottmann and Wallner s textbook [25, Chapters 5,6] for a thorough exposition of this representation, and to Selig [32, Chapter 6], for a more concise exposition. This viewpoint was also used by Rudnev [29]. Here, though, we stick to the other point of view, thinking of a ruled surface as the surface itself (the point set), and not as its Fano variety. We note again that our analysis is carried out in the complex projective setting, which makes it simpler, and facilitates the application of numerous tools from algebraic geometry that are developed in this setting. In the particular context discussed here, the passage from the complex projective setup back to the real affine one is straightforward the former is a generalization of the latter. Given a plane π by a homogeneous equation A 0 x 0 + A 1 x 1 + A 2 x 2 + A 3 x 3 = 0, and a line l not contained in π, given in Plücker coordinates as (π 0,1, π 0,2, π 0,3, π 2,3, π 3,1, π 1,2 ), their point of intersection is given in homogeneous coordinates by (A m, A m A 0 d), where d = (π 0,1, π 0,2, π 0,3 ), m = (π 2,3, π 3,1, π 1,2 ), and where stands for the scalar product, and for the vector product; see, e.g., [43, p. 29]. This, together with the continuity argument stated above, implies that, if the Fano variety F (V ) is one-dimensional, and l is a line represented by a non-singular point of F (V ), then the cross section of the union of the lines that lie near l in F (V ) with a generic plane π is a simple arc 13. When l is a singular point of F (V ), then the corresponding cross section is a union of simple arcs meeting at l π, where some of these arcs might appear with multiplicity; the number of these arcs is determined by the multiplicity of the singularity of l. 13 In this context, a simple arc is a connected piece of an algebraic curve that is irreducible as a topological space. For more details, we refer the reader to any classical textbook on the geometry of curves, such as Beltarmetti et al. [2, Section 3]. 14

173 Singly ruled surfaces. Ruled surfaces that are neither planes nor reguli are called singly ruled surfaces (a terminology justified by Theorem 2.7, given below). A line l, contained in an irreducible singly ruled surface V, such that every point of l is doubly ruled, i.e., every point on l is incident to another line contained in V, is called an exceptional line 14 of V. A point p V V that is incident to infinitely many lines contained in V is called an exceptional point of V. The following result is another folklore result in the theory of ruled surfaces, used in many studies (such as Guth and Katz [15]). It justifies the terminology singly-ruled surface, by showing that the surface is generated by a one-dimensional family of lines, and that each point on the surface, with the possible exception of points lying on some curve, is incident to exactly one generator line. It also shows that there are only finitely many exceptional lines; the property that their number is at most two (see [15]) is presented later. We give a detailed and rigorous proof, to make our presentation as self-contained as possible; we are not aware of any similarly detailed argument in the literature. Theorem 2.7. (a) Let V be an irreducible ruled two-dimensional surface of degree D > 1 in R 3 (or in C 3 ), which is not a regulus. Then, except for finitely many exceptional lines, the lines that are contained in V are parameterized by an irreducible algebraic curve Σ 0 (in the parametric Plücker space P 5, or rather in the Klein quadric contained in that space, that represents lines in 3-space), and thus yielding a 1-parameter family of lines (referred to as generators) l(t), for t Σ 0, that depend continuously on the real or complex parameter t. Moreover, if t 1 t 2, and l(t 1 ) l(t 2 ), then there exist sufficiently small and disjoint neighborhoods 1 of t 1 and 2 of t 2, such that all the lines l(t), for t 1 2, are distinct. (b) There exists a one-dimensional curve C V, such that any point p in V \ C is incident to exactly one line contained in V. Remark. For a detailed description of the algebraic representation of V by generators, as in part (a) of the theorem, see Edge [5, Section II] (and see also the remark made earlier). Proof. Assume first that we are working over C. Consider the Fano variety F (V ) of V, as defined above. We claim that all the irreducible components of F (V ) are at most onedimensional. Informally, if any component Σ 0 of F (V ) were two-dimensional, then the set {(p, l) V F (V ) p l} would be three-dimensional, so, on average, the set of lines of F (V ) incident to a point p V would be one-dimensional, implying that most points of V are incident to infinitely many lines that are contained in V, which can happen only when V is a plane (or a non-planar cone, which cannot arise with a non-singular point p as an apex), contrary to assumption. To make this argument formal, consider the set (already mentioned above) W := {(p, l) p l, l F (V )} V F (V ), and the two projections Ψ 1 : W V, Ψ 2 : W F (V ) 14 In Guth and Katz [15], a line l contained in an irreducible singly ruled surface V, is called exceptional if it contains infinitely many doubly ruled points, each incident to another line contained in V. Our definition appears to be stricter, but, as the proof below will reveal, the two notions are equivalent. 15

174 to the first and second factors of the product V F (V ), respectively. W can formally be defined as the zero set of suitable homogeneous polynomials; briefly, with the Plücker parameterization of lines in P 3, and putting the point p into homogeneous coordinates, the condition p l can be expressed as the vanishing of two suitable homogeneous polynomials, and the other defining polynomials of W are those that define the projective variety F (V ). Therefore, W is a projective variety. Consider an irreducible component Σ 0 of F (V ) (which is also a projective variety); put W 0 := Ψ 1 2 (Σ 0) = {(p, l) W l Σ 0 }. Since W and Σ 0 are projective varieties, so is W 0. As is easily verified, Ψ 2 (W 0 ) = Σ 0 (that is, Ψ 2 is surjective). We claim that W 0 is irreducible. Indeed, for any l Σ 0, the fiber of the map Ψ 2 W0 : W 0 Σ 0 over l is {(p, l) p l} which is (isomorphic to) a line, and is therefore irreducible of dimension one. As Σ 0 is irreducible, Lemma 2.5 implies that W 0 is also irreducible, as claimed. For a point p Ψ 1 (W 0 ), consider the set Σ 0,p = Ψ 1 1 W 0 ({p}), put λ(p) = dim(σ 0,p ), and let λ 0 := min p Ψ1 (W 0 ) λ(p). By the Theorem of the Fibers (Theorem 2.3), applied to the map Ψ 1 W0 : W 0 V, we have dim(w 0 ) = dim(ψ 1 (W 0 )) + λ 0. (7) We claim that λ 0 = 0. In fact, λ(p) = 0 for all points p V, except for at most one point. Indeed, if λ(p) 1 for some point p V, then Σ 0,p is (at least) one-dimensional, and V, being irreducible, is thus a cone with apex at p; since V can have at most one apex, the claim follows. Hence λ 0 = 0, and therefore dim(w 0 ) = dim(ψ 1 (W 0 )) dim(v ) = 2. (8) Next, assume, for a contradiction, that dim(σ 0 ) = 2. For a point (i.e., a line in P 3 ) l Ψ 2 (W 0 ), the set Ψ 2 1 W 0 ({l}) = {(p, l) p l} is one-dimensional (the equality follows from the way W 0 is defined). ( Conforming ) to the notations in the Theorem of the Fibers, we have µ(l) := dim Ψ 2 1 W 0 ({l}) = 1, and thus µ 0 := min l Ψ2 (W 0 ) µ(l) = 1. Also, by assumption, dim(ψ 2 (W 0 )) = dim(σ 0 ) = 2. By the Theorem of the Fibers, applied this time to Ψ 2 W0 : W 0 Σ 0, we thus have dim(w 0 ) = dim(ψ 2 (W 0 )) + µ 0 = 3, (9) contradicting Equation (8). Therefore, every irreducible component of F (V ) is at most one-dimensional, as claimed. Let Σ 0 be such an irreducible component, and let W 0 := Ψ 1 2 (Σ 0), as above. As argued, for every p V, the fiber of Ψ 1 W0 over p is non-empty and finite, except for at most one point p (the apex of V if V is a cone). Since W 0 is irreducible, Theorem 2.4 implies that there exists a Zariski open set O V, such that for any point p O, the fiber of Ψ 1 W0 over p has fixed cardinality c f. Put C := V \ O. Being the complement of a Zariski open subset of the two-dimensional irreducible variety V, C is (at most) a one-dimensional variety. If c f 2, then, by Lemma 2.6, V is a regulus. Otherwise, c f = 1 (c f cannot be zero for a ruled surface), meaning that, for every p V \ C, there is exactly one line l, such that (p, l) W 0, i.e., Σ 0 contains exactly one line incident to p and contained in V. 16

175 Moreover, we observe that the union of lines of Σ 0 is the entire variety V. Indeed, by Equations (7) and (8), we have dim(w 0 ) = dim(ψ 1 (W 0 )) = 2. That is, the variety Ψ 1 (W 0 ), which is the union of the lines of Σ 0, must be the entire variety V, because it is two-dimensional and is contained in the irreducible variety V. To recap, we have proved that if Σ 0 is a one-dimensional component of F (V ), then the union of lines that belong to Σ 0 covers V, and that there exists a one-dimensional subvariety (a curve) C V such that, for every p V \ C, Σ 0 contains exactly one line incident to p and contained in V. Since V is a ruled surface, some component of F (V ) has to be one-dimensional, for otherwise we would only have a finite number of lines contained in V. We claim that there is exactly one irreducible component of F (V ) which is one-dimensional. Indeed, assume to the contrary that Σ 0, Σ 1 are two (distinct) one-dimensional irreducible components of F (V ). As we observed, the union of lines parameterized by Σ 0 (resp., Σ 1 ) covers V. Let C 0, C 1 V denote the respective excluded curves, so that, for every p V \ C 0 (resp., p V \ C 1 ) there exists exactly one line in Σ 0 (resp., Σ 1 ) that is incident to p and contained in V. Next, notice that the intersection Σ 0 Σ 1 is a subvariety strictly contained in the irreducible one-dimensional variety Σ 0 (since Σ 0 and Σ 1 are two distinct irreducible components of F (V )), so it must be zero-dimensional, and thus finite. Let C 01 denote the union of the finitely many lines in Σ 0 Σ 1, and put C := C 0 C 1 C 01. For any point p V \ C, there are two (distinct) lines incident to p and contained in V (one belongs to Σ 0,p and the other to Σ 1,p ). Lemma 2.6 (with C as defined above) then implies that V is a regulus, contrary to assumption. In other words, the unique one-dimensional irreducible component Σ 0 of F (V ) serves as the desired 1-parameter family of generators for V. The local parameterization of Σ 0 can be obtained, e.g., by using a suitable Plücker coordinate to represent its lines. In addition to Σ 0, there is a finite number of zero-dimensional components (i.e., points) of F (V ). They correspond to a finite number of lines, contained in V, and not parameterized by Σ 0. Since the union of the lines in Σ 0 covers V, any of these additional lines l is exceptional, since each point on l is also incident to a generator (different from l), and is thus doubly ruled. This establishes part (a) of the theorem, when V is defined over C. We remark that Guth and Katz [15, Corollary 3.6] argue that there are at most two such exceptional lines, so there are at most two zero-dimensional components of F (V ). For the sake of completeness, we sketch a proof of our own of this fact, in Lemma 2.8 below. If V is defined over R, we proceed as above, i.e., consider instead the complex variety V C corresponding to V. As we have just proven, the unique one-dimensional irreducible component Σ 0 of F (V ) (regarded as a complex variety) is a (complex) 1-parameter family of generators for the set of complex points of V. Since V is real, the (real) Fano variety of V consists of the real points of F (V ), i.e., it is F (V ) P 5 (R). As we have mentioned above, the (complex) F (V ) is the union of Σ 0 with at most two other points. If Σ 0 R := Σ 0 P 5 (R) were zero-dimensional, the real F (V ) would also be discrete, as there is only one onedimensional component Σ 0, so V would contain only finitely many (real) lines, contradicting the assumption that V is ruled by real lines. Therefore, Σ 0 R is a one-dimensional irreducible component of the real Fano variety of V. It is irreducible, since otherwise the complex Σ 0 would be reducible too, as is easily checked. 17

176 Summarizing, we have shown that there exists exactly one irreducible one-dimensional component Σ 0 of F (V ), and a corresponding one-dimensional subvariety C V, such that, for each point p V \ C, Σ 0 contains exactly one line that is incident to p (and contained in V ). In addition to Σ 0, F (V ) might also contain up to two zero-dimensional (i.e., singleton) components, whose elements are the exceptional lines mentioned above. Let D denote the union of C and of the at most two exceptional lines; D is clearly a one-dimensional subvariety of V. Then, for any point p V \ D, there is exactly one line incident to p and contained in V, as claimed. This establishes part (b), and thus completes the proof of the theorem. Exceptional lines on a singly ruled surface. In view of the proofs of Theorem 2.7 and Lemma 2.2, every point on a singly ruled surface V is incident to at least one generator. Hence an exceptional (non-generator) line is a line l V such that every point on l is incident to a generator (which is different from l). Lemma 2.8. Let V be an irreducible ruled surface in R 3 or in C 3, which is neither a plane nor a regulus. Then (i) V contains at most two exceptional lines, and (ii) V contains at most one exceptional point. Proof. (i) We use the property, established in [35] and already used in the proof of Lemma 2.6, that for a line l contained in V, the union τ(l) of the lines that meet l and are contained in V is a variety in the complex projective space P 3 (C). Moreover, if l is an exceptional line of V, then it follows by [35, Lemma 8] that τ(l) = V. Indeed, τ(l) must be two-dimensional, since otherwise it would consist of only finitely many lines. Since V is irreducible, τ(l) must then be equal to V. If V contained three exceptional lines, l 1, l 2 and l 3, then V would have to be either a plane or a regulus. Indeed, otherwise, by Theorem 2.7 (whose proof does not depend on the number of exceptional lines), there would exist a one-dimensional curve C V (that includes l 1 l 2 l 3 ), such that every point p V \ C is incident to exactly one line l p contained in V. As p V \ C and σ(l i ) = V, for i = 1, 2, 3, it follows that l p intersects l 1, l 2, and l 3. If l 1, l 2, and l 3 are pairwise skew, p belongs to the regulus R l1,l 2,l 3 of all lines intersecting l 1, l 2, and l 3. We have thus proved that V \ C is contained in R l1,l 2,l 3, and as R l1,l 2,l 3 is irreducible, it follows that V = R l1,l 2,l 3. If l 1, l 2, and l 3 are concurrent but not coplanar then, arguing similarly, V is a cone with their common intersection point as an apex. Since a (non-planar) cone has no exceptional lines, as is easily checked, we may ignore this case. Finally if any pair among l 1, l 2, l 3, say l 1, l 2, are parallel then V must be the plane that they span, contrary to assumption. If l 1 and l 2 intersect at a point ξ, disjoint from l 3, then V is the union of the plane spanned by l 1 and l 2 and the plane spanned by ξ and l 3, again a contradiction. Having exhausted all possible cases, the proof of (i) is complete. (ii) By Theorem 2.7 and (i), all the lines that are contained in V, except for possibly two such lines, are parameterized by an irreducible algebraic curve Σ 0 in the Plücker space P 5. Let p be an exceptional point of V. The set Σ of lines incident to p is an algebraic curve contained in the irreducible curve Σ 0, implying that Σ = Σ 0. This clearly implies that 18

177 there is at most one exceptional point (and then it does not contain any exceptional line), and the proof of (ii) is complete too. Remark. We refer the reader to Guth and Katz [15, Lemma 3.5, Corollary 3.6], for yet another (somewhat more compact) proof of this lemma. Generic projections preserve non-planarity. Most of the analysis in Section 3 handles two-dimensional varieties embedded in R 3, and then handles the general case in R d, for d > 3, by projecting R d onto some generic 3-flat, so that non-coplanar triples of lines do not project to coplanar triples. This is easily achieved by repeated applications of the following technical result, reducing the dimension one step at a time. Lemma 2.9. Let l 1, l 2, l 3 be three non-coplanar lines in R d, for d 4. Then, under a generic projection of R d onto some hyperplane H, the respective images l 1, l 2, l 3 of these lines are still non-coplanar. Proof. Assume without loss of generality that the (generic) hyperplane H onto which we project passes through the origin of R d, and let w denote the unit vector normal to H. The projection h : R d H is then given by h(v) = v (v w)w. Assume first that two of the three given lines, say l 1, l 2, are skew (i.e., not coplanar). Let l 1, l 2 denote their projection onto H. If l 1, l 2 are coplanar they are either intersecting or parallel. If they are intersecting, then there are points p 1 l 1, p 2 l 2 that project to the same point, i.e., p 1 p 2 has the same direction as w. Then w belongs to the set { p 1 p 2 p 1 p 2 p 1 l 1, p 2 l 2 }. Since this is a two-dimensional set, it will be avoided for a generic choice of w, which is a generic point in S d 1, a set that is at least three-dimensional. If l 1, l 2 are parallel, let v 1, v 2 denote the directions of l 1, l 2. Since v 1 (v 1 w)w and v 2 (v 2 w)w are vectors in the directions of l 1, l 2, and are thus parallel, it follows that w must be a linear combination of v 1 and v 2. Since w = 1, the resulting set of possible directions is only one-dimensional, and, again, it will be avoided with a generic choice of w. We may therefore assume that every pair of lines among l 1, l 2, l 3 are coplanar. Since these three lines are not all coplanar, the only two possibilities are that either they are all mutually parallel, or all concurrent. Assume first that they are concurrent, say they all pass through the origin (even though the origin belongs to H, this still involves no less of generality). Their projections are in the directions v i (v i w)w, for i = 1, 2, 3. If these projections are coplanar then there exist coefficients α 1, α 2, α 3, not all zero, such that i α i(v i (v i w)w) = 0. That is, putting u := i α iv i, we have u = (u w)w, so u is parallel to w. In this case w belongs to the set { i α iv i } i α iv i α 1, α 2, α 3 R or C. Again, being a two-dimensional set, it will be avoided by a generic choice of w. In the remaining case, the lines l 1, l 2, l 3 are mutually parallel, i.e., they all have the same direction v. Put, for i = 1, 2, 3, l i = {p i + tv} t R, and choose p i so that p i v = 0. The plane π 0 spanned by p 1, p 2, p 3 is projected to the plane π spanned by the points p i = p i (p i w)w, for i = 1, 2, 3 (since p 1, p 2, p 3 are not collinear, they will not project into collinear points in a generic projection), and the three lines project into a common plane if and only if their projections are contained in π, meaning that the projection v = v (v w)w is parallel 19

178 to π, so it must be a linear combination of p 1, p 2, and p 3. A similar argument to those used above shows that a generic choice of w will avoid the resulting two-dimensional set of forbidden directions. This completes the proof. 3 Proof of Theorem 1.3 In most of the analysis in this section, we will consider the case d = 3. The reduction from an arbitrary dimension to d = 3 will be presented at the end of the section. We will prove both parts of the theorem hand in hand, bifurcating (in a significant manner) only towards the end of the analysis. For a point p on an irreducible singly ruled surface V, which is not the exceptional point of V (see Section 2 for its definition), we let Λ V (p) denote the number of generator lines incident to p and contained in V (so if p is incident to an exceptional line, we do not count that line in Λ V (p)). We also put Λ V (p) := max{0, Λ V (p) 1}. Finally, if V is a cone and p V is its exceptional point (that is, apex), we put Λ V (p V ) = Λ V (p V ) := 0. We also consider a variant of this notation, where we are also given a finite set L of lines (where not all lines of L are necessarily contained in V ), which does not contain any of the (at most two) exceptional lines of V. For a point p V, we let λ V (p; L) denote the number of lines in L that pass through p and are contained in V, with the same provisions as above, namely that we do not count incidences with exceptional lines, nor do we count incidences occurring at an exceptional point, and put λ V (p; L) := max{0, λ V (p; L) 1}. If V is a cone with apex p V, we put λ V (p V ; L) = λ V (p V ; L) = 0. We clearly have λ V (p; L) Λ V (p) and λ V (p; L) Λ V (p), for each point p. Lemma 3.1. Let V be an irreducible singly ruled two-dimensional surface of degree D > 1 in R 3 or in C 3. Then, for any line l, except for the (at most) two exceptional lines of V, we have Λ V (p) D if l is not contained in V, p l V p l V Λ V (p) D if l is contained in V. Proof. To streamline the analysis and avoid degenerate situations that might arise over the reals, we confine ourselves to the complex case; as already mentioned, the incidence bounds that we will obtain will automatically hold over the reals too. That is, every real line that we need to count is also a complex line, and if the real line is contained in the real part of the variety, its complex counterpart is contained in the complex variety, and the arguent then carries over. We note that the difference between the two cases arises because we do not want to count l itself the former sum would be infinite when l is contained in V. Note also that if V is a cone and p V l, we ignore in the sum the infinitely many lines incident to p V and contained in V. The proof is a variant of an observation due to Salmon [30] and repeated in Guth and Katz [15] over the real numbers, and later in Kollár [20] over the complex field and other general fields. 20

179 By Theorem 2.7(a), excluding the exceptional lines of V, the set of lines contained in V can be parameterized as a (real or complex) 1-parameter family of generator lines l(t), represented by the irreducible curve Σ 0 F (V ). Let V (2) denote the locus of points of V that are incident to at least two generator lines contained in V. By Theorem 2.7(b), V (2) is contained in some one-dimensional curve C V. Let p V l be a point incident to k generator lines of V, other than l, for some k 1. In case V is a cone, we assume that p p V. Denote the generator lines incident to p (other than l, if l V, in which case it is assumed to be a generator) as l i = l(t i ), for t i Σ 0 and for i = 1,..., k. If l i is a singular point of F (V ), it may arise as l(t i ) for several values of t i, and we pick one arbitrary such value. Let π be a generic plane containing l, and consider the curve γ 0 = V π, which is a plane curve of degree D. Since V (2) C is one-dimensional, a generic choice of π will ensure that V (2) π is a discrete set (since l is non-exceptional, it too meets V (2) in a discrete set). There are two cases to consider: If l is contained in V (and is thus a generator), then γ 0 contains l. In this case, let γ denote the closure of γ 0 \l; it is also a plane algebraic curve, of degree at most D 1. In case l is not contained in V, we put γ := γ 0. By Theorem 2.7(a), we can take, for each i = 1,..., k, a sufficiently small open (real or complex) neighborhood i along Σ 0 containing t i, so that all the lines l(t), for t k i=1 i are distinct. Put V i := t i l(t). It follows from the discussion in Section 2 (see Lines on a variety there) that V i π is either a simple arc or a union of simple arcs meeting at p (depending on whether or not l i is a regular point of Σ 0 ); in the latter case, take γ i to be any one of these arcs. Each of the arcs γ i is incident to p and is contained in γ. Moreover, since π is generic, the arcs γ i are all distinct. Indeed, for any i j, and any point q γ i γ j, there exist t i i, t j j such that l(t i ) π = l(t j ) π = q, and l(t i ) l(t j ) (by the properties of these neighborhoods). Therefore, any point in γ i γ j is incident to (at least) two distinct generator lines contained in V. Again, the generic choice of π ensures that γ i γ j V (2) π is a discrete set, so, in particular, γ i and γ j are distinct. We have therefore shown that (i) if l is not contained in V then p is a singular point of γ of multiplicity at least k (for k 2; when k = 1 the point does not have to be singular), and (ii) if l is contained in V then p is singular of multiplicity at least k + 1. We have k Λ V (p) (resp., k Λ V (p)) if l is not contained (resp., is contained) in V. As argued at the beginning of Section 2, the line l can intersect γ in at most D points, counted with multiplicity, and the result follows. We also need the following result (see, e.g., Guth and Katz [15, Corollary 3.3]), which is an immediate consequence of the Cayley Salmon Monge theorem (Theorem 2.1) and a suitable extension of Bézout s theorem for intersecting surfaces (see Fulton [10, Proposition 2.3]). Proposition 3.2. Let V be an irreducible two-dimensional variety in C 3 of degree D. If V contains more than 11D 2 24D lines then V is ruled by (complex) lines. Corollary 3.3. Let V be a two-dimensional variety in C 3 of degree D. Then the number of lines that are contained in the union of the non-ruled components of V is O(D 2 ). Proof. Let V 1,..., V k denote those irreducible components of V that are not ruled by lines. By Proposition 3.2, for each i, the number of lines contained in V i is at most 11deg(V i ) 2 21

180 24deg(V i ). Summing over i = 1,..., k, the number of lines contained in the union of the non-ruled components of V is at most k i=1 11deg(V i) 2 = O(D 2 ). The following theorem, which we believe to be of independent interest in itself, is the main technical ingredient of our analysis in this section. Note that it holds over both real and complex fields. Theorem 3.4. Let V be a possibly reducible algebraic surface of degree D > 1 in R 3 or in C 3, with no planar components. Let P be a set of m points on V and let L be a set of n lines contained in V. Then there exists a subset L 0 L of at most O(D 2 ) lines, such that the number of incidences between P and L \ L 0 satisfies ( ) I(P, L \ L 0 ) = O m 1/2 n 1/2 D 1/2 + m + n. (10) Remarks. (a) We can always assume that D = O(n 1/2 ), since there always exists a trivariate polynomial of degree O(n 1/2 ) that identically vanishes on all the lines of L. Note that Theorem 3.4 becomes vacuous when D = Ω(n 1/2 ), as then L 0 could be all of L. In the real case we indeed assume that D n 1/2 (see Remark (b) following the statement of Theorem 1.3), for otherwise the bound in Theorem 1.3 is inferior to that in Theorem 1.1. In the complex case we do allow D to be Θ(n 1/2 ) and as just observed, there is no need to consider larger values of D. (b) An important feature of the theorem, already noted for the more general Theorem 1.3, and discussed in more detail later on, is that the bound in (10) avoids the term nd, which arises naturally in many earlier works, e.g., when bounding the number of incidences between points on V and lines not contained in V. This is significant when D is large. Proof. We first present a quick informal sketch of the proof, to help the reader navigating through it. Skipping over the definition of the subset L 0, we consider incidences with the surviving set L 1 := L \ L 0. The key technical step is Lemma 3.5, which roughly asserts that each line l L 1 is incident to at most O(D) other lines of L 1. We want to count the incidences of l with the points of P, so we introduce a parameter ξ, and distinguish between ξ-rich points, incident to more than ξ lines of L 1, and ξ-poor points, incident to at most ξ lines. The overall number of incidences with the poor points is O(mξ), and the number of incidences of a line with the rich points is O(D/ξ), for a total of O(mξ + nd/ξ). Optimizing ξ gives the main term in the bound we seek. We now proceed to the full proof. As in the proof of Lemma 3.1, we only work over C, and the results are then easily transported to the real case too. Consider the irreducible components W 1,..., W k of V. By Corollary 3.3, the number of lines contained in the union of the non-ruled components of V is O(D 2 ), and we place all these lines in the exceptional set L 0. In what follows we thus consider only ruled components of V. For simplicity, continue to denote them as W 1,..., W k, and note that k D/2. We further augment L 0 as follows. We first dispose of lines of L that are contained in more than one ruled component W i. We claim that their number is O(D 2 ). Indeed, for any pair W i, W j of distinct components, the intersection W i W j is a curve of degree (at most) deg(w i )deg(w j ), which can therefore contain at most deg(w i )deg(w j ) lines (by the generalized version of Bézout s theorem [10, Proposition 2.3], already mentioned in 22

181 connection with Proposition 3.2). Since k i=1 deg(w i) D, we have ( 2 deg(w i )deg(w j ) deg(w i )) = O(D 2 ), i j as claimed. We add to L 0 all the O(D 2 ) lines in L that are contained in more than one ruled component, and all the exceptional lines of all singly ruled components. The number of lines of the latter kind is at most 2k 2 (D/2) = D, so the size of L 0 is still O(D 2 ). Hence, each line of L 1 := L\L 0 is contained in a unique (singly or doubly) ruled component of V, and is a generator of that component. The strategy of the proof is to consider each line l of L 1, and to estimate the number of its incidences with the points of P in an indirect manner, via Lemma 3.1, applied to l and to each of the ruled components W j of V. We recall that l is contained in a unique component W i, and treat that component in a somewhat different manner than the treatment of the other components. In more detail, we proceed as follows. We first ignore, for each singly ruled conic component W i, the incidences between its apex (exceptional point) p Wi and the lines of L 1 that are contained in W i. We refer to these incidences as conical point-line incidences and to the other incidences as non-conical. When we talk about a line l incident to another line l at a point p, we will say that l is conically incident to l (at p) if p is the apex of some conic component W i and l is contained in W i (and thus incident to p). In all other cases, we will say that l is non-conically incident to l (at p). Note that this definition is asymmetric in l and l ; in particular, l does not have to lie in the cone W i. We also note that the number of conical point-line incidences is at most n, because each line of L 1 is contained in a unique component W i, so it can be involved in at most one conical incidence (at the apex of W i, when W i is a cone). We next prune away points p P that are non-conically incident to at most three lines of L 1. Note that p might be an apex of some conic component(s) of V ; in this case p is removed if it is incident to at most three lines of L 1 that are not contained in any of these components. We lose O(m) (non-conical) incidences in this process. Let P 1 denote the subset of the remaining points. Lemma 3.5. Each line l L 1 is non-conically incident, at points of P 1, to at most 4D other lines of L 1. Proof. Fix a line l L 1 and let W i denote the unique ruled component that contains l. Let W j be any of the other ruled components. We estimate the number of lines of L 1 that are non-conically incident to l and are contained in W j. If W j is a regulus, there are at most four such lines, since l meets the quadratic surface W j in at most two points, each incident to exactly two generators (and to no other lines contained in W j ). In this case, we write the bound 4 as deg(w j ) + 2. Assume then that W j is singly ruled. By Lemma 3.1, we have λ Wj (p; L 1 ) Λ Wj (p) deg(w j ). p l W j p l W j Note that, by definition, the above sum counts only non-conical incidences (and only with generators of W j, but the exceptional lines of W j have been removed from L 1 anyway). 23 i

182 We sum this bound over all components W j W i, including the reguli. Denoting the number of reguli by ρ, which is at most D/2, we obtain a total of deg(w j ) + 2ρ D + 2ρ 2D. j i Consider next the component W i containing l. Assume first that W i is a regulus. Each point p P 1 l can be incident to at most one other line of L 1 contained in W i (the other generator of W i through p). Since p is in P 1, it is non-conically incident to at least 3 2 = 1 other line of L 1, contained in some other ruled component of V. That is, the number of lines that are (non-conically) incident to l and are contained in W i, which apriorily can be arbitrarily large, is nevertheless at most the number of other lines (not contained in W i ) that are non-conically incident to l, which, as shown above, is at most 2D. If W i is not a regulus, Lemma 3.1 implies that p l W i Λ W i (p) deg(w i ) D, where again only non-conical incidences are counted in this sum (and only with generators). That is, the number of lines of L 1 that are non-conically incident to l (at points of P 1 ) and are contained in W i is at most D. Adding the bound for W i, which has just been shown to be either D or 2D, to the bound 2D for the other components, the claim follows. To proceed, choose a threshold parameter ξ 3, to be determined shortly. Each point p P 1 that is non-conically incident to at most ξ lines of L contributes at most ξ (nonconical) incidences, for a total of at most mξ incidences. Recall that the overall number of conical incidences is at most n. For the remaining non-conical incidences, let l be a line in L 1 that is incident to t points of P 1, so that each such point p is non-conically incident to at least ξ + 1 lines of L 1 (one of which is l). It then follows from Lemma 3.5 that t 4D/ξ. Hence, summing this over all l L 1, we obtain a total of at most 4nD/ξ incidences. We can now bring back the removed points of P \ P 1, since the non-conical incidences that they are involved in are counted in the bound mξ. That is, we have I(P, L 1 ) mξ + n + 4nD. ξ We now choose ξ = (nd/m) 1/2. For this choice to make sense, we want to have ξ 3, which will be the case if 9m nd. In this case we get the bound O ( m 1/2 n 1/2 D 1/2 + n ). If 9m > nd we take ξ = 3 and obtain the bound O(m). Combining these bounds, and adding the at most n conical incidences, the theorem follows. Having established Theorem 3.4, we now proceed to the proof of Theorem 1.3. The final stretch for non-singular points: The real case. To complete this proof, we need to bound the number I(P, L 0 ) of incidences involving the lines in the exceptional set L 0 yielded by Theorem 3.4. We remark that, in both the real and the complex cases, no special properties need to be assumed in the forthcoming analysis for the lines of L 0 ; the only property that matters is that the size of L 0 is small, i.e., L 0 = min{n, O(D 2 )}. In 24

183 the real case, we estimate I(P, L 0 ) using Guth and Katz s bound ([15]; see Theorem 1.1), recalling that no plane contains more than s lines of L 0. We thus obtain ( ) I(P, L 0 ) = O m 1/2 L 0 3/4 + m 2/3 L 0 1/3 s 1/3 + m + L 0 (11) ( ) = O m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + m + n. Combining the bounds in Theorem 3.4 and in (11) yields the asserted bound on I(P, L). We remark that in the first term in (11) we have estimated L 3/4 0 by O(n 1/2 D 1/2 ) instead of the sharper estimate O(D 3/2 ). This is because the term O(m 1/2 n 1/2 D 1/2 ) appears in Theorem 3.4 anyway, so the sharper estimate has no effect on the overall asymptotic bound. The final stretch for non-singular points: The complex case. We next estimate I(P, L 0 ) in the complex case. Again, we have n 0 := L 0 = O(D 2 ). We may ignore the points of P that are incident to fewer than three lines of L 0, as they contribute altogether only O(m) incidences. Continue to denote the set of surviving points as P. A point p that is incident to at least three lines of L 0 is either a singular point of V (when not all its incident lines are coplanar) or a flat point of V ; see Section 2, and also Guth and Katz [15], and Elekes et al. [7]. We decompose V into its irreducible components, sort them in some arbitrary fixed order, and assign each point p P (resp., line l L 0 ) to the first component that contains it. Similar to what has been observed above, the number of cross-incidences, between points and lines assigned to different components of V, is O(n 0 D) = O(D 3 ). We therefore assume, as we may, that V is irreducible (over C), and write V = Z(f), for an irreducible trivariate polynomial f of degree D. As in Section 2, we call a line flat if all its non-singular points are flat. As argued in Section 2 and in earlier works (see, e.g., [7, 15]), since Z(f) is not a plane, there exists a certain polynomial Π satisfying (i) deg(π) 3D 4, (ii) Z(f, Π) is a curve, and (iii) the flat points of P and the flat lines of L 0 are contained in Z(f, Π). We need the following lemma, adapted (with almost the same proof, which we omit) from a similar lemma that was established in our earlier work [37] for the four-dimensional case (see also Section 4). Lemma 3.6 ([37, Lemma 2.15]). Let f C[x, y, z] be an irreducible polynomial. If a line l Z(f) is flat, then all the tangent planes T p Z(f), for all the non-singular points p l, coincide. All this implies, arguing as in previous works [7, 15], that a line l L 0 that is nonsingular and non-flat contains at most 4D 4 points of P (each of which is either singular or flat). We prune away these lines from L 0, losing at most (4D 4) L 0 = O(D 3 ) incidences with the points of P. Continue to denote the subset of surviving lines as L 0. Therefore, it remains to bound the number of incidences between the surviving points and the surviving lines, each of which is either singular or flat. Write P as the union of the subset P f of flat points and the subset P s of singular points. Similarly, write L 0 as the union of the subset L f of flat lines and the subset L s of singular lines. A singular line 25

184 contains no flat points, and a flat line contains at most D 1 singular points. Thus, I(P, L 0 ) I(P f, L f ) + I(P s, L s ) + n 0 D. By Lemma 3.6, all the non-singular points of a flat line have the same tangent plane. Assign each point p P f (resp., line in L f ) to its tangent plane T p Z(f) (resp., T p Z(f) for some non-singular point p P f l; we only consider lines in L f that are incident to at least one point in P f ). We have therefore partitioned the points in P f and the lines in L f among planes in some finite set H = {h 1,..., h k }, and we only need to count incidences between points and lines assigned to the same plane. Within each h H, we have a set P h P f of m h points in h, and a set L h L f of n h lines contained in h. Using the planar bound (1), which also holds in the complex plane [45, 47], the number of incidences within h is O(m 2/3 h n2/3 h + m h + n h ). Summing these bounds over h H, and using Hölder s inequality, and the fact that n h s for each h, we obtain a total of O(m 2/3 n 1/3 0 s 1/3 +m+n 0 ) incidences. Bounding incidences involving singular points and lines. To complete the proof, when the ambient space is three-dimensional, we continue with both the real and the complex cases. We actually work over C, and obtain the results over the reals as an immediate consequence. Bounding the number of incidences between the singular points and lines is done via a procedure that may be referred to as degree reduction, to be described next. Assuming, without loss of generality, that f x (namely, the partial derivative of f with respect to x) does not vanish identically on Z(f), the points of P s and the lines of L s are then contained in Z(f x ), and deg(f x ) D 1. We thus construct a sequence of partial derivatives of f that are not identically zero on Z(f). For this we assume, as we may, that f, and each of its derivatives, are square-free; whenever this fails, we replace the corresponding derivative by its square-free counterpart before continuing to differentiate. Without loss of generality, assume that this sequence is f, f x, f xx, and so on, and that no square-free reduction is ever needed. Denote the j-th element in this sequence as f j, for j = 0, 1,... (so f 0 = f, f 1 = f x, and so on). Assign each point p P to the first polynomial f j in the sequence for which p is non-singular; more precisely, we assign p to the first f j for which f j (p) = 0 but f j+1 (p) 0 (recall that f 0 (p) is always 0 by assumption). Similarly, assign each line l to the first polynomial f j in the sequence for which l is contained in Z(f j ) but not contained in Z(f j+1 ) (again, by assumption, there always exists such a polynomial f j ). If l is assigned to f j then it can only contain points p that were assigned to some f k with k j. Indeed, if l contained a point p assigned to f k with k < j then f k+1 (p) 0 but l is contained in Z(f k+1 ), since k + 1 j; this is a contradiction that establishes the claim. Fix a line l L, which is assigned to some f j. An incidence between l and a point p P, assigned to some f k, for k > j, can be charged to the intersection of l with Z(f j+1 ) at p (by construction, p belongs to Z(f j+1 )). The number of such intersections is at most deg(f j+1 ) D j 1 D, so the overall number of incidences of this sort, over all lines l L, is O(n 0 D) = O(D 3 ). It therefore suffices to consider only incidences between points and lines that are assigned to the same zero set Z(f i ). The reductions so far have produced a finite collection of up to D polynomials, each of degree at most D, so that the points of P are partitioned among the polynomials and so are the lines of L, and we only need to bound the number of incidences between points and lines assigned to the same polynomial. Moreover, for each j, all the points assigned to f j are non-singular, by construction. For each j, let P j and L j denote the subsets of P and 26

185 of L 0, respectively, that are assigned to f j, and put m j := P j and n j := L j. We have j m j m and j n j n 0. We would like to apply the preceding analysis to P j and L j, but we face the technical issue that f j might be reducible and have some linear factors. The theorem does not require the variety to be irreducible, but forbids it to have linear components. We therefore proceed as follows. We first consider only those points and lines that are contained in some nonlinear component of Z(f j ). We apply the preceding analysis to these sets, and obtain the incidence bound O ( m 2/3 j n 1/3 j ) s 1/3 + m j + n j D. Summing these bounds over all f j s, using Hölder s inequality, we get the overall bound ( ) O m 2/3 n 1/3 0 s 1/3 + m + n 0 D. To bound the number of incidences involving points and lines in the linear components of Z(f j ), for any fixed j, we order arbitrarily the planar components of Z(f j ), assign each point to first component that contains it, and assign each line to the first component that contains it. As Z(f j ) has at most D such components, the number of cross incidences, between points and lines assigned to different components, is O(n j D). For the number of same component incidences, we use the Tóth-Zahl extension of the Szemerédi-Trotter bound [45, 47] in each plane, and sum them up, exactly as in the case of flat points and lines, and get a total bound of O ( m 2/3 j n 1/3 j ) s 1/3 + m j + n j D, and, summing these bounds over all f j s, as above, we get the overall bound ( ) I(P s, L s ) = O m 2/3 n 1/3 0 s 1/3 + m + n 0 D, thereby completing the proof of part (b) (for the case where the ambient space is threedimensional). Reduction to three dimensions. To complete the analysis, we need to consider the case where V is a two-dimensional variety embedded in R d or in C d, for d > 3. Let H be a generic 3-flat, and denote by P, L, and V the respective orthogonal projections of P, L, and V onto H. Since H is generic, we may assume that all the projected points in P are distinct, and so are all the projected lines in L. Clearly, every incidence between a point of P and a line of L corresponds to an incidence between the projected point and line. Since no 2-flat contains more than s lines of L, and H is generic, repeated applications of Lemma 2.9 imply that no plane in H contains more than s lines of L. One subtle point is that the set-theoretic projection V of V does not have to be a real algebraic variety (in general, it is only a semi-algebraic set), but it is always contained in a two-dimensional real algebraic variety Ṽ, which we call, as we did in an earlier work [36], the algebraic projection of V. The property that deg(ṽ ) D is well known, and follows by standard arguments in algebraic geometry; see, e.g., Beltrametti et al. [2, Proposition 3.4.8], 27

186 and also Harris [16]. That Ṽ does not contain a 2-flat follows by a suitable adaptation of the argument in [36, Lemma 2.1] (which is stated there for d = 4 over the reals), that applies for general d and over the complex field too. In conclusion, we have I(P, L) I(P, L ), where P is a set of m points and L is a set of n lines, all contained in the two-dimensional algebraic variety Ṽ, embedded in 3-space, which is of degree at most D and does not contain any plane, and no plane contains more than s lines of L. The preceding analysis thus implies that the bound asserted in the theorem applies in any dimension d 3. 4 Proof of Theorem 1.5 In most of this section we assume that the ambient space is four-dimensional, and work, with a few exceptions, over the complex field. The reduction from higher dimensions to four dimensions is handled as the reduction to three dimensions at the end of the previous section. We exploit the following useful corollary of Theorem 1.3 (recall that we are now in four dimensions). Note that in the bound given below, the term nd does not appear yet. Corollary 4.1. Let f and g be two 4-variate polynomials, over R or C, of degree O(D), such that Z(f, g) is two-dimensional over C. Let P be a set of m points and L a set of n lines, such that all the points of P and all the lines of L are contained in the union of the irreducible components of Z(f, g) that are not 2-flats. Assume also that no 2-flat contains more than s lines of L. Then we have (a) in the real case: ( ) I(P, L) = O m 1/2 n 1/2 D + m 2/3 D 4/3 s 1/3 + m + n, (12) (b) and in the complex case: ( ) I(P, L) = O m 1/2 n 1/2 D + m 2/3 D 4/3 s 1/3 + D 6 + m + n. (13) Proof. Let Z(f, g) = s i=1 V i be the decomposition of Z(f, g) into its irreducible components. By the generalized version of Bézout s theorem [10], we have s i=1 deg(v i) deg(f)deg(g) = O(D 2 ). Assume that V 1,..., V k are the components that are not 2-flats, for some k s, and let W denote their union. As just observed, deg(w ) = O(D 2 ). Applying to W Theorem 1.3(a) (over the reals) or Theorem 1.3(b) (over the complex field) thus completes the proof. The application of Theorem 1.3(b) goes smoothly in the complex case. In the real case, it is possible that Z C (f, g) is two-dimensional while some components of its real part Z(f, g) = Z C (f, g) R 2 are only one- (or zero-)dimensional. In the latter case, we apply Theorem 3.4 to Z C (f, g), viewed as a complex algebraic variety of degree O(D 2 ), and deduce that there exists a subset L 0 L of at most O(D 4 ) lines, such that the number of incidences between P and L\L 0 satisfies I(P, L\L 0 ) = O ( m 1/2 n 1/2 D + m + n ). To bound the number of incidences between P and L 0, we notice that L 0 is a set of real lines, and proceed precisely as in the final stretch for the real case in the proof of Theorem 1.3(a), thus completing the proof. 28

187 Remark. Corollary 4.1(a) is significant when D n 1/4. For larger values of D, we can project P, L, and Z(f, g) onto some generic 3-flat, and apply the incidence bound of Guth and Katz, as given in Theorem 1.1, within that 3-flat. When D n 1/4, the resulting bound is better than the one in (12). The proof of Theorem 1.5 revisits the proof of Theorem 1.2, as presented in [37], and applies Corollary 4.1 as a major technical tool. The only difference between the real and the complex cases is in the application of that corollary. Except for this application, we will work over the complex domain, but the analysis carries over, in a straightforward manner, to the real case too. Each line l L that is not contained in V contributes at most D incidences, for a total of O(nD) incidences. We thus assume, as we may, that all the lines of L are contained in V. Let V = t i=1 V i be the decomposition of V into its irreducible components, and assign each point p P, (resp., line l L) to the first V i (the one with the smallest index i) that contains it (such a V i always exists for a line l). It is easy to verify that points and lines that are assigned to different V i s contribute at most nd incidences. Indeed, any such incidence (p, l) can be charged to an intersection point of l with the component V i that p is assigned to, and thus there are at most i deg(v i) = D such incidences for each l, for an overall number of O(nD) such incidences. Therefore, it suffices to establish the bound in (5) or in (6) for the number of incidences between points and lines assigned to the same component. Moreover, once we establish (5) or (6) for each irreducible component V i separately, the bound for the entire V follows by an easy application of Hölder s inequality, as detailed below. We thus assume that V is irreducible, and write V = Z(f), for some real or complex irreducible polynomial f of degree D. We assume for now that P consists exclusively of non-singular points of the irreducible variety Z(f). The treatment of the singular points, similar to the handling of singular points in the proof of Theorem 1.3(b), will be given towards the end of the proof. We recall the definition of the four-dimensional flecnode polynomial FL 4 f of f, as given in Section 2. That is, FL 4 f vanishes at each flecnode p Z(f), namely, points p for which there exists a line that is incident to p and osculates to Z(f) up to the fourth order. Clearly, FL 4 f vanishes identically on every line of L, and thus also on P (assuming that each point of P is incident to at least one line of L). As noted in Section 2, its degree is O(D). If FL 4 f does not vanish identically on Z(f), then Z(f, FL4 f ) := Z(f) Z(FL4 f ) is a twodimensional variety that contains P and all the lines of L, and is of degree deg(f) deg(fl 4 f ) = O(D 2 ) (so we are in the setup assumed in Corollary 4.1). The other possibility is that FL 4 f vanishes identically on Z(f), and then Landsberg s theorem [22], mentioned in Section 2 (see also [37] for details), implies that Z(f) is ruled by (real or complex) lines. First case: Z(f, FL 4 f ) is two-dimensional. Put g = FL4 f and apply Corollary 4.1 to f and g. In the real case we obtain the bound ( ) O m 1/2 n 1/2 D + m 2/3 D 4/3 s 1/3 + m + n, and in the complex case we obtain the bound ( ) O m 1/2 n 1/2 D + m 2/3 D 4/3 s 1/3 + D 6 + m + n, over all components of Z(f, FL 4 f ) that are not 2-flats. 29

188 Incidences within 2-flats contained in Z(f, FL 4 f ). From this point on, throughout most of the proof, there is no distinction between the real and complex cases, so we work strictly over the complex domain; the differences between the real and complex cases will surface again when we discuss incidences with singular points and the case where V is embedded in higher dimensions. The strategy here is to distribute the points of P and the lines of L among the 2-flats that contain them (lines not contained in any 2-flat are contained in some other component of Z(f, FL 4 f ) and are dealt with, as above, within that component). See also below for a more detailed account of this strategy. Points that belong to at most two such 2-flats get duplicated at most twice, and we bound the number of incidences with these points by applying the planar bound (1) (which, as we recall, also holds in the complex plane [45, 47]) to each 2-flat separately, and sum up the bounds, to get O(m 2/3 n 1/3 s 1/3 + m + n), using Hölder s inequality, combined with the assumption that no 2-flat contains more than s lines of L. For the remaining points in P (points that belong to at least three 2-flats contained in Z(f, FL 4 f )), we proceed as follows. We first recall from Section 2 that a non-singular point p of Z(f) is linearly flat, if it is incident to at least three 2-flats that are contained in Z(f) (and thus also in the tangent hyperplane T p Z(f)). Linearly flat points are also flat, meaning that the second fundamental form of f vanishes at them (see Section 2 and [37]). This property, at a point p, is expressed by several polynomials of degree at most 3D 4 vanishing at p (see Section 2 and [37, Section 2.5]). We also recall that a line is flat if all its non-singular points are flat. Each line of L that is not flat contains at most O(D) flat points, and thus the non-flat lines contribute a total of at most O(nD) incidences with flat points, so we assume in what follows that the points of P and the lines of L are all flat. Since Z(f) is not a hyperplane, the second fundamental form does not vanish identically on Z(f) (this property holds in any dimension; see, e.g. [18, Exercise ]), and it then follows from the characterization of flat points that there exists a certain polynomial Π satisfying (i) deg(π) 3D 4, (ii) Z(f, Π) is two-dimensional, and (iii) the (flat) points of P and the (flat) lines of L are contained in Z(f, Π). Similar to Lemma 3.6, we have the property that all the (flat) points that lie on the same flat line have the same tangent hyperplane to Z(f) (see [37, Lemma 2.15]). Using this property, we partition the points and lines among tangent hyperplanes of Z(f), so that each point p (assumed to be non-singular on Z(f)) is assigned to its tangent hyperplane to Z(f), and each line l is assigned to the common hyperplane that is tangent to all non-singular points on l. Let H denote the resulting set of tangent hyperplanes. Clearly, it suffices to bound the number of incidences within each hyperplane in H. For each hyperplane h H, we now have a set P h P of m h points on h, a set L h L of n h lines contained in h, and a set F h of 2-flats, each of which is a 2-flat component of Z(f, FL 4 f ) that is contained in h. Each point p P h is contained in at least one 2-flat in F h, and each line l L h is contained in at least one 2-flat in F h. Note that we have h m h m and h n h n. Notice also that each 2-flat in F h is also contained in the two-dimensional surface Z(f) h (it is indeed two-dimensional since Z(f) is assumed not to have any hyperplane component), which is of degree D, so, by the generalized version of Bézout s theorem [10], it can contain at most D 2-flats, so we have F h D. We assign each point p P h (resp., line l L h ) to the first 2-flat in F h (in some arbitrary fixed order) that contains it. Similar to what has been observed above, the number of crossincidences, between points and lines assigned to different 2-flats within h, is at most n h D, 30

189 for a total, over the hyperplanes h H, of at most nd incidences. Again, using the (complex version of the) planar bound (1) and Hölder s inequality, the numbers of incidences within the 2-flats of h sum up to O(m 2/3 h n2/3 h + m h + n h D), and, summing over the hyperplanes h H, using Hölder s inequality once again, and the fact that n h s, we obtain a total of O(m 2/3 n 1/3 s 1/3 + m + nd) incidences. Remark. The novel feature of this step of the proof, as compared with the analogous argument used in [37], is that the number of 2-flats in F h, for any fixed h, is at most D. This allows us to bound the number of incidences within each hyperplane h H separately, so that, within each such hyperplane, instead of using the Guth-Katz bound (in the real case), we partition the points and lines among at most D planes, and then use the Szemerédi- Trotter bound in the real case, or the Tóth-Zahl bound in the complex case. The fact that there are at most D planes within each hyperplane h H guarantees that the number of cross-incidences (within h) is at most n h D, for a total of nd incidences (notice that, in contrast, the total number of planes (over all h H) can be arbitrarily large). Second case: Z(f) is ruled by (complex) lines. We next consider the case where the four-dimensional flecnode polynomial FL 4 f vanishes identically on Z(f). By Landsberg s theorem [22], this implies that Z(f) is ruled by (complex) lines. Similar to the notations for the three-dimensional case treated in Section 2, we denote by Σ 3 p (resp., Σ p ), for p Z(f), the set of all lines that are incident to p and osculate to Z(f) to order 3 at p (resp., are contained in Z(f)). We put Σ 3 := p Z(f) Σ3 p, and Σ := p Z(f) Σ p. Σ is the Fano variety of (lines contained in) Z(f), now represented in a higher-dimensional projective space. In [37], we proved that, for each p P, either Σ p 6 or Σ p is infinite. In the interest of completeness, we sketch here the outline of this argument. The analysis provides an algebraic characterization of points p of the latter kind, which uses an auxiliary polynomial U = U(p; u 0, u 1, u 2, u 3 ), called the u-resultant, defined in terms of f and its derivatives at p (see [37] and also [3] for details), where (u 0,..., u 3 ) denotes the direction of a line incident to p (in homogeneous coordinates). The polynomial U is of degree O(D) in p and is a homogeneous polynomial of degree 15 six in u. The characterization is that Σ 3 p is infinite if and only if U(p; u 0, u 1, u 2, u 3 ) 0, as a polynomial of u, at p. In the complementary case, Bézout s theorem [10] can be used to show that there are only six lines in Σ 3 p, and thus at most six lines in Σ p. Pruning away points p P with Σ 3 p 6 (the number of incidences involving these points is at most 6m = O(m)), we may then assume that Σ 3 p is infinite for every p P. If U(p; u 0, u 1, u 2, u 3 ) does not vanish identically (as a polynomial in u 0, u 1, u 2, u 3 ) at every point p Z(f), then at least one of its coefficients, call it c U, which is a polynomial in p, of degree O(D), does not vanish identically on Z(f). In this case, as U vanishes identically at every point of P (as a polynomial in u 0, u 1, u 2, u 3 ), we have P Z(f, c U ), which is a two-dimensional variety. The machinery developed in the first case can then be applied here (with g = c U ), and the bounds and properties derived for that case hold here too. We may therefore assume that U(p; u 0, u 1, u 2, u 3 ) 0 at every non-singular point p 15 The system of three homogeneous polynomial equations of degree 1, 2, 3, respectively, in the variable (u 0 : u 1 : u 2 : u 3) have either at most = 6 distinct solutions or infinitely many. 31

190 Z(f) (as a polynomial in u 0, u 1, u 2, u 3 ). By the aforementioned characterization via u- resultants, it follows that Σ 3 p is then infinite at each such point. We now use another theorem of Landsberg [18, Theorem 3.8.7]: Theorem 4.2 (Landsberg). Let f be an irreducible polynomial over P 4 (C), such that there exists an irreducible component Σ 3 0 Σ3 = Σ 3 (Z(f)) with the property that, for every point p in a Zariski-open and dense set O Z(f), dim Σ 3 0,p 1, where Σ3 0,p is the set of lines in Σ 3 0 incident to p. Then, for every point p O, all lines in Σ3 0,p are contained in Z(f); that is, for each p O, Σ 3 0,p is equal to the set Σ 0,p of lines incident to p and contained in Z(f). Since Σ 3 p is infinite at each non-singular point p Z(f), its dimension is 1 at each such point. As shown in [37], the main condition in Landsberg s theorem, about the existence of a component Σ 3 0 of Σ3 with the required properties, is satisfied too. One can then argue that the conclusion of Landsberg s theorem holds at every point of Z(f); see Lemma 2.2 in Section 2 for a similar claim concerning two-dimensional surfaces. That is, Z(f) is infinitely ruled by (complex) lines, in the sense that each point p Z(f) is incident to infinitely many (complex) lines that are contained in Z(f), and, moreover, Σ 3 0,p = Σ 0,p at each p. That is, Σ 3 0 is contained in Σ. Denoting this set as Σ 0, it is shown (in full detail) in [37] that the union of the lines in Σ 0 is equal to Z(f), and that dim(σ 0 ) 3. Severi s theorem. The following theorem was already used in [37], and we make a similar use thereof here too. It has been obtained by Severi [33] in A variant of this result has also been obtained by Segre [31]; see also the more recent works [23, 27, 28]. The reader can find a (sketch of a) proof of this theorem in [37] (as suggested to us by A. J. de Jong). We state here a special case of the theorem that we need. Theorem 4.3 (Severi s Theorem [33]; special case). Let X P 4 (C) be a three-dimensional irreducible variety, and let Σ 0 be a component of maximal dimension of the Fano variety Σ = Σ(X) of X, such that the lines of Σ 0 cover X. Then the following holds. (i) If dim(σ 0 ) = 4, then X is a hyperplane. (ii) If dim(σ 0 ) = 3, then either X is a quadric, or X is ruled by 2-flats. Informally, dim(σ 0 ) = 3 corresponds to the case where X is infinitely ruled by lines of Σ 0 : There are four degrees of freedom to specify a line in Σ 0, three to specify p X, and one to specify the line in Σ 0,p. We can assume that Σ 0,p is one-dimensional, because if it were two-dimensional, then X would have been a hyperplane. However, one degree of freedom has to be removed, to account for the fact that the same line (being contained in X) arises at each of its points. Severi s theorem asserts, again informally, that in this case the infinite family of lines of Σ 0,p must form a 2-flat, unless X is a quadric or a hyperplane. Moreover, by [37, Theorem 3.9] (whose proof is based on Theorem 2.3, Σ 0 has maximal dimension. Applying the second case in Severi s theorem to Z(f), which is justified by the preceding arguments, we conclude that either Z(f) is a quadric or it is ruled by 2-flats. The cases where Z(f) is a quadric or a hyperplane are ruled out by our assumption, so we only need to consider the case where Z(f) is ruled by (complex) 2-flats. The case where Z(f) is ruled by 2-flats. Handling this last step is somewhat intricate; it resembles the analysis of flat points and lines in the first case, where here points and lines 32

191 are partitioned among the ruling 2-flats. In this case, every point p Z(f) is incident to at least one 2-flat τ p Z(f). Let D p denote the set of 2-flats that are incident to p and are contained in Z(f). For a non-singular point p Z(f), if D p > 2, then p is a (linearly flat and thus) flat point of Z(f). Recall that we have bounded the number of incidences involving flat points (and lines) by partitioning them among a finite number of containing hyperplanes, and by bounding the incidences within each hyperplane. Lines incident to fewer than 3D 4 points of P have been pruned away, losing only O(nD) incidences, and the remaining lines are all flat. Repeating this argument here, noticing that here too, the number of 2-flats contained in a hyperplane is at most D, we obtain the bound ( ) O m 2/3 n 1/3 s 1/3 + m + nd. In what follows we therefore assume that all points of P are non-singular and non-flat (call these points, as in [37], ordinary for short), and therefore D p = 1 or 2, for each such p. Put H 1 (p) (resp., H 1 (p), H 2 (p)) for the 2-flat (resp., two 2-flats) in D p, when D p = 1 (resp., D p = 2). Clearly, each line in L, containing at least one ordinary point p Z(f), is contained in at most two 2-flats contained in Z(f) (namely, the 2-flats of D p ). Assign each ordinary point p P to each of the at most two 2-flats in D p, and assign each line l L that is incident to at least one ordinary point to the at most two 2-flats that contain l and are contained in Z(f) (it is possible that l is not assigned to any 2- flat see below). Changing the notation, enumerate these 2-flats, over all ordinary points p P, as U 1,..., U k, and, for each i = 1,..., k, let P i and L i denote the respective subsets of points and lines assigned to U i, and let m i and n i denote their cardinalities. We then have i m i 2m and i n i 2n, and the total number of incidences within the 2-flats U i (excluding lines not assigned to any 2-flat) is at most k i=1 I(P i, L i ). This incidence count can be obtained exactly as in the first case of the analysis, with the aid of Hölder s inequality, and yields the bound k i=1 ( ) I(P i, L i ) = O m 2/3 n 1/3 s 1/3 + m + n. As noted, this bound does not take into account incidences involving lines which are not contained in any of the 2-flats U i (and are therefore not assigned to any such 2-flat). It suffices to consider only lines of this sort that are non-singular and non-flat, since singular or flat lines are only incident to singular or flat points, and we assumed above that all the points of P are ordinary points. If l is a non-singular and non-flat line, and is not contained in any of the U i, we call it a piercing line of Z(f). We need the following lemma from [37]. Lemma 4.4 ([37, Lemma 3.13]). Let p Z(f) be an ordinary point. Then p is incident to at most one piercing line. Therefore, each ordinary point p P is incident to at most one piercing line, and the total contribution of incidences involving ordinary points and piercing lines is at most m. In conclusion, combining the bounds that we have obtained for the various subcases of the second case, we get that the number of incidences accounted for so far, involving only non-singular points, satisfies the desired bound in (5). 33

192 Incidences involving singular points of Z(f). The forthcoming reasoning is very similar to the handling of singular points and lines in the proof of Theorem 1.3(b), although it is somewhat more involved because we need to ensure that the resulting polynomials that we construct be irreducible; we present the analysis in detail, for the sake of clarity. In the analysis presented so far, we have assumed that the points of P are non-singular points of Z(f). To reduce the general setup to this situation we proceed as follows, similar to the way singularities were handled in Section 3; an identical reduction has also been used in [37]. We only handle lines that are contained in Z(f), because the other lines contribute at most O(nD) incidences. We construct a sequence of partial derivatives of f that are not identically zero on Z(f). For this we assume, as Section 3, that f, and each of its derivatives, are square-free. As before, assume that this sequence is obtained by always differentiating with respect to x, and denote the j-th element in this sequence as f j, for j = 0, 1,.... Assign each point p P to the first polynomial f j in the sequence for which p is non-singular; more precisely, we assign p to the first f j for which f j (p) = 0 but f j+1 (p) 0 (recall that f 0 (p) is always 0 by assumption). Similarly, assign each line l to the first polynomial f j in the sequence for which l is contained in Z(f j ) but not contained in Z(f j+1 ). If l is assigned to f j then it can only contain points p that were assigned to some f k with k j. Fix a line l L, which is assigned to some f j. An incidence between l and a point p P, assigned to some f k, for k > j, can be charged to the intersection of l with Z(f j+1 ) at p (by construction, p belongs to Z(f j+1 )). The number of such intersections is at most deg(f j+1 ) D j 1 D, so the overall number of incidences of this sort, over all lines l L, is O(nD). It therefore suffices to consider only incidences between points and lines that are assigned to the same zero set Z(f i ). The reductions so far have produced a finite collection of up to O(D) polynomials, each of degree at most D, so that the points of P are partitioned among the polynomials and so are the lines of L, and each point p is non-singular with respect to the polynomial it is assigned to, and we only need to bound the number of incidences between points and lines assigned to the same polynomial. This is not the end yet, because the various (reduced forms of the) partial derivatives might be reducible, which we want to avoid. Thus, in a final decomposition step, we split each derivative polynomial f j into its irreducible factors, and reassign the points and lines that were assigned to Z(f j ) to the various factors, by the same first come first served rule used above. The overall number of incidences that are lost in this process is again O(nD). The overall number of polynomials is O(D 2 ), as can easily be checked. Note also that the last decomposition step preserves non-singularity of the points in the special sense defined above; that is, as is easily verified, a point p Z(f j ) with f j+1 (p) 0, continues to be a non-singular point of the irreducible component it is reassigned to. We now fix one such final polynomial, call it f j, denote its degree by D j (which is upper bounded by the original degree D), and denote by P j and L j the subsets of the original sets of points and lines that are assigned to f j, and by m j and n j their respective cardinalities. By construction, Z(f j ) is irreducible, and P j consists exclusively of non-singular points of the irreducible Z(f j ). The preceding analysis, concerning incidences with non-singular points, yields, for each j, either the bound (5) ( ) I(P j, L j ) = O m 1/2 j n 1/2 j D j + m 2/3 j n 1/3 j s 1/3 + n j D j + m j 34

193 in the real case, or the bound (6) ( ) I(P j, L j ) = O m 1/2 j n 1/2 j D j + m 2/3 j n 1/3 j s 1/3 + D 6 + n j D j + m j in the complex case. Summing these bounds, upper bounding D j by D, and using Hölder s inequality for the first two terms, we get the bound (5) or (6) for the entire sets P and L. This completes the proof for the case where the variety containing P and the lines of L are embedded in 4-space. Reduction to the four-dimensional case. To complete the analysis, we need to consider the case where V is a three-dimensional variety embedded in d-dimensional space, for d > 4. The analysis follows closely the one at the end of the proof of Theorem 1.3, in Section 3. Concretely, let H be a generic 4-flat, and denote by P, L, and V the respective projections of P, L, and V onto H. Since H is generic, we may assume that all the projected points in P are distinct, and so are all the projected lines in L. Clearly, every incidence between a point of P and a line of L corresponds to an incidence between the projected point and line. Since no 2-flat contains more than s lines of L, and H is generic, repeated applications of Lemma 2.9 imply that no 2-flat in H contains more than s lines of L. As in Section 3, the set-theoretic projection V of V does not have to be a real algebraic variety, so we use instead the algebraic projection Ṽ of V that contains V. That Ṽ does not contain a hyperplane or quadric is argued by a suitable adaptation of the preceding argument (see [36, Lemma 2.1]). Indeed, the case of a hyperplane is straightforward (reasoning as at the end of the preceding section). For quadrics we have: Claim. Let X be a three-dimensional algebraic variety in d dimensions, for d 5, such that a generic (real or complex) algebraic projection of V on 4-space is a quadric. Then X is a quadratic variety in d dimensions. Proof. Assume to the contrary that X is not a quadratic variety. This implies that, for a generic (d 2)-flat h, the curve C h = X h is not a quadratic curve. For any 2-flat g h, let C h,g denote the projection of C h onto g. This implies that for a generic choice of g and a (d 2)-flat h satisfying g h, the curve C h,g is not a quadratic planar curve (that is, a conic section) in g. Next, by taking a suitable rotation of the coordinate frame, we may assume that g is the x 1 x 2 -flat, and h is the x 1 x 2... x d 2 -flat. In these coordinates, it is easy to verify that C h,g can be obtained by first projecting X onto the x 1 x 2 x d 1 x d -flat, and then cutting it with the x 1 x 2 -plane. But the projection of X onto the x 1 x 2 x d 1 x d -flat (which is actually a generic 4-flat) is a 3-quadric by assumption, and then cutting it with any 2-flat yields a quadratic planar curve, a contradiction that completes the proof. In conclusion, we have I(P, L) I(P, L ), where P is a set of m points and L is a set of n lines, all contained in the three-dimensional algebraic variety Ṽ, embedded in 4-space, which is of degree at most D and does not contain any hyperplane or quadric component, and no 2-flat contains more than s lines of L. The preceding analysis thus implies that the bound asserted in the theorem applies in any dimension d 4. 35

194 5 Discussion (1) As already emphasized, most of the analysis in the proof of Theorem 1.3(a) is carried out over the complex domain. The only place where the proofs of (a) and (b) bifurcate is in the final step. Over the reals we bound I(P, L 0 ) using the bound of Guth and Katz (which only holds over the reals, because of the polynomial partitioning that it employs), as a black box. Over the complex field, we use a variant of the analysis to bypass this step, and obtain more or less the same bound, except for the additional term O(D 3 ), which becomes insignificant for D m 1/3, say. For three-dimensional varieties, the proof of both parts of Theorem 1.5 are more or less the same, with the main difference being the application of the real or complex version of Corollary 4.1 (which in itself is Theorem 1.3 in disguise ). Another difference is in the application of the planar point-line incidence bound the bound is the same in both cases, but the sources (Szemerédi-Trotter or Tóth-Zahl) are different. The derivation of fairly sharp point-line incidence bounds over the complex domain in higher dimensions constitutes, in our opinion, significant progress in this theory. (2) In view of the lower bound constructions in [15, 37, 40], the new bounds in Theorems 1.3 and 1.5 do not hold without the assumption that the points lie on a variety of relatively small degree. We also note that, for a three-dimensional variety, we also get rid of the term m 1/2 n 1/2 q 1/4 ; this term may arise only when we consider points on hyperplanes or quadrics, but in our case the variety does not contain any such components. Therefore, our theorems indicate that these terms may only arise if the variety contains such components. (3) As mentioned in the introduction, Corollary 1.4 can be extended to the case where V, which is of constant degree D, also contains planes. Here too, we assume that no plane contains more than s lines of L, but this time it is not necessarily the case that s D. Let π 1,..., π k denote the planar components of V, where k D = O(1). For each i = 1,..., k, the number of incidences within π i, namely, between the set P i of points contained in π i and the set L i of lines contained in π i, in both real and complex cases, is ( ) ( ) I(P i, L i ) = O P i 2/3 L i 2/3 + P i + L i = O m 2/3 s 2/3 + m + s. Summing these bounds over the k = O(1) planes, we get the same asymptotic bound for the overall number of the incidences within these planes. Any other incidence between a point p lying in one of these planes π i and a line l not contained in π i can be uniquely identified with the intersection of l with π i. The overall number of such intersections is at most nk = O(n). This leads to the following extension of Corollary 1.4. Corollary 5.1. Let P be a set of m points and L a set of n lines in R d or in C d, for any d 3, and let s n be a parameter, such that all the points and lines lie in a common two-dimensional algebraic surface of constant degree, and no 2-flat contains more than s lines of L. Then ( ) I(P, L) = O m 2/3 s 2/3 + m + n, where the constant of proportionality depends on the degree of the surface. 36

195 (4) As already noted, one of the significant achievements of the analysis in Theorem 1.3 is that the bound there does not include the term O(nD). Such a term arises naturally, when one considers incidences between points lying in some irreducible component of V and lines not contained in that component. These incidences can be bounded by nd, by charging them, as above, to line-component intersections. When D is large, eliminating the term nd can be crucial for the analysis, as demonstrated in our earlier work [37]. (5) An interesting challenge is to establish a similar bound for I(P, L), for the case where the points of P lie on a two-dimensional variety V, but the lines need not be contained in V. A trivial extension of the proof adds the term O(nD) to the bound. The challenge is to avoid this term (if possible); see also Remark (4) above. (6) Similar to item (2), Theorem 1.5(a) can be extended to the case where V also contains hyperplane and quadric components, albeit only for the real case. Here, as in Theorem 1.2, we add the condition that no hyperplane or quadric contains more than q lines of L. Note that here we do not assume that D is a constant. Let H 1,..., H k denote the hyperplane and quadric components of V, where k D. Assign, whenever applicable, each point (resp., line) to the first H i that contains it. As observed above, the number of cross-incidences is O(nD). By [37, Proposition 3.6], the total number of incidences within the hyperplanes and quadrics H i, for i = 1,..., k, is O(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + n). This leads to the following extension of Theorem 1.5(a). Corollary 5.2. Let P be a set of m points and L a set of n lines in R d, for any d 4, and let s q n be parameters, such that all the points and lines lie in a common three-dimensional algebraic surface of degree D, and assume that (i) no 3-flat or 3-quadric contains more than q lines of L, and (ii) no 2-flat contains more than s lines of L. Then ( ) I(P, L) = O m 1/2 n 1/2 (D + q 1/4 ) + m 2/3 n 1/3 s 1/3 + nd + m. (14) (7) An interesting offshoot of Lemma 3.5 is the following result. Proposition 5.3. Let V be a possibly reducible two-dimensional algebraic surface of degree D > 1 in R 3 or in C 3, with no plane or regulus components, and let L be a set of n lines contained in V. Then the number of 2-rich points (points incident to at least two lines of L) is O(nD). Proof. Partition L into the subsets L 1 and L 0, as in the proof of Theorem 1.3. Recall that L 0 is the set of all lines that are either contained in non-ruled components of V, or contained in more than one component, or are exceptional lines on ruled components. By Lemma 3.5, each line of L 1 is non-conically incident to only O(D) other lines of L 1, for a total of O(nD) 2-rich points of this sort. Note that we now carry out the analysis without pruning any point (we do not want to do that), because V does not contain any plane or regulus component. The number of 2-rich points that are exceptional points is at most the number of irreducible components of V, that is, at most D, so this number is negligible. 37

196 The number of lines in L 0 is O(D 2 ). Let h be a plane or a regulus. The number of lines of L 0 contained in h is at most deg(v h) 2D = O( L 0 ) (this holds if we assume, as we may, that L 0 = Θ(D 2 )). It therefore follows from Guth and Katz [15] that the number of 2-rich points involvoing the lines of L 0 is O( L 0 3/2 ) = O( L 0 D). It remains to consider 2-rich points that are intersection points of a line in L 0 and a line in L 1. By construction, each line l L 1 is contained in precisely one (ruled) component W of V. If p l is also incident to a line l L 0 then, again by construction, l is contained in another component W of V, which does not contain l. Hence l intersects W in at most deg(w ) points (one of which is p), for a total of at most deg(v ) = D points. Therefore, the number of 2-rich points involving one line in L 1 and another in L 0 is at most n 1 D. As we have exhausted all cases, the assertion follows. (8) Challenging directions for further research are (a) to bound the number of incidences between points and lines on (d 1)-dimensional varieties in R d (or in higher dimensions), for d 5, (b) to bound the number of r-rich points, for any r 2, in a finite set of lines contained in such a variety, and (c) to bound the number of incidences between points on a variety and k-flats (under suitable restrictions) in three, four, or higher dimensions. Acknowledgement. We are deeply grateful to Martin Sombra and to two anonymous referees for their critical and thorough reading of the paper, and for many valuable comments. References [1] A. Beauville, Complex Algebraic Surfaces, Vol. 34, Cambridge University Press, Cambridge, [2] M. Beltrametti, E. Carletti, D. Gallarati and G. M. Bragadin, Lectures on Curves, Surfaces and Projective Varieties: A Classical View of Algebraic Geometry, European Mathematical Society, [3] D. Cox, J. Little and D. O Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer Verlag, Heidelberg, [4] D. Cox, J. Little and D. O Shea, Using Algebraic Geometry, Springer Verlag, Heidelberg, [5] W. L. Edge, The Theory of Ruled Surfaces, Cambridge University Press, Cambridge, [6] G. Elekes, Sums versus products in number theory, algebra and Erdős geometry A survey, in Paul Erdős and his Mathematics II, Bolyai Math. Soc., Stud. 11, Budapest, 2002, pp [7] G. Elekes, H. Kaplan and M. Sharir, On lines, joints, and incidences in three dimensions, J. Combinat. Theory, Ser. A 118 (2011), Also in arxiv:

197 [8] P. Erdős, On sets of distances of n points, Amer. Math. Monthly 53 (1946), [9] D. Fuchs and S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, Amer. Math. Soc. Press, Providence, RI, [10] W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, Expository Lectures from the CBMS Regional Conference Held at George Mason University, June 27 July 1, 1983, Vol. 54. AMS Bookstore, [11] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Vol. 52, John Wiley & Sons, New York, [12] L. Guth, Ruled surface theory and incidence geometry, in arxiv: [13] L. Guth, Polynomial Methods in Combinatorics, University Lecture Series, Vol. 64, Amer. Math. Soc. Press, Providence, RI, [14] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem, Advances Math. 225 (2010), Also in arxiv: [15] L. Guth and N. H. Katz, On the Erdős distinct distances problem in the plane, Annals Math. 181 (2015), Also in arxiv: [16] J. Harris, Algebraic Geometry: A First Course, Vol Springer-Verlag, New York, [17] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, [18] T. A. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics, volume 61, Amer. Math. Soc., Providence, RI, [19] N. H. Katz, The flecnode polynomial: A central object in incidence geometry, in arxiv: [20] J. Kollár, Szemerédi Trotter-type theorems in dimension 3, Advances Math. 271 (2015), Also in arxiv: [21] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Springer- Verlag, Heidelberg, [22] J. M. Landsberg, Is a linear space contained in a submanifold? On the number of derivatives needed to tell, J. Reine Angew. Math. 508 (1999), [23] E. Mezzetti and D. Portelli, On threefolds covered by lines, in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Vol. 70, No. 1, Springer Verlag, Heidelberg, [24] G. Monge, Application de l Analyse á la Geometrie, Bernard, Paris, [25] H. Pottmann and J. Wallner, Computational Line Geometry, Springer Science and Business Media, Berlin Heidelberg, [26] A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer Verlag, London,

198 [27] S. Richelson, Classifying Varieties With Many Lines, a senior thesis, Harvard university, [28] E. Rogora, Varieties with many lines, Manuscripta Mathematica 82.1 (1994), [29] M. Rudnev, On the number of incidences between planes and points in three dimensions, in arxiv: (2014). [30] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2, 5th edition, Hodges, Figgis and co. Ltd., Dublin, [31] B. Segre, Sulle V n contenenti più di n k S k, Nota I e II, Rend. Accad. Naz. Lincei 5 (1948), , [32] J. M. Selig, Geometric Fundamentals of Robotics, Springer Science and Business Media, [33] F. Severi, Intorno ai punti doppi impropri etc., Rend. Cir. Math. Palermo 15 (10) (1901), [34] I. R. Shafarevich, Basic Algebraic Geometry, Vol. 197, Springer-Verlag, New York, [35] M. Sharir and N. Solomon, Incidences between points and lines in three dimensions, in Intuitive Geometry (J. Pach, Ed.), to appear. Also in Proc. 31st Sympos. Comput. Geom. (2015), , and in arxiv: [36] M. Sharir and N. Solomon, Incidences between points and lines in R 4, Proc. 30th Annu. Sympos. Comput. Geom., 2014, [37] M. Sharir and N. Solomon, Incidences between points and lines in R 4, Discrete Comput. Geom. 57 (2017), Also in Proc. 56th IEEE Sympos. Foundations of Computer Science 2015, , and in arxiv: [38] M. Sharir and N. Solomon, Incidences with curves and surfaces and applications to distinct and repeated distances, Proc. 28th ACM-SIAM Sympos. Discrete Algorithms, 2017, Also in arxiv: [39] A. Sheffer, E. Szabó and J. Zahl, Point-curve incidences in the complex plane, Combinatorica, to appear. Also in arxiv: [40] N. Solomon and R. Zhang, Highly incidental patterns on a quadratic hypersurface in R 4, Discrete Math. 340(4) (2017), Also in arxiv: [41] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput. Geom. 48 (2012), [42] J. Solymosi and F. de Zeeuw, Incidence bounds for complex algebraic curves on Cartesian products, in arxiv: [43] M. R. Spiegel, S. Lipschutz, and D. Spellman, Vector Analysis, 2nd edition, Schaum s outlines, McGraw-Hill,

199 [44] E. Szemerédi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), [45] C.D. Tóth, The Szemerédi-Trotter theorem in the complex plane, Combinatorica 35 (2015), Also in arxiv: (2003). [46] B. L. van der Waerden, E. Artin, and E. Noether, Modern Algebra, Vol. 2, Springer Verlag, Heidelberg, [47] J. Zahl, A Szemerédi-Trotter type theorem in R 4, Discrete Comput. Geom. 54 (2015), Also in arxiv:

200 Part IV Incidences between points and curves and points and surfaces 191

201

202 7 Incidences with curves in R d 193

203 Incidences with curves in R d Micha Sharir Adam Sheffer Noam Solomon Submitted: Jan 4, 2016; Accepted: Oct 8, 2016 ; Published: Oct 28, 2016 Mathematics Subject Classifications: 52C10 Abstract We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in R d is d 1 O m k dk d+1 +ε n dk d dk d+1 + m k jk j+1 +ε n d(j 1)(k 1) (d j)(k 1) (d 1)(jk j+1) (d 1)(jk j+1) q + m + n, j=2 for any ε > 0, where the constant of proportionality depends on k, ε and d, provided that no j-dimensional surface of degree c j (k, d, ε), a constant parameter depending on k, d, j, and ε, contains more than q j input curves, and that the q j s satisfy certain mild conditions. This bound generalizes the well-known planar incidence bound of Pach and Sharir to R d. It generalizes a recent result of Sharir and Solomon [21] concerning point-line incidences in four dimensions (where d = 4 and k = 2), and partly generalizes a recent result of Guth [9] (as well as the earlier bound of Guth and Katz [11]) in three dimensions (Guth s three-dimensional bound has a better dependency on q 2 ). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl [8], in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi [5] and by Hablicsek and Scherr [13] concerning rich lines in high-dimensional spaces. Our bound is not known to be tight in most cases. Work on this paper was partially supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), and by the Hermann Minkowski MINERVA Center for Geometry at Tel Aviv University. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM) at UCLA, which is supported by the National Science Foundation. A preliminary version of this paper appeared in Proc. European Sympos. Algorithms, 2015, pages Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel; michas@post.tau.ac.il Dept. of Mathematics, California Institute of Technology, Pasadena, CA, USA; adamsh@gmail.com Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel; noamsolomon@post.tau.ac.il j the electronic journal of combinatorics 23(4) (2016), #P4.16 1

204 1 Introduction Let C be a set of curves in R d. We say that C has k degrees of freedom with multiplicity s if (i) for every k points in R d there are at most s curves of C that are incident to all k points, and (ii) every pair of curves of C intersect in at most s points. The bounds that we derive depend more significantly on k than on s see below. In this paper we derive general upper bounds on the number of incidences between a set P of m points and a set C of n bounded-degree algebraic curves that have k degrees of freedom (with some constant multiplicity s). We denote the number of these incidences by I(P, C). Before stating our results, let us put them in context. The basic and most studied case involves incidences between points and lines. In two dimensions, writing L for the given set of n lines, the classical Szemerédi Trotter theorem [28] yields the worst-case tight bound I(P, L) = O ( m 2/3 n 2/3 + m + n ). (1) In three dimensions, in the 2010 groundbreaking paper of Guth and Katz [11], an improved bound has been derived for I(P, L), for a set P of m points and a set L of n lines in R 3, provided that not too many lines of L lie in a common plane. Specifically, they showed: Theorem 1.1 (Guth and Katz [11]) Let P be a set of m distinct points and L a set of n distinct lines in R 3, and let q 2 n be a parameter, such that no plane contains more than q 2 lines of L. Then ( ) I(P, L) = O m 1/2 n 3/4 + m 2/3 n 1/3 q 1/3 2 + m + n. This bound was a major step in the derivation of the main result of [11], an almost-linear lower bound on the number of distinct distances determined by any set of n points in the plane, a classical problem posed by Erdős in 1946 [7]. Their proof uses several nontrivial tools from algebraic and differential geometry. This machinery comes on top of the main innovation of Guth and Katz, the introduction of the polynomial partitioning technique; see below. In four dimensions, Sharir and Solomon [22] have obtained the following sharp pointline incidence bound: Theorem 1.2 (Sharir and Solomon [22]) Let P be a set of m distinct points and L a set of n distinct lines in R 4, and let q 2, q 3 n be parameters, such that (i) no hyperplane or quadric contains more than q 3 lines of L, and (ii) no 2-flat contains more than q 2 lines of L. Then I(P, L) 2 ( c log m m 2/5 n 4/5 + m ) ( ) + A m 1/2 n 1/2 q 1/4 3 + m 2/3 n 1/3 q 1/3 2 + n, (2) where A and c are suitable absolute constants. When m n 6/7 or m n 5/3, we get the sharper bound ( ) I(P, L) A m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 3 + m 2/3 n 1/3 q 1/3 2 + n. (3) the electronic journal of combinatorics 23(4) (2016), #P4.16 2

205 In general, except for the factor 2 c log m, the bound is tight in the worst case, for any values of m, n, with corresponding suitable ranges of q 2 and q 3. This improves, in several aspects, an earlier treatment of this problem in Sharir and Solomon [21]. Another way to extend the Szemerédi Trotter bound is for curves in the plane with k degrees of freedom (for lines, k = 2). This has been done by Pach and Sharir, who showed: 1 Theorem 1.3 (Pach and Sharir [18]) Let P be a set of m points in R 2 and let C be a set of bounded-degree algebraic curves in R 2 with k degrees of freedom and with multiplicity s. Then ) I(P, C) = O (m k 2k 2 2k 1 n 2k 1 + m + n, where the constant of proportionality depends on k and s. Several special cases of this result, such as the cases of unit circles and of arbitrary circles, have been considered separately [4, 26]. Unlike the Szemerédi-Trotter result (which arises as a special case of Theorem 1.3 with k = 2), the bound in Theorem 1.3 is not known to be tight for any k 3. In fact, it is known not to be tight for the case of arbitrary circles; see [1]. Here too one can consider the extension of these bounds to higher dimensions. Excluding this paper, the following theorem states the current best bound for this case Theorem 1.4 (Fox et al. [8]) Let P be a set of m points and let V be a set of n constant-degree algebraic varieties, both in R d, such that the incidence graph of P V does not contain a copy of K s,t (here we think of s, t, and d as being fixed constants, and m and n as large). Then for every ε > 0, we have ) I(P, V) = O (m (d 1)s ds 1 +ε n d(s 1) ds 1 + m + n, where the constant of proportionality depends on ε, s, t, d, and the maximum degree of the varieties. While the bound of Theorem 1.4 holds for varieties of any dimension, in this paper we only consider the case of curves. Several better bounds are known for specific types of curves. The case of lines is studied in several papers, such as [5, 13]. It is also worth mentioning here the work of Sharir, Sheffer and Zahl [20] on incidences between points and circles in three dimensions; an earlier study of this problem by Aronov et al. [2] gives a different, dimension-independent bound. A very recent result of Sharir and Zahl [24] gives an improved bound (over the one in Theorem 1.3) for curves in the plane. The bounds given in Theorem 1.1 and Theorem 1.2 include a leading term that depends only on m and n (the terms m 1/2 n 3/4 and 2 c log n m 2/5 n 4/5, respectively), and, 1 Their result holds for more general families of curves, not necessarily algebraic, but, since algebraicity will be assumed in higher dimensions, we assume it also in the plane. the electronic journal of combinatorics 23(4) (2016), #P4.16 3

206 except for the two-dimensional case, a series of lower-dimensional terms (like the term m 2/3 n 1/3 q 1/3 2 in Theorem 1.1 and the terms m 1/2 n 1/2 q 1/4 3 and m 2/3 n 1/3 q 1/3 2 in Theorem 1.2). The leading terms, in the case of lines, become smaller as d increases (when m is not too small and not too large with respect to n). Informally, by placing the lines in a higherdimensional space, it should become harder to create many incidences on them. Nevertheless, this is true only if the setup is truly d-dimensional. This means that not too many lines or curves are allowed to lie in a common lower-dimensional space. The lower-dimensional terms handle incidences within such lower-dimensional spaces. There is such a term for every dimension j = 2,..., d 1, and the j-dimensional term handles incidences within j-dimensional subspaces (which, as the quadrics in the case of lines in four dimensions in Theorem 1.2, are not necessarily linear and might be algebraic of low constant degree). Comparing the bounds for lines in two, three, and four dimensions, we see that the j-dimensional term in d dimensions, for j < d, is a sharper variant of the leading term in j dimensions. More concretely, if that leading term in j dimensions is m a n b then its counterpart in the d-dimensional bound, for d > j, is of the form m a n t q b t j, where q j is the maximum number of lines that can lie in a common j-dimensional flat or low-degree variety, and t depends on j and d. Our results. In this paper we consider a generalization of these results, to the case where C is a family of bounded-degree algebraic curves with k degrees of freedom (and some multiplicity s) in R d. This is a very ambitious and difficult project, and the challenges that it faces seem to be enormous. Here we make the first step in this direction, and obtain the following bounds. As the exponents in the bounds are rather cumbersome expressions in d, k, and j, we first state the special case of d = 3 (and prove it separately), and then give the general bound in d dimensions. Theorem 1.5 (Curves in R 3 ) Let k 2 be an integer, and let ε > 0. Then there exists a constant c(k, ε) that depends on k and ε, such that the following holds. Let P be a set of m points and C a set of n irreducible algebraic curves of constant degree with k degrees of freedom (and some multiplicity s) in R 3, such that every algebraic surface of degree at most c(k, ε) contains at most q 2 curves of C. Then ( ) I(P, C) = O m k 3k 2 +ε n 3k 3 k 3k 2 + m 2k 1 +ε n 3k 3 k 1 4k 2 q 4k m + n, where the constant of proportionality depends on k, s, and ε (and on the degree of the curves). The corresponding result in d dimensions is as follows. Theorem 1.6 (Curves in R d ) Let d 3 and k 2 be integers, and let ε > 0. Then there exist constants c j (k, d, ε), for j = 2,..., d 1, that depend on k, d, j, and ε, such that the following holds. Let P be a set of m points and C a set of n irreducible algebraic curves of constant degree with k degrees of freedom (and some multiplicity s) in R d. Moreover, assume that, for j = 2,..., d 1, every j-dimensional algebraic variety of degree at most the electronic journal of combinatorics 23(4) (2016), #P4.16 4

207 c j (k, d, ε) contains at most q j curves of C, for given parameters q 2 q d 1 n. Then we have ( ) d 1 I(P, C) = O m k dk d+1 +ε n dk d dk d+1 + m k jk j+1 +ε n d(j 1)(k 1) (d j)(k 1) (d 1)(jk j+1) q (d 1)(jk j+1) + m + n, j=2 where the constant of proportionality depends on k, s, d, and ε (and on the degree of the curves), provided that, for any 2 j < l d, we have (with the convention that q d = n) j q j ( ql 1 q l ) l(l 2) q l 1. (4) Note that the constraints (4) for l = j +1 simply require that the sequence q 2,..., q d 1 be (weakly) increasing, as already stipulated. Discussion. The advantages of our results are obvious: They provide the first nontrivial bounds for the general case of curves with any number of degrees of freedom in any dimension (with the exception of one previous study of Fox et al. [8], in which weaker bounds are obtained, albeit for arbitrary varieties instead of algebraic curves). Apart for the ε in the exponents, the leading term is best possible, in the sense that (i) the polynomial partitioning technique [11] that our analysis employs (and that has been used in essentially all recent works on incidences in higher dimensions) yields a recurrence that solves to this bound, and, moreover, (ii) it is (nearly) worst-case tight for lines in two, three, and four dimensions (as shown in the respective works cited above), and in fact is likely to be tight for lines in higher dimensions too, using a suitable extension of a construction, due to Elekes and used in [11, 22]. Nevertheless, our bounds are not perfect, and tightening them further is a major challenge for future research. Specifically: (i) While it seems likely that the leading terms in our bounds are tight for lines in R d, they are probably not tight for most constant-degree algebraic curves. Sharir and Zahl [24] recently proved better bounds 2 for the case of d = 2 and k 3, and it seems likely that better bounds also exist in higher dimensions. One common conjecture suggests that in R 2 the number of incidences between any n points and any n constant-degree curves (with no common components) should be O(n 4/3 ); the conjecture does not always hold when the number of points and the number of curves are significantly different. (ii) The bounds involve the factor m ε. As the existing works indicate, getting rid of this factor is no small feat. Although the factor does not show up in the cases of lines in two and three dimensions, it already shows up (sort of) in four dimensions (Theorem 1.2), as well as in the case of circles in three dimensions [20]. (A recent study of Guth [9] also pays this factor for the case of lines in three dimensions, in order to simplify the original analysis in the Guth Katz paper [11]. Another recent simplified proof, due to Sharir and 2 To be precise, it is assumed in [24] that the curves come from a k-dimensional family of curves, which is a similar constraint, albeit not quite the same, as having k degrees of freedom. the electronic journal of combinatorics 23(4) (2016), #P4.16 5

208 Solomon [23], manages to get rid of this factor, except for some narrow range of m and n.) See the proofs and comments below for further elaboration of this issue. (iii) The condition that no surface of degree c j (k, d, ε) contains too many curves of C, for j = 2,..., d 1, is very restrictive, especially since the actual values of these constants that arise in the proofs can be quite large. Again, earlier works also suffer from this handicap, such as Guth s work [9] mentioned above, as well as an earlier version of Sharir and Solomon s four-dimensional bound [21]. A recent interesting study of Guth and Zahl [12] may offer some tools for better controlling these parameters. (iv) Finally, the lower-dimensional terms that we obtain are not best possible. For example, the bound that we get in Theorem 1.5 for the case of lines in R 3 (k = 2) is O(m 1/2+ε n 3/4 + m 2/3+ε n 1/2 q 1/6 2 + m + n). When q 2 n, the two-dimensional term m 2/3+ε n 1/2 q 1/6 2 in that bound is worse than the corresponding term m 2/3 n 1/3 q 1/3 2 in Theorem 1.1 (even when ignoring the factor m ε ). Since the statement of Theorem 1.6 is rather involved, we also present two simplified versions thereof. The first is a straightforward corollary as a simpler case. Corollary 1.7 Let d 3, k 2 be integers, and let ε > 0. Then there exists a constant c(k, d, ε) that depends on k, d, and ε, such that the following holds. Let P be a set of m points and C a set of n irreducible algebraic curves of some constant maximum degree with k degrees of freedom (and some multiplicity s) in R d, such that m = O(n d/(d 1) ). Moreover, assume that every algebraic variety of degree at most c(k, d, ε) contains a constant number of curves of C, where this constant may depend on d, k, and ε. Then we have ) I(P, C) = O (m k dk d+1 +ε n dk d dk d+1 + n, where the constant of proportionality depends on k, s, d, ε, and the maximum degree of the curves. The second simplification replaces the sequence of constraints on the number of curves in lower-dimensional varieties of constant degrees by a single constraint involving only (d 1)-dimensional varieties (hypersurfaces). Its proof is similar to that of Theorem 1.6, and will be briefly discussed later. Theorem 1.8 Let d 3, k 2 be integers, and let ε > 0. Then there exists a constant c(k, d, ε) that depends on k, d, and ε, such that the following holds. Let P be a set of m points, let C be a set of n irreducible algebraic curves of some constant maximum degree and with k degrees of freedom, both in R d, and let q n be another parameter, such that every hypersurface of degree at most c(k, d, ε) contains at most q curves of C. Then ) I(P, C) = O (m k dk d+1 +ε n dk d k dk d+1 + m 2k 1 +ε n dk d (k 1)(d 2) (d 1)(2k 1) q (d 1)(2k 1) + m + n, where the constant of proportionality depends on ε, k, d, and the maximum degree of the curves. the electronic journal of combinatorics 23(4) (2016), #P4.16 6

209 Our results are also related to recent works by Dvir and Gopi [5] and by Hablicsek and Scherr [13], that study rich lines in high dimensions. Specifically, let P be a set of n points in R d and let L be a set of r-rich lines (that is, each line of L contains at least r points of P). If L = Ω(n 2 /r d+1 ) then there exists a hyperplane containing Ω(n/r d 1 ) points of P. Our bounds are relevant for extending this result to rich curves. Concretely, for a set P of n points in R d and a collection C of r-rich constant-degree algebraic curves with k degrees of freedom, if C is too large then the incidence bound becomes larger than the leading term in Theorem 1.8, indicating that some hypersurface must contain many curves of C, which would then imply that such a surface has to also contain many points of P. We omit the rather routine, albeit fairly tedious, calculations. As in the classical work of Guth and Katz [11], and in the numerous follow-up studies of related problems, here too we use the polynomial partitioning method, as pioneered in [11]. The reason why our bounds suffer from the aforementioned handicaps is that we use a partitioning polynomial of (large but) constant degree. (The idea of using constantdegree partitioning polynomials for problems of this kind is due to Solymosi and Tao [25].) When using a polynomial of a larger, non-constant degree, we face the difficult task of bounding incidences between points and curves that are fully contained in the zero set of the polynomial, where the number of curves of this kind can be large, because the polynomial partitioning technique has no control over this value. We remark that for lines we have the classical Cayley Salmon theorem (see, e.g., Guth and Katz [11]), which essentially bounds the number of lines that can be fully contained in an algebraic surface of a given degree, unless the surface is ruled by lines. However, such a property has not been known for more general curves. Nevertheless, Nilov and Skopenkov [17] have recently established such a result involving lines and circles in R 3, and, very recently, Guth and Zahl [12] have done the same for general algebraic curves in three dimensions. Handling these incidences requires heavy-duty machinery from algebraic geometry, and leads to profound new problems in that domain that need to be tackled. In contrast, using a polynomial of constant degree makes this part of the analysis much simpler, as can be seen below, but then handling incidences within the cells of the partition becomes non-trivial, and a naive approach yields a bound that is too large. To handle this part, one uses induction within each cell of the partitioning, and it is this induction process that is responsible for the weaker aspects of the lower-dimensional terms in the resulting bound, as well as the extra m ε factor in the leading term. Nevertheless, with these sacrifices we are able to obtain a general purpose bound that holds for a broad spectrum of instances. It is our hope that this study will motivate further research on this problem that would improve our results along the handicaps mentioned above. Recalling how inaccessible were these kinds of problems prior to Guth and Katz s breakthroughs eight and six years ago, it is quite gratifying that so much new ground can be gained in this area, including the progress made in this paper. Background. Incidence problems have been a major topic in combinatorial and computational geometry for the past thirty years, starting with the aforementioned Szemerédi- Trotter bound [28] back in 1983 (and even earlier). Several techniques, interesting in their own right, have been developed, or adapted, for the analysis of incidences, including the the electronic journal of combinatorics 23(4) (2016), #P4.16 7

210 crossing-lemma technique of Székely [27], and the use of cuttings as a divide-and-conquer mechanism (e.g., see [4]). Connections with range searching and related algorithmic problems in computational geometry have also been noted and exploited, and studies of the Kakeya problem (see, e.g., [29]) indicate the connection between this problem and incidence problems. See Pach and Sharir [19] for a comprehensive (albeit a bit outdated) survey of the topic. The landscape of incidence geometry has dramatically changed in the past eight years, due to the infusion, in two groundbreaking papers by Guth and Katz [10, 11], of new tools and techniques drawn from algebraic geometry. Although their two direct goals have been to obtain a tight upper bound on the number of joints in a set of lines in three dimensions [10], and a near-linear lower bound for the classical distinct distances problem of Erdős [11], the new tools have quickly been recognized as useful for incidence bounds. See [6, 14, 15, 20, 25, 30, 31] for a sample of recent works on incidence problems that use the new algebraic machinery. The present paper continues this line of research, and aims at extending the collection of instances where nontrivial incidence bounds in higher dimensions can be obtained. 2 The three-dimensional case Proof of Theorem 1.5. We fix ε > 0, and prove by induction on m + n that ( ) I(P, C) α 1 m k 3k 2 +ε n 3k 3 k 3k 2 + m 2k 1 +ε n 3k 3 k 1 4k 2 q 4k 2 + α 2 (m + n), (5) where α 1, α 2 are sufficiently large constants, α 1 depends on ε and k (and s), and α 2 depends on k (and s). For the induction basis, the case where m, n are sufficiently small constants can be handled by choosing sufficiently large values of α 1, α 2. Another base case is m = O(n 1/k ). Since the incidence graph, as a subgraph of P C, does not contain K k,s+1 as a subgraph, the Kővári-Sós-Turán theorem (e.g., see [16, Section 4.5]) implies that I(P, C) = O(mn 1 1/k +n), where the constant of proportionality depends on k (and s). When m = O(n 1/k ), this implies the bound I(P, C) = O(n), which is subsumed in (5) if we choose α 2 sufficiently large. We may thus assume that n cm k, for some absolute constant c, and that m and n are at least some sufficiently large constants. Applying the polynomial partitioning technique. We construct an r-partitioning polynomial f for P, for a sufficiently large constant r (depending on ε). That is, as established in Guth and Katz [11], f is of degree O(r 1/3 ) (the constant in the O notation is an absolute constant), and the complement of its zero set Z(f) is partitioned into u = O(r) open connected cells, each containing at most m/r points of P. Denote the (open) cells of the partition as τ 1,..., τ u. For each i = 1,..., u, let C i denote the set of curves of C that intersect τ i and let P i denote the set of points that are contained in τ i. We set m i = P i and n i = C i, for i = 1,..., u, and m = i m i, and notice that m i m/r for each i (and m m). An obvious property (which is a consequence of the electronic journal of combinatorics 23(4) (2016), #P4.16 8

211 Bézout s theorem, see, e.g., [25, Theorem A.2]) is that every curve of C intersects O(r 1/3 ) cells of R 3 \ Z(f). Therefore, i n i bnr 1/3, for a suitable constant b > 1 (that depends on the degree of the curves in C). Using Hölder s inequality, we have i i n 3k 3 3k 2 i n 3k 3 4k 2 i ( ) 3k 3 ( 3k 2 n i 1 i ( ) 3k 3 ( 4k 2 n i 1 i i i i ) 1 3k 2 ) k+1 4k 2 n 3k 3 i b (nr 1 3 b (nr 1 3 i ) 3k 3 3k 2 r 1 3k 2 ) 3k 3 4k 2 r k+1 4k 2 n 3k 3 i = b n 3k 3 3k 2 r k 3k 2, = b n 3k 3 4k 2 r k 2k 1, for another absolute constant b. Combining the above with the induction hypothesis, applied within each cell of the partition, implies ( ) ) (α 1 m k 3k 2 +ε 3k 2 + m k 2k 1 +ε 4k 2 q k 1 4k α 2 (m i + n i ) I(P i, C i ) i i α 1 m k 3k 2 +ε r k 3k 2 +ε α 1 b i n 3k 3 3k 2 i m k 3k 2 +ε n 3k 3 3k 2 r ε + m k + m 2k 1 +ε q k 1 4k 2 2 r k 2k 1 +ε k 2k 1 +ε n 3k 3 k 1 4k 2 q 4k 2 2 r ε i n 3k 3 4k 2 i + i α 2 (m i + n i ) + α 2 ( m + bnr 1/3). ) Our assumption that n = O(m k ) implies that n = O (m k 3k 3 3k 2 n 3k 2 (with an absolute constant of proportionality). Thus, when α 1 is sufficiently large with respect to r, k, and α 2, we have I(P i, C i ) 2α 1 b m k 3k 2 +ε n 3k 3 3k 2 + m k 2k 1 +ε n 3k 3 k 1 4k 2 q 4k α r ε r ε 2 m. i When r is sufficiently large, such that r ε 6b, we have I(P i, C i ) α ( 1 m k 3k 2 +ε n 3k 3 k 3k 2 + m 2k 1 +ε n 3k 3 3 i k 1 4k 2 q 4k 2 2 ) + α 2 m. (6) Incidences on the zero set Z(f). It remains to bound incidences with points that lie on Z(f). Set P 0 := P Z(f) and m 0 = P 0 = m m. Let C 0 denote the set of curves that are fully contained in Z(f), and set C := C \ C 0, n 0 := C 0, and n := C = n n 0. Since every curve of C intersects Z(f) in O(r 1/3 ) points, we have, taking α 1 to be sufficiently large, and arguing as above, I(P 0, C ) = O(nr 1/3 ) α 1 3 m k 3k 2 +ε n 3k 3 3k 2. (7) the electronic journal of combinatorics 23(4) (2016), #P4.16 9

212 Finally, we consider the number of incidences between points of P 0 and curves of C 0. For this, we set c(k, ε) to be the degree of f, which is O(r 1/3 ), and can be taken to be O((6b ) 1/(3ε) ). Then, by the assumption of the theorem, we have C 0 q 2. We consider a generic plane π R 3 and project P 0 and C 0 onto two respective sets P and C on π. Since π is chosen generically, we may assume that no two points of P 0 project to the same point in π, and that no pair of distinct curves in C 0 have overlapping projections in π. Moreover, the projected curves still have k degrees of freedom, in the sense that, given any k points on the projection γ of a curve γ C 0, there are at most s 1 other projected curves that go through all these points. This is argued by lifting each point p back to the point p on γ in R 3, and by exploiting the facts that the original curves have k degrees of freedom, and that, for a sufficiently generic projection, any curve that does not pass through p does not contain any point that projects to p. The number of intersection points between a pair of projected curves may increase but it must remain a constant since these are intersection points between constant-degree algebraic curves with no common components. By applying Theorem 1.3, we obtain I(P 0, C 0 ) = I(P, C ) = O(m k 2k 1 0 q 2k 2 2k m 0 + q 2 ), where the constant of proportionality depends on k (and s). Since q 2 n and m 0 m, we have m k 3k 3 k 1 2k 1 2k 1 2k 1 n 4k 2 q 4k 2 0 q 2k 2 O 2 m k 2. We thus get that I(P 0, C 0 ) is at most ) k 1 4k 2 q 4k n + m 0 α 1 3 m k 3k 3 k 1 2k 1 n 4k 2 q 4k b 2 n + α 2 m 0, (8) ( m k 2k 1 n 3k 3 for sufficiently large α 1 and α 2 ; the constant b 2 comes from Theorem 1.3, and is independent of ε and of the choices for α 1, α 2 made so far. By combining (6), (7), and (8), including the case m = O(n 1/k ), and choosing α 2 sufficiently large, we obtain ( ) I(P, C) α 1 m k 3k 2 +ε n 3k 3 k 3k 2 + m 2k 1 +ε n 3k 3 k 1 4k 2 q 4k 2 + α 2 (m + n). This completes the induction step and thus the proof of the theorem. Example 1: The case of lines. Lines in R 3 have k = 2 degrees of freedom, and we almost get the bound of Guth and Katz in Theorem 1.1. There are three differences that make this derivation somewhat inferior to that in Guth and Katz [11], as detailed in items (i) (iii) in the discussion in the introduction. We also recall the two follow-up studies of point-line incidences in R 3, of Guth [9] and of Sharir and Solomon [23]. Guth s bound suffers from weaknesses (i) and (ii), but avoids (iii), using a fairly sophisticated inductive argument. Sharir and Solomon s bound avoids (i) and (iii), and almost avoids (ii), in a sense that we do not make explicit here. In both cases, considerably more sophisticated machinery is needed to achieve these improvements. Example 2: The case of circles. Circles in R 3 have k = 3 degrees of freedom, and we get the bound ( ) I(P, C) = O m 3/7+ε n 6/7 + m 3/5+ε n 3/5 q 1/5 2 + m + n. 2 the electronic journal of combinatorics 23(4) (2016), #P

213 The leading term is the same as in Sharir et al. [20], but the second term is weaker, because it relies on the general bound of Pach and Sharir (Theorem 1.3), whereas the bound in [20] exploits an improved bound for point-circle incidences, due to Aronov et al. [2], which holds in any dimension. If we plug that bound into the above scheme, we obtain an exact reconstruction of the bound in [20]. In addition, considering the items (i) (iii) discussed earlier, we note: (i) The requirements in [20] about the maximum number of circles on a surface are weaker, and are only for planes and spheres. (ii) The m ε factors are present in both bounds. (iii) Even after the improvement noted above, the bounds still seem to be weak in terms of their dependence on q 2, and improving this aspect, both here and in [20], is a challenging open problem. Theorem 1.5 can easily be restated as bounding the number of rich points. Corollary 2.1 For each ε > 0 there exists a parameter c(k, ε) that depends on k and ε, such that the following holds. Let C be a set of n irreducible algebraic curves of constant degree and with k degrees of freedom (with some multiplicity s) in R 3. Moreover, assume that every surface of degree at most c(k, ε) contains at most q 2 curves of C. Then there exists some constant r 0 (k, ε) depending on ε, k (and s), such that for any r r 0 (k, ε), the number ( of points that are incident ) to at least r curves of C (so-called r-rich points), is n 3/2+ε O r + n3/2+εq 1/2+ε 2 + n, where the constant of proportionality depends on k, s 3k 2 2k 2 +ε r 2k 1 k 1 +ε r and ε. Proof. Denoting by m r the number of r-rich points, the corollary is obtained by combining the upper bound in Theorem 1.5 with the lower bound rm r. 3 Incidences in higher dimensions Proof of Theorem 1.6. Again, we fix ε > 0, and prove, by double induction, where the outer induction is on the dimension d and the inner induction is on m + n, that I(P, C) is at most ( ) d 1 α 1,d m k dk d+1 +ε n dk d dk d+1 + m k jk j+1 +ε n d(j 1)(k 1) (d j)(k 1) (d 1)(jk j+1) q (d 1)(jk j+1) + α 2,d (m + n), (9) j=2 where α 1,d, α 2,d are sufficiently large constants, α 1,d depends on k (and s), ε, d, and the maximum degree of the curves, and α 2,d depends only on k (and s), d, and the maximum degree of the curves. For the outer induction basis, the case d = 2 follows by Theorem 1.3, and the case d = 3 is just Theorem 1.5, proved in the previous section. We assume therefore that the claim holds up to dimension d 1, and prove it in dimension d 4. The base cases of the inner induction (that is, when d is fixed, we induct over m+n) is when m, n are sufficiently small constants, and when m = O(n 1/k ). The bound in (9) can be enforced in the former case by choosing sufficiently large values of α 1,d, α 2,d, and in the latter case exactly as for d = 3, so we may assume, as before, that n cm k for some absolute constant c. j the electronic journal of combinatorics 23(4) (2016), #P

214 Applying the polynomial partitioning technique. The analysis is somewhat repetitive and resembles the one in the previous section, although many details are different; it is given in detail for the convenience of the reader, and in the interest of completeness. Let f be an r-partitioning polynomial, for a sufficiently large constant r. According to the polynomial partitioning theorem [11], we have degf = O(r 1/d ). Denote the (open) cells of the partition as τ 1,..., τ u, where u = O(r). For each i = 1,..., u, let C i denote the set of curves of C that intersect τ i and let P i denote the set of points that are contained in τ i. We set m i = P i, and n i = C i, for i = 1,..., u, and m = i m i, and notice that m i m/r for each i (and m m). Arguing as before, every curve of C intersects at most deg(f) = O(r 1/d ) cells of R d \ Z(f). Therefore, i n i b d nr 1/d, for a suitable constant b d > 1 that depends on d and the degree of the curves. Using Hölder s inequality, we have n i i n dk d dk d+1 i d(j 1)(k 1) (d 1)(jk j+1) i b d b d ( ) nr 1 dk d dk d+1 d r 1 dk d+1 b d n dk d ( ) d(j 1)(k 1) nr 1 (d 1)(jk j+1) d r dk jk+j 1 (d 1)(jk j+1) k dk d+1 r dk d+1, and k (d 1)(jk j+1) r jk j+1, b d n d(j 1)(k 1) for each j = 2,..., d 1, where b d is another constant parameter that depends on d. Combining the above with the induction hypothesis implies that i I(P i, C i ) is at most ( (α 1,d m i α 1,d m r m k dk d+1 +ε i n dk d d 1 dk d+1 i + j=2 k dk d+1 +ε k dk d+1 +ε i k jk j+1 +ε i n dk d d 1 dk d+1 i + j=2 m n d(j 1)(k 1) (d 1)(jk j+1) i q (d j)(k 1) (d 1)(jk j+1) j k jk j+1 +ε (d j)(k 1) (d 1)(jk j+1) qj k jk j+1 +ε r ) n i + α 2,d (m i + n i ) d(j 1)(k 1) (d 1)(jk j+1) i ) + i α 2,d (m i + n i ) α 1,d b d m k dk d+1 +ε n dk d dk d+1 r ε + d 1 j=2 m k jk j+1 +ε n d(j 1)(k 1) (d j)(k 1) (d 1)(jk j+1) q (d 1)(jk j+1) j r ε ( +α 2,d m + b d nr 1/d). ) Since we assume that n = O(m k ), we have n = O (m k dk d dk d+1 n dk d+1, with a constant of proportionality that depends only on d. Thus, when α 1,d is sufficiently large with respect to r, d, and α 2,d, we have (m k dk d+1 +ε n dk d dk d+1 I(P i, C i ) 2α 1,d b i r ε + m k 2k 1 +ε n dk d (k 1)(d 2) (d 1)(2k 1) q (d 1)(2k 1) r ε ) + α 2,d m. the electronic journal of combinatorics 23(4) (2016), #P

215 When r is sufficiently large, such that r ε 6b, the bound is at most ( ) α 1,d d 1 m k dk d+1 +ε n dk d dk d+1 + m k jk j+1 +ε n d(j 1)(k 1) (d j)(k 1) (d 1)(jk j+1) q (d 1)(jk j+1) j + α 2,d m. (10) 3 j=2 Incidences on the zero set Z(f). It remains to bound incidences with points that lie on Z(f). Set P 0 = P Z(f) and m 0 = P 0 = m m. Let C 0 denote the set of curves that are fully contained in Z(f), and set C = C \ C 0, n 0 = C 0, and n = C = n n 0. Since every curve of C intersects Z(f) in O(r 1/d ) points, we have, arguing as above, I(P 0, C ) b d n r 1/d = O(nr 1/d ) α 1,d 3 m k dk d+1 +ε n dk d dk d+1, (11) provided that α 1,d is chosen sufficiently large. Finally, we consider the number of incidences between points of P 0 and curves of C 0. For this, we set c d 1 (k, d, ε) to be the degree of f, which is O(r 1/d ) = O((6b ) 1/(εd) ). Then, by the assumption of the theorem, we have C 0 q d 1. We consider a generic hyperplane H R d and project P 0 and C 0 onto two respective sets P and C on H. Arguing as in the three-dimensional case, we can enforce that I(P 0, C 0 ) = I(P, C ), that the projected curves have k degrees of freedom, and that, for j < d 1, the pairs (q j, c j ) remain unchanged for P and C within H. Applying the induction hypothesis for dimension d 1, and recalling that C 0 q d 1, we obtain ( d 1 I(P 0, C 0 ) = I(P, C ) α 1,d 1 m j=2 k jk j+1 +ε qd 1 (d 1)(j 1)(k 1) (d 2)(jk j+1) q (d j 1)(k 1) (d 2)(jk j+1) j ) + α 2,d 1 (m + n). As is easily verified, Equation (4) with l = d (and q d = n) implies that, for each j, (d 1)(j 1)(k 1) (d 2)(jk j+1) qd 1 q (d j 1)(k 1) (d 2)(jk j+1) j n d(j 1)(k 1) (d j)(k 1) (d 1)(jk j+1) q (d 1)(jk j+1) j. By choosing α 1,d 3α 1,d 1 and α 2,d α 2,d 1, we have that I(P 0, C 0 ) is at most α 1,d 3 ( d 1 m j=2 k jk j+1 +ε n d(j 1)(k 1) (d j)(k 1) (d 1)(jk j+1) q (d 1)(jk j+1) j ) + α 2,d (m + n). (12) By combining (10), (11), and (12), including the case m = O(n 1/k ), and choosing α 2,d sufficiently large, we obtain ( ) I(P, C) α 1,d m k dk d+1 +ε n dk d k dk d+1 + m 2k 1 +ε n dk d (k 1)(d 2) (d 1)(2k 1) q (d 1)(2k 1) + α 2,d (m + n). This completes the induction step and thus the proof of the theorem. Proof of Theorem 1.8. The proof is similar to that of Theorem 1.6, except that, when handling incidences between points and curves on Z(f), we simply project the points and the electronic journal of combinatorics 23(4) (2016), #P

216 curves onto some generic 2-plane, argue that the projected curves also have k degrees of freedom (and degree at most D), and apply the Pach Sharir planar bound, given in Theorem 1.3 to the projected points and curves. Both terms in the bound go through the induction controlled by the polynomial partitioning. This is clear for the leading term, and follows for the second term in much the same way as in the preceding proof. As a consequence of Theorem 1.6, we have: Example: incidences between points and lines in R 4. In the earlier version [21] of Sharir and Solomon s study of point-line incidences in four dimensions, we have obtained the following weaker version of Theorem 1.2. Theorem 3.1 For each ε > 0, there exists an integer c ε, so that the following holds. Let P be a set of m distinct points and L a set of n distinct lines in R 4, and let q, s n be parameters, such that (i) for any polynomial f R[x, y, z, w] of degree c ε, its zero set Z(f) does not contain more than q lines of L, and (ii) no 2-plane contains more than s lines of L. Then, I(P, L) A ε ( m 2/5+ε n 4/5 + m 1/2+ε n 2/3 q 1/12 + m 2/3+ε n 4/9 s 2/9) + A(m + n), where A ε depends on ε, and A is an absolute constant. This result follows from our main Theorem 1.6, if we impose Equation (4) on q 2 = s, q 3 = q, and n, which in this case is equivalent to s q n and q9 < s. This illustrates n 8 how the general theory developed in this paper extends similar results obtained earlier for isolated instances. Nevertheless, as already mentioned earlier, the bound for lines in R 4 has been improved in Theorem 1.2 of [22], in its lower-dimensional terms. Discussion. We first notice that similarly to the three-dimensional case, Theorem 1.6 implies an upper bound on the number of k-rich points in d dimensions (see Corollary 2.1 in three dimensions), and the proof thereof applies verbatim, with the appropriate modifications of the various exponents that now depend also on d. We leave it to the reader to work out the precise (and, admittedly, somewhat cumbersome) statement. Second, we note that Theorems 1.5 and 1.6 have several weaknesses. The obvious ones are the items (i) (iii) discussed in the introduction. Another, less obvious weakness, has to do with the way in which the q j -dependent terms in the bounds are derived. Specifically, these terms facilitate the induction step, when the constraining parameters q j are passed unchanged to the inductive subproblems. Informally, since the overall number of lines in a subproblem goes down, one would expect the various parameters q j to decrease too, but so far we do not have a clean mechanism for doing so. This weakness is manifested, e.g., in Corollary 2.1, where one would like to replace the second term by one with a smaller exponent of n and a larger one of q = q 2. Specifically, for lines in R 3, one would like to get a term close to O(nq 2 /k 3 ). This would yield O(n 3/2 /k 3 ) for the important special case q 2 = O(n 1/2 ) considered in [11]; the present bound is weaker. A final remark concerns the relationships between the parameters q j, as set forth in Equation (4). These conditions are forced upon us by the induction process. As noted above, for incidences between points and lines in R 4, the bound derived in our the electronic journal of combinatorics 23(4) (2016), #P

217 main Theorem 1.6 is (asymptotically) the same as that of the main result of Sharir and Solomon in [21]. The difference is that there, no restrictions on the q j are imposed. The proof in [21] is facilitated by the so called second partitioning polynomial (see [14, 21]). Recently, Basu and Sombra [3] proved the existence of a third partitioning polynomial (see [3, Theorem 3.1]), and conjectured the existence of a k-th partitioning polynomial for general k > 3 (see [3, Conjecture 3.4] for an exact formulation); for completeness we refer also to [8, Theorem 4.1], where a weaker version of this conjecture is proved. Building upon the work of [3], the proof of Sharir and Solomon [22] is likely to extend and yield the same bound as in our main Theorem 1.6, for the more general case of incidences between points and bounded degree algebraic curves in dimensions at most five, and, if Conjecture 3.4 of [3] holds, in every dimension, without any conditions on the q j. References [1] P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky, Lenses in arrangements of pseudocircles and their applications, J. ACM 51 (2004), [2] B. Aronov, V. Koltun, and M. Sharir, Incidences between points and circles in three and higher dimensions, Discrete Comput. Geom. 33 (2005), [3] S. Basu and M. Sombra, Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions, Discrete Comput. Geom. 55 (2016), [4] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), [5] Z. Dvir and S. Gopi, On the number of rich lines in truly high dimensional sets, Proc. 31st Annu. Sympos. Comput. Geom., 2015, [6] G. Elekes, H. Kaplan and M. Sharir, On lines, joints, and incidences in three dimensions, J. Combinat. Theory, Ser. A 118 (2011), Also in arxiv: [7] P. Erdős, On sets of distances of n points, Amer. Math. Monthly 53 (1946), [8] J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl, A semi-algebraic version of Zarankiewicz s problem, J. European Math. Soc., to appear. [9] L. Guth, Distinct distance estimates and low-degree polynomial partitioning, Discrete Comput. Geom. 53 (2015), Also arxiv: [10] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem, Advances Math. 225 (2010), Also arxiv: [11] L. Guth and N. H. Katz, On the Erdős distinct distances problem in the plane, Annals Math. 181 (2015), Also arxiv: [12] L. Guth and J. Zahl, Algebraic curves, rich points, and doubly ruled surfaces, arxiv: [13] M. Hablicsek and Z. Scherr, On the number of rich lines in high dimensional real vector spaces, Discrete Comput. Geom. 55 (2016), the electronic journal of combinatorics 23(4) (2016), #P

218 [14] H. Kaplan, J. Matoušek, Z. Safernová and M. Sharir, Unit distances in three dimensions, Combinat. Probab. Comput. 21 (2012), Also arxiv: [15] H. Kaplan, J. Matoušek and M. Sharir, Simple proofs of classical theorems in discrete geometry via the Guth Katz polynomial partitioning technique, Discrete Comput. Geom. 48 (2012), Also arxiv: [16] J. Matoušek, Lectures on Discrete Geometry, Springer Verlag, Heidelberg, [17] F. Nilov and M. Skopenkov, A surface containing a line and a circle through each point is a quadric, Geometria Dedicata (2013), [18] J. Pach and M. Sharir, On the number of incidences between points and curves, Combinat. Probab. Comput. 7 (1998), [19] J. Pach and M. Sharir, Geometric incidences, in Towards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics, Vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp [20] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between points and circles, Combinat. Probab. Comput. 24 (2015), Also in Proc. 29th ACM Symp. on Computational Geometry (2013), , and arxiv: [21] M. Sharir and N. Solomon, Incidences between points and lines in four dimensions, Proc. 30th ACM Sympos. on Computational Geometry (2014), [22] M. Sharir and N. Solomon, Incidences between points and lines in R 4, Proc. 56th IEEE Symp. on Foundations of Computer Science (2015), Also arxiv: [23] M. Sharir and N. Solomon, Incidences between points and lines in three dimensions, in Intuitive Geometry (J. Pach, Ed.), to appear. Also in Proc. 31st ACM Sympos. on Computational Geometry (2015), , and arxiv: [24] M. Sharir and J. Zahl, Cutting algebraic curves into pseudo-segments and applications, arxiv: [25] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput. Geom. 48 (2012), [26] J. Spencer, E. Szemerédi and W.T. Trotter, Unit distances in the Euclidean plane, In: Graph Theory and Combinatorics (B. Bollobás, ed.), Academic Press, New York, 1984, [27] L. Székely, Crossing numbers and hard Erdős problems in discrete geometry, Combinat. Probab. Comput. 6 (1997), [28] E. Szemerédi and W.T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), [29] T. Tao, From rotating needles to stability of waves: Emerging connections between combinatorics, analysis, and PDE, Notices AMS 48(3) (2001), [30] J. Zahl, An improved bound on the number of point-surface incidences in three dimensions, Contrib. Discrete Math. 8(1) (2013), Also arxiv: [31] J. Zahl, A Szemerédi-Trotter type theorem in R 4, Discrete Comput. Geom. 54 (2015), Also arxiv: the electronic journal of combinatorics 23(4) (2016), #P

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220 8 Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances 211

221 Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances Micha Sharir Noam Solomon August 13, 2017 Abstract We study a wide spectrum of incidence problems involving points and curves or points and surfaces in R 3. The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [38, 39], requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies [40], by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for numerous incidence problems in R 3. In broad terms, we consider two kinds of problems, those involving points and constant-degree algebraic curves, and those involving points and constant-degree algebraic surfaces. In some variants we assume that the points lie on some fixed constant-degree algebraic variety, and in others we consider arbitrary sets of points in 3-space. The case of points and curves has been considered in several previous studies, starting with Guth and Katz s work on points and lines [39]. Our results, which are based on a recent work of Guth and Zahl [40] concerning surfaces that are doubly ruled by curves, provide a grand generalization of all previous results. We reconstruct the bound for points and lines, and improve, in certain signifcant ways, recent bounds involving points and circles (in [54]), and points and arbitrary constant-degree algebraic curves (in [53]). While in these latter instances the bounds are not known (and are strongly suspected not) to be tight, our bounds are, in a certain sense, the best that can be obtained with this approach, given the current state of knowledge. In the case of points and surfaces, the incidence graph between them can contain large complete bipartite graphs, each involving points on some curve and surfaces containing this curve (unlike earlier studies, we do not rule out this possibility, which makes our approach more general). Our bounds estimate the total size of the vertex sets in such a complete bipartite graph decomposition of the incidence graph. In favorable cases, our bounds translate into actual incidence bounds. Overall, here too our results provide a grand generalization of most of the previous studies of (special instances of) this problem. As an application of our point-curve incidence bound, we consider the problem of bounding the number of similar triangles spanned by a set of n points in R 3. We obtain the bound O(n 15/7 ), which improves the bound of Agarwal et al. [1]. As applications of our point-surface incidence bounds, we consider the problems of distinct and repeated distances determined by a set of n points in R 3, two of the most celebrated open problems in combinatorial geometry. We obtain new and improved bounds for two special cases, one in which the points lie on some algebraic variety of constant degree, and one involving incidences between pairs in P 1 P 2, where P 1 is contained in a variety and P 2 is arbitrary. Work on this paper by Noam Solomon and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S. Israel Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), by the Blavatnik Research Fund in Computer Science at Tel Aviv University and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. A preliminary version of the paper has appeared in Proc. 28th ACM-SIAM Symposium on Discrete Algorithms (2017), School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. michas@tau.ac.il School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. noam.solom@gmail.com 1

222 Keywords. Combinatorial geometry, incidences, the polynomial method, algebraic geometry, distinct distances, repeated distances. 1 Introduction 1.1 The setups Incidences between points and curves in three dimensions. Let P be a set of m points and C a set of n irreducible algebraic curves of constant degree in R 3. We consider the problem of obtaining sharp incidence bounds between the points of P and the curves of C. This is a major topic in incidence geometry since the groundbreaking work of Guth and Katz [39] on point-line incidences in R 3, with many follow-up studies, some of which are reviewed below. Building on the recent work of Guth and Zahl [40], which bounds the number of 2-rich points determined by a set of bounded-degree algebraic curves in R 3 (i.e., points incident to at least two of the given curves), we are able to generalize Guth and Katz s pointline incidence bound to a general bound on the number of incidences between points and bounded-degree irreducible algebraic curves that satisfy certain natural assumptions, discussed in detail below. Incidences between points and surfaces in three dimensions. Let P be a set of m points, and S a set of n two-dimensonal algebraic varieties of constant maximum degree in R 3. Here too we impose certain natural assumptions on the surfaces in S, discussed in detail below. Let G(P, S) P S denote the incidence graph of P and S; its edges connect all pairs (p, σ) P S such that p is incident to σ. In general, I(P, S) := G(P, S) might be as large as the maximum possible value mn, by placing all the points of P on a suitable curve, and make all the surfaces of S contain that curve, 1 in which case G(P, S) = P S. The bound that we are going to obtain will of course acknowledge this possibility, and will in fact bypass it altogether. Concretely, rather than bounding I(P, S), our basic approach will represent G(P, S) as a union of complete bipartite subgraphs ( ) Pγ S γ, and of a leftover subgraph G 0 (P, S) (which, in certain cases, might be empty), and derive an upper bound for the overall size of their vertex sets, namely, a bound on J(P, S) := γ Γ 0 ( Pγ + S γ ), γ Γ 0 where the decomposition is over a set Γ 0 of constant-degree algebraic curves γ so that, for each γ Γ 0, P γ = P γ and S γ is the set of the surfaces of S that contain γ. (In some cases we will derive different bounds on γ P γ and on γ S γ.) For the residual subgraph G 0 (P, S), we derive a sharp bound on the actual number of incidences that it encodes (namely, the number of its edges). This generalizes previous results in which one had to require that G(P, S) does not contain some fixed-size complete bipartite graph, or (only for spheres or planes) that the surfaces in S be non-degenerate ([5, 26]; see below). Incidences between points on a variety and surfaces. An interesting special case is where P is contained in some two-dimensional algebraic variety (surface) V of constant degree. Besides being, as we believe, a problem of independent interest, it arises as a key subproblem in our analysis of the general case discussed above. 1 This situation can arise in many instances, for example in the case of planes (where many of them can intersect in a common line), or spheres (where many can intersect in a common circle), but there are also many cases where this is impossible. In this latter situation, which we do not yet know how to characterize in a simple and general manner, our analysis becomes sharper see below. 2

223 We assume that the surfaces of S are taken from an s-dimensional family of surfaces, meaning that each of them can be represented by a constant number of real parameters (e.g., by the coefficients of the monomials of the polynomial whose zero set is the surface), so that, in this parametric space, the points representing the surfaces of S lie in an s-dimensional algebraic variety F of some constant degree (to which we refer as the complexity of F). This assumption, which holds in practically all applications, extends, in an obvious manner, to lower-dimensional varieties (e.g., curves) and to higher dimensions; see Sharir and Zahl [60] for a more thorough study of this notion. 1.2 Background Points and curves, the planar case. The case of incidences between points and curves has a rich history, starting with the aforementioned case of points and lines in the plane [23, 65, 66], where the worstcase tight bound on the number of incidences is Θ(m 2/3 n 2/3 + m + n), where m is the number of points and n is the number of lines. Still in the plane, Pach and Sharir [49] extended this bound to incidence bounds between points and curves with k degrees of freedom. These are curves with the property that, for each set of k points, there are only µ = O(1) curves that pass through all of them, and each pair of curves intersect in at most µ points; µ is called the multiplicity (of the degrees of freedom). Theorem 1.1 (Pach and Sharir [49]). Let P be a set of m points in R 2 and let C be a set of n bounded-degree algebraic curves in R 2 with k degrees of freedom and with multiplicity µ. Then ) I(P, C) = O (m k 2k 2 2k 1 n 2k 1 + m + n, where the constant of proportionality depends on k and µ. Remark. The result of Pach and Sharir holds for more general families of curves, not necessarily algebraic, but, since algebraicity will be assumed in higher dimensions, we assume it also in the plane. Except for the case k = 2 (lines have two degrees of freedom), the bound is not known, and strongly suspected not to be tight in the worst case. Indeed, in a series of papers during the 2000 s [2, 9, 46], an improved bound has been obtained for incidences with circles, parabolas, or other families of curves with certain properties (see [2] for the precise formulation). Specifically, for a set P of m points and a set C of n circles, or parabolas, or similar curves [2], we have I(P, C) = O(m 2/3 n 2/3 + m 6/11 n 9/11 log 2/11 (m 3 /n) + m + n). (1) Some further (slightly) improved bounds, over the bound in Theorem 1.1, for more general families of curves in the plane have been obtained by Chan [18, 19] and by Bien [12]. They are, however, considerably weaker than the bound in (1). Recently, Sharir and Zahl [60] have considered general families of constant-degree algebraic curves in the plane that belong to an s-dimensional family of curves. Similarly to the case of surfaces, discussed above, this means that each curve in that family can be represented by a constant number of real parameters, so that, in this parametric space, the points representing the curves lie in an s-dimensional algebraic variety F of some constant degree (to which we refer, as above, as the complexity of F). See [60] for more details. Theorem 1.2 (Sharir and Zahl [60]). Let C be a set of n algebraic plane curves that belong to an s- dimensional family F of curves of maximum constant degree E, no two of which share a common irreducible component, and let P be a set of m points in the plane. Then, for any ε > 0, the number I(P, C) of incidences between the points of P and the curves of C satisfies ) I(P, C) = O (m 2s 5s 6 5s 4 n 5s 4 +ε + m 2/3 n 2/3 + m + n, where the constant of proportionality depends on ε, s, E, and the complexity of the family F. 3

224 Except for the factor O(n ε ), this is a significant improvement over the bound in Theorem 1.1 (for s 3), in cases where the assumptions in Theorem 1.2 imply (as they often do) that C has k = s degrees of freedom. Concretely, when k = s, we obtain an improvement, except for the factor n ε, for the entire meaningful range n 1/s m n 2, in which the bound is superlinear. The factor n ε makes the bound in [60] slightly weaker only when m is close to the lower end n 1/s of that range. Note also that for circles (where s = 3), the bound in Theorem 1.2 nearly coincides with the slightly more refined bound (1). Incidences with curves in three dimensions. The seminal work of Guth and Katz [39] establishes the sharper bound O(m 1/2 n 3/4 + m 2/3 n 1/3 q 1/3 + m + n) on the number of incidences between m points and n lines in R 3, provided that no plane contains more than q of the given lines. This has lead to many recent works on incidences between points and lines or other curves in three and higher dimensions; see [17, 40, 53, 54, 55, 59] for a sample of these results. Most relevant to our present study are the works of Sharir, Sheffer, and Solomon [53] on incidences between points and curves in any dimension, the work of Sharir, Sheffer, and Zahl [54] on incidences between points and circles in three dimensions, and the work of Sharir and Solomon [55] on incidences between points and lines in four dimensions, as well as several other studies of point-line incidences by the authors [56, 59]. Of particular significance is the recent work of Guth and Zahl [40] on the number of 2-rich points in a collection of curves, namely, points incident to at least two of the given curves. For the case of lines, Guth and Katz [39] have shown that the number of such points is O(n 3/2 ), when no plane or regulus contains more than O(n 1/2 ) lines. Guth and Zahl obtain the same asymptotic bound for general algebraic curves, under analogous (but stricter) restrictive assumptions. The new bounds that we will derive require the extension to three dimensions of the notions of having k degrees of freedom and of being an s-dimensional family of curves. The definitions of these concepts, as given above for the planar case, extend, more or less verbatim, to three (or higher) dimensions, but, even in typical situations, these two concepts do not coincide anymore. For example, lines in three dimensions have two degrees of freedom, but they form a 4-dimensional family of curves (this is the number of parameters needed to specify a line in R 3 ). Points and surfaces. Many of the earlier works on point-surface incidences have only considered special classes of surfaces, most notably planes and spheres (see below). The case of more general surfaces has barely been considered, till the recent work of Zahl [69], who has studied the general case of incidences between m points and n bounded-degree algebraic surfaces in R 3 that have k degrees of freedom. More precisely, in analogy with the case of curves, one needs to assume that for any k points there are at most µ = O(1) of the given surfaces that pass through all of them. Zahl s bound is O(m 2k 3k 3 3k 1 n 3k 1 + m + n), with the constant of proportionality depending on k, µ, and the maximum degree of the surfaces. By Bézout s theorem, if we require every triple of the given surfaces to have finite intersection, the number of intersection points would be at most E 3, where E is the degree of the surfaces. In particular, E points would then have at most two of the given surfaces passing through all of them. In many instances, though, the actual number of degrees of freedom can be shown to be much smaller. Zahl s bound was later generalized by Basu and Sombra [11] to incidences between points and boundeddegree hypersurfaces in R 4 satisfying certain analogous conditions. Points and planes. Although we will not specifically address this special case, we refer the reader to the earlier works on this problem, going back to Edelsbrunner, Guibas and Sharir [25]. More recently, Apfelbaum and Sharir [4] (see also Brass and Knauer [14] and Elekes and Tóth [26]) have shown that if the incidence graph, for a set P of m points and a set H of n planes, does not contain a copy of K r,s, for constant parameters r and s, then I(P, H) = O(m 3/4 n 3/4 + m + n). In more generality, Apfelbaum and 4

225 Sharir [4] have shown that if I = I(P, H) is significantly larger than this bound, then G(P, H) must contain a large complete bipartite subgraph P H, such that P H = Ω(I 2 /(mn)) O(m + n). Moreover, as also shown in [4] (slightly improving a similar result of Brass and Knauer [14]), G(P, H) can be expressed as the union of complete bipartite graphs P i H i so that i ( P i + H i ) = O(m 3/4 n 3/4 + m + n). (This is a specialization to the case d = 3 of a similar result of [4, 14] in any dimension d, and it concurs with the approach followed in this paper for more general scenarios.) Recently, Solomon and Sharir [57] improved this bound substantially when all the points of P lie on a constant-degree variety V. Points and spheres. Earlier works on the special case of point-sphere incidences have considered the general setup, where the points of P are arbitrarily placed in R 3. Initial partial results go back to Chung [22] and to Clarkson et al. [23], and continue with the work of Aronov et al. [7]. Later, Agarwal et al. [1] have bounded the number of non-degenerate spheres with respect to a given point set; their bound was subsequently improved by Apfelbaum and Sharir [5]. 2 The aforementioned recent work of Zahl [69] can be applied in the case of spheres if one assumes that no three, or any larger but constant number, of the spheres intersect in a common circle. In this case the family has k = 3 degrees of freedom any three points determine a unique circle that passes through all of them, and, by assumption, only O(1) spheres contain that circle. Zahl s bound then becomes O(m 3/4 n 3/4 +m+n). In particular, this bound holds for congruent (unit) spheres (where three such spheres can never contain a common circle). The case of incidences with unit spheres have also been studied in Kaplan et al. [43], with the same upper bound; see also [58]. If many spheres of the family can intersect in a common circle, the bound does no longer hold. The only earlier work that handled this situation is by Apfelbaum and Sharir [4], where it was assumed that the given spheres are non-degenerate. In this case the bound obtained in [4] is O(m 8/11 n 9/11 + m + n). Interestingly, this is also the bound that Zahl s result would have yielded if the sphere had k = 4 degrees of freedom, which however they only almost have : four generic points determine a unique sphere that passes through all of them, but four co-circular points determine an infinity of such spheres. Distinct and repeated distances in three dimensions. The case of spheres is of particular interest, because it arises, in a standard and natural manner, in the analysis of distinct and repeated distances determined by n points in three dimensions (see Section 6, where we use these well-known reductions in our analysis). After Guth and Katz s almost complete solution of the number of distinct distances in the plane [39], the three-dimensional case has moved to the research forefront. The prevailing conjecture is that the lower bound is Ω(n 2/3 ) (the best possible in the worst case), but the current record, due to Solymosi and Vu [63], is still far smaller 3 (close to Ω(n 3/5 )), and the problem seems much harder than its two-dimensional counterpart. Obtaining lower bounds for distinct distances using circles or spheres has in general been suboptimal when compared with more effective methods (such as in [39]), but here we use it effectively to obtain new lower bounds (larger than Ω(n 2/3 )) when the points lie on a variety of fixed degree. The status of the case of repeated distances is also far from being satisfactory. The planar case is stuck with the upper bound O(n 4/3 ) of Spencer et al. [64] from the 1980 s. This bound also holds for points on the 2-sphere, and there it is tight in the worst case (when the repeated distance is 1, say, and the radius of the sphere is 1/ 2) [29], but it is strongly believed that in the plane the correct bound is close to linear. In three dimensions, the aforementioned bound of [43, 69] immediately implies the upper bound O(n 3/2 ) on the number of repreated distances (a slight improvement over the earlier bound of Clarkson et 2 Given a finite point set P R 3 and a constant 0 < η < 1, a sphere σ R 3 is called η-degenerate (with respect to P ), if there exists a circle c σ such that c P η σ P. 3 This follows by substituting the new lower bound Ω(n/ log n) of Guth and Katz for distinct distances in the plane, in the recursive analysis of [63]. 5

226 al. [23]), but the best known lower bound is only Ω(n 4/3 log log n) [28]. 1.3 Our results Incidences with curves. We first consider the problem of incidences between points and algebraic curves. Before we state our results, we discuss three notions that are used in these statements. These are the notions of k degrees of freedom (already mentioned above), of constructibility, and of surfaces infinitely ruled by curves. k degrees of freedom. Let C 0 be an infinite family of irreducible algebraic curves of constant degree E in R 3. Formally, in complete analogy with the planar case, we say that C 0 has k degrees of freedom with multiplicity µ, where k and µ are constants, if (i) for every tuple of k points in R 3 there are at most µ curves of C 0 that are incident to all k points, and (ii) every pair of curves of C 0 intersect in at most µ points. As in [49], the bounds that we derive depend more significantly on k than on µ see below. We remark that the notion of k degrees of freedom gets more involved for surfaces, and raises several annoying technical issues. For example, how many points does it take to define, say, a sphere (up to a fixed multiplicity)? As already observed earlier, four generic points do the job (they define a unique sphere passing through all four of them), but four co-circular points do not. While it seems possible to come up with some sort of working definition, we bypass this issue in this paper, by defining this notion, for a family F of surfaces, only with respect to a given surface V, by saying that F has k degrees of freedom with respect to V if the family of the irreducible components of the curves {σ V σ F}, counted without multiplicity, has k degrees of freedom, in the sense just defined. In the case of spheres, for example, this definition gives, as is easily checked, four degrees of freedom when V is neither a plane nor a sphere, but only three when V is a plane or a sphere. Constructibility. In the statements of the following theorems, we also assume that C 0 is a constructible family of curves. This notion generalizes the notion of being algebraic, and is discussed in detail in Guth and Zahl [40]. Informally, a set Y C d is constructible if it is a Boolean combination of algebraic sets. The formal definition goes as follows (see, e.g., Harris [42, Lecture 3]). For z C, define v(0) = 0 and v(z) = 1 for z 0. Then Y C d, for some fixed d, is a constructible set if there exist a finite set of polynomials f j : C d C, for j = 1,..., J Y, and a subset B Y {0, 1} J Y, so that x Y if and only if (v(f 1 (x)),..., v(f JY (x))) B Y. When we apply this definition to a set of curves, we think of them as points in some parametric (complex) d-space, where d is the number of parameters needed to specify a curve. When J Y = 1 we get all the algebraic hypersurfaces (that admit the implied d-dimensional representation) and their complements. An s-dimensional family of curves, for s < d, is obtained by taking J Y = d s and B Y = {0} J Y. In doing so, the curves that we obtain are complete intersections. Following Guth and Zahl (see also a comment to that effect in the appendix), this involves no loss of generality, because every curve is contained in a curve that is a complete intersection. In what follows, when we talk about constructible sets, we implicitly assume that the ambient dimension d is constant. The constructible sets form a Boolean algebra. This means that finite unions and intersections of constructible sets are constructible, and the complement of a constructible set is constructible. Another fundamental property of constructible sets is that, over C, the projection of a constructible set is constructible; this is known as Chevalley s theorem (see Harris [42, Theorem 3.16] and Guth and Zahl [40, Theorem 2.3]). If Y is a constructible set, we define the complexity of Y to be min(deg f deg f JY ), where the minimum is taken over all representations of Y, as described above. As just observed, constructibility of a family C 0 of curves extends the notion of C 0 being s-dimensional. One of the main 6

227 motivations for using the notion of constructible sets (rather than just s-dimensionality) is the fact, established by Guth and Zahl [40, Proposition 3.3], that the set C 3,E of irreducible curves of degree at most E in complex 3-dimensional space (either affine or projective) is a constructible set of constant complexity that depends only on E. Moreover, Theorem 1.13, one of the central technical tools that we use in our analysis (see below for its statement and proof), holds for constructible families of curves. The connection between degrees of freedom and constructibility/dimensionality. Loosely speaking, in the plane the number of degrees of freedom and the dimensionality of a family of curves tend to be equal. In three dimensions the situation is different. This is because the constraint that a curve γ passes through a point p imposes two equations on the parameters defining γ. We therefore expect the number of degrees of freedom to be half the dimensionality. A few instances where this is indeed the case are: (i) Lines in three dimensions have two degrees of freedom, and they form a 4-dimensional family of curves (this is the number of parameters needed to specify a line in R 3 ). (ii) Circles in three dimensions have three degrees of freedom, and they form a 6-dimensional family of curves (e.g., one needs three parameters to specify the plane containing the circle, two additional parameters to specify its center, and a sixth parameter for its radius). (iii) Ellipses have five degrees of freedom, but they form an 8-dimensional family of curves, as is easily checked. (This discrepancy (for ellipses) is explained by noting that four points are not sufficient to define the ellipse because the first three determine the plane containing it, so the fourth point, if at all coplanar with the first three, only imposes one constraint on the parameters of the ellipse.) Remark. The definition of constructibility is given over the complex field C. This is in accordance with most of the basic algebraic geometry tools, which have been developed over the complex field. Some care has to be exercised when applying them over the reals. For example, Theorem 1.13, one of the central technical tools that we use in our analysis, as well as the results of Guth and Zahl [40], apply over the complex field, but not over the reals. On the other hand, when we apply the partitioning method of [39] (as in the proofs of Theorems 1.4) and 1.12 or when we use Theorem 1.2, we (have to) work over the reals. It is a fairly standard practice in algebraic geometry that handles a real algebraic variety V, defined by real polynomials, by considering its complex counterpart V C, namely the set of complex points at which the polynomials defining V vanish. The rich toolbox that complex algebraic geometry has developed allows one to derive various properties of V C, which, with some care, can usually be transported back to the real variety V. This issue arises time and again in this paper. Roughly speaking, we approach it as follows. We apply the polynomial partitioning technique to the given sets of points and of curves or surfaces, in the original real (affine) space, as we should. Within the cells of the partitioning we then apply some field-independent argument, based either on induction or on some ad-hoc combinatorial argument. Then we need to treat points that lie on the zero set of the partitioning polynomial. We can then switch to the complex field, when it suits our purpose, noting that this step preserves all the real incidences; at worst, it might add additional incidences involving the non-real portions of the variety and of the curves or surfaces. Hence, the bounds that we obtain for this case transport, more or less verbatim, to the real case too. Surfaces infinitely ruled by curves. Back in three dimensions, a surface V is (singly, doubly, or infinitely) ruled by some family Γ of curves of degree at most E, if each point p V is incident to (at least one, at least two, or infinitely many) curves of Γ that are fully contained in V. The connection between ruled surface theory and incidence geometry goes back to the pioneering work of Guth and Katz [39] and shows up in many subsequent works. See Guth s recent survey [36] and recent book [37], and Kollár [44] for details. In most of the previous works, only singly-ruled and doubly-ruled surfaces have been considered. Looking at infinitely-ruled surfaces adds a powerful ingredient to the toolbox, as will be demonstrated in this 7

228 paper. We recall that the only surfaces that are infinitely ruled by lines are planes (see, e.g., Fuchs and Tabachnikov [31, Corollary 16.2]), and that the only surfaces that are infinitely ruled by circles are spheres and planes (see, e.g., Lubbes [45, Theorem 3] and Schicho [52]; see also Skopenkov and Krasauskas [62] for recent work on celestials, namely surfaces doubly ruled by circles, and Nilov and Skopenkov [48], proving that a surface that is ruled by a line and a circle through each point is a quadric). It should be noted that, in general, for this definition to make sense, it is important to require that the degree E of the ruling curves be much smaller than deg(v ). Otherwise, every variety V is infinitely ruled by, say, the curves V h, for hyperplanes h, having the same degree as V. A challenging open problem is to characterize all the surfaces that are infinitely ruled by algebraic curves of degree at most E (or by certain special classes thereof). However, the following result of Guth and Zahl provides a useful sufficient condition for this property to hold. Theorem 1.3 (Guth and Zahl [40]). Let V be an irreducible surface, and suppose that it is doubly ruled by curves of degree at most E. Then deg(v ) 100E 2. In particular, an irreducible surface that is infinitely ruled by curves of degree at most E is doubly ruled by these curves, so its degree is at most 100E 2. Therefore, if V is irreducible of degree D larger than this bound, V cannot be infinitely ruled by curves of degree at most E. This leaves a gray zone, in which the degree of V is between E and 100E 2. We would like to conjecture that in fact no irreducible variety with degree in this range is infinitely ruled by degree-e curves. Being unable to establish this conjecture, we leave it as a challenging open problem for further research. Finally, we remark that the notion of surfaces infinitely ruled by curves also plays a crucial role in one of our results on point-surface incidences (see Theorem 1.8). Our results: points and curves. We can now state our main results on point-curve incidences. Theorem 1.4 (Curves in R 3 ). Let P be a set of m points and C a set of n irreducible algebraic curves of constant degree E, taken from a constructible family C 0, of constant complexity, with k degrees of freedom (and some multiplicity µ) in R 3, such that no surface that is infinitely ruled by curves of C 0 contains more than q curves of C, for a parameter q < n. Then ) I(P, C) = O (m k 3k 3 k k 1 k 1 3k 2 n 3k 2 + m 2k 1 n 2k 1 q 2k 1 + m + n, (2) where the constant of proportionality depends on k, µ, E, and the complexity of the family C 0. Remarks. (1) In certain favorable situations, such as in the cases of lines or circles, discussed above, the surfaces that are infinitely ruled by curves of C 0 have a simple characterization. In such cases the theorem has a stronger flavor, as its assumption on the maximum number of curves on a surface has to be made only for this concrete kind of surfaces. For example, as already noted, for lines (resp., circles) we only need to require that no plane (resp., no plane or sphere) contains more than q of the curves. In general, as mentioned, characterizing infinitely-ruled surfaces by a specific family of curves is a difficult task. Nevertheless, we can overcome this issue by replacing the assumption in the theorem by a more restrictive one, requiring that no surface that is infinitely ruled by curves of degree at most E contain more than q curves of C. By Theorem 1.3, any infinitely ruled surface of this kind must be of degree at most 100E 2. Hence, an even simpler (albeit weaker) formulation of the theorem is to require that no surface of degree at most 100E 2 contains more than q curves of C. This can indeed be much weaker: In the case of circles, say, instead of making this requirement only for planes and spheres, we now have to make it for every surface of degree at most

229 (2) In several recent works (see [34, 53, 54]), the assumption in the theorem is replaced by a much more restrictive assumption, that no surface of degree at most c ε contains more than q given curves, where c ε is a constant that depends on another prespecified parameter ε > 0 (where ε appears in the exponents in the resulting incidence bound), and is typically very large (and increases as ε becomes smaller). Getting rid of such an ε-dependent constant (and of the ε in the exponent) is a significant feature of Theorem 1.4. (3) Theorem 1.4 generalizes the incidence bound of Guth and Katz [39], obtained for the case of lines. In this case, lines have k = 2 degrees of freedom, they certainly form a constructible (in fact, a 4-dimensional) family of curves, and, as just noted, planes are the only surfaces in R 3 that are infinitely ruled by lines. Thus, in this special case, both the assumptions and the bound in Theorem 1.4 are identical to those in Guth and Katz [39]. That is, if no plane contains more than q input lines, the number of incidences is O(m 1/2 n 3/4 + m 2/3 n 1/3 q 1/3 + m + n). Improving the bound. The bound in Theorem 1.4 can be further improved, if we also throw into the analysis the dimensionality s of the family C 0. Actually, as will follow from the proof, the dimensionality that will be used is only that of any subset of C 0 whose members are fully contained in some variety that is infinitely ruled by curves of C 0. As just noted, such a variety must be of constant degree (at most 100E 2, or smaller as in the cases of lines and circles), and the additional constraint that the curves be contained in the variety can typically be expected to reduce the dimensionality of the family. For example, if C 0 is the collection of all circles in R 3, then, since the only surfaces that are infinitely ruled by circles are spheres and planes, the subfamily of all circles that are contained in some sphere or plane is only 3-dimensional (as opposed to the entire C 0, which is 6-dimensional). We capture this setup by saying that C 0 is a family of reduced dimension s if, for each surface V that is infinitely ruled by curves of C 0, the subfamily of the curves of C 0 that are fully contained in V is s-dimensional. In this case we obtain the following variant of Theorem 1.4. Theorem 1.5 (Curves in R 3 ). Let P be a set of m points and C a set of n irreducible algebraic curves of constant degree E, taken from a constructible family C 0 with k degrees of freedom (and some multiplicity µ) in R 3, such that no surface that is infinitely ruled by curves of C 0 contains more than q of the curves of C, and assume further that C 0 is of reduced dimension s. Then ) ) I(P, C) = O (m k 3k 3 3k 2 n 3k 2 + O ε (m 2/3 n 1/3 q 1/3 + m 2s 3s 4 2s 2 5s 4 n 5s 4 q 5s 4 +ε + m + n, (3) for any ε > 0, where the first constant of proportionality depends on k, µ, s, E, and the maximum complexity of any subfamily of C 0 consisting of curves that are fully contained in some surface that is infinitely ruled by curves of C 0, and the second constant also depends on ε. Remarks. (1) Theorem 1.5 is an improvement of Theorem 1.4 when s k and m > n 1/k, in cases where q is sufficiently large so as to make the second term in (2) dominate the first term; for smaller values of m the bound is always linear. This is true except for the term q ε, which affects the bound only when m is very close to n 1/k (when s = k). When s > k we get a threshold exponent β = 5s 4k 2 ks 4k+2s (which becomes 1/k when s = k), so that the bound in Theorem 1.5 is stronger (resp., weaker) than the bound in Theorem 1.4 when m > n β (resp., m < n β ), again, up to the extra factor q ε. (2) The bounds in Theorems 1.4 and 1.5 improve, in three dimensions, the recent result of Sharir, Sheffer, and Solomon [53], in three significant ways: (i) The leading terms in both bounds are essentially the same, but our bound is sharper, in that it does not include the factor O(n ε ) appearing in [53]. (ii) The assumption here, concerning the number of curves on a low-degree surface, is much weaker than the one made in [53], where it was required that no surface of some (constant but potentially very large) degree c ε, that depends on ε, contains more than q curves of C. (See also Remark (2) following Theorem 1.4.) 9

230 (iii) The two variants of the non-leading terms here are significantly smaller than those in [53], and, in a certain sense (that will be elaborated following the proof of Theorem 1.5) are best possible. Point-circle incidences in R 3. Theorem 1.5 yields a new bound for the case of incidences between points and circles in R 3, which improves over the previous bound of Sharir, Sheffer, and Zahl [54]. Specifically, we have: Theorem 1.6. Let P be a set of m points and C a set of n circles in R 3, so that no plane or sphere contains more than q circles of C. Then ( ) I(P, C) = O m 3/7 n 6/7 + m 2/3 n 1/3 q 1/3 + m 6/11 n 5/11 q 4/11 log 2/11 (m 3 /q) + m + n. Here too we have the three improvements noted in Remark (2) above. In particular, in the sense of part (iii) of that remark, the new bound is best possible with respect to the best known bound (1) for the planar or spherical cases. See Section 3 for details. Theorem 1.6 has an interesting application to the problem of bounding the number of similar triangles spanned by a set of n points in R 3. It yields the bound O(n 15/7 ), which improves the bound of Agarwal et al. [1]. See Section 3 for details. Incidence graph decomposition, for points on a variety and surfaces. Our first main result on point-surface incidences deals with the special case where the points of P lie on some algebraic variety V of constant degree. Besides being of independent interest, this is a major ingredient of the analysis for the general case of an arbitrary set of points in R 3 and surfaces. In the statements of the following theorems we assume that the set S of the given surfaces is taken from some infinite family F that either has k degrees of freedom with respect to V (with some multiplicity µ), as defined earlier, for suitable constant parameters k (and µ), or is of reduced dimension s with respect to V, for some constant parameter s, meaning that the family Γ := {σ V σ F } is an s-dimensional family of curves (this is reminiscent of the notion of reduced dimension defined above for curves). Theorem 1.7. Let P be a set of m points on some algebraic surface V of constant degree D in R 3, and let S be a set of n algebraic surfaces in R 3 of maximum constant degree E, taken from some family F of surfaces, which either has k degrees of freedom with respect to V (with some multiplicity µ), or is of reduced dimension s with respect to V, for some constant parameters k (and µ) or s. We also assume that the surfaces in S do not share any common irreducible component (which certainly holds when they are irreducible). Then the incidence graph G(P, S) can be decomposed as G(P, S) = γ (P γ S γ ), (4) where the union is over all irreducible components of curves γ of the form σ V, for σ S, and, for each such γ, P γ = P γ and S γ is the set of surfaces in S that contain γ. If F has k degrees of freedom then P γ = O (m k 2k 2 2k 1 n 2k 1 + m + n ), (5) γ and if F is s-dimensional then we have, for any ε > 0, ) P γ = O (m 2s 5s 6 5s 4 n 5s 4 +ε + m 2/3 n 2/3 + m + n, (6) γ where the constants of proportionality depends on D, E, and the complexity of the family F, and either on k and µ in the former case, or on ε and s in the latter case. 10

231 Moreover, in both cases we have γ S γ = O(n), where the constant of proportionality depends on D and E. Remark. A major feature of this result is that it does not impose any restrictions on the incidence graph, such as requiring it not to contain some fixed complete bipartite graph K r,r, for r a constant, as is done in the preceding studies [11, 43, 69]. We re-iterate that, to allow for the existence of large complete bipartite graphs, the bounds in (5) and (6), as well as the bound γ S γ = O(n), are not on the number of incidences (that is, on the number of edges in G(P, S), which could be as high as mn) but on the overall size of the vertex sets of the subgraphs in the complete bipartite graph decomposition of G(P, S). This would lead to the same asymptotic bound on G(P, S) itself, if one assumes that this graph does not contain K r,r as a subgraph, for a constant r. This kind of compact representation of incidences has already been used in the previous studies of Brass and Knauer [14], Apfelbaum and Sharir [4], and our recent works [57, 58], albeit only for the special cases of planes or spheres. Remark. Another way of bypassing the possible presence of large complete bipartite graphs in G(P, S), used in several earlier works [1, 4, 26], is to assume that the surfaces in S are non-degenerate. These studies, already mentioned earlier, only considered the cases of planes and spheres (or of hyperplanes and spheres in higher dimensions) [1, 26]. For spheres, for example, this means that no more than some fixed fraction of the points of P on any given sphere can be cocircular. Although large complete bipartite graphs can exist in G(P, S) in this case, the non-degeneracy assumption allows us to control, in a sharp form, the number of incidences (and shows that the resulting complete bipartite graphs are not so large after all). It would be interesting (and, as we believe, doable) to extend our analysis to the case of (suitably defined) more general non-degenerate surfaces. These remarks also apply to the general case (involving points anywhere in R 3 ), given in Theorem 1.12 below. A mixed incidence bound (for points on most varieties and general surfaces). Our second result is an improvement of Theorem 1.7, still for the case where the points of P lie on some algebraic variety V of constant degree, where we now also assume that V is not infinitely ruled by the (irreducible components of the) intersection curves of pairs of members of the given family F of surfaces. In this case we obtain an improved, mixed bound, in which G(P, S) can be split into two subgraphs, G 0 (P, S) and G 1 (P, S), where the bound in (5) or in (6) now holds for G 0 (P, S), i.e., for the actual number of incidences that it represents, and where G 1 (P, S) admits a complete bipartite graph decomposition, as above, for which the sum of the vertex sets is only 4 O(m + n). The actual bound is slightly sharper see below. Specializing the theorem to the case of spheres, as is done later on (in Section 6), leads to interesting implications to distinct and repeated distances in three dimensions. Theorem 1.8. Let P be a set of m points on some irreducible algebraic surface V of constant degree D in R 3, and let S be a set of n algebraic surfaces in R 3 of constant degree E, which do not share any common irreducible component, taken from some infinite constructible family F of surfaces that either has k degrees of freedom with respect to V (with some multiplicity µ) or is s-dimensional with respect to V, for some constant parameters k (and µ) or s. Assume further that V is not infinitely ruled by the family C 0 of the irreducible components of the intersection curves of pairs of surfaces 5 in F. Then the incidence graph G(P, S) can be decomposed as G(P, S) = G 0 (P, S) (P γ S γ ), (7) γ 4 In fact, many bad things must happen for G 1(P, S) to be nontrivial, and in many situations one would expect G 1(P, S) to be empty; see below. 5 A stricter assumption is that V is not infinitely ruled by algebraic curves of degree at most E 2, which will hold if we assume that each irreducible component of V has degree larger than 100E 4. 11

232 where the union is over all irreducible curves γ contained in (one-dimensional) intersections of the form σ σ V, for σ σ S, and, for each such γ, P γ P γ (for some points on some curves, their incident pairs are moved to, and counted in G 0 (P, S)), and S γ is the set (of size at least two) of surfaces in S that contain γ. Moreover, if F has k degrees of freedom with respect to V (with some multiplicity µ) then G 0 (P, S) = O (m k 2k 2 2k 1 n 2k 1 + m + n ), (8) and if F is s-dimensional with respect to V then, for any ε > 0, ) G 0 (P, S) = O (m 2s 5s 6 5s 4 n 5s 4 +ε + m 2/3 n 2/3 + m + n, (9) where the constants of proportionality depends on D, E, and the complexity of the family F, and either on k and µ in the former case, or on ε and s in the latter case. In either case we also have P γ = O(m), and S γ = O(n), γ where the constants of proportionality depend on D, E, and the complexity of the family F, and either on k (and µ) in the former case, or on ε and s in the latter case. γ Remarks. (1) As already alluded to, we note that, typically, one would expect the complete bipartite decomposition part of (7) to be empty or trivial. To really be significant, (a) many surfaces of S would have to intersect in a common curve, and, in cases where the multiplicity of these curves is not that large, (b) many curves of this kind would have to be fully contained in V. Thus, in many cases, in which (a) and (b) do not hold, the bounds in (8) or in (9) in Theorem 1.8 are for the overall number of incidences. Note also that both Theorem 1.7 and Theorem 1.8 yield a decomposition of (the whole or a portion of) G(P, S) into complete bipartite subgraphs. The major difference is that the bound γ P γ on the overall P -vertex sets size of these graphs is (relatively) large in Theorem 1.7, but it is only linear in m and n (if at all nonzero) in Theorem 1.8. (The bound on γ S γ remains O(n) in both cases.) (2) We note that if V is infinitely ruled by our curves the results break down. For a simple example, take m points and N lines in the plane which form Θ(m 2/3 N 2/3 ) incidences between them. Now pick any surface V in R 3, say the paraboloid z = x 2 + y 2 for specificity, and lift up each of the N lines to a vertical parabola on V. Clearly, V is infinitely ruled by such parabolas, and we get a system of m points and n parabolas with Θ(m 2/3 N 2/3 ) incidences between them. It is also easy to turn this construction into a point-surface incidence structure, in which γ P γ is equal to this bound, which is larger than the lower bound O(m+N) asserted in the theorem. The line y = ax+b in the plane is lifted to the parabola γ a,b = {(x, y, z) R 3 : y = ax + b, z = x 2 + y 2 } contained in the paraboloid V. Define a family S of quadratic surfaces parameterized by a, b, c 0, c 1, c 2 R by S a,b,c0,c 1,c 2 := {(x, y, z) R 3 (z x 2 y 2 ) + (y ax b)(c 0 + c 1 x + c 2 y) = 0}. For any c 0, c 1, c 2 R, the quadric S a,b,c0,c 1,c 2 contains the parabola γ a,b, i.e., many surfaces in S intersect in a common parabola. Incidences between points on a variety and spheres. A particular case of interest is when S is a set of spheres. The intersection curves of spheres are circles, and, as already noted, the only surfaces that are infinitely ruled by circles are spheres and planes. Hence, to apply Theorem 1.8, we need to assume that the constant-degree surface V that contains the points of P has no planar or spherical components, thereby ensuring that V is not infinitely ruled by circles. Clearly, as already noted, spheres in R 3 have four degrees of freedom with respect to any constant-degree variety with no planar or spherical components, and they form a four-dimensional family of surfaces, with respect to any such variety (and also in general). We can therefore apply Theorem 1.8, with s = 4, and conclude: 12

233 Theorem 1.9. Let P be a set of m points on some algebraic surface V of constant degree D in R 3, which has no linear or spherical components, and let S be a set of n spheres, of arbitrary radii, in R 3. The incidence graph G(P, S) can be decomposed as G(P, S) = G 0 (P, S) γ Γ(P γ S γ ), (10) where Γ is the set of circles that are contained in V and in at least two spheres of S, and such that, for each γ Γ, P γ = P γ and S γ is the set of all spheres in S that contain γ. We have ( ) G 0 (P, S) = O m 1/2 n 7/8+ε + m 2/3 n 2/3 + m + n, (11) P γ = O(m), and S γ = O(n),, γ for any ε > 0, where the constant of proportionality depends on D and ε. Remark. Since V does not contain a planar or spherical component, the number of circles in Γ is O(D 2 ), as follows by Guth and Zahl [40]. That is, the union in (11) is only over a constant number of circles. On the other hand, there might also be incidence edges contained in complete bipartite graphs corresponding to circles that are not contained in V, whose number might be quite large. These incidences are recorded in G 0 (P, S) and their number is bounded in (11). Zahl s assumption that G(P, S) does not contain K r,3, for some (arbitrary) constant r (that is, by assuming that every triple of spheres intersect in at most r points of P ), leads to the bound I(P, S) = O(m 3/4 n 3/4 + m + n); our bound is better for m > n 1/2 (ignoring the n ε factor in our bound). Except for this rather restrictive assumption, Zahl s result is more general, as it does not require the points to lie on a constant-degree variety. We also note that if we assume that G(P, S) does not contain any K r,r, for r > 3 a constant, the bound in the second part of (11) becomes a bound on the number of incidences, so, under this somewhat weaker assumption (than that of Zahl), we improve Zahl s bound for points on a variety and for m > n 1/2. The bound in (11) further improves when either (i) the centers of the spheres of S lie on V (or on some other constant-degree variety), or (ii) the spheres of S have the same radius. In both cases, S is only three-dimensional, so the bound improves to ( ) G 0 (P, S) = O m 6/11 n 9/11+ε + m 2/3 n 2/3 + m + n, (12) for any ε > 0. When both conditions hold the spheres are congruent and their centers lie on V S is only two-dimensional with respect to V, and the bound improves still further to ( ) G 0 (P, S) = O m 2/3 n 2/3+ε + m + n. Using a slightly refined machinery, developed in a companion paper [58], the latter bound can be actually improved further to O(m 2/3 n 2/3 + m + n). (13) γ Applications of Theorem 1.9 and (12), (13): Distinct distances. As already mentioned, and as will be detailed in the proofs of the following results, the new bounds on point-sphere incidences have immediate applications to the study of distinct and repeated distances determined by a set of n points in R 3, when the points (or a subset thereof see below) lie on some fixed-degree algebraic variety. Specifically, for distinct distances, we have the following results. 13

234 Theorem (a) Let P be a set of n points on an algebraic surface V of constant degree D in R 3, with no linear or spherical components. Then the number of distinct distances determined by P is Ω(n 7/9 ε ), for any ε > 0, where the constant of proportionality depends on D and ε. (b) Let P 1 be a set of m points on a surface V as in (a), and let P 2 be a set of n arbitrary points in R 3. Then the number of distinct distances determined by pairs of points in P 1 P 2 is ( { }) Ω min m 4/7 ε n 1/7 ε, m 1/2 n 1/2, m, for any ε > 0, where the constant of proportionality depends on D and ε. Remark. In a recent work [58], we have obtained slightly improved bounds, without the ε in the exponents, using a more refined space decomposition technique, which can be applied for arrangements of spheres. While we believe that the bounds in the theorem are not tight, we note that the bounds in both (a) and (b) (with, say, m = n) are significantly larger than the conjectured best-possible lower bound Ω(n 2/3 ) for arbitrary point sets in R 3. Repeated distances. As another application, we bound the number of unit (or repeated) distances involving points on a surface V, as above. Theorem (a) Let P be a set of n points on some algebraic surface V of constant degree D in R 3, which does not contain any planar or spherical components. Then P determines O(n 4/3 ) unit distances, where the constant of proportionality depends on D. (b) Let P 1 be a set of m points on a surface V as in (a), and let P 2 be a set of n arbitrary points in R 3. Then the number of unit distances determined by pairs of points in P 1 P 2 is ( ) O m 6/11 n 9/11+ε + m 2/3 n 2/3 + m + n, for any ε > 0, where the constant of proportionality depends on D and ε. In part (a) we extend, to the case of general constant-degree algebraic surfaces, the known bound O(n 4/3 ), which is worst-case tight when V is a sphere [29]. Part (b) gives (say, for the case m = n) an intermediate bound between O(n 4/3 ) and the best known upper bound O(n 3/2 ) for a arbitrary set of points in R 3 [43, 69]. Another thing to notice is that, for distinct distances, the situation is quite different when V is (or contains) a plane or a sphere, in which case the bound goes up to Ω(n/ log n) [39, 67] (see also Sheffer s survey [61] for details). Incidence graph decomposition (for arbitrary points and surfaces). Our final main result on point-surface incidences deals with the general setup involving a set S of constant-degree algebraic surfaces and an arbitrary set of points in R 3. The analysis in this general setup proceeds by a recursive argument, based on the polynomial partitioning technique of Guth and Katz [39], in which Theorem 1.7 plays a central role 6. This result extends a recent result in preliminary work by the authors [58, Theorem 1.4] from spheres to general surfaces, and extends the aforementioned result of Zahl [69], for general algebraic surfaces, to the case where no constraints are imposed on G(P, S). 6 Ideally, applying Theorem 1.8 would yield a better estimate, but, unfortunately, we cannot control the polynomial generated by the polynomial partitioning technique. 14

235 Theorem Let P be a set of m points in R 3, and let S be a set of n surfaces from some s-dimensional family 7 F of surfaces, of constant maximum degree E in R 3. Then the incidence graph G(P, S) can be decomposed as G(P, S) = G 0 (P, S) (P γ S γ ), (14) γ where the union is now over all curves γ of intersection of at least two of the surfaces of S, and, for each such γ, P γ = P γ and S γ is the set (of size at least two) of surfaces in S that contain γ. Moreover, we have, for any ε > 0, J(P, S) := γ ( Pγ + S γ ) = O (m 2s 3s 1 n 3s 3 3s 1 +ε + m + n ), and G 0 (P, S) = O(m + n), (15) where the constants of proportionality depend on ε, s, D, E, and the complexity of the family F. As already noted, this result extends Zahl s bound [69] to the case where no restrictions are imposed on the incidence graph (see the remark following Theorem 1.7). Zahl s bound is the same as ours, except for the extra factor n ε in our bound. We also note that Theorem 1.12 only applies to s-dimensional families F, and not to families with k degrees of freedom. The main issue here is that in Theorems 1.7 and 1.8, the notion of k degrees of freedom (and that of s-dimensionality) is applied to the intersection curves of the surfaces from F with some constant-degree variety, whereas here it has to hold for the surfaces themselves in the entire threedimensional space. So far we are lacking a good definition of this notion that will facilitate certain steps in the proof. See a discussion of this issue following the proof, in Section The main techniques There are three main ingredients used in our approach. The first ingredient, already mentioned in the context of planar point-curve incidences, is the techniques of Pach and Sharir [49] (given in Theorem 1.1), and of Sharir and Zahl [60] (Theorem 1.2) concerning incidences between points and algebraic curves in the plane. The latter bound will be used in the analysis of incidences both between points and curves, and between points and surfaces. The second ingredient, relevant to the proofs of Theorems 1.4 and 1.12, is the polynomial partitioning technique of Guth and Katz [39], and its more recent extension by Guth [35], which yields a divide-andconquer mechanism via space decomposition by the zero set of a suitable polynomial. This will produce subproblems that will be handled recursively, and will leave us with the overhead of analyzing the incidence pattern involving the points that lie on the zero set itself. The latter step will be accomplished by a straightforward application of Theorem 1.7. We assume familiarity of the reader with these results; more details will be given in the applications of this technique in the proofs of the aforementioned theorems. The third ingredient arises in the proof of Theorems 1.4 and 1.5, where we argue that a generic point on a variety V, that is not infinitely ruled by constant-degree curves of some given family, as in the statement of the theorems, is incident to at most a constant number of the given curves that are fully contained in V. Moreover, we can also control the number and structural properties of non-generic points. Before formally stating, in detail, the technical properties that we need, we review a few notations. Fix a constructible set C 0 C 3,E of irreducible curves of degree at most E in 3-dimensional space, and a trivariate polynomial f. Following Guth and Zahl [40, Section 9], we call a point p Z(f) a (t, C 0, r)- flecnode, if there are at least t curves γ 1,..., γ t C 0, such that, for each i = 1,..., t, (i) γ i is incident to p, 7 Here we use the general notion of s-dimensionality, not confined to points on a variety. 15

236 (ii) p is a non-singular point of γ i, and (iii) γ i osculates to Z(f) to order r at p. This is a generalization of the notion of a flecnodal point, due to Salmon [51, Chapter XVII, Section III] (see also [39, 55] for more details). Our analysis requires the following theorem. It is a consequence of the analysis of Guth and Zahl [40, Corollary 10.2], which itself is a generalization of the Cayley Salmon theorem on surfaces ruled by lines (see, e.g., Guth and Katz [39]), and is closely related to Theorem 1.3 (also due to Guth and Zahl [40]). The novelty in this theorem is that it addresses surfaces that are infinitely ruled by certain families of curves, where the analysis in [40] only handles surfaces that are doubly ruled by such curves. Theorem (a) For given integer parameters c and E, there are constants c 1 = c 1 (c, E), r = r(c, E), and t = t(c, E), such that the following holds. Let f be a complex irreducible polynomial of degree D E, and let C 0 C 3,E be a constructible set of complexity at most c. If there exist at least c 1 D 2 curves of C 0, such that each of them is contained in Z(f) and contains at least c 1 D points on Z(f) that are (t, C 0, r)-flecnodes, then Z(f) is infinitely ruled by curves from C 0. (b) In particular, if Z(f) is not infinitely ruled by curves from C 0 then, except for at most c 1 D 2 exceptional curves, every curve in C 0 that is fully contained in Z(f) is incident to at most c 1 D points that are incident to at least t curves in C 0 that are also fully contained in Z(f). Note that, by making c 1 sufficiently large (specifically, choosing c 1 > E), the assumption that each of the c 1 D 2 curves in the premises of the theorem is fully contained in Z(f) follows (by Bézout s theorem) from the fact that each of them contains at least c 1 D points on Z(f). Although the theorem is a corollary of the work of Guth and Zahl in [40], we review (in the appendix) the machinery needed for its proof, and sketch a brief version of the proof itself, for the convenience of the reader and in the interest of completeness. 2 Proofs of Theorems 1.4 and 1.5 (points and curves) The proofs of both theorems are almost identical, and they differ in only one step in the analysis. We will give a full proof of Theorem 1.4, and then comment on the few modifications that are needed to establish Theorem 1.5. Proof of Theorem 1.4. Since the family C has k degrees of freedom with multiplicity µ, the incidence graph G(P, C), as a subgraph of P C, does not contain K k,µ+1 as a subgraph. The Kővári-Sós-Turán theorem (e.g., see [47, Section 4.5]) then implies that I(P, C) = O(mn 1 1/k + n), where the constant of proportionality depends on k (and µ). We refer to this as the naive bound on I(P, C). In particular, when m = O(n 1/k ), we get I(P, C) = O(n). We may thus assume that m a n 1/k, for some absolute constant a. The proof proceeds by double induction on n and m, and establishes the bound ) I(P, C) A (m k 3k 3 k k 1 k 1 3k 2 n 3k 2 + m 2k 1 n 2k 1 q 2k 1 + m + n, (16) for a suitable constant A that depends on k, µ, E, and the complexity of C 0. The base case for the outer induction on n is n n 0, for a suitable sufficiently large constant n 0 that will be set later. The bound (16) clearly holds in this case if we choose A n 0. The base case for the inner induction on m is m a n 1/k, in which case the naive bound implies that I(P, C) = O(n), so (16) holds with a sufficiently large choice of A. Assume then that the bound (16) holds for all sets P, C with C < n or with C = n and P < m, and let P and C be sets of sizes P = m, C = n, such that n > n 0, and m > a n 1/k. 16

237 It is instructive to notice that the two terms m k 3k 3 3k 2 n 3k 2 and m in (16) compete for dominance; the former (resp., latter) dominates when m n 3/2 (resp., m n 3/2 ). One therefore has to treat these two cases somewhat differently; see below and also in earlier works [39, 56]. Applying the polynomial partitioning technique. We construct a partitioning polynomial f for the set C of curves, as in the recent variant of the polynomial partitioning technique, due to Guth [35]. Specifically, we choose a degree D = {cm k 3k 2 /n 1 3k 2, for a n 1/k m an 3/2, cn 1/2, for m > an 3/2, (17) for suitable constants c, a, and a (whose values will be set later), and obtain a polynomial f of degree at most D, such that each of the O(D 3 ) (open) connected components of R 3 \ Z(f) is crossed by at most O(n/D 2 ) curves of C, where the former constant of proportionality is absolute, and the latter one depends on E. Note that in both cases 1 D n 1/2, if a, a, and c are chosen appropriately. Denote the cells of the partition as τ 1,..., τ u, for u = O(D 3 ). For each i = 1,..., u, let C i denote the set of curves of C that intersect τ i, and let P i denote the set of points that are contained in τ i. We set m i = P i and n i = C i, for i = 1,..., u, put m = i m i m, and notice that n i = O(n/D 2 ), for each i. An obvious property (which is a consequence of the generalized version of Bézout s theorem [32]) is that every curve of C intersects at most ED + 1 cells of R 3 \ Z(f). When a n 1/k m an 3/2, within each cell τ i of the partition, for i = 1,..., u, we use the naive bound ( I(P i, C i ) = O(m i n 1 1/k i + n i ) = O m i (n/d 2 ) 1 1/k + n/d 2), and, summing over the O(D 3 ) cells, we get a total of ( ) mn 1 1/k O + nd. D2(1 1/k) With the above choice of D, we deduce that the total number of incidences within the cells is ) O (m k 3k 3 3k 2 n 3k 2. When m > an 3/2, within each cell τ i of the partition we have n i = O(n/D 2 ) = O(1), so the number of incidences within τ i is at most O(m i n i ) = O(m i ), for a total of O(m) incidences. Putting these two alternative bounds together, we get a total of ) O (m k 3k 3 3k 2 n 3k 2 + m (18) incidences within the cells. Incidences within the zero set Z(f). It remains to bound incidences with points that lie on Z(f). Set P := P Z(f) and m := P = m m. Let C denote the set of curves that are fully contained in Z(f), and set C := C \ C, n := C, and n := C = n n. Since every curve of C intersects Z(f) in at most ED = O(D) points, we have (for either choice of D) ) I(P, C ) = O(nD) = O (m k 3k 3 3k 2 n 3k 2 + m. (19) Finally, we consider the number of incidences between points of P and curves of C. Decompose f into (complex) irreducible components f 1,..., f t, for t D, and assign each point p P (resp., curve γ C ) to the first irreducible component f i, such that Z(f i ) contains p (resp., fully contains γ; such 17

238 a component always exists). The number of cross-incidences, between points and curves assigned to different components, is easily seen, arguing as above, to be O(nD), which satisfies our bound. In what follows, we recycle the symbols m i (resp., n i ), to denote the number of points (resp., curves) assigned to f i, and put D i = deg(f i ), for i = 1,..., t. We clearly have i m i = P = m, i n i = C = n, and i D i = deg(f) = D. For each i = 1,..., t, there are two cases to consider. Case 1: Z(f i ) is infinitely ruled by curves of C 0. By assumption, there are at most q curves of C on Z(f i ), implying that n i q. We project the points of P i and the curves of C i onto some generic plane π 0. A suitable choice of π 0 guarantees that (i) no pair of intersection points or points of P i project to the same point, (ii) if p is not incident to γ then the projections of p and of γ remain non-incident, (iii) no pair of curves in C i have overlapping projections, and (iv) no curve of C i contains any segment orthogonal to π 0. Moreover, the number of degrees of freedom does not change in the projection (see Sharir et al. [53]). The number of incidences for the points and curves assigned to Z(f i ) is equal to the number of incidences between the projected points and curves, which, by Theorem 1.1, is ( O m k 2k 1 i ) 2k 2 2k 1 ni + m i + n i = O ( m k 2k 1 i n k 1 2k 1 i Summing over i = 1,..., t, and using Hölder s inequality, we get the bound ) O (m k k 1 k 1 2k 1 n 2k 1 q 2k 1 + m + n, which, by making A sufficiently large, is at most A 4 q k 1 2k 1 + mi + n i ). (m k 2k 1 n k 1 2k 1 q k 1 2k 1 + m + n ). (20) Remark. This is the only step in the proof where being of reduced dimension s, for s sufficiently small, might yield an improved bound (over the one in (20)); see below, in the follow-up proof of Theorem 1.5, for details. Case 2: Z(f i ) is not infinitely ruled by curves of C 0. In this case, Theorem 1.13(b) implies that there exist suitable constants c 1, t that depend on E and on the complexity of C 0, such that there are at most c 1 Di 2 exceptional curves, namely, curves that contain at least c 1D i points that are incident to at least t curves from C. Therefore, by choosing c (in the definition of D) sufficiently small, we can ensure that, in both cases (of small m and large m), i D2 i ( i D i) 2 = D 2 n. This allows us to apply induction on the number of curves, to handle the exceptional curves. Concretely, we have an inductive instance of the problem involving m i points and at most c 1 Di 2 n curves of C. By the induction hypothesis, the corresponding incidence bound is at most ( k 3k 2 A m (c 1 Di 2 ) 3k 3 k 3k 2 + m i 2k 1 i ) (c 1 Di 2 ) k 1 k 1 2k 1 q 2k 1 + mi + c 1 Di 2. We now sum over i. For the first and fourth terms, we bound each m i by m, and use the fact that i Dα i D α for any α 1. For the second terms, we use Hölder s inequality. Overall, we get the incidence bound ( ) 3k 3 3k 2 A c 3k 2 (D 2 ) 3k 3 k 1 3k 2 + c 2k 1 2k 1 (D 2 ) k 1 k 1 2k 1 q 2k 1 + m + c1 D 2, 1 m k 1 m k which, with a proper choice of c (in (17)), can be upper bounded by A 4 (m k 3k 2 n 3k 3 3k 2 + m k 2k 1 n k 1 2k 1 q k 1 2k 1 + m + n ). (21) 18

239 Except for these incidences, for each f i, each non-exceptional curve in C that is assigned to Z(f i ) is incident to at most c 1 D i points that are incident to at least t curves from C ; the total number of incidences of this kind involving the n i curves assigned to Z(f i ) and their incident points is O(n i D i ). Other incidences involving the non-exceptional curves in C that are assigned to Z(f i ) only involve points assigned to Z(f i ) that are incident to at most t = O(1) curves from C ; the number of such point-curve incidences is thus O(m i t) = O(m i ). Therefore, when Z(f i ) is not infinitely ruled by curves of C, the number of incidences assigned to Z(f i ) is O(m i +n i D i ), plus terms that are accounted for by the induction. Summing over these components Z(f i ), we get the bound O(m + nd), plus the inductive bounds in (21), and, choosing A to be sufficiently large, these bounds will collectively be at most A 2 (m k 3k 2 n 3k 3 3k 2 + m k 2k 1 n k 1 2k 1 q k 1 2k 1 + m + n ). (22) In summary, by choosing A sufficiently large, the number of incidences is well within the bound of (16), thus establishing the induction step, and thereby completing the proof. Proof of Theorem 1.5. The proof proceeds by the same double induction on n and m, and establishes the bound, for any prespecified ε > 0, ( ) I(P, C) Am k 3k 3 3k 2 n 3k 2 + Aε m 2s 3s 4 2s 2 5s 4 n 5s 4 q 5s 4 +ε + m 2/3 n 1/3 q 1/3 + m + n, (23) for a suitable constant A that depends on k, µ, s, E, and the complexity of C 0, and another constant A ε that also depends on ε. The flow of the proof is very similar to that of the preceding proof. The main difference is in the case where some component Z(f i ) of Z(f) is infinitely ruled by curves from C 0. Again, in this case it contains at most q curves of C. We take the points of P and the curves of C that are assigned to Z(f i ), and project them onto some generic plane π 0 (the same plane can be used for all such components), as in the proof of Theorem 1.4 and get the same properties (i) (iv) of the projected points and curves. Let P i and C i denote, respectively, the set of projected points and the set of projected curves; the latter is a set of n i plane irreducible algebraic curves of constant maximum degree 8 DE. Moreover, as in the preceding proof, the contribution of Z(f i ) to I(P, C ) is equal to the number I(P i, C i ) of incidences between P i and C i. We can now apply Theorem 1.2 to P i and C i. To do so, we first note: Lemma 2.1. C i is contained in an s-dimensional family of curves. Proof. Here it is more convenient to work over the complex field C (see the general remark in the introduction). Let Π 0 denote the projection of C 3 onto π 0. Let C 0 (f i ) denote the family of the curves of C 0 that are contained in Z(f i ), and let C 0 (f i ) denote the family of their projections onto π 0 (under π 0 ). Define the mapping ψ : C 0 (f i ) C 0 (f i ), by ψ(γ) = Π 0 (γ), for γ C 0 (f i ). By Green and Morrison [33] (see also [42, Lecture 21] and Ellenberg et al. [27, Section 2]), C 0 (f i ) and C 0 (f i ) are algebraic varieties and ψ is a (surjective) morphism from C 0 (f i ) to C 0 (f i ). In general, if ψ : X Y is a surjective morphism of algebraic varieties, then the dimension of X is at least as large as the dimension of Y. Indeed, Definition 11.1 in [42] defines the dimension via such a morphism, provided that it is finite-to-one. A complete proof of the general case is given in [70]. Therefore, C 0 (f i ) is of dimension at most dim(c 0 (f i )) = s, and the proof of the lemma is complete. Applying Theorem 1.2 to the projected points and curves, we conclude that the number of incidences for the points and curves assigned to Z(f i ) is at most B ε (m 2s 5s 4 i ) 5s 6 5s 4 n +ε i + m 2/3 i n 2/3 i + m i + n i B ε (m 2s 5s 4 i 3s 4 5s 4 ni q 2s 2 5s 4 +ε + m 2/3 i n 1/3 i q 1/3 + m i + n i ), 8 A projection preserves irreducibility and does not increase the degree; see, e.g., Harris [42] for a reference to these facts. 19

240 with a suitable constant of proportionality B ε that depends on s and on ε. By Hölder s inequality, summing this bound over all such components Z(f i ), we get the bound ) (m 2s 3s 4 2s 2 5s 4 n 5s 4 q 5s 4 +ε + m 2/3 n 1/3 q 1/3 + m + n, B ε for another constant B ε proportional to B ε. By making A ε sufficiently large, this bound is at most ( ) A ε 4 m 2s 3s 4 2s 2 5s 4 n 5s 4 q 5s 4 +ε + m 2/3 n 1/3 q 1/3 + m + n. The rest of the proof proceeds as the previous proof, more or less verbatim, except that we need a more careful (albeit straightforward) separate handling of the leading term, multiplied by A, and the other terms, multiplied by A ε. The induction step then establishes the bound in (23) in much the same way as above. Remarks. (1) As already mentioned in the introduction, the lower-order terms ) O (m 2s 3s 4 2s 2 5s 4 n 5s 4 q 5s 4 +ε + m 2/3 n 1/3 q 1/3 + m + n in the bound are best possible in the following sense. If the bound in Theorem 1.2 were optimal, or nearly optimal, in the worst case, for points and curves of C 0 that lie in a constant-degree surface V that is infinitely ruled by such curves, the same would also hold for the lower-order terms in the bound in Theorem This is shown by a simple packing argument, in which we take n/q generic copies of V, and place on each of them mq/n points and q curves, so as to obtain ) Ω ((mq/n) 2s 5s 6 5s 4 q 5s 4 + (mq/n) 2/3 q 2/3 + mq/n + q incidences on each copy, for a total of ) (n/q) Ω ((mq/n) 2s 5s 6 5s 4 q 5s 4 + (mq/n) 2/3 q 2/3 + mq/n + q ) = Ω (m 2s 3s 4 2s 2 5s 4 n 5s 4 q 5s 4 + m 2/3 n 1/3 q 1/3 + m + n incidences. (This construction works when m > n/q. Otherwise, the bound is linear, and clearly best possible. Also, we assume that the lower bound does not involve the factor q ε, to simplify the reasoning.) In particular, this remark applies to the case of points and circles, as discussed in Theorem 1.6. (2) There is an additional step in the proof in which the fact that C 0 is of some constant (not necessarily reduced) dimension s could lead to an improved bound. This is the base case m = O(n 1/k ), where we use the Kővári-Sós-Turán theorem to obtain a linear bound on I(P, C). Instead, we can use the result of Fox et al. [30, Corollary 2.3], and the fact that the incidence graph does not contain K k,µ+1 as a subgraph, to show that, when m = O(n 1/s ), the number of incidences is linear. The problem is that here we need to use the dimension s of the entire C 0, rather than the reduced dimension s (which, as we recall, applies only to subsets of C 0 on a variety that is infinitely ruled by curves of C 0 ). Typically, as already noted, s is larger than k (generally twice as large as k), making this bootstrapping bound inferior to what we have. Still, in cases where s happens to be smaller than k, this would lead to a further improved incidence bounds, in which the leading term is also smaller. Rich points. Theorems 1.4 and 1.5 can easily be restated as bounding the number of r-rich points for a set C of curves with k degrees of freedom (and or reduced dimension s) in R 3, when r is at least some sufficiently large constant. The case r = 2 is treated in Guth and Zahl [40], and the same bound that they obtain holds for larger values of r (albeit without an explicit dependence on r), smaller than the threshold in the following corollary. 9 Theorem 1.2 is formulated, and proved in [60], only for plane curves. Nevertheless, it also holds for curves contained in a variety V of constant degree, simply by projecting the points and curves onto some generic plane, as done in the proofs. 20

241 Corollary 2.2. (a) Let C be a set of n irreducible algebraic curves, taken from some constructible family C 0 of irreducible curves of degree at most E and with k degrees of freedom (with some multiplicity µ) in R 3, and assume that no surface that is infinitely ruled by curves of C 0, or, alternatively, by curves of degree at most E, contains more than q curves of C (e.g., make this assumption for all surfaces of degree at most 100E 2 ). Then there exists some constant r 0, depending on k (and µ) and on C 0, or, more generally, on E, such that, for any r r 0, the number of points that are incident to at least r curves of C (so-called r-rich points) is O ( n 3/2 r 3k 2 2k 2 + nq r 2k 1 k 1 + n r where the constant of proportionality depends on k and E (and on µ). (b) If C 0 is also of reduced dimension s, the bound on the number of r-rich points becomes ( ) O n 3/2 r 3k 2 2k 2 + nq 2s 2 3s 4 +ε where the constant of proportionality now also depends on s and ε. (Actually, the first term comes with a constant that is independent of ε.) r 5s 4 3s 4 Proof. Denoting by m r the number of r-rich points, the corollary is obtained by combining the upper bound in Theorem 1.4 or Theorem 1.5 with the lower bound rm r. The bound in (b) is an improvement, for s = k, when q > r k+ε, for another arbitrarily small parameter ε, which is linear in the prespecified ε. (To be more precise, this is an improvement at all only when the second term dominates the bound.) It would be interesting to close the gap, by obtaining an r-dependent bound also for values of r between 3 and r 0. It does not seem that the technique in Guth and Zahl [40] extends to this setup. ) + n r,, 3 Incidences between points and circles and similar triangles in R 3 We first briefly discuss the fairly straightforward proof of Theorem 1.6. As already discussed in the introduction, we have k = s = 3, for the case of circles, so we can apply Theorem 1.5 in the context of circles, and obtain the bound ( ) I(P, C) = O m 3/7 n 6/7 + m 2/3 n 1/3 q 1/3 + m 6/11 n 5/11 q 4/11+ε + m + n, for any ε > 0, where q is the maximum number of the given circles that are coplanar or cospherical. In fact, the extension of the planar bound (1) to higher dimensions, due to Aronov et al. [6], asserts that, for any set C of circles in any dimension, we have ( ) I(P, C) = O m 2/3 n 2/3 + m 6/11 n 9/11 log 2/11 (m 3 /n) + m + n, (24) which is slightly better than the general bound of Sharir and Zahl [60] (given in Theorem 1.2). If we use this bound, instead of that in Theorem 1.2, in the proof of Theorem 1.5 (specialized for the case of circles), we get the slight improvement (in which the two constants of proportionality are now absolute) ( ) I(P, C) = O m 3/7 n 6/7 + m 2/3 n 1/3 q 1/3 + m 6/11 n 5/11 q 4/11 log 2/11 (m 3 /q) + m + n, which establishes Theorem

242 The number of similar triangles. Theorem 1.6 has the following interesting application. Let P be a set of n points in R 3, and let = abc be a fixed given triangle. The goal is to bound the number, denoted as S (P ), of triangles spanned by P and similar to. The best known upper bound for S (P ), obtained by Agarwal et al. [1], is O(n 13/6 ), and the proof that establishes this bound in [1] is fairly involved. Using Theorem 1.6, we obtain the following simple and fairly straightforward improvement. Theorem 3.1. S (P ) = O(n 15/7 ). Proof. Following a standard strategy, fix a pair p, q of points in P, and consider the locus γ pq of all points r such that the triangle pqr is similar to (when p, q, r are mapped to a, b, c, respectively). Clearly, γ pq is a circle whose axis (line passing through the center of γ pq and perpendicular to its supporting plane) passes through p and q. Moreover, there exist at most two (ordered) pairs p, q and p, q for which γ pq = γ p q. Let C denote the set of all these circles (counted without multiplicity). Then S (P ) is at most two thirds of the number I(P, C) of incidences between the n points of P and the N = O(n 2 ) circles of C. By Theorem 1.6 we thus have ( S (P ) = O n 3/7 (n 2 ) 6/7 + n 2/3 (n 2 ) 1/3 q 1/3 + n 6/11 (n 2 ) 5/11 q 4/11 log 2/11 n + n 2), where q is the maximum number of circles in C that are either coplanar or cospherical. That is, we have ( S (P ) = O n 15/7 + n 4/3 q 1/3 + n 16/11 q 4/11 log 2/11 n + n 2). (25) We claim that q = O(n). This is easy for coplanarity, because, for any fixed plane π, each point p P can generate at most one circle γ pq in C that is contained in π. Indeed, the axis of such a circle is perpendicular to π and passes through p. This fixes the center of γ pq, and it is easily checked that the radius is also fixed. A similar argument holds for cospherical circles. Here too, for a fixed sphere σ, each point p P that is not the center o of σ can generate at most one circle γ pq in C that is contained in σ. This is because the axis of such a circle must pass through o, which fixes the center of the circle, and the radius is also fixed, as an easy calculation shows. For p = o there are at most n 1 additional such circles. Hence, plugging q = O(n) into (25), we get S (P ) = O(n 15/7 ), as asserted. 4 Proof of Theorem 1.7 (points on a variety and surfaces) Let P, V, S, F, m, and n be as in the statement of the theorem. We first restrict the analysis to the case where V is irreducible. This involves no loss of generality, because, when V is reducible, we can decompose it into its irreducible components, assign each point of P to each component that contains it, and assign the surfaces of S to all the components. This decomposes the problem into at most D subproblems, each involving an irreducible surface, and it thus follows that the original vertex set count is at most D = O(1) times the bound for the irreducible case. In the remainder of this section we thus assume that V is irreducible. To obtain the bound in (5) or in (6) on γ P γ, we reduce this problem to the case of incidences between points and algebraic curves in the plane, and then apply either Theorem 1.1 or Theorem 1.2, as appropriate. Surfaces with k degrees of freedom. Recall that a family F of surfaces is said to have k degrees of freedom with respect to a constant-degree variety V, if the family of the irreducible components of the intersection curves {σ V σ F}, counted without multiplicity, has k degrees of freedom, with some constant multiplicity µ, as defined for curves in R 3 in Section 1.3. Note that this definition means that, for any k points on V there are at most µ curves of the form σ V, for σ F, that pass through all the points; the number of surfaces that pass through all the points 22

243 could be much larger, even infinite. For example, spheres in R 3 have four degrees of freedom with respect to any variety that is neither a sphere nor a plane, because four non-cocircular points determine a unique sphere that passes through all four, whereas four cocircular points (over-)determine a unique circle that passes through all of them, but an infinity of spheres with this property. Interestingly, when V is a sphere or a plane, the number of degrees of freedoms goes down to three. The reduction. Consider the intersection curves γ σ := σ V, for σ S. These are algebraic curves of degree O(DE) = O(1). These curves are not necessarily distinct, and pairs of distinct curves can have common irreducible components. We denote by Γ the multiset of the irreducible components of these curves, where each component appears with multiplicity equal to the number of surfaces that contain it; we also denote by Γ 0 the underlying set of distinct irreducible components of the curves, obtained by removing duplications from Γ. We may assume that V does not fully contain any surface of S. Indeed, since V is irreducible, it can contain (that is, coincide with) at most one such surface, which contributes at most m to γ P γ. Ignoring this surface, we have that each γ σ is at most one-dimensional; it can be empty, and it may have isolated points. To treat these points, we note that each such curve has only O(1) such points, 10 so the isolated points contribute a total of at most O(n) incidences with their corresponding surfaces. For uniformity, we simply record these incidences, as well as those involving the surface fully contained in (coinciding with) V, if any, as trivial (complete) bipartite graphs, with total vertex set size O(m + n) on the P -side, and O(n) on the S-side. We represent (the remainder of) G(P, S) simply as the union γ Γ 0 (P γ S γ ), where, for each γ Γ 0, P γ = P γ and S γ is the set of all surfaces in S that contain γ. This representation is not necessarily edge disjoint, but a pair (p, σ) can appear in this union at most O(DE) times, because σ V can have at most O(DE) irreducible components, and (p, σ) appears in the union once for each of these components that contains p. The argument just offered also shows that γ S γ = O(n). The corresponding sum γ P γ (excluding the special cases treated above, which only add O(m + n) to the count) is the number of incidences I(P, Γ 0 ) between the points of P and the curves of Γ 0 (counted without multiplicity). We therefore proceed to estimate I(P, Γ 0 ). We follow an argument very similar to the one in the proofs of Theorems 1.4 and 1.5; due to certain differences, some of which are rather nontrivial, we spell it out for clarity. We take a generic plane π 0 and project the points of P and the curves of Γ 0 onto π 0. As before, a suitable choice of π 0 guarantees that (i) no pair of intersection points or points of P project to the same point, (ii) if p is not incident to γ then the projections of p and of γ remain non-incident, (iii) no pair of curves in Γ 0 have overlapping projections, and (iv) no curve of Γ 0 contains any segment orthogonal to π 0. Let P and Γ 0 denote, respectively, the set of projected points and the set of projected curves; the latter is a set of n plane irreducible algebraic curves of constant maximum degree O(DE) (see a previous footnote). Moreover, I(P, Γ 0 ) is equal to the number I(P, Γ 0 ) of incidences between P and Γ 0. We now bifurcate according to which property F is assumed to satisfy. Consider first the case where F is of reduced dimension s with respect to V. We have the following lemma, which is an extension of the simpler variant given in Lemma 2.1 above. Lemma 4.1. Γ 0 is contained in an s-dimensional family of curves. Proof. As in the preceding proof, we work over the complex field C. Let Π 0 denote the projection of C 3 onto π 0. Define the family of curves Γ F = {σ V σ F}, and let Γ F denote the family of the projections of the curves in Γ F under Π 0. Define mappings ϕ : F Γ F and ψ : Γ F Γ F, by ϕ(σ) = σ V, and 10 The number of isolated points on a curve is easily seen to be quadratic in its degree. In our case, this degree is O(DE) = O(1), and the claim follows. 23

244 ψ(γ) = Π 0 (γ). As above, by [27, 33, 42], Γ F and Γ F are algebraic varieties and ψ is a morphism from Γ F to Γ f. By Fulton [32, Section 3.4], ϕ is a morphism from F to Γ F, and since both ϕ and ψ are surjective, it follows that their composition Φ = ψ ϕ is a surjective morphism from F to Γ F. As in the proof of Lemma 2.1, the definition in Harris [42, Definition 11.1], and its extension in [70], imply that the dimension of F is at least as large as the dimension of Γ F. Therefore, Γ F is of dimension at most dim(f) = s, and the proof of the lemma is complete. Remark. It might be the case that Γ 0 is of smaller dimension than s. As will follow from the proof, the incidence bound depends on the dimension of Γ 0 and not on the dimension of F, so the bound will improve if Γ 0 is indeed of smaller dimension. In addition, the curves of Γ 0 are of constant maximum degree. Applying Theorem 1.2 to our setup, we get the bound in (6). Adding the counts obtained separately for the preceding special cases completes the proof of Theorem 1.7 when F is of reduced dimension s with respect to V. Consider next the case where F has k degrees of freedom with respect to V. Then Theorem 1.1 is applicable to P and Γ 0, and we have ) I(P, Γ 0) = O (m k 2k 2 2k 1 n 2k 1 + m + n, and adding to this the bounds obtained in the other cases yields the bound asserted in (5). 5 Proof of Theorem 1.8 (points on a variety and general surfaces) Let P, V, S, F, k, µ, s, m, and n be as in the statement of the theorem. We first restrict the analysis to the case where V is irreducible. The general case can be handled, as in the case of Theorem 1.7, by repeating the analysis to each of the O(1) irreducible components of V, and summing up the resulting bounds, to obtain the same asymptotic bound (multiplied by an extra factor of D = O(1)). Let then f be an irreducible complex polynomial such that V = Z(f), and let C denote the set of irreducible curves that (i) are fully contained in V, (ii) are contained in at least two surfaces of S, and (iii) contain at least one point of P. By Bézout s theorem [32] and condition (ii), we have deg(γ) E 2 for each γ C. For each curve γ C we form the bipartite subgraph P γ S γ of G(P, S), where P γ = P γ (the actual sets P γ for some of the curves will be smaller see below), and S γ is the set of the surfaces of S that contain γ. To estimate γ P γ and γ S γ, we argue as follows. First, γ P γ is the number of incidences between the points of P and the curves of C, counted without multiplicity. By assumption, V is not infinitely ruled by the irreducible components of the intersection curves of pairs of surfaces from F, and C is contained in this family of curves. To apply Theorem 1.13, it remains to argue that C is constructible. Although the proof of this property is not too hard, it is rather technical, and we will present it in Lemma A.4 in the Appendix. We then conclude that, for a suitable constant t = t(e, c) (where c is the complexity of the family C), except for possibly O(D 2 ) exceptional curves, every curve in C contains only O(D) points that are incident to at least t curves of C. Consider first incidences between the non-exceptional curves in C and the rich points (those incident to at least t curves of C). Each surface σ S intersect V in a curve of degree at most DE, and can therefore contain at most O(DE) curves from C. Each of these curves contains at most O(D) t-rich points, for a total of O(D 2 E) incidences for each surface σ S, so the overall number of incidences of this kind is O(nD 2 E) = O(n). Note that this is a bound on the actual number of point-surface incidences. We include the incident pairs of this kind in G 0 (P, S), and the resulting bound O(n) is clearly subsumed by the asserted bound in (8) or (9). Removing these edges from the complete bipartite decomposition, the remaining incidences counted in I(P, C) are estimated as follows. The number of incidences with the t-poor points (each lying on at most 24

245 t curves of C) is at most mt = O(m), and the number of incidences between the O(D 2 ) exceptional curves and the points of P is at most O(mD 2 ) = O(m), for a total of O(m) incidences. In summary, the complete bipartite decomposition that we end up with is of the form γ P γ S γ, where γ ranges over the curves of C, and (i) for each of the O(D 2 ) exceptional curves γ we have P γ = P γ, and (ii) for each of the non-exceptional curves γ, P γ is the set of the t-poor points of P that lie on γ. We thus obtain that γ C P γ = O(m). As V is irreducible and does not contain any of the surfaces σ S (except possibly for at most one, which then coincides with V and which we may ignore, as before), the preceding argument implies again that each σ S generates at most DE curves of C, so γ S γ nde = O(n). This gives us the complete bipartite graph decomposition portion of the representation in (7), which satisfies all the properties asserted in the theorem. Let G 0 (P, S) denote the remainder of the incidence graph. For the moment, ignore the pairs involving the t-rich points on the curves of C, which are also part of the final G 0 (P, S). For each σ S, put γ σ := (σ V ) \ C. As just noted, each γ σ is at most one-dimensional (i.e., a curve). By construction, it does not contain any curve in C, and it might also be empty (for this or for other reasons). Note that if σ V does contain a curve γ in C, then the incidences between σ and the points of P on γ are all already recorded in P γ S γ, or are the O(n) special incidences with t-rich points on the curves in C, so ignoring them is safe. Finally, we may ignore the isolated points of γ σ, because, as already argued, each curve γ σ can contain at most O(1) such points, which contribute a total of at most O(n) incidences with their corresponding surfaces. Let G 0 (P, S) continue to denote the remaining portion of G(P, S), after pruning away all the incidences already accounted for. Put I 0 (P, S) = G 0 (P, S). Let Γ denote the set of the n curves γ σ, for σ S (and notice that this time it is an actual set, not a multiset). The curves of Γ are algebraic curves of degree at most DE, and, as is easily checked, any pair of curves γ σ, γ σ Γ intersect in at most min{de 2, E 4 } = O(1) points. Note that I 0 (P, S) is equal to the number I(P, Γ) of incidences between the points of P and the curves of Γ. To bound the latter quantity, we proceed exactly as in the preceding proof, bifurcating according to whether F is of reduced dimension s with respect to V or has k degrees of freedom with respect to V. In both cases we project the points and curves onto some generic plane π 0, and bound the number of incidences between the projected points and curves, using either Theorem 1.1 (for families with k degrees of freedom) or Theorem 1.2 (for s-dimensional families), obtaining the respective bounds asserted in (8) or in (9). 6 Distinct and repeated distances in three dimensions In this section we prove Theorems 1.10 and 1.11, the applications of Theorems 1.7 and 1.8 to distinct and repeated distances in three dimensions; see also our earlier work [58] that handles these problems in a somewhat different manner. The theorems present four results, in each of which the problem is reduced to one involving incidences between spheres and points on a surface V. However, except for Theorem 1.10(b), the spheres that arise in the other three cases are restricted, by requiring their centers to lie on V and / or to have a fixed radius. This makes the number of degrees of freedom (with respect to the variety) and the dimensionality of the corresponding families of spheres go down to 3 or 2. The case of two degrees of freedom (in Theorem 1.11(a)) is the simplest, and requires very little of the machinery developed here (see below). The cases of three degrees of freedom (in Theorem 1.10(a) and Theorem 1.11(b)) yield improved in-between bounds. Proof of Theorem 1.10 (distinct distances). We will first establish the more general bound in (b); handling (a) will be done later, in a similar, somewhat simpler manner. (b) Let t denote the number of distinct distances in P 1 P 2. For each q P 2, draw t spheres centered at 25

246 q and having as radii the t distinct distances. We get a collection S of nt spheres, a set P 1 of m points on V, which we relabel as P, to simplify the notation, and exactly mn incidences between the points of P and the spheres of S. Let C denote the set of intersection circles of pairs of spheres from S that are contained in V, counted without multiplicity; we keep in C only circles that contain points of P 1. For each γ C, let µ(γ) 2 denote its multiplicity, namely the number of spheres containing γ; note that µ(γ) is equal to the number of points of P 2 that lie on the axis of γ, namely the line that passes through the center of γ and is orthogonal to the plane containing γ. The maximum possible multiplicity of a circle is at most 2t, because the distances of the corresponding centers in P 2 to the points of P 1 γ are all distinct, up to a possible multiplicity of 2. For each k, let C k (resp., C k ) denote the subset of circles in C of multiplicity exactly (resp., at least) k, and put N k := C k, N k := C k. In order to effectively apply the bound in Theorem 1.9, we first have to control the term γ P γ S γ, which arises since we deal here with the actual number of incidences. Specifically, we claim that most of the mn incidences do not come from this part of the incidence graph, unless t = Ω(n). Indeed, write this sum as P γ S γ = k P γ. k 2 γ C k Putting E k := γ C k P γ, and E k := γ C k P γ, we then have By Theorem 1.8, we have E k = O(m), so we have γ P γ S γ = ke k = 2E 2 + E k. γ k 2 k 3 P γ S γ = 2E 2 + ( ) 2t E k = O m + m = O(mt). k 3 γ If this would have accounted for more than, say, half the incidences, we would get mn = O(mt), or t = Ω(n), as claimed, and then (a much larger lower bound than) the bound in the theorem would follow. We may thus ignore this term, and write ( ) mn = O m 1/2 (nt) 7/8+ε + m 2/3 (nt) 2/3 + m + nt, k=3 for any ε > 0, or ( { }) t = Ω min m 4/(7+8ε) n (1 8ε)/(7+8ε), m 1/2 n 1/2, m, which, by replacing ε by another, still arbitrarily small ε, becomes the bound asserted in the theorem (and is also smaller than the bound for the complementary situation treated above). (a) Here we are in a more favorable situation, because the spheres in S belong to a three-dimensional family of surfaces a family that can be represented simply as V R. We can therefore apply Theorem 1.8 with dimensionality s = 3 (which is the actual dimensionality of the family, not the reduced one with respect to V ), arguing first that, as in the proof of (b), we may ignore the term γ P γ S γ in the bound on I(P, S), which is negligible unless t = Ω(n). We thus get the inequality ( ) n 2 = O n 6/11 (nt) 9/11+ε + n 2/3 (nt) 2/3 + nt, for any ε > 0, which yields t = Ω(n 7/(9+11ε) ), which, by replacing ε, as in the proof of (b), can easily be massaged into the bound asserted in the theorem. 26

247 Proof of Theorem 1.11 (repeated distances). Consider (a) first. Following the standard approach to problems involving repeated distances, we draw a unit sphere s p around each point p P, and seek an upper bound on the number of incidences between these spheres and the points of P ; this latter number is exactly twice the number of unit distances determined by P. This instance of the problem has several major advantages over the general analysis in Theorem 1.9. First, in this case the incidence graph G(P, S) cannot contain K 3,3 as a subgraph, eliminating altogether the complete bipartite graph decomposition in (10) (or, rather, bounding the overall number of edges in these subgraphs by O(n)). More importantly, the family of the fixed-radius spheres whose centers lie on V is 2-dimensional and has two degrees of freedom, which leads to the standard Szemerédi-Trotter-like bound I(P, S) = O( P 2/3 S 2/3 + P + S ) = O(n 4/3 ), so the number of repeated distances in this case is O(n 4/3 ), as claimed. (The last bound can be obtained by applying Theorem 1.7, but it can also be obtained more directly, e.g., via Székely s technique [65].) We remark that the last bound does not use much of the machinery developed in this paper. Still, we are not aware of any previous claim of the bound for the general case of points on a surface; see Brass et al. [15, Section 5.2] and Brass [13] for a discussion of closely-related problems. We now consider (b). Again, we reduce the problem to that of bounding the number of incidences between the m points of P 1, which lie on V, and the n unit spheres centered at the points of P 2. Here too the overall number of edges in the complete bipartite graph decomposition is O(m + n), so we can ignore this part of the bound. In this case, the family of unit spheres is 3-dimensional. Applying the same reasoning as in the proof of Theorem 1.7, we conclude that the number of unit distances in this case is ( ) O m 6/11 n 9/11+ε + m 2/3 n 2/3 + m + n, for any ε > 0, as claimed. 7 Proof of Theorem 1.12 (surfaces and arbitrary points) The proof establishes the bound in (15), via induction on m, with a prespecified fixed parameter ε > 0. Concretely, we claim that, for any such choice of ε > 0, we can write G(P, S) = G 0 (P, S) γ Γ 0 (P γ S γ ), where Γ 0 is a collection of distinct constant-degree irreducible algebraic curves in R 3, and, for each γ Γ 0, P γ = P γ and S γ is the set of surfaces in S that contain γ. We only include in Γ 0 curves γ with S γ 2 (as the cases where S γ = 1 will be swallowed in G 0 (P, S)), so each γ Γ 0 is an irreducible component of an intersection curve of at least two surfaces from S, and its degree is therefore at most E 2. We then claim that J(P, S) := ( Pγ + S γ ) ) A (m 2s 3s 3 3s 1 n 3s 1 +ε + m + n, (26) γ Γ 0 and G 0 (P, S) A(m + n), for a suitable constant A that depends on ε, s, E, D, and the complexity of the family F. The base cases are when m n 1/s or when m m 0, for some sufficiently large constant m 0 that will be set later. Consider first the case m n 1/s. Note that in this case the right-hand side of (26) is O(n 1+ε ). We 27

248 will actually establish the bound O(n) on both J(P, S) and G 0 (P, S), as follows. Since the surfaces of S come from an s-dimensional family, a suitable extension of the analysis in Sharir and Zahl [60, Lemma 3.2] shows that there exists an s-dimensional real parametric space R s, and a duality mapping that sends each surface σ S to a point σ R s, and sends each point p P to a constant-degree algebraic hypersurface p in that space, so that if p is incident to σ then σ is incident to p. This holds with the exception of at most O(1) bad points in P and at most O(1) bad surfaces in S, and the constants depend on s, E, and the complexity of F. The contribution to J(P, S), or rather to I(P, S), of the bad points and surfaces is only O(m + n), so we can ignore it, or, rather, place these incidences in the remainder subgraph G 0 (P, S). Construct the arrangement of the dual surfaces p, for p P. Its complexity is O(m s ) = O(n), and this bound also holds if we count each face of the arrangement, of any dimension, with multiplicity equal to the number of surfaces that contain it. For each such (relatively open) face 11 f, we form the complete bipartite graph P f S f, where S f is the set of all surfaces σ such that σ f, and P f is the set of all points p whose dual surface p contains f. We have f S f n, and f P f = O(m s ) = O(n). Back in primal space, if S f 2, then all the points of P f lie in the intersection γ f := σ S f σ, which, by Bézout s theorem, is either one-dimensional, i.e., a curve in Γ 0 as in the theorem, or a discrete set of at most E 3 points. In the latter case P f E 3 = O(1), implying that f P f S f = O( f S f ) = O(n). Similarly, if S f = 1, then we also have f P f S f = O( f P f ) = O(m s ) = O(n). Clearly, f P f S f, over faces f for which either S f 1 or γ f is discrete, counts the number of incidences between P and S that fall into these special cases, so the number of these incidences is only O(n). We are left with a portion of the incidence graph that can be written as the union of complete bipartite graphs f P f S f, over faces f for which S f 2 and γ f is a curve in Γ 0. This union is of the form asserted in the theorem, and the corresponding J(P, S) is O(n). The asserted bound thus holds by choosing A sufficiently large. The case m m 0 follows easily (since in this case we have I(P, S) m 0 n) if we choose A sufficiently large. This holds for any choice of m 0 (and a corresponding choice of A); the value that we choose is specified later. Suppose then that (26) holds for all sets P, S, with P < m, and consider the case where the sets P, S are of respective sizes m, n, and we have m > n 1/s and m > m 0. Before continuing, we also dispose of the case m n 3. In this case we consider the arrangement A(S) (in R 3 ) of the surfaces in S. The complexity of A(S) is O(n 3 ) = O(m). More precisely, this bound holds, and is asymptotically tight, for surfaces in general position. In our case, S is likely not to be in general position, and then the complexity of A(S) might be smaller, because vertices and edges might be incident to many surfaces. Nevertheless, if we count each vertex and edge of A(S) with its multiplicity, we still get the complexity upper bound O(n 3 ). (Here we reason in complete analogy with the dual s-dimensional case treated above.) This means that the number of incidences with points that are either vertices or lie on the (relatively open) 2-faces of A(S) is O(n 3 + m) = O(m). Incidences with points that lie on the (relatively open) edges of A(S) (note that each such edge is a portion of some curve of intersection between at least two surfaces of S) are recorded, as usual, by a complete bipartite graph decomposition γ (P γ S γ ), where the curves γ are as stipulated in the theorem, and where, as just argued, we have γ P γ m and γ S γ = O(n 3 ) = O(m). This implies that (26) holds in this case. Thus, in what follows, we assume that m n 3. Since we also assume that m > m 0, we have n m 1/3 > n 0 := m 1/3 0. Applying the polynomial partitioning technique. We fix a sufficiently large constant parameter D m 1/3, whose concrete choice will be specified later, and apply the polynomial partitioning technique 11 Technically, rather than considering individual faces f, we should consider the full varieties that contain these faces and are obtained by intersecting subsets of the surfaces p, where each such intersection might contain many faces f. However, in doing so, we want to exclude faces that lie on such an intersection and are of dimension smaller than that of the intersection (because they lie on other surfaces too), and treat them separately. To simplify the presentation, we ignore this modification, which does not affect the asymptotic bound that is derived here. 28

249 of Guth and Katz [39]. We obtain a polynomial f R[x, y, z] of degree at most D, whose zero set Z(f) partitions 3-space into O(D 3 ) (open) connected components (cells), and each cell contains at most O(m/D 3 ) points. By duplicating cells if necessary 12, we may also assume that each cell is crossed by at most O(n/D) surfaces of S; this duplication keeps the number of cells O(D 3 ) (because each surface crosses only O(D 2 ) cells, a well known property that follows, e.g., from Warren s theorem [68]). 13 That is, we obtain at most ad 3 subproblems, for some absolute constant a, each associated with some cell of the partition, so that, for each i ad 3, the i-th subproblem involves a subset P i P and a subset S i S, such that m i := P i b 0 m/d 3 and n i := S i bn/d, for another absolute constant b 0 and a constant b that depends on E. Set P 0 := P Z(f) and P = P \ P 0. We have J(P, S) J(P 0, S) + J(P, S). (27) We first bound J(P 0, S). Similarly to what was done earlier, decompose Z(f) into its O(D) irreducible components, assign each point of P 0 to every component that contains it, and assign the surfaces of S to all components. We now fix a component, and bound the vertex count in the complete bipartite graph decomposition involving incidences between the points and surfaces assigned to that component; J(P 0, S) is at most D times the bound that we get. We may therefore assume that Z(f) is irreducible. Since deg(z(f)) D is a constant, Theorem 1.7 implies that we can write G(P 0, S) as γ P 0γS γ, over a suitable set of curves γ V, with P 0γ = P 0 γ and S γ is the set of surfaces in S containing γ, and we have ) J(P 0, S) = O (m 2s 5s 6 5s 4 n 5s 4 +ε + m 2/3 n 2/3 + m + n, where the constant of proportionality depends on ε, s, D, E, and the complexity of F. (Here, as in Theorem 1.7, the contribution of S to this bound, namely to γ S γ, is only O(n). Unfortunately, this no longer holds in the estimation of J(P, S), given, so we ignore this improvement). As is easily checked, this bound is subsumed in (26) for m n 1/s (and s 3), if we choose A sufficiently large (so here, as in all the other steps, A depends on all the parameters just listed). Finally, we estimate By the induction hypothesis, we get, for each i, ( J(P i, S i ) A m J(P, S) J(P i, S i ). 2s 3s 1 i ad 3 i=1 ) 3s 3 3s 1 n +ε i + m i + n i. Summing this over i, we get ( ) J(P, S) A ad 3 (b 0 m/d 3 ) 2s 3s 3 3s 1 (bn/d) 3s 1 +ε + (b 0 m/d 3 ) + (bn/d) = Aab 2s 3s 1 0 b 3s 3 D ε 3s 1 +ε m 2s 3s 1 n 3s 3 3s 1 +ε + Aab 0 m + AabD 2 n. We note that m 2s 3s 3 3s 1 n 3s 1 +ε n ε m and m 2s 3s 3 3s 1 n 3s 1 +ε n ε n for n 1/s m n 3. We choose D sufficiently large so that D ε 4Aab 2s 3s 1 0 b 3s 3 3s 1 +ε, and then the bound is at most ( A 4 + Aab ) 0 n ε + AabD2 n ε m 2s 3s 3 3s 1 n 3s 1 +ε. 12 By using Guth s recent technique for partitioning sets of varieties [35], already mentioned earlier, we could do without this cell duplication step. 13 In actuality, the bound is O(D 2 E 2 ), because the intersection curve σ Z(f), for any σ S, is of degree at most DE. Here we treat E as a much smaller quantity than D, and bear in mind that the relevant constants may depend on E. 29

250 Choosing n 0 (that is, m 0 ) sufficiently large, so that n ε 0 4a max{b 0, bd 2 }, we ensure that, for n n 0, A 4 + Aab 0 n ε + AabD2 n ε A 4 + A 4 + A 4 = 3A 4. Adding the bounds for J(P 0, S), and choosing A sufficiently large, we get that (26) holds for P and S. This establishes the induction step and thereby completes the proof. Remark. The dimensionality s of S is used in the proof in two different steps, once in establishing a linear bound when m < n 1/s, and once in deriving the bound on J(P 0, S), using Theorem 1.7. As in the remark following Theorem 1.7, the values of s used in these two steps need not be the same, and the latter one is typically smaller (because it is reduced, with respect to intersection curves with a variety). Unfortunately, this in itself does not lead to an improvement in the bound, because the leading term m 2s 3s 3 3s 1 n 3s 1 +ε depends on the former value of s. If this value of s could also be improved, say by additional assumptions on the points and/or the surfaces, the bound in the theorem would improve too. 8 Discussion In this paper we have made significant progress on major incidence problems involving points and curves and points and surfaces in three dimensions. We have also obtained several applications of these results to problems involving repeated and distinct distances in three dimensions, with significantly improved lower and upper bounds, in cases where the points, or in the bipartite versions, the points in one of the two given sets, lie on a constant-degree algebraic surface. The study in this paper raises several interesting open problems. (i) A long-standing open problem is that of establishing the lower bound of Ω(n 2/3 ) for the number of distinct distances determined by a set of n points in R 3, without assuming them to lie on a constant-degree surface. The best known lower bound, close to Ω(n 3/5 ), which follows from the work of Solymosi and Vu [63], still falls short of this bound. In the present study we have obtained some partial results (with better lower bounds) for cases where (all or some of) the points do lie on such a surface. We hope that some of the ideas used in this work could be applied in more general contexts, or in other special situations. (ii) Another major long-standing open problem is that of improving the upper bound O(n 3/2 ), established in [43, 69], on the number of unit distances determined by a set of n points in R 3, again without assuming (all or some of) them to lie on a constant-degree surface. It would be interesting to make progress on this problem. (iii) As remarked above, a challenging open problem is to characterize all the surfaces that are infinitely ruled by algebraic curves of degree at most E (or by certain classes thereof), extending the known characterizations for lines and circles. A weaker, albeit still hard problem is to reduce the upper bound 100E 2 on the degree of such a surface, perhaps all the way down to E, or at least to O(E). (iv) It would also be interesting to find additional applications of the results of this paper, like the one with an improved bound on the number of similar triangles in R 3, given in Section 3. One direction to look at is the analysis of other repeated patterns in a point set, such as higher-dimensional congruent or similar simplices, which can sometimes be reduced to point-sphere incidence problems; see [1, 3]. (v) Concerning Theorem 1.12, we note that it is stated only for families F of surfaces of a given dimensionality s. It would be interesting to obtain a variant in which we assume instead that F has k degrees of freedom, after one comes up with a definition of this notion that is both (a) natural and simple to state, and (b) makes (a suitable variant of) the analysis work. We are currently studying such an extension. 30

251 (vi) A potentially weak issue in our analysis, manifested in the proof of all our main theorems, is that in order to bound the number of incidences between points and curves on some variety V of constant degree, we project the points and curves on some generic plane and use a suitable planar bound, from Theorem 1.1 or Theorem 1.2, to bound the number of incidences between the projected points and curves. It would be very interesting if one could obtain an improved bound, exploiting the fact that the points and curves lie on a variety V in R 3, under suitable (natural) assumptions on V. (vii) Finally, it would be challenging to extend the results of this paper to higher dimensions. References [1] P. K. Agarwal, R. Apfelbaum, G. Purdy and M. Sharir, Similar simplices in a d-dimensional point set, Proc. 23rd Annu. ACM Sympos. Comput. Geom. (2007), [2] P. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky, Lenses in arrangements of pseudocircles and their applications, J. ACM 51 (2004), [3] P. Agarwal and M. Sharir, On the number of congruent simplices in a point set, Discrete Comput. Geom. 28 (2002), [4] R. Apfelbaum and M. Sharir, Large bipartite graphs in incidence graphs of points and hyperplanes, SIAM J. Discrete Math. 21 (2007), [5] R. Apfelbaum and M. Sharir, Non-degenerate spheres in three dimensions, Combinat. Probab. Comput. 20 (2011), [6] B. Aronov, V. Koltun and M. Sharir, Incidences between points and circles in three and higher dimensions, Discrete Comput. Geom (2005), [7] B. Aronov, J. Pach, M. Sharir and G. Tardos, Distinct distances in three and higher dimensions, Combinat. Probab. Comput. 13 (2004), [8] B. Aronov, M. Pellegrini and M. Sharir, On the zone of a surface in a hyperplane arrangement, Discrete Comput. Geom. 9 (1993), [9] B. Aronov and M. Sharir, Cutting circles into pseudo-segments and improved bounds for incidences, Discrete Comput. Geom. 28 (2002), [10] A. Basit and A. Sheffer, Incidences with k-non-degenerate sets and their applications, J. Comput. Geom. 5 (2014), [11] S. Basu and M. Sombra, Polynomial partitioning on varieties of codimension two and pointhypersurface incidences in four dimensions, Discrete Comput. Geom (2016), [12] L. Bien, Incidences between points and curves in the plane, M.Sc. Thesis, School of Computer Science, Tel Aviv University, [13] P. Brass, Exact point pattern matching and the number of congruent triangles in a threedimensional point set, Proc. European Sympos. Algorithms, 2000, Springer LNCS 1879, pp [14] P. Brass and Ch. Knauer, On counting point-hyperplane incidences, Comput. Geom. Theory Appls. 25 (2003), [15] P. Brass, W. Moser and J. Pach, Research Problems in Discrete Geometry, Springer Verlag, New York,

252 [16] P. Bürgisser, M. Clausen and A. Shokrollahi, Algebraic Complexity Theory, Vol. 315, Springer Science and Business Media, Berlin-Heidelberg, [17] J. Cardinal, M. Payne, and N. Solomon, Ramsey-type theorems for lines in 3-space, Discrete Math. Theoret. Comput. Sci. 18 (2016). [18] T.M. Chan, On levels in arrangements of curves, Discrete Comput. Geom. 29 (2003), [19] T.M. Chan, On levels in arrangements of curves, II: A simple inequality and its consequences, Discrete Comput. Geom. 34 (2005), [20] B. Chazelle, Cuttings, in Handbook of Data Structures and Applications, (D. P. Mehta and S. Sahni, Eds.), Chapman and Hall/CRC, [21] B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geometry, Combinatorica 10 (1990), [22] F. R. K. Chung, Sphere-and-point incidence relations in high dimensions with applications to unit distances and furthest-neighbor pairs, Discrete Comput. Geom. 4 (1989), [23] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), [24] D. Cox, J. Little and D. O Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer Verlag, Heidelberg, [25] H. Edelsbrunner, L. Guibas and M. Sharir, The complexity of many cells in arrangements of planes and related problems, Discrete Comput. Geom. 5 (1990), [26] Gy. Elekes and Cs. D. Tóth, Incidences of not-too-degenerate hyperplanes, Proc. 21st Annu. ACM Sympos. Comput. Geom., 2005, [27] J. Ellenberg, J. Solymosi, and J. Zahl, New bounds on curve tangencies and orthogonalities, Discrete Analysis, 22 (2016), [28] P. Erdős, On sets of distances on n points in Euclidean space, Magyar Tud. Akad. Mat. Kutató Int. Kozl. 5 (1960), [29] P. Erdős, D. Hickerson, and J. Pach, A problem of Leo Moser about repeated distances on the sphere, Amer. Math. Monthly 96 (1989), [30] J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl, A semi-algebraic version of Zarankiewicz s problem, J. European Math. Soc., to appear. Also in arxiv: [31] D. Fuchs and S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, Amer. Math. Soc. Press, Providence, RI, [32] W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, Expository Lectures from the CBMS Regional Conference Held at George Mason University, June 27 July 1, 1983, Vol. 54. AMS Bookstore, [33] M. Green and I. Morrison, The equations defining Chow varieties, Duke Math. J. 53 (1986), [34] L. Guth, Distinct distance estimates and low-degree polynomial partitioning, Discrete Comput. Geom. 53 (2015), [35] L. Guth, Polynomial partitioning for a set of varieties, Math. Proc. Cambridge Phil. Soc. 159 (2015), Also in arxiv:

253 [36] L. Guth, Ruled surface theory and incidence geometry, in arxiv: [37] L. Guth, Polynomial Methods in Combinatorics, University Lecture Series, Vol. 64, Amer. Math. Soc. Press, Providence, RI, [38] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem, Advances Math. 225 (2010), Also in arxiv: v1. [39] L. Guth and N. H. Katz, On the Erdős distinct distances problem in the plane, Annals Math. 181 (2015), Also in arxiv: [40] L. Guth and J. Zahl, Algebraic curves, rich points, and doubly-ruled surfaces, Amer. J. Math., in press. Also in arxiv: [41] C. G. A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), [42] J. Harris, Algebraic Geometry: A First Course, Vol. 133, Springer-Verlag, New York, [43] H. Kaplan, J. Matoušek, Z. Safernová and M. Sharir, Unit distances in three dimensions, Combinat. Probab. Comput. 21 (2012), Also in arxiv: [44] J. Kollár, Szemerédi Trotter-type theorems in dimension 3, Advances Math. 271 (2015), Also in arxiv: [45] N. Lubbes, Families of circles on surfaces, Contributions to Algebra and Geometry, accepted. Also in arxiv: [46] A. Marcus and G. Tardos, Intersection reverse sequences and geometric applications, J. Combinat. Theory Ser. A 113 (2006), [47] J. Matoušek, Lectures on Discrete Geometry, Springer Verlag, Heidelberg, [48] F. Nilov and M. Skopenkov, A surface containing a line and a circle through each point is a quadric, Geom. Dedicata 163:1 (2013), [49] J. Pach and M. Sharir, On the number of incidences between points and curves, Combinat. Probab. Comput. 7 (1998), [50] O. E. Raz, M. Sharir, and F. de Zeeuw, Polynomials vanishing on Cartesian products: The Elekes Szabó theorem revisited, Duke Math. J (2016), [51] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2, 5th edition, Hodges, Figgis and co. Ltd., Dublin, [52] J. Schicho, The multiple conical surfaces, Contributions to Algebra and Geometry 42.1 (2001), [53] M. Sharir, A. Sheffer, and N. Solomon, Incidences with Curves in R d, Electronic J. Combinat. 23(4): (2016). Also in Proc. European Sympos. Algorithms (2015), Springer LNCS 9294, pp , and in arxiv: [54] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between points and circles, Combinat. Probab. Comput. 24 (2015), Also in arxiv: [55] M. Sharir and N. Solomon, Incidences between points and lines in R 4, Discrete Comput. Geom. 57(3) (2017), Also in Proc. 56th IEEE Sympos. Foundations of Computer Science 2015, , and in arxiv: (A preliminary version appeared in Proc. 30th Symp. on Computational Geometry (2014), ) 33

254 [56] M. Sharir and N. Solomon, Incidences between points and lines in three dimensions, in Intuitive Geometry (J. Pach, Ed.), to appear. Also in Proc. 31st Annu. Sympos. Computat. Geom. (2015), , and in arxiv: [57] M. Sharir and N. Solomon, Incidences between points on a variety and planes in three dimensions, in arxiv: [58] M. Sharir and N. Solomon, Distinct and repeated distances on a surface and incidences between points and spheres, in arxiv: [59] M. Sharir and N. Solomon, Incidences between points and lines on two- and three- dimensional varieties, in arxiv: [60] M. Sharir and J. Zahl, Cutting algebraic curves into pseudo-segments and applications, J. Combinat. Theory Ser. A 150 (2017), [61] A. Sheffer, Distinct distances: Open problems and current bounds, in arxiv:1406:1949. [62] M. Skopenkov and R. Krasauskas, Surfaces containing two circles through each point and Pythagorean 6-tuples, in arxiv: [63] J. Solymosi and V. H. Vu, Near optimal bounds for the Erdős distinct distances problem in high dimensions, Combinatorica 28 (2008), [64] J. Spencer, E. Szemerédi, and W. T. Trotter, Unit distances in the Euclidean plane, in: Graph Theory and Combinatorics (B. Bollobás, ed.), Academic Press, New York, 1984, [65] L. Székely, Crossing numbers and hard Erdős problems in discrete geometry, Combinat. Probab. Comput. 6 (1997), [66] E. Szemerédi and W.T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), [67] T. Tao, [68] H. E. Warren, Lower bound for approximation by nonlinear manifolds, Trans. Amer. Math. Soc. 133 (1968), [69] J. Zahl, An improved bound for the number of point-surface incidences in three dimensions, Contrib. Discrete Math. 8(1) (2013), [70] surjective-morphism-of-affine-varieties-and-dimension. A On surfaces ruled by curves In this appendix we review, and sketch the proofs, of several tools from algebraic geometry that are required in our analysis, the main one of which is Theorem These tools are presented in Guth and Zahl [40, Section 6], but we reproduce them here, in a somewhat sketchy form, for the convenience of the reader and in the interest of completeness. We work over the complex field C, but the results here also apply to our setting over the real numbers (see [40, 55] and a preceding remark for discussions of this issue). A subset of C N described by some polynomial equalities and one non-equality, of the form {p C N f 1 (p) = 0,..., f r (p) = 0, g(p) 0}, for f 1,..., f r, g C[x 1,..., x N ], 34

255 is called locally closed. We recall that the (geometric) degree of an algebraic variety V C N is defined as the number of intersection points of V with the intersection of N dim(v ) hyperplanes in general position (see, e.g., Harris [42, Definition 18.1]). Locally closed sets have the following property. Theorem A.1 (Bézout s inequality; Bürgisser et al. [16, Theorem 8.28]). Let V be a nonempty locally closed set in C N, and let H 1,..., H r be algebraic hypersurfaces in C N. Then deg(v H 1 H r ) deg(v ) deg(h 1 ) deg(h r ). A constructible set C is easily seen to be a union of locally closed sets. Moreover, one can decompose C uniquely as the union of irreducible locally closed sets (namely, sets that cannot be written as the union of two nonempty and distinct locally closed sets). By Bürgisser et al. [16, Definition 8.23]), the degree of C is the sum of the degrees of its irreducible locally closed components. Theorem A.1 implies that when a constructible set C has complexity O(1), its degree is also O(1). We also have the following corollaries. Corollary A.2. Let X = {p C N f 1 (p) = 0,..., f r (p) = 0, g(p) 0}, for f 1,..., f r, g C[x 1,..., x N ]. If X contains more than deg(f 1 ) deg(f r ) points, then X is infinite. Proof. Assume that X is finite. Put V = {p C n g(p) 0}. By Bézout s inequality (Theorem A.1), we have deg(x) deg(v ) deg(f 1 ) deg(f r ) = deg(f 1 ) deg(f r ), where the equality deg(v ) = 1 follows by the definition of the degree of locally closed sets (see, e.g., Bürgisser et al. [16, Definition 8.23]). When X is finite, i.e., zero-dimensional, its degree is equal to the number of points in it, counted with multiplicities. This implies that the number of points in X is at most deg(f 1 ) deg(f r ), contradicting the assumption of the theorem. Therefore, X is infinite. As an immediate consequence, we also have: Corollary A.3. Let C C N be a constructible set and write it as the union of locally closed sets t i=1 X i, where X i = {p C N f i 1(p) = 0,..., f i r i (p) = 0, g i (p) 0}, for f i 1,..., f i r i, g i C[x 1,..., x N ]. If C contains more than t i=1 deg(f i 1 ) deg(f i r i ) points, then C is infinite. For a constructible set C, let d(c) denote the minimum of t i=1 deg(f i 1 ) deg(f i r i ), as in Corollary A.3, over all possible decompositions of C as the union of locally closed sets. By Bézout s inequality (Theorem A.1), it follows that deg(c) d(c). Corollary A.3 implies that if C contains more than d(c) points, then it is infinite. Following Guth and Zahl [40, Section 4], we call an algebraic curve γ C 3 a complete intersection if γ = Z(P, Q) for some pair of polynomials P, Q. We let C[x, y, z] E denote the space of complex trivariate polynomials of degree at most E, and choose an identification of C[x, y, z] E with 14 C (E+3 3 ). We use the variable α to denote an element of (C[x, y, z] E ) 2, and write α = (P α, Q α ) (C[x, y, z] E ) 2 = (C (E+3 3 ) ) Here ( ) E+3 3 is the maximum number of monomials of the polynomials that we consider. For obvious reasons, the actual representation should be in the complex projective space CP (E+3 3 ), but we use the many-to-one representation in C ( E+3 3 ) for convenience. 35

256 Given an irreducible curve γ, we associate with it a choice of α (C (E+3 3 ) ) 2 such that γ is contained in Z(P α, Q α ), and the latter is a curve (one can show that such an α always exists; see Guth and Zahl [40, Lemma 4.2] and also Basu and Sombra [10]). Let x γ be a non-singular point 15 of γ; we say that α is associated to γ at x, if α is associated to γ, and P α (x) and Q α (x) are linearly independent. We refer the reader to [40, Definition 4.1 and Lemma 4.2] for details. This is analogous to the works of Guth and Katz [39] and of Sharir and Solomon [55] for the special cases of parameterizing lines in three and four dimensions, respectively. Before applying this machinery to derive the main result of this appendix, we fill in the missing part in the proof of Theorem 1.8. Lemma A.4. Let V be some irreducible algebraic surface in C 3, and let F be a constructible family of algebraic surfaces in C 3 of constant degree E. Let C be the family of the irreducible components of the intersection curves of pairs of surfaces in F. Then C is a constructible family of curves. Proof. We use the preceding identification of C[x, y, z] E with C (E+3 3 ), and use the variable α to denote an element of (C[x, y, z] E ) 2, writing, as before Since F is constructible, the following set α = (P α, Q α ) (C[x, y, z] E ) 2 = (C (E+3 3 ) ) 2. W = {α = (P α, Q α ) (C[x, y, z] E ) 2 P α, Q α F} is also contructible. By definition, recalling that C 3,D denotes the set of all irreducible algebraic curves of degree at most D in C 2, we have C = {γ C 3,E 2 α W, γ P α Q α }. It then follows that C is irreducible by a simple variant of Lemma 4.3 of Guth and Zahl [40]. We now go on to the main result of the appendix. In what follows, we fix a constructible set C 0 C 3,E of irreducible curves of degree at most E in 3-dimensional space (recall that the entire family C 3,E is constructible). Following [40, Section 9], we call a point p Z(f), for a given polynomial f C[x, y, z], a (t, C 0, r)-flecnode, if there are at least t curves γ 1,..., γ t C 0, such that, for each i = 1,..., t, (i) γ i is incident to p, (ii) p is a non-singular point of γ i, and (iii) γ i osculates to Z(f) to order r at p. This is a generalization of the notion of a flecnodal point, due to Salmon [51, Chapter XVII, Section III] (see also [39, 55] for details). With all this machinery, we can now present a (sketchy) proof of Theorem The theorem is stated in Section 1, and we recall it here. It is adapted from Guth and Zahl [40, Corollary 10.2], serves as a generalization of the Cayley Salmon theorem on surfaces ruled by lines (see, e.g., Guth and Katz [39]), and is closely related to Theorem 1.3 (also due to Guth and Zahl [40]). Theorem For given integer parameters c, E, there are constants c 1 = c 1 (c, E), r = r(c, E), and t = t(c, E), such that the following holds. Let f be a complex irreducible polynomial of degree D E, and let C 0 C 3,E be a constructible set of complexity at most c. If there exist at least c 1 D 2 curves of C 0, such that each of them is contained in Z(f) and contains at least c 1 D points on Z(f) that are (t, C 0, r)-flecnodes, then Z(f) is infinitely ruled by curves from C 0. In particular, if Z(f) is not infinitely ruled by curves from C 0 then, except for at most c 1 D 2 exceptional curves, every curve in C 0 that is fully contained in Z(f) is 15 Given an irreducible curve in R 3, a point x γ is non-singular if there are polynomials f 1, f 2 that vanish on γ such that f 1(x) and f 2(x) are linearly independent. 36

257 incident to at most c 1 D points that are incident to at least t curves in C 0 that are also fully contained in Z(f). Proof. For the time being, let r be arbitrary. By Guth and Zahl [40, Lemma 8.3 and Equation (8.1)], since f is irreducible, there exist r polynomials 16 h j (α, p) C[α, x, y, z], for j = 1,..., r of degree at most b j in α (where b j is a constant depending on j and on E), and of degree O(D) in p = (x, y, z), with the following property: let γ be an irreducible curve, let p be a non-singular point of γ, and let α be associated to γ at p, then γ osculates to Z(f) to order r at p if and only if h j (α, p) = 0 for j = 1,..., r. (These polynomials are suitable representations of the first r terms of the Taylor expansion of f at p along γ; see [40, Section 6.2] for this definition, and also [39, 55] for the special cases of lines in R 3 and R 4, respectively.) Regarding p as fixed, the system h j (α, p) = 0, for j = 1,..., r, in conjunction with ( the constructible r condition that α C 0, defines a constructible set C p. By definition, we have d(c p ) j=1 j) b d(c 0 ), which is a constant that depends only on r and E. By Corollary A.3, C p is either infinite or contains at most d(c p ) = O(1) points. By Guth and Zahl [40, Corollary 12.1], there exist a Zariski open set O, and a sufficiently large constant r 0, that depend on C 0 and E (see [40, Theorem 8.1] for the way r 0 is obtained), such that if p O is a (t, C 0, r)-flecnode, with r r 0, there are at least t curves that are incident to p and are fully contained in Z(f). Since, by assumption, there are at least c 1 D 2 curves, each containing at least c 1 D (t, C 0, r)-flecnodes, it follows from [40, Proposition 10.2] that there exists a Zariski open subset O of Z(f), all of whose points are (t, C 0, r)-flecnodes. As noted above, [40, Corollary 12.1] then implies that every point of O is incident ( to at least t curves of degree at most E that are fully contained in Z(f). r ) As observed above, when t j=1 b j d(c 0 ), a constant depending only on C 0 and E, Z(f) is infinitely ruled on this Zariski open set. By a simple argument (a variant of which is given in [59, Lemma 6.1]), we can conclude that Z(f) is infinitely ruled (everywhere) by curves from C 0, thus completing the proof. 16 To say that h j is a ploynomial in α (and p) means that it is a polynomial in the 2 ( ) E+3 3 coefficients of the monomials of the two polynomials in the pair α (and in the coordinates (x, y, z) of p). 37

258 Part V Conclusion 249

259

260 9 Discussion 9.1 Summary In this thesis we have studied several Erdős-type questions in combinatorial geometry, involving incidences between points and lines, curves, or surfaces, in three or higher dimensions, by applying tools from algebra and algebraic geometry. In Chapter 2 we presented an almost tight bound on the number of incidences between points and lines in R 4. We then presented, in Chapter 3, a configuration of points and lines on a quadratic hypersurface in R 4, having more incidences than does the previous bound, under its assumptions, showing that the relevant assumptions made in Chapter 2 are essential. In Chapter 4 we presented an elementary proof of the Guth-Katz bound on the number of incidences between points and lines in R 3. We then presented, in Chapter 5, applications of the Guth-Katz bound to Ramsey-type theorems on the intersection pattern of lines in R 3. In Chapter 6, we presented bounds on the number of incidences between points and lines on a two- or three-dimensional variety. The results in this chapter extend to the field of complex numbers. Moreover, when the configuration of points is strongly two- or three-dimensional (in the sense made precise in that chapter), the number of incidences becomes linear. In Chapter 7, we presented general bounds on the number of incidences between points and algebraic curves in R d, with k degrees of freedom. Then, in Chapter 8, we improved these bounds for the three-dimensional case, extending, and then sharpening several earlier works on similar problems. In particular, we obtained a refined bound for point-circle incidences in three dimensions. Still in Chapter 8, we presented bounds on the number of incidences between points and constant-degree algebraic surfaces, also in three dimensions (such as planes, spheres, paraboloids, etc.). As a corollary of the point-surface bounds, we deduced new bounds on the number of distinct and repeated distances determined by points on a constant-degree variety in R 3. In addition to the new bounds, a significant contribution of the thesis is in the algebraic tools needed to obtain these results. These tools are adaptations of various techniques in algebraic geometry, differential geometry, and topology, as well as development of new specially-tailored 251

261 machinery from these areas. Some of these techniques go back to the 19th or early 20th centuries, while others are fairly recent. 9.2 Future challenges Incidences between points and curves in higher dimensions and k-rich points. The thesis constitutes significant progress in the study of incidence geometry in higher dimensions and in several related areas, continuing the thrust set forth by Guth and Katz almost a decade ago. Still, there are many further challenges in this area, some inspired by the work in this thesis, and some of broader interest. Here is a representative sample of such challenges. (i) Our bound for incidences between points and lines in R 4 is almost optimal. It remains open to get rid of the factor 2 c logm in the bound. We have achieved this improvement when m is not too close to n 4/3, so to speak, allowing us to use weak but non-inductive bounds, and complete the analysis in one step. See the discussion in Chapter 2 for further details. (ii) Extend (and sharpen) the bound on the number of k-rich points in a set of n lines in R 4, for any value of k. In particular, is it true that the number of intersection points of the lines (this is the case k = 2; the intersection points are also known as 2-rich points) is O(n 4/3 + nq 1/2 + ns)? We conjecture that this is indeed the case. Recently, Guth and Zahl proved, for every ε > 0, that the number of 2-rich points is O ε (n 4/3+3ε ), provided that at most O(n 2/3+ε ) lines lie on a common hypersurface of constant degree (depending on ε), and at most O(n 1/3+ε ) lines lie on a common plane. The goal is to weaken these fairly restrictive assumptions, in the spirit of similar recent attempts in [57, 102]. A deeper question, extending a similar open problem in three dimensions that has been posed by Guth and others (see, e.g., Katz s expository note [68]), is whether the above conjectured bound can be improved when q = o(n 2/3 ) and s = o(n 1/3 ), that is, when the second and third terms in the conjectured bound are subsumed by the term n 4/3. We also note that if we could establish such a bound for the number of k-rich points, for any constant k (when q and s are not too large), then the case of large m (that is, m = Ω(n 4/3 )) would become vacuous, as only O(n 4/3 ) points could be incident to more than k lines. (iii) Extend our study of point-line incidences in R 4 to point-curve incidences in higher dimensions. In Chapter 7, we achieved a general bound in arbitrary dimension, using a constant-degree partitioning polynomial, with the disadvantages discussed in that chapter and in the introduction (slightly weaker bounds, significantly more restrictive assumptions, and inferior lowerdimensional terms). The leading terms in the resulting bounds, for points and curves in R d, are O(m 2/(d+1)+ε n d/(d+1) + m 1+ε ), for any ε > 0. See also Dvir and Gopi [35], Zahl [127], for recent studies on the number of rich lines determined by points sets in C d, for arbitrary dimension d, and an improved bound by 252

262 Hablicsek and Scherr [59] when the points lie in R d, for arbitrary dimension d. Obtaining sharper results in such general settings, like the ones obtained in Chapters 4 and 2, is quite challenging algebraically, although some of the tools developed in this thesis seem promising for higher dimensions too. Most of the techniques used in the thesis only work in three or four dimensions. Extending them to higher dimensions is a major challenge in algebraic and differential geometry. Incidences between points and varieties, and distinct and repeated distances in higher dimensions. (i) Our bounds on incidences between points and constant-degree algebraic surfaces are not sharp. A major challenge is to improve these bounds, even only for the special cases of planes or spheres. Such improvements have important implications, some of which have been noted in Chapter 8. It is our hope that this thesis, and other related studies in incidence geometry, would motivate researchers in these fields to tackle these problems, interesting in their own right. (ii) A long-standing open problem, which has moved to the research front after Guth and Katz s nearly complete solution of the planar case, is that of establishing the lower bound of Ω(n 2/3 ) for the number of distinct distances determined by a set of n points in R 3, without assuming them to lie on a constant-degree surface. The best known lower bound, close to Ω(n 3/5 ), which follows from the work of Solymosi and Vu [114], still falls short of this bound. In the present study, in Chapter 8, we have obtained some partial results (with better lower bounds) for cases where (all or some of) the points do lie on such a surface. We hope that some of the ideas used in this work could be applied in more general contexts, or in other special situations. This is perhaps the most significant open problem in the research front of this area, and appears to be an extremely hard problem. Some recent work of Bardwell-Evans and Sheffer [13] reduces this problem to an incidence problem involving points and 2-planes in R 5. The reduction is interesting and promising (and extends to arbitrary dimensions as well), but the resulting incidence problems still seem to be out of reach with the current arsenal of techniques. (iii) Another major long-standing open problem is that of improving the upper bound O(n 3/2 ), established in [65, 125], on the number of unit distances determined by a set of n points in R 3, again without assuming (all or some of) them to lie on a constant-degree surface. It would be interesting to make progress on this difficult problem. (A small breakthrough progress in this direction was achieved, very recently, by Zahl [128].) (iv) In the context of studying incidences between points and spheres or surfaces in R 3, as presented in Chapter 8, a challenging open problem (already noted there) is to characterize all the surfaces that are infinitely ruled by algebraic curves of degree at most E (or by certain classes thereof), extending the known characterizations for lines and circles, where the only such surfaces are planes (for lines) and planes and spheres (for circles). A weaker, albeit still hard problem is to reduce the upper bound 100E 2 on the degree of such a surface, perhaps all the way down to E, or 253

263 at least to O(E). Algebraic and other tools. Following Guth and Katz, many, if not most, of the subsequent papers (including most chapters of this thesis) use partitioning polynomials. In many cases, the difficult situation, is when all the points lie on the zero set of a partitioning polynomial f of large degree. Then, one has to construct a second-level partitioning polynomial g, when the goal is to partition Z( f ) \ Z(g), in an appropriate analogous manner (see [65, 125]). Then, a difficult case is when all the points lie on the variety of co-dimension two Z( f,g) = Z( f ) Z(g). This is fine in three dimensions, as Z( f,g) is then a curve. In four dimensions, Basu and Sombra [16] showed how to construct a third-level partitioning polynomial h that partitions Z( f,g) \ Z(h), in an appropriate manner, using bounds on the Hilbert function of a variety. In other recent works, Fox et al. [47] and Matoušek and Patáková [79] showed how to construct higher-level partitioning polynomials, which have weaker properties that do not yield in general the desired conjectured bound, and it remains a major challenge to generalize the result of Basu and Sombra [16] to higher dimensions, with the conjectured properties. In Chapter 2, we extended to four dimensions the flecnode polynomial, defined in three dimensions by Salmon [90], and introduced again by Guth and Katz [56]. In a more recent work, Guth and Zahl [57] extended the notion of the flecnode polynomial, from lines in three dimensions, to constant-degree algebraic curves in three dimensions. The next natural steps would be to extend the notion of flecnode polynomials to constant-degree algebraic varieties (or at least just for curves) in higher dimensions, and using these polynomials for solving other fundamental problems in combinatorial geometry. In Chapter 8, we extended the notion, for a given family of curves, of having k degrees of freedom, from the plane to three dimensions. Using this notion, we were able to generalize and improve previous bounds on the number of incidences between points and curves. The important feature of having k degrees of freedom is that, given k points, there are only O(1) curves from the given family that pass through the k points. In some problems, though, for a generic choice of the k points, there is no curve of the family that passes through all of them. For example, the family of unit circles in R 3 certainly has three degrees of freedom, but most triples of points do not span a unit circle. On the other hand, any pair of points (at distance at most 2) determine infinitely many unit circles. We can say (informally and incorrectly) that unit circles in R 3 have 2.5 degrees of freedom. The goal is to exploit this property and derive smaller incidence bounds for such families (as has been done for unit circles in R 3 in [98]). It is also appealing to extend this definition from curves to higher-dimensional varieties (like 2-surfaces in three dimensions), even though the extension is not straightforward. Algorithmic questions. Finally, the results in this thesis, as well as the overwhelming portion of results obtained since the Guth-Katz breakthroughs, are combinatorial in nature. Nevertheless, the space decomposition yielded by partitioning polynomials appears to be a useful tool for many 254

264 algorithmic questions, extending the arsenal of space partitioning techniques from the early 1990 s (based on cuttings or simplicial partitionings [27, 76]) in a significant way. The problem is that the existence of partitioning polynomials is a consequence of polynomial Ham-Sandwich cuts, or more generally, of the the Borsuk-Ulam theorem in very high dimensions, and there are no known efficient (polynomial-time) algorithms for these constructs. Knauer et al. [70] showed that the decision version of the Ham-Sandwich problem is NP-hard and W[1]- hard. One of the very few algorithmic applications of this machinery, by Agarwal et al. [3], proposes a randomized technique that constructs approximate partitioning polynomials for point sets, but no similar construction is known for Guth s partitioning polynomial [52] (for higher-dimensional varieties). This is a major open problem whose solution is likely to have many algorithmic implications. Matoušek and Patáková [79] provide a polynomial partitioning method with up to d polynomials in dimension d, which allows for a complete decomposition for a given point set. They then apply it to obtain a new algorithm for the semialgebraic range-searching problem, which has similar running time bounds as in [3]. 255

265 256

266 References [1] P. K. Agarwal, R. Apfelbaum, G. Purdy and M. Sharir, Similar simplices in a d-dimensional point set, Proc. 23rd Annu. ACM Sympos. Comput. Geom. (2007), [2] P. K. Agarwal and J. Matoušek, On range searching with semialgebraic sets, Discrete Comput. Geom., 11 (1994), [3] P.K. Agarwal, J. Matoušek and M. Sharir, On range searching with semialgebraic sets. II, SIAM J. Comput. 42 (2013), [4] P.K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky, Lenses in arrangements of pseudocircles and their applications, J. ACM 51 (2004), [5] P. Agarwal and M. Sharir, On the number of congruent simplices in a point set, Discrete Comput. Geom. 28 (2002), [6] R. Apfelbaum and M. Sharir, Large bipartite graphs in incidence graphs of points and hyperplanes, SIAM J. Discrete Math. 21 (2007), [7] R. Apfelbaum and M. Sharir, Non-degenerate spheres in three dimensions, Combinat. Probab. Comput. 20 (2011), [8] B. Aronov, V. Koltun and M. Sharir, Incidences between points and circles in three and higher dimensions, Discrete Comput. Geom. 33 (2005), [9] B. Aronov, J. Pach, M. Sharir and G. Tardos, Distinct distances in three and higher dimensions, Combinat. Probab. Comput. 13 (2004), [10] B. Aronov, M. Pellegrini and M. Sharir, On the zone of a surface in a hyperplane arrangement, Discrete Comput. Geom. 9 (1993),

267 [11] B. Aronov and M. Sharir, Cutting circles into pseudo-segments and improved bounds for incidences, Discrete Comput. Geom. 28 (2002), [12] L. Barba, J. Cardinal, J. Iacono, S. Langerman, A. Ooms, and N. Solomon, Subquadratic algorithms for algebraic generalizations of 3SUM, Proc. 33rd Annu. Sympos. Comput. Geom., 2017, 13:1 13:15. Also in arxiv: [13] S. Bardwell-Evans and A. Sheffer, A Reduction for the Distinct Distances Problem in R d, in arxiv: (2017). [14] S. Barone and S. Basu, Refined bounds on the number of connected components of sign conditions on a variety, Discrete Comput. Geom. 47 (2012), [15] A. Basit and A. Sheffer, Incidences with k-non-degenerate sets and their applications, J. Comput. Geom. 5 (2014), [16] S. Basu and M. Sombra, Polynomial partitioning on varieties of codimension two and pointhypersurface incidences in four dimensions, Discrete Comput. Geom. 55 (2016), [17] A. Beauville, Complex Algebraic Surfaces, No. 34, Cambridge University Press, Cambridge, [18] L. Bien, Incidences between points and curves in the plane, M.Sc. Thesis, School of Computer Science, Tel Aviv University, [19] J. Bourgain, A. Gamburd, and P. Sarnak, Markoff surfaces and strong approximation: 1, in arxiv: (2016). [20] P. Brass, Exact point pattern matching and the number of congruent triangles in a threedimensional point set, Proc. European Sympos. Algorithms, 2000, Springer LNCS 1879, pp [21] P. Brass and Ch. Knauer, On counting point-hyperplane incidences, Comput. Geom. Theory Appls. 25 (2003), [22] P. Brass, W. Moser and J. Pach, Research Problems in Discrete Geometry, Springer Verlag, New York, [23] P. Bürgisser, M. Clausen and A. Shokrollahi, Algebraic Complexity Theory, Vol. 315, Springer Science and Business Media, Berlin-Heidelberg, [24] J. Cardinal, M. Payne, and N. Solomon, Ramsey-type theorems for lines in 3-space, Discrete Math. Theoret. Comput. Sci. 18 (2016). 258

268 [25] T.M. Chan, On levels in arrangements of curves, Discrete Comput. Geom. 29 (2003), [26] T.M. Chan, On levels in arrangements of curves, II: A simple inequality and its consequences, Discrete Comput. Geom. 34 (2005), [27] B. Chazelle, Cuttings, in Handbook of Data Structures and Applications, (D. P. Mehta and S. Sahni, Eds.), Chapman and Hall/CRC, Boca Raton, [28] B. Chazelle, H. Edelsbrunner, L. Guibas, R. Pollack, R. Seidel, M. Sharir and J. Snoeyink, Counting and cutting cycles of lines and rods in space, Comput. Geom. Theory Appl. 1 (1992), [29] B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geometry, Combinatorica 10 (1990), [30] F. R. K. Chung, Sphere-and-point incidence relations in high dimensions with applications to unit distances and furthest-neighbor pairs, Discrete Comput. Geom. 4 (1989), [31] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), [32] D. Cox, J. Little and D. O Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer Verlag, Heidelberg, [33] D. Cox, J. Little and D. O Shea, Using Algebraic Geometry, Springer Verlag, Heidelberg, [34] Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), [35] Z. Dvir and S. Gopi, On the number of rich lines in truly high dimensional sets, in Proc. 30th Annu. ACM Sympos. Comput. Geom., 2015, [36] H. Edelsbrunner, L. Guibas and M. Sharir, The complexity of many cells in arrangements of planes and related problems, Discrete Comput. Geom. 5 (1990), [37] W. L. Edge, The Theory of Ruled Surfaces, Cambridge University Press, Cambridge, [38] Gy. Elekes, Sums versus products in number theory, algebra and Erdős geometry a survey, Paul Erdős and his Mathematics II, Bolyai Math. Soc. Stud. 11, Budapest, 2002,

269 [39] Gy. Elekes, H. Kaplan and M. Sharir, On lines, joints, and incidences in three dimensions, J. Combinat. Theory, Ser. A 118 (2011), Also in arxiv: [40] Gy. Elekes and M. Sharir, Incidences in three dimensions and distinct distances in the plane, Combinat. Probab. Comput. 20 (2011), [41] Gy. Elekes and Cs. D. Tóth, Incidences of not-too-degenerate hyperplanes, Proc. 21st Annu. ACM Sympos. Comput. Geom., 2005, [42] J. Ellenberg, J. Solymosi, and J. Zahl, New bounds on curve tangencies and orthogonalities, Discrete Analysis, 22 (2016), [43] P. Erdős, On a set of distances of n points, Amer. Math. Monthly 53 (1946), [44] P. Erdős, On sets of distances on n points in Euclidean space, Magyar Tud. Akad. Mat. Kutató Int. Kozl. 5 (1960), [45] P. Erdős, D. Hickerson, and J. Pach, A problem of Leo Moser about repeated distances on the sphere, Amer. Math. Monthly 96 (1989), [46] S. Feldman and M. Sharir, An improved bound for joints in arrangements of lines in space, Discrete Comput. Geom. 33 (2005), [47] J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl, A semi-algebraic version of Zarankiewicz s problem, J. European Math. Soc., to appear. Also in arxiv: [48] D. Fuchs and S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, Amer. Math. Soc. Press, Providence, RI, [49] W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, Expository Lectures from the CBMS Regional Conference Held at George Mason University, June 27 July 1, 1983, Vol. 54. AMS Bookstore, [50] M. Green and I. Morrison, The equations defining Chow varieties, Duke Math. J. 53 (1986), [51] L. Guth, Distinct distance estimates and low-degree polynomial partitioning, Discrete Comput. Geom. 53 (2015), [52] L. Guth, Polynomial partitioning for a set of varieties, Math. Proc. Cambridge Phil. Soc. 159 (2015), Also in arxiv:

270 [53] L. Guth, Ruled surface theory and incidence geometry, in arxiv: [54] L. Guth, Polynomial Methods in Combinatorics, University Lecture Series, Vol. 64, Amer. Math. Soc. Press, Providence, RI, [55] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem, Advances Math. 225 (2010), Also in arxiv: v1. [56] L. Guth and N. H. Katz, On the Erdős distinct distances problem in the plane, Annals Math. 181 (2015), Also in arxiv: [57] L. Guth and J. Zahl, Algebraic curves, rich points, and doubly-ruled surfaces, Amer. J. Math., to appear. Also in arxiv: [58] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Vol. 52, John Wiley & Sons, New York, [59] M. Hablicsek and Z. Scherr, On the number of rich lines in high dimensional real vector spaces, Discrete Comput. Geom. 55 (2014), 1 8. Also in arxiv: [60] C. G. A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), [61] J. Harris, Algebraic Geometry: A First Course, Vol Springer-Verlag, New York, [62] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, [63] T. A. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics, volume 61, Amer. Math. Soc., Providence, RI, [64] S. Kalia, M. Sharir, N. Solomon and B. Yang, Generalizations of Szemerédi-Trotter Theorem, Discrete Comput. Geom. 55 (2016), [65] H. Kaplan, J. Matoušek, Z. Safernová and M. Sharir, Unit distances in three dimensions, Combinat. Probab. Comput., 21(2012), [66] H. Kaplan, J. Matoušek and M. Sharir, Simple proofs of classical theorems in discrete geometry via the Guth-Katz polynomial partitioning technique, Discrete Comput. Geom., 48 (2012),

271 [67] H. Kaplan, M. Sharir and E. Shustin, On lines and joints, Discrete Comput. Geom. 44 (2010), Also in arxiv: [68] N. H. Katz, The flecnode polynomial: A central object in incidence geometry, in arxiv: [69] N. H. Katz and G. Tardos, A new entropy inequality for the Erdős distance problem, in Towards a Theory of Geometric Graphs (J. Pach, ed.), vol. 342 of Contemporary Mathematics, AMS, Providence, RI, 2004, [70] C. Knauer, H.R. Tiwary and D. Werner, On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems, Sympos. Theor. Aspects Computer Science, 9 (2011). [71] J. Kollár, Szemerédi Trotter-type theorems in dimension 3, Advances Math. 271 (2015), Also in arxiv: [72] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Springer-Verlag, Heidelberg, [73] J. M. Landsberg, Is a linear space contained in a submanifold? On the number of derivatives needed to tell, J. Reine Angew. Math. 508 (1999), [74] N. Lubbes, Families of circles on surfaces, Contributions to Algebra and Geometry, accepted. Also in arxiv: [75] A. Marcus and G. Tardos, Intersection reverse sequences and geometric applications, J. Combinat. Theory Ser. A 113 (2006), [76] J. Matoušek, Efficient partition trees, Discrete Comput. Geom. 8 (1992), [77] J. Matoušek, Lectures on Discrete Geometry, Springer Verlag, Heidelberg, [78] J. Matoušek, and B. Gärtner, Understanding and Using Linear Programming (Universitext), Springer-Verlag, New York, [79] J. Matoušek and Z. Patáková, Multilevel polynomial partitions and simplified range searching, Discrete Comput. Geom. 54 (2015), [80] E. Mezzetti and D. Portelli, On threefolds covered by lines, in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Vol. 70, No. 1, Springer Verlag, Heidelberg,

272 [81] G. Monge, Application de l Analyse á la Geometrie, Bernard, Paris, [82] F. Nilov and M. Skopenkov, A surface containing a line and a circle through each point is a quadric, Geom. Dedicata 163:1 (2013), [83] J. Pach and M. Sharir, Geometric incidences, in Towards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics, Vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp [84] J. Pach, G. Tardos and G. Tóth, Disjointness graphs of segments, in arxiv: (2017). [85] A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer Verlag, London, [86] R. Quilodrán, The joints problem in R n, SIAM J. Discrete Math. 23 (2010), Also in arxiv: [87] O. E. Raz, M. Sharir, and F. de Zeeuw, Polynomials vanishing on Cartesian products: The Elekes Szabó theorem revisited, Duke Math. J. 165 (2016), [88] S. Richelson, Classifying Varieties With Many Lines, a senior thesis, Harvard University, [89] E. Rogora, Varieties with many lines, Manuscripta Mathematica 82 (1994), [90] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2, 5th edition, Hodges, Figgis and co. Ltd., Dublin, [91] H. Schenck, Computational Algebraic Geometry, Cambridge University Press, Cambridge, [92] M. R. Spiegel, S. Lipschutz, and D. Spellman, Vector Analysis, 2nd edition, Schaum s Outlines, McGraw-Hill, [93] J. Schicho, The multiple conical surfaces, Contributions to Algebra and Geometry 42 (2001), [94] B. Segre, Sulle V n contenenti più di n k S k, Nota I e II, Rend. Accad. Naz. Lincei 5 (1948),

273 [95] F. Severi, Intorno ai punti doppi impropri etc., Rend. Cir. Math. Palermo 15 (10) (1901), [96] I. R. Shafarevich, Basic Algebraic Geometry, Vol. 197, Springer-Verlag, New York, [97] M. Sharir, A. Sheffer, and N. Solomon, Incidences with curves in R d, Electronic J. Combinat., 23 (2016), P4.16. Also in Proc. European Sympos. Algorithms (2015), Springer LNCS 9294, pp , and in arxiv: [98] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between points and circles, Combinat. Probab. Comput. 24 (2015), Also in arxiv: [99] M. Sharir and N. Solomon, Incidences between points and lines in R 4, Proc. 30th Annu. Sympos. Computational Geometry, 2014, [100] M. Sharir and N. Solomon, Incidences between points and lines in three dimensions, in Intuitive Geometry (J. Pach, Ed.), to appear. Also in Proc. 31st Annu. Sympos. Computat. Geom. (2015), , and in arxiv: [101] M. Sharir and N. Solomon, Incidences between points and lines on a two-dimensional variety, in arxiv [102] M. Sharir and N. Solomon, Incidences between points and lines in R 4, Discrete Comput. Geom. 57 (2017): Also in Proc. 56th IEEE Sympos. Foundations of Computer Science 2015, , and in arxiv: [103] M. Sharir and N. Solomon, Incidences between points and lines on two- and threedimensional varieties, Discrete Comput. Geom., in press. Also in arxiv: [104] M. Sharir and N. Solomon, Incidences between points on a variety and planes in three dimensions, in arxiv: [105] M. Sharir and N. Solomon, Distinct and repeated distances on a surface and incidences between points and spheres, in arxiv: [106] M. Sharir and N. Solomon, Incidences with curves and surfaces and applications to distinct and repeated distances, Proc. 28th ACM-SIAM Sympos. Discrete Algorithms, 2017, Also in arxiv: [107] M. Sharir and J. Zahl, Cutting algebraic curves into pseudo-segments and applications, J. Combinat. Theory Ser. A 150 (2017),

274 [108] A. Sheffer, Distinct distances: Open problems and current bounds, in arxiv:1406:1949. [109] A. Sheffer, E. Szabó, and J. Zahl, Point-curve incidences in the complex plane, Combinatorica, to appear. [110] M. Skopenkov and R. Krasauskas, Surfaces containing two circles through each point and Pythagorean 6-tuples, in arxiv: [111] N. Solomon and R. Zhang, Highly incidental patterns on a quadratic hypersurface in R 4, Discrete Math., 340(4) (2017), Also in arxiv: [112] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput. Geom., 48 (2012), [113] J. Solymosi and F. de Zeeuw, Incidence bounds for complex algebraic curves on Cartesian products, in arxiv: [114] J. Solymosi and V. H. Vu, Near optimal bounds for the Erdős distinct distances problem in high dimensions, Combinatorica 28 (2008), [115] J. Spencer, E. Szemerédi and W. T. Trotter, Unit distances in the Euclidean plane, In: Graph Theory and Combinatorics (Proc. Cambridge Conf. on Combinatorics, B. Bollobas, ed.), , Academic Press, [116] A. H. Stone and J. W. Tukey, Generalized sandwich theorems, Duke Math. J. 9 (1942), [117] L. Székely, Crossing numbers and hard Erdős problems in discrete geometry, Combinat. Probab. Comput. 6 (1997), [118] E. Szemerédi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), [119] T. Tao, From rotating needles to stability of waves: Emerging connections between combinatorics, analysis, and PDE, Notices AMS 48(3) (2001), [120] T. Tao, The Cayley-Salmon theorem via classical differential geometry, the-cayley-salmon-theorem-via-classical-differential-geometry/ [121] C.D. Tóth, The Szemerédi-Trotter theorem in the complex plane, Combinatorica 35 (2015), Also in arxiv: (2003). 265

275 [122] P. Valtr, Strictly convex norms allowing many unit distances and related touching questions, manuscript, [123] B. L. van der Waerden, E. Artin, and E. Noether, Modern Algebra, Vol. 2, Springer Verlag, Heidelberg, [124] H. E. Warren, Lower bound for approximation by nonlinear manifolds, Trans. Amer. Math. Soc. 133 (1968), [125] J. Zahl, An improved bound on the number of point-surface incidences in three dimensions, Contrib. Discrete Math., 8 (2013), Also in arxiv: [126] J. Zahl, A Szemerédi-Trotter type theorem in R 4, Discrete Comput. Geom. 54 (2015), [127] J. Zahl, A note on rich lines in truly high dimensional sets, Forum of Mathematics, Sigma 4, 2016, [128] J. Zahl, Breaking the 3/2 barrier for unit distances in three dimensions, in arxiv: (2017). [129] surjective-morphism-of-affine-varieties-and-dimension. 266

276 תקציר גיאומטריה קומבינטורית הוא תחום העוסק בבעיות בעלות תוכן גיאומטרי. התחום נהגה ופותח ע"י פאול ארדש, החל מראשית המאה ה- 20. בעוד שבעיות אלו (בעיות "מטיפוס ארדש") בדרך כלל קלות מאוד לניסוח, חלקן קשות מאוד לפתרון ובבסיסן עומדת תיאוריה עמוקה, וחלקן פתוחות (או היו פתוחות) עשורים רבים. בעשור החולף, פני הנוף של הגיאומטריה הקומבינטורית השתנו מהותית, בעקבות שתי פריצות דרך של גוט וכץ ([55] ב ו-[ 56 ] ב ). הם הציגו שיטות פשוטות יחסית בגיאומטריה אלגברית שאפשרו את פתרונן המוצלח של מספר בעיות מרכזיות בגיאומטריה קומבינטורית. מאמרם הראשון השיג פתרון מלא ל"בעיית המחברים",(Joints) בעיה המערבת חילות בין נקודות וישרים בשלושה מימדים, שהוצעה על ידי שאזל ושות' [28] ב מאמרם השני של גוט וכץ הציג פתרון כמעט מלא ל"בעיית המרחקים השונים" של ארדש [43] במישור, שהייתה בעיה פתוחה מרכזית החל מהצגתה בשנת שתי הבעיות הללו (ובמיוחד השנייה שבהן) נחקרו רבות לאורך השנים, באמצעות שיטות מסורתיות יותר, ולאחר מכן בשיטות מורכבות יותר בגיאומטריה קומבינטורית, שהניבו פתרונות חלקיים ולא מספקים. ה"נישואין" הללו, בין גיאומטריה אלגברית וגיאומטריה קומבינטורית וחישובית מציגים אתגרים משמעותיים לשני התחומים, וביניהם הבנה מחודשת וזיקוק כלים קיימים בגיאומטריה אלגברית ופיתוח כלים חדשים, מוכוונים לקראת צד ה"לקוח" החדש, ושימוש בארגז הכלים החדש לפתרון מגוון רחב ועשיר של בעיות בגיאומטריה קומבינטורית וחישובית. בתזה זו, אנו בונים גשרים נוספים בין שתי הדיסציפלינות, מאמצים ומפתחים טכניקות וכלים אלגברים חדשים, ומפעילים אותם בפתרונן של מספר בעיות בגיאומטריה קומבינטורית. אנו מניחים ידע בסיסי בגיאומטריה אלגברית, וכל אימת שנדרש ידע יותר מתקדם, נרחיב וניתן רקע מתאים. רוב הכלים המתקדמים יותר מוצגים בסעיף 1.3, אך גם בפרקים נוספים בתזה. התזה בנויה כאסופת מאמרים, ומורכבת משבעה מאמרים, המוצגים בארבעה חלקים אותם נסקור כעת בארבעת הסעיפים הבאים. 1. חילות בין נקודות וישרים בארבעה מימדים בפרק 2, אנחנו מרחיבים את המחקר של גוט וכץ [56], מחילות בין נקודות וישרים בשלושה מימדים, לחילות בין נקודות וישרים בארבעה מימדים, ומוכיחים חסמים הדוקים או כמעט הדוקים על מספר החילות הללו במקרה הגרוע ביותר. בעיה זו, שהינה יותר קשה משמעותית מהגרסה התלת-ממדית, דורשת פיתוח של טכניקות וכלים נוספים בגיאומטריה אלגברית, שאת רובם נסקור בסעיף 1.3 בהקדמה, ושימוש במגוון שיטות קומבינטוריות שפיתחנו בכדי לפתור את הבעיות הרלוונטיות. בצורה חופשית, ניתן לתאר את המחקר בסעיף זה, כחקר התבניות שבהן הישרים יכולים לגעת האחד ברעהו 1

277 < < כאשר הם "נזרקים" לחלל הארבע-מימדי, כאשר תת-בעיה משמעותית היא להבין את אותן תבניות כאשר הישרים הללו מוכלים ביריעה אלגברית תלת-ממדית מדרגה קבועה. המשפט המרכזי אותו הוכחנו, שהופיע ב- [102] (לאחר עבודה מקדימה [99]), הינו המשפט הבא. R 4 משפט 1. תהא >P קבוצה של < m נקודות ו- L < קבוצה של < n ישרים ב >, ויהיו <,q s n פרמטרים, כך ש-( i ) כל על-מישור או על-משטח ריבועי מכיל לכל היותר >q ישרים מתוך >, L ו-( ii ) כל 2 -מישור מכיל לכל היותר >s ישרים של >. L אז מתקיים I(P, L) 2 c logm (m 2/5 n 4/5 + m) + A(m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n), כאשר >A ו- c < הם קבועים מתאימים. כאשר < m n 5/3 או >, m n 6/7 אנו מקבלים את החסם החזק יותר I(P, L) A(m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n). באופן כללי, פרט לגורם < 2 c logm בחסם הראשון, החסם הדוק במקרה הגרוע ביותר, לכל טווח הערכים של, m, n < ועבור ערכים מתאימים של.< q, s בפתרון שלנו, השתמשנו בכלים כבדים מגיאומטריה אלגברית, חלקם בני מעל 150 שנים. למשל, משפט מ של קיילי וסלמון (שהתקבל באופן בלתי-תלוי גם ע"י מונז') [81,90], מבטיח שמשטח אלגברי בשלושה מימדים יכול להכיל לכל היותר מספר קבוע של ישרים, אלא אם כן הוא "נשלט" על ידי ישרים. המחקר של משטחים נשלטים surfaces),(ruled כולל האופן בו ניתן לשכנם במימדים ארבע ומעלה, הינו נושא מרכזי במחקרנו. משפט קשור אחר שאנו משתמשים בו, משנת 1901, של סגרה וסברי [94,95], מאפיין על-משטחים במרחב המרוכב ה- 4 -מימדי שהינם "נשלטים אינסוף-פעמים" ע"י ישרים. הכלים הללו מוצגים גם בסעיף 1.3, וגם בפרק 2. מספר רב של ישרים על משטח ריבועי עשוי להגדיל את מספר החילות. בעבודת המשך עם רויז'יאנג ז'אנג [111], המוצגת בפרק 3, אנו מראים שההגבלות שנעשו במשפט 1 הינן הכרחיות, כלומר שאם נוותר עליהן, נקבל מספר חילות גדול יותר. קונקרטית, אנו מראים שההנחה שעשינו בתנאי (i) במשפט 1, על כך שאף על-משטח ריבועי לא יכיל יותר מדי ישרים, הינה הכרחית, על ידי כך שאנו מציגים, לכל זוג טבעיים,,m n < קבוצה של < m נקודות ו- n < ישרים המוכלים במשטח הריבועי < S ב- < R 4 המוגדר על ידי,< S = { (x 1, x 2, x 3, x 4) R 4 x 1 = x x 2 3 x 2 4 } כך שמספר החילות בין הנקודות והישרים הינו n).< Ω(m 2/3 n 1/2 + m + כאשר >, n 9/8 < m < n 3/2 מתקבל מספר חילות גדול יותר מהחסם שמופיע במשפט 1. 2

278 2. חילות בין נקודות וישרים בשלושה מימדים, עם שימושים בפרק 4, בפרק 4, שהתפרסם ב-[ 100 ], אנחנו מציגים הוכחה אלמנטרית ופשוטה יחסית לחסם של גוט וכץ בשלושה מימדים, כלומר, אנחנו מראים שמספר החילות ביןm < נקודות ל- >n ישרים ב- >, R 3 כך שאף מישור לא מכיל יותר מאשר < s ישרים, הוא (בניסוח גרסת המשפט המדויקת, קבועי הפרופורציה במחוברים הראשון והשלישי, תלויים, בצורה חלשה, בקשר בין >m ו- n >). משפטי רמזי עבור ישרים ב- 7. O(m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n), R 3 בפרק 5, אנחנו מוכיחים, במשותף עם מייקל פיין וז'אן קרדינל [24], משפטים מטיפוס רמזי עבור קבוצת ישרים במרחב התלת-מימדי. המשפטים הללו מבטיחים קיום של קליקה או קבוצה בלתי-תלויה עבור ה(היפר)גרפים המושרים על ידי חילות בין ישרים, נקודות, ומשטחים ריבועיים מטיפוס רגולוס (משטח ריבועי הנשלט באופן כפול ע"י ישרים) במרחב התלת-מימדי. בין יתר הדברים, אנחנו מוכיחים את התוצאות הבאות: Ω(n 1/3 ) א. ב. גרף החיתוכים של n ישרים ב- R 3 מכיל קליקה או קבוצה בלתי-תלויה בגודל. כל קבוצה של n ישרים ב- R 3 מכילה תת-קבוצה של ישרים שכולם נדקרים ע"י Ω ( n ) ישר אחד, או תת קבוצה של כולם ע"י ישר אחד. Ω ( (n /logn) 1/5 ) ישרים כך שאין בה 6 ישרים שנדקרים ג. כל קבוצה של n ישרים ב- R 3 במצב כללי מכילה תת-קבוצה של ישרים שמוכלים ברגולוס, או תת-קבוצה של ברגולוס. Ω(n 2/3 ) Ω(n 1/3 ) ישרים, כך שאין בה 4 ישרים המוכלים כולם 3. חילות בין נקודות וישרים על יריעות כאשר >m הנקודות נמצאות על יריעה דו-ממדית בתלת מימד, או יריעה תלת-ממדית במרחב הארבע-מימדי, אשר דרגתן אינה גדולה מדי, אנו מראים, בפרק 6 של התזה, שהתפרסם ב-[ 103 ], שמספר החילות בין נקודות אלו ו- n < ישרים משמעותית קטן יותר מהחסמים של גוט וכץ והחסם במשפט 1, בהתאמה. יתירה מכך, אין צורך להניח שהיריעות הללו משוכנות במרחבים ה- 3 או ה- 4 מימדיים, בהתאמה, והחסמים שלנו תקפים גם כאשר היריעות הללו משוכנות במרחבים ממימדים גבוהים יותר. המשפט המרכזי הראשון בפרק זה של התזה הוא המשפט הבא. 3

279 < < משפט 2. R d א. המקרה הממשי: תהי >P קבוצה של >m נקודות ו- >L קבוצה של >n ישרים ב- >, עבור 3 d < כלשהוא, ויהיו s D 2 < שני פרמטרים טבעיים, כך שכל הנקודות והישרים נמצאים על יריעה דו-מימדית >V מדרגה >D אשר לא מכילה אף 2 -מישור, וכך שאף 2 -מישור לא מכיל יותר מאשר >s ישרים של >. L אז מתקיים I(P, L) = O(m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + m + n). ב. המקרה המרוכב: תחת אותם תנאים בדיוק, כאשר המרחב הוא המרחב המרוכב, C d עבור 3 d כלשהוא, מתקיים I(P, L) = O(m 1/2 n 1/2 D 1/2 + m 2/3 D 2/3 s 1/3 + D 3 + m + n). בשני המקרים, כאשר < s ו- D < קבועים, מתקבל החסם הלינארי (n >. O(m + המשפט המרכזי השני בפרק זה של התזה הוא המשפט הבא. משפט 3. R d n L m P P s, D d 4 תלת-ממדית מדרגה D אף 2 -מישור לא מכיל יותר מאשר s ישרים של. L אז מתקיים א. המקרה הממשי: תהי קבוצה של נקודות ותהי קבוצה של ישרים ב-, עבור כלשהוא, ויהיו פרמטרים כך ש-( i ) כל הנקודות של נמצאות על יריעה, שאינה מכילה 3 -מישורים או משטחים ריבועיים תלת-ממדיים, ו-( ii ) I(P, L) = O(m 1/2 n 1/2 D + m 2/3 n 1/3 s 1/3 + nd + m). ב. המקרה המרוכב: תחת אותם תנאים בדיוק, כאשר המרחב הוא המרחב המרוכב, C d עבור 4 d כלשהוא, מתקיים I(P, L) = O(m 1/2 n 1/2 D + m 2/3 n 1/3 s 1/3 + D 6 + nd + m). בשני המקרים, כאשר < s ו- D < קבועים, מתקבל החסם (n >. O(m + מאפיין מרכזי של הניתוח שלנו הוא שהחסמים המתקבלים תקפים גם מעל המרוכבים (כפי שמצוין בסעיף ב' של משפטים 2 ו- 3, עם תוספת קטנה שנהיית זניחה אם הדרגה < D קטנה מספיק). מנגד, החסמים הכלליים יותר עבור חילות בין נקודות וישרים בשלושה וארבעה מימדים (כפי שנסקרו לעיל) תקפים אך ורק מעל הממשיים. ההדדיות בין המקרה הממשי והמקרה המרוכב תידון בהרחבה בסעיף

280 < 4. חילות בין נקודות ועקומים וחילות בין נקודות ומשטחים חילות בין נקודות ועקומים בשלושה מימדים ומעלה. R 3 התוצאות שהוצגו עד כה מערבות חילות בין נקודות וישרים. אוסף התוצאות הבאות שנציג, המוצגות בחלק 4, מערבות חילות בין נקודות ועקומים אלגבריים מדרגה קבועה בשלושה מימדים ומעלה. שני מאמרים שונים מתייחסים לבעיה זו. ראשית, אנחנו חוקרים את הבעיה הכללית, במימד כלשהוא. מאמר זה, במשותף עם אדם שפר, מוצג בפרק 7 והופיע ב-[ 97 ]. המאמר השני, המוצג בפרק 8, הופיע ב-[ 106 ]. במאמר זה אנחנו מתייחסים למקרה התלת-מימדי בלבד, ובו אנו מקבלים תוצאות חזקות יותר משמעותית (לעומת התוצאות בפרק 7, במקרה הכללי יותר). פרק 8 גם מכיל מחקר על חילות בין נקודות ומשטחים ב- >, אותם נסקור בהמשך הסעיף. R d חילות בין נקודות ועקומים אלגבריים ב- 7. בפרק 7, אנו מוכיחים, יחד עם אדם שפר, חסם על מספר החילות בין נקודות ועקומים אלגבריים ב- >. R d ובאופן מפורש, אנו מוכיחים כי מספר החילות בין < m נקודות ובין < n עקומים אלגבריים מדרגה קבועה עם < k דרגות חופש ב- < R d הוא O m dk k d + 1 +ε n dk dk d + d d j=2 m jk k j + 1 +ε d(j 1)(k 1) (d j)(k 1) n (d 1)( jk j + 1) q j (d 1)( jk j + 1) + m + n q j q j לכל > 0 ε >, כאשר קבוע הפרופורציונליות תלוי ב- d,,k,ε < בהינתן שאף יריעה j >-מימדית לא מכילה יותר מאשר < עקומי קלט, ושה- < מקיימים אי-אלו קשרים חלשים ביניהם. המושג של >k דרגות חופש במימד כלשהוא מכליל את המושג המקביל במישור, שהוגדר ב-[ 83 ] ע"י פאך ושריר. הדרישה היא שיהיו לכל היותר (1)O < μ = עקומים מהמשפחה הנתונה שעוברים דרך >k נקודות כלשהן, ושכל זוג עקומים ייחתכו ב- >μ נקודות לכל היותר. נגדיר מושג זה במדויק בפרק 8 של התזה. החסם הזה מכליל את החסם של פאך ושריר [83] מהמישור ל- רבות (עם הגבלות מסוימות), כולל תוצאות מסוימות שמוצגות בתזה זו. >. והוא גם מכליל תוצאות קודמות הכלליות של תוצאה זו טומנת בחובה גם מגבלות, ובמקרים רבים ידוע כי החסם הכללי שקיבלנו במקרה זה אינו הדוק, והתנאים הנדרשים הם מגבילים מאוד במקרים מסוימים. דיון מלא בנושאים אלה מוצג בפרקים 7 ו- 8. חילות בין נקודות ועקומים אלגבריים בשלושה מימדים. R d כעת נעבור לסקור את התוצאות בפרק 8, הפרק האחרון בתזה. בפרק זה נדרשת היכרות עם מספר מושגים מתקדמים יותר בגיאומטריה אלגברית, בהם אנו דנים בהרחבה בהקדמה באנגלית, ובפרק 8 עצמו (אך לא כאן). נעיר כאן שמשטח אלגברי >V נשלט אינסופית ע"י משפחת עקומים >Γ מדרגה >E לכל היותר, אם כל נקודה < p V הינה סמוכה לאינסוף עקומים של >Γ שמוכלים בשלמותם ב- V >. כפי שכבר הוסבר, הקשר בין משטחים נשלטים לתחום החילות בגיאומטריה קומבינטורית התגלה בעבודה החלוצית 5

281 של גוט וכץ [56], ושב והופיע בעבודות רבות מאז. סקירה מפורטת של נושא זה מופיעה בסעיף 1.3, כמו כן גם במאמר הסקירה של גוט [53] ובספרו החדש [54], ובעבודתו של קולאר [71]. עוד נעיר שמשפחה נוצרת (constructible) מסיבוכיות קבועה היא משפחה שמקיימת תנאי אלגברי שקל לתארו באמצעות מספר קבוע של שוויונות ואי-שוויונות אלגבריים. משפט 4 (עקומים ב- >). תהי >P קבוצה של < m נקודות ותהי < קבוצה של < n עקומים אלגבריים אי-פריקים מדרגה מקסימלית >, E הלקוחים מתוך משפחה נוצרת < מסיבוכיות קבועה של עקומים אלגבריים, בעלת < k דרגות חופש (ו- μ < קבוע כלשהוא) ב- >, כך שאף משטח הנשלט ע"י אינסוף עקומים של < לא מכיל יותר מאשר < q עקומים של >, עבור פרמטר >. q < n אז מתקיים כאשר קבוע הפרופורציונליות תלוי ב- E,,k,μ ובסיבוכיות של 0. הערות. 0 I(P, ) = O ( m k 3k 2 n 3k 3 3k 2 + m k 2k 1 n k 1 2k 1 q k 1 2k 1 + m + n ) 0 R 3 R 3 במקרים מסוימים, כמו למשל במקרים של ישרים ומעגלים, משטחים הנשלטים ע"י אינסוף עקומים של 0 הם בעלי תיאור פשוט (משטחים הנשלטים ע"י אינסוף ישרים הינם מישורים, ומשטחים הנשלטים ע"י אינסוף מעגלים הינם מישורים או ספרות). במקרים אלו, משפט 4 מתחזק, היות וההנחה היא בעצם שלסוג מאוד קונקרטי של משטחים אסור להכיל יותר מאשר מספר מקסימלי של עקומים. משפט 4 מכליל את התוצאה של גוט וכץ [56], שהתקבלה במקרה של ישרים..1.2 שיפור החסם. ניתן לשפר את החסם במשפט 4, אם אנחנו גם מוסיפים הנחה על המימד של משפחת העקומים 0 >. לא זו אף זו, כפי שיעלה מן ההוכחה, המימד שנשתמש בו הוא רק זה של תת משפחה של > 0 שאיבריה מוכלים בשלמותם במשטח נשלט אינסופית ע"י עקומים של >. אנו נאמר שהמשפחה < היא בעלת מימד מצומצם < s אם, לכל משטח >V הנשלט אינסופית ע"י עקומים של >, תת-המשפחה של < של עקומים המוכלים בשלמותם ב- V >הינה s >-מימדית (כלומר ניתן לייצג את עקומי < המוכלים ב- V < כנקודות במרחב פרמטרי s >-מימדי). במקרה זה אנו מקבלים את המשפט הבא. משפט 5 (עקומים ב- >) R 3 R 3 תהי >P קבוצה של < m נקודות ותהי < קבוצה של < n עקומים אלגבריים אי-פריקים מדרגה מקסימלית >, E הלקוחים מתוך משפחה נוצרת < מסיבוכיות קבועה, של עקומים אלגבריים, בעלת >k דרגות חופש (ו- μ < 0 קבוע כלשהוא) ב- >, כך שאף משטח הנשלט ע"י אינסוף עקומים של < לא מכיל יותר מאשר < q 0 עקומים של >, עבור פרמטר >, q < n ונניח בנוסף ש- < הינה בעלת מימד מצומצם >. s אז מתקיים I(P, ) = O ( m k 3k 2 n 3k 3 3k 2 ) + O ε( m2/3 n 1/3 q 1/3 + m 2s 5s 4 n 3s 4 5s 4 q 2s 2 5s 4 +ε + m + n ) 6

282 כאשר קבועי הפרופורציונליות תלוי ב- E >,,k,μ,s ובסיבוכיות המקסימלית של תת-משפחה כלשהיא של < שמורכבת מעקומים המוכלים בשלמותם במשטח כלשהוא הנשלט אינסופית ע"י עקומים של >, והקבוע השני תלוי גם ב- ε >. הערות. 0 0 משפט 5 הינו שיפור של משפט 4 כאשר וגם, במקרים בהם מספיק גדול כך שהביטוי השני בחסם של משפט 4 שולט על הביטוי הראשון באותו חסם; לערכים קטנים יותר של m הביטוי כולו לינארי (זה נכון פרט לביטוי ). q ε פרטים נוספים על ההדדיות בין משפטים 4 ו- 5 ניתן למצוא בפרק 8. החסמים במשפטים 4 ו- 5 משפרים, בשלושה מימדים, את התוצאה מפרק 7, בשלושה אופנים משמעותיים: (i) הגורמים המובילים בכל הביטויים הינם פחות או יותר זהים, אך החסמים במשפטים 4 ו- 5 לא כוללים את הגורם q m > n 1/k k s O(n ε ) q שמופיע בפרק 7. (ii) ההנחה כאן, לגבי מספר העקומים המוכלים במשטח מדרגה נמוכה, חלשות משמעותית לעומת ההנחה בפרק 7, אשר דורשת שאף משטח מדרגה (קבועה אך פוטנציאלית מאוד גבוהה), c ε התלויה ב- 2ε, לא יכיל יותר מאשר עקומים של. (iii) הגורמים הלא מובילים במשפטים 4 ו- 5 קטנים יותר משמעותית מאשר אלו בפרק 7, ובמובן מסוים (שיובהר בהרחבה בפרק 8), הם הטובים ביותר האפשריים..1.2 חילות בין נקודות ומשטחים אלגבריים בשלושה מימדים. כעת נסקור את אוסף התוצאות האחרון בתזה זו, המערבות חילות בין נקודות ומשטחים אלגבריים מדרגה קבועה בשלושה מימדים. הסקירה תהיה חלקית, והתוצאות מוצגות בפרק 8 בתזה. כאמור, גם התוצאות האלה התפרסמו ב-[ 106 ]. נשים לב, שבמקרה של חילות בין נקודות ומשטחים, גרף החילות בין הנקודות והמשטחים יכול להכיל גרפים דו-צדדיים שלמים, כך שכל אחד מהם מערב נקודות רבות שמוכלות בעקום אלגברי, ומשטחים רבים המכילים את אותו עקום.במקרים כאלה מספר החילות עשוי להיות רב, ואף להגיע למקסימום האפשרי >. mn שלא כמו מחקרים קודמים בתחום, אנחנו לא שוללים אפשרות כזו, מה שהופך את הגישה שלנו לכללית יותר. החסמים שלנו מעריכים את הגודל הכולל של קבוצת הקודקודים בפירוק של גרף החילות לתתי-גרפים דו-צדדיים שלמים. במקרים מסוימים, החסמים שלנו מיתרגמים לחסמים על מספר החילות ממש. בסך הכול, גם במקרה הזה התוצאות שלנו מספקות הכללה גורפת של תוצאות קודמות בנושא. פירוק גרף החילות, עבור נקודות על יריעה ומשטחים. התוצאה המרכזית הראשונה שלנו, היא כאשר הנקודות נמצאות על יריעה אלגברית < V מדרגה קבועה. מלבד העובדה שלתוצאה זו יש עניין עצמאי, היא גם משמשת לניתוח המקרה הכללי בו הנקודות לא חייבות להיות על יריעה אלגברית. 7

283 במשפטים הבאים נניח שקבוצת המשטחים >S נלקחת מתוך משפחה אינסופית < F של משטחים אלגבריים, שהיא או בעלת < k דרגות חופש ביחס ל- V >(עם < μ כלשהוא), או שהיא ממימד מצומצם >s ביחס ל- V >, עבור פרמטר קבוע כלשהוא,< s כשהכוונה היא שהמשפחה F} Γ< {σ V σ היא משפחה -< s מימדית של עקומים. משפט 6. R 3 תהי >P קבוצה של < m נקודות על יריעה אלגברית >V מדרגה קבועה >D ב- >, ותהי >S קבוצה של < n משטחים אלגבריים מדרגה מקסימלית >, E הלקוחים מתוך משפחה < F של משטחים אלגבריים, שהיא בעלת < k דרגות חופש ביחס ל- V < (ו- μ < קבוע כלשהוא), או שהיא ממימד מצומצם >s ביחס ל- V >, עבור פרמטרים קבועים כלשהם >, k (ו- μ >), או >. s אנחנו מניחים בנוסף שהמשטחים של >S לא מכילים רכיבים אי-פריקים משותפים (וזה ודאי קורה אם כולם אי-פריקים). אז ניתן לפרק את גרף החילות ) S < G(P, כ- G(P, S) = γ P γ S γ כאשר האיחוד הוא על-פני כל העקומים < γ שהם רכיבים אי-פריקים של עקומים מהצורה >, σ V עבור < S ו- < מייצג את כל המשטחים ב- >, γ המוכלות ב- >P וכאשר < מייצג את כל הנקודות ב- >, σ S המכילים את >. γ P γ אם < F היא בעלת >k דרגות חופש ביחס ל- >V אזי S γ γ P γ = O ( m k 2k 1 n 2k 2 2k 1 + m + n ) ואילו אם < F בעלת מימד מצומצם >s ביחס ל- V >, אז, לכל > 0 ε >, γ P γ = O ( m 2s 5s 4 n 5s 6 5s 4 +ℇ + m 2/3 n 2/3 + m + n ) כאשר קבועי הפרופורציונליות תלויים ב- E >,,D ובסיבוכיות של >, F וב- μ,,k < במקרה הראשון, או ב- s < וב- ε < במקרה השני. יתירה מכך, בשני המקרים, >, S כאשר קבוע הפרופורציונליות תלוי ב- >D וב- E. < γ = O(n) γ הערה. תכונה חשובה של תוצאה זו היא שאיננו מניחים שום הנחות על גרף החילות, כמו למשל הדרישה שלא יכיל עותק של הגרף הדו-צדדי השלם, K r,r עבור קבוע, r כפי שהניחו במחקרים קודמים [17,66,125]. התוצאה שלנו כללית יותר, ופירוט נרחב על כך ניתן למצוא בפרק 8 בתזה. 8

284 חסם חילות משולב (עבור נקודות על יריעה "כללית" ומשטחים כלליים). התוצאה הבאה שלנו היא שיפור של משפט 6, כאשר הנקודות נמצאות על יריעה < V מדרגה קבועה, שאינה נשלטת ע"י אינסוף עקומי חיתוך של משטחים מתוך >. F במקרה זה, אנו מקבלים חסם "משולב" משופר, שבו ניתן לפצל את גרף החילות ) S < G(P, לשני תתי-גרפים ) S < G 0,P) ו- ) S >, G 1,P) והחסמים על תת-הגרף הראשון הם חסמים על מספר החילות ממש. הוכחנו את המשפט הבא. משפט 7. R 3 תהי >P קבוצה של < m נקודות על יריעה אלגברית >V מדרגה קבועה >D ב- >, ותהי >S קבוצה של < n משטחים אלגבריים מדרגה מקסימלית >, E הלקוחים מתוך משפחה < F של משטחים אלגבריים, שהיא בעלת < k דרגות חופש ביחס ל- V < (ו- >μ קבוע כלשהוא), או שהיא ממימד מצומצם >s ביחס ל- V >, עבור פרמטרים קבועים כלשהם >, k (ו- μ >), או >. s אנחנו מניחים בנוסף שהמשטחים של >S לא מכילים רכיבים אי-פריקים משותפים (וזה ודאי קורה אם כולם אי-פריקים), וש- < V אינה נשלטת ע"י (הרכיבים האי-פריקים של) עקומי החיתוך של משטחים מתוך >. F אז ניתן לפרק את גרף החילות ) S G(P, < כ- G(P, S) = G 0 (P, S) γ P γ S γ P γ כאשר האיחוד הוא על-פני כל העקומים < γ שהם רכיבים אי-פריקים של עקומים מהצורה >, σ σ V עבור >, σ σ S וכאשר < מייצג חלק מהנקודות ב- >P המוכלות ב- >, γ ו- < מייצג את כל המשטחים ב- >S המכילים את >. γ אם < F היא בעלת >k דרגות חופש ביחס ל- V >אזי S γ G 0 (P, S) = O ( m k 2k 1 n 2k 2 2k 1 + m + n ), G 0 (P, S) ואם < F בעלת מימד מצומצםs < ביחס ל- V >, אז, לכל > 0 ε >, = O ( m 2s 5s 4 n 5s 6 5s 4 +ℇ + m 2/3 n 2/3 + m + n ) כאשר קבועי הפרופורציונליות תלויים ב- E >,,D ובסיבוכיות של >, F וב- μ,,k < במקרה הראשון, או ב- s >וב- ε < במקרה השני. יתירה מכך, בשני המקרים מתקיים < P ו- >, S כאשר קבוע γ = O(m) γ = O(n) γ γ הפרופורציונליות תלוי ב- D <,E, ובסיבוכיות של >, F וב- μ,,k < במקרה הראשון, או ב- s >וב- ε < במקרה השני. R 3 ניתן להשתמש במשפט 7 כאשר לוקחים בתור המשפחה >F את משפחת הספרות ב- >. 9

285 R 3 משפט זה מופיע בתזה (בפרק 8). נכלול כאן שתי מסקנות מעניינות שנובעות מהמקרה בו < F היא משפחת הספרות ב- >. משפט 8 (מרחקים שונים). א. ב. תהי P קבוצה של m נקודות על יריעה אלגברית V מדרגה קבועה D ב-, R 3 אז, לכל > 0 ε >, מספר המרחקים השונים הנקבעים ע"י הנקודות ב- הינו, כאשר קבוע הפרופורציונליות תלוי ב- וב- ε >. Ω(n 7/9 ε ) P 2 P D R 3 P 1 תהי < קבוצה של >m נקודות על יריעה אלגברית < V (כמו בסעיף א'), ותהי < קבוצה של < n נקודות כלשהן ב- >. אז, לכל > 0 ε >, מספר המרחקים השונים הנקבעים ע"י זוגות נקודות ב- < P 1 P 2 הינו m}),< Ω ( min{m 4/7 ε n 1/7 ε, m 1/2 n 1/2, כאשר קבוע הפרופורציונליות תלוי ב- D < וב- ε >. משפט 9 (מרחקים נשנים). א. ב. תהי P קבוצה של m נקודות על יריעה אלגברית V מדרגה קבועה D ב-, R 3 אז, מספר מרחקי היחידה הנקבעים ע"י הנקודות ב- הינו, כאשר קבוע הפרופורציונליות תלוי ב-. P 2 O(n 4/3 ) P R 3 D P 1 תהי < קבוצה של >m נקודות על יריעה אלגברית < V (כמו בסעיף א'), ותהי < קבוצה של < n נקודות כלשהן ב- >. אז, לכל > 0 ε >, מספר מרחקי היחידה הנקבעים ע"י זוגות נקודות ב- < P 1 P 2 הינו n),< O(m 6/11 n 9/11+ε + m 2/3 n 2/3 + m + כאשר קבוע הפרופורציונליות תלוי ב- D < וב- ε >. R 3 R 3 התוצאה האחרונה שאנו מציגים בתזה מתייחסת למקרה הכללי המערב קבוצת משטחים < S וקבוצת נקודות כלשהיא ב- >. הטיפול במקרה כללי זה הינו באמצעות טיעון אינדוקטיבי המתבסס על שיטת החלוקה הפולינומיאלית של גוט וכץ [56], שבו משפט 6 משחק תפקיד מרכזי. תוצאה זו מרחיבה תוצאה בעבודה מקדימה שלנו [105, משפט 1.4], מספירות למשטחים כלליים, ומרחיבה תוצאה חדשה של זאל, עבור משטחים כלליים, כאשר בניסוח המשפט שלנו, אין כלל אילוצים מוקדמים על גרף החילות ) S >. G(P, הוכחנו את המשפט הבא. משפט 10. תהי >P קבוצה של < m נקודות ב- >, ותהי < S קבוצה של < n משטחים אלגבריים מדרגה מקסימלית >, E הלקוחים מתוך משפחה s >-מימדית < F של משטחים אלגבריים. אז ניתן לפרק את גרף החילות ) S < G(P, כ- G(P, S) = G 0 (P, S) γ P γ S γ 10

286 < כאשר האיחוד הוא על-פני כל העקומים < γ המוכלים בחיתוכים של לפחות שני משטחים ב- S >, כאשר < P γ מייצג את כל הנקודות ב- >P המוכלות ב- >, γ ו- < מייצג את כל (אך לפחות שני) המשטחים ב- < S J(P, S ) γ ( P γ + S γ ) = O ( m 2s S γ 3s 1 n 3s 3 3s 1 +ε + m + n ) המכילים את >. γ יתירה מכך, לכל > 0 ε >, מתקיים ו- n) < G 0 (P, S ) = O(m + כאשר קבועי הפרופורציונליות תלויים ב- E <,ε,s,d, ובסיבוכיות של. F 11

287 תמצית בתזה זו אנחנו חוקרים מספר בעיות בגיאומטריה קומבינטורית, העוסקות בחילות בין נקודות ואובייקטים גיאומטריים אחרים, החל בישרים בשלושה וארבעה מימדים, עבור לעקומים אלגבריים, ועד למשטחים אלגבריים בשלושה מימדים. אנחנו מפתחים ומשתמשים במגוון כלים "תשתיתיים" באלגברה ובגיאומטריה אלגברית כדי להתמודד עם בעיות אלו, כלים שעשויים למצוא שימושים גם במגוון בעיות קומבינטוריות אחרות. בנוסף, אנו גם מיישמים חסמים אלו בפתרון בעיות נוספות בגיאומטריה קומבינטורית. בחלק הראשון של התזה, אנו מתייחסים לבעיה של קבלת חסמים הדוקים אסימפטוטית על מספר החילות בין נקודות וישרים במימדים גבוהים, ובכך מכלילים את החסם של סמרדי וטרוטר משנת 1983 עבור המקרה המישורי, ואת התוצאה החדשה יותר אך מרעישה לא פחות של גוט וכץ )משנת 2010( עבור המקרה התלת-מימדי. העבודה האחרונה הציגה כלים מתחום האלגברה המתקדמת ובמיוחד מתחום הגיאומטריה האלגברית, אשר לא נעשה בהם שימוש בתחום הקומבינטוריקה בעבר. בכך התאפשר לגוט וכץ לפתור באופן )כמעט( מלא את בעיית המרחקים השונים של ארדש, ולהשיג גבול תחתון על מספר המרחקים השונים הנקבעים על ידי קבוצה כלשהיא בת n נקודות במישור, בעיה פתוחה שעמדה עיקשת בסירובה מעל 60 שנה, למרות ניסיונות אמיצים רבים לפותרה. העבודה של גוט וכץ הוסיפה מומנטום משמעותי לתחום גיאומטריית החילות, ואפשרה פתרון של בעיות רבות, וביניהן הבעיות שחקרנו בתזה, אשר נדמו כבלתי פתירות לפני פריצת הדרך שלהם, באמצעות הכלים והטכניקות האלגבריות החדשות שפותחו. אנחנו מרחיבים את מחקר החילות בין נקודות וישרים לארבעה מימדים, ולאחר מכן לנקודות וישרים שמוכלים ביריעות דו- ותלת-מימדיות. בנוסף גם מצאנו הוכחה אלמנטרית לחסם של גוט-כץ עבור חילות בין נקודות וישרים בשלושה מימדים. אנו גם מציגים חסמים תחתונים עבור נקודות וישרים על משטח ריבועי תלת-מימדי ב- R, 4 ומקבלים משפטים מטיפוס רמזי עבור גרף המגעים של ישרים בשלושה מימדים. בחלק השני של התזה, אנו מרחיבים את מחקרנו על חילות בין נקודות וישרים לחקר חילות בין נקודות ועקומים אלגבריים מדרגה קבועה בשלושה מימדים ובמימדים גבוהים יותר. כמקרה פרטי של מחקר זה, אנחנו מקבלים חסם עליון חדש ומשופר על מספר החילות בין נקודות ומעגלים בשלושה מימדים. לאחר מכן אנו חוקרים חילות בין נקודות ומשטחים אלגבריים מדרגה קבועה בשלושה מימדים, כמו למשל מישורים, ספרות, ועוד. כתוצאה ממחקר זה, אנו מקבלים מספר חסמים חדשים על מספר המרחקים השונים והנשנים הנקבעים על ידי קבוצת נקודות המוכלת ביריעה דו-מימדית מדרגה קבועה בשלושה מימדים.

288 אוניברסיטת תל אביב הפקולטה למדעים מדוייקים ע"ש ריימונד ובברלי סאקלר ביה"ס למדעי המחשב ע"ש בלבטניק שיטות אלגבריות בגיאומטריה קומבינטורית וחישובית חיבור לשם קבלת תואר "דוקטור לפילוסופיה" מאת נועם סולומון עבודת המחקר בוצעה בהדרכתו של פרופסור מיכה שריר הוגש לסנאט של אוניברסיטת תל אביב אלול תשע"ז