MANIN S CONJECTURE FOR DEL PEZZO SURFACES

Transcription

1 MANIN S CONJECTURE FOR DEL PEZZO SURFACES Daniel Thomas Loughran School of Mathematics September 2011 A dissertation submitted to the University of Bristol in accordance with the requirements of the degree of Doctor of Philosophy in the Faculty of Science

2 Abstract This thesis concerns the study of the number of solutions of bounded height to some certain diophantine equations. We achieve asymptotic formulae for the corresponding counting problems for two surfaces over the rational numbers. One is a split singular del Pezzo surface of degree six with singularity type A 2, the other a split singular del Pezzo surface of degree four with singularity type 2A 1 and eight lines. These results are in accordance with Manin s conjecture for Fano varieties. Along the way we derive an asymptotic formula for a certain restricted divisor problem for binary linear forms, and we also classify those singular del Pezzo surfaces that are equivariant compactifications of the additive group G 2 a. This latter work is joint with Ulrich Derenthal. i

3 Acknowledgements I would like to begin by thanking my supervisor, Dr. Tim Browning for his continued support and near constant open door during my time at Bristol. I have gained much from having conversations with many people, in particular Régis de la Bretèche, Ulrich Derenthal, Emmanuel Peyre, Thomas Preu, Per Salberger, Tomer Schlank, Tony Scholl, Trevor Wooley and my fellow PhD students at Bristol. Part of this work was completed while visiting l Université Paris VII - Diderot, and whilst attending a trimester on diophantine equations at the Hausdorff Research Institute for Mathematics in Bonn. I am grateful to these institutions for their support. Thanks also goes to EPSRC for funding my PhD, and the University of Bristol for looking after me so well and providing me with funding to attend many conferences. Final thanks goes to Rag Morris and Folk club, for allowing me to have lots of fun whilst being as eccentric and silly as possible. ii

4 Author s Declaration I declare that the work in this dissertation was carried out in accordance with the requirements of the University s Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate s own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of the author. Daniel Thomas Loughran Date: iii

5 Contents Abstract Acknowledgements Author s Declaration Notation i ii iii vi 1 Introduction 1 2 The geometry of del Pezzo surfaces Divisors on algebraic varieties Blow-ups Del Pezzo surfaces Singular del Pezzo surfaces Split surfaces Singular del Pezzo surfaces and equivariant compactifications Introduction Preliminaries Strategy Proof of the main result An equivariant compactification of G a G m A restricted divisor problem 39 iv

6 Contents 4.1 Introduction Some multiplicative functions Proof of the main theorem Proof of corollary Counting rational points on varieties Classical results Manin s conjecture Peyre s constant Manin s conjecture for a singular sextic del Pezzo surface Introduction Preliminary steps The proof Manin s conjecture for a singular quartic del Pezzo surface Introduction Useful results The proof Bibliography 121 v

7 Notation Given a, b N, we write (a, b) for their greatest common divisor and [a, b] = ab/(a, b) for their least common multiple. For any arithmetic function f : N C, we sometimes consider it as a function on Z 0 by extending the definition of f to include f(0) = 0. We say that an arithmetic function f : N n C is multiplicative if for all d, e N n such that (d 1... d n, e 1... e n ) = 1, we have f(d 1 e 1,..., d n e n ) = f(d 1,..., d n )f(e 1,..., e n ). We also write ν p for the p-adic valuation of a rational number, and define 2 N = {2 n : n N}. We use the symbols O, o,,, and in accordance with their standard meaning in analytic number theory, as in the book [IK04]. For any function f : R C, we also use the shorthand n x f(n) to denote 0<n x f(n), and write n x f(n) to mean n cx f(n) for some unspecified constant c. Throughout this thesis ε is any positive real number which all implied constants are allowed to depend upon. We use the common practice that ε can take different values throughout the course of an argument. For any scheme X over a field k, we write X k = X k k for the base-change of X to an algebraic closure of k. We say that X is geometrically P if X k has property P. Then by a variety over a field k we mean a geometrically integral separated scheme of finite type over k. vi

8 Chapter 1 Introduction This thesis concerns the study of solutions to diophantine equations, something which has occupied mathematicians for thousands of years. Diophantine equations can be thought of as systems of polynomial equations with integer coefficients, where we seek to find solutions in the rational numbers or the integers. There are three logical possibilities, either the set of solutions is empty, finite but non-empty, or infinite. Trying to discern which of the above possibilities occurs can be a very difficult problem in general. However, if one knows that the set of solutions is infinite, then it is natural to try to quantify the density of such solutions. It is this problem which we will consider in this thesis. Our approach is to get approximate formulae for the number of solutions of bounded height, the height of a solution being a measure of its size. First note that it became clear throughout the course of the twentieth century that a very natural language and setting in which to study diophantine equations is that of algebraic geometry; whereby we can think of the study of diophantine equations as the study of rational points on algebraic varieties. We adopt this geometric formalism throughout this thesis. If X P n is a projective variety defined over the field of rational numbers, 1

9 Chapter 1. Introduction then we define the height of a rational point x X(Q) to be H(x) = max{ x 0, x 1,..., x n }, where x = (x 0 : : x n ) is a representative chosen so that (x 0,..., x n ) is a primitive integer vector. For any B > 0, we define the counting function associated to X to be N X (B) = #{x X(Q) : H(x) B}. Then it is clear that by studying this function, and in particular its asymptotic behaviour as B, one can hope to get an understanding of the distribution of the rational points on X. A common mantra in modern diophantine geometry is Geometry determines Arithmetic. Adhering to this principle, Manin and his collaborators (see [FMT89] and [BM90]) formulated a series of conjectures on the behaviour of counting functions for Fano varieties. We shall be particularly interested in the case of del Pezzo surfaces, the Fano varieties of dimension two. Important classical examples of del Pezzo surfaces include non-singular cubic surfaces in P 3 and non-singular intersections of two quadrics in P 4. These are the del Pezzo surfaces of degree three and four respectively. For such surfaces, Manin s conjecture takes the following form. Conjecture. Let X P d be a del Pezzo surface of degree d. Then there is a Zariski open subset U X, and constants c X and r X such that N U (B) = c X B(log B) r X (1 + o(1)), as B. Some remarks are in order. Firstly, the above formula is trivially true when X(Q) =, taking simply c X = 0. The conjecture therefore only has any real content as soon as X(Q). Moreover for del Pezzo surfaces, as soon as there is a rational point not lying on an exceptional curve we know that X(Q) is in fact Zariski dense in X [Man86, Thm. 29.4]. 2

10 Also in general, a variety may contain accumulating subvarieties, whose contribution will dominate the counting problem. For example, any line defined over Q inside a smooth cubic surface will contribute roughly B 2 rational points to the problem, so it is natural to remove these (finitely many) lines in this case. Moreover one believes that r X should be a subtle geometrical invariant, namely r X = ρ X 1 where ρ X is the rank of the Picard group of X. For example, if X is a smooth cubic surface all of whose lines are defined over Q, then one should have r X = 6. There is a programme (see [BB07], [DT07] or [LB10a] for example) to try to prove this conjecture for all (possibly singular) del Pezzo surfaces. One allows singularities, since in general singular del Pezzo surfaces are more rigid and have simpler arithmetic than their non-singular counterparts. For example Manin s conjecture is not known for a single non-singular cubic surface, however it is known (see e.g [BBD07]) for some singular del Pezzo surface of degree three (these are cubic surfaces with mild singularities). Moreover, only recently has Manin s conjecture been proved for a single example of a non-singular del Pezzo surface of degree four [BB08]. There are some more general results. For example Manin s conjecture is known for all toric varieties [BT98a], and for any variety which is an equivariant compactification of the additive group [CLT02]. Varieties of this latter type are those varieties which possess an open dense subset isomorphic to G n a, such that the natural translation action of G n a on itself extends to the whole variety. The proofs of these results make essential use of the underlying group action, by the means of harmonic analysis. When there is no group structure present, each proof of Manin s conjecture, for even a single variety, offers more insight into the problem. Table 5.2, which can be found at the end of Chapter 5, contains an up to date account of the progress so far for del Pezzo surfaces. In this thesis, we prove versions of Manin s conjecture for two specific singular del Pezzo surfaces. The first is a split singular del Pezzo surface of degree six with singularity type A 2. It given as the intersection of 9 quadrics 3

11 Chapter 1. Introduction in P 6. Here, we get a significant saving on the error term, and achieve an asymptotic formula of the shape N U (B) = BP (log B) + O(B 7/8+ε ), where P R[x] is a cubic polynomial. The proof relies on transferring the counting problem for this surface to the associated universal torsor. This can be thought of as a kind of descent, whereby we parameterise the rational points on the surface by a simpler variety. We also manage to get a meromorphic continuation of the associated height zeta function Z U (s) = x U(Q) 1 H(x) s, to the half-plane Re(s) > 5/6, by finding an expression for Z U (s) in terms of the Riemann zeta function and some other well-behaved functions. This work has been published in [Lou10]. The second surface is a singular del Pezzo surface of degree four with singularity type 2A 1, and has equations S : x 0 x 1 = x 2 2, x 3 x 4 = x 2 (x 1 x 0 ), inside P 4. Here we prove Manin s conjecture, but with no saving in the error term. The method of proof is quite different to that of the previous surface, and utilises the fact that S comes equipped with a conic bundle structure. This transforms the problem of counting the rational points on S, to one of counting rational points on a family of conics. In order to count the rational points on each conic, we prove an auxiliary result on the asymptotic behaviour of a certain restricted divisor problem for binary linear forms. Namely, for a lattice Λ Z 2, a suitable region R R 2, and binary linear forms L 1, L 2, L 3, L 4 Q[x] such that L i (x) Z for all x Λ, (i = 1, 2, 3, 4), we show that x Λ XR τ(l 1 (x), L 2 (x), L 3 (x), L 4 (x); V ) = cx 2 (log X) 4 (1 + O Li,Λ 4 ( log log X log X )),

12 as X. The function that we consider is a kind of restricted divisor function, whereby we only count those divisors which satisfy certain relations of the shape d a 1 1 d a 2 2 d a 3 3 d a 4 4 C for some a 1, a 2, a 3, a 4, C R, determined by a set V [0, 1] 4. If V = [0, 1] 4 for example, then we simply recover a fourfold product of the usual divisor function. Sums of the shape x Z 2 XR τ(f (x)), for some binary form F, have been considered before. Daniel [Dan99] proved an asymptotic formula in the case where F is an irreducible quartic form, and in [BB10] an asymptotic formula was proved in the case where Q factors as two linear forms times a quadratic form, also considering a restricted divisor function as we do here. There is also recent work of Matthiesen [Mat11] in which she proves an asymptotic formula in the case where F is the product of an arbitrary number of linear forms. We note that this work does not imply ours, and moreover the fact that we consider a restricted divisor function is essential to our proof of Manin s conjecture for the surface S. As noted above, Manin s conjecture is known for all toric varieties and equivariant compactifications of the additive group. Therefore, it is useful to know which del Pezzo surfaces are already covered by these results. Derenthal [Der06] has classified those singular del Pezzo surfaces which occur as toric varieties, and the final piece of original research in this thesis is a classification of those singular del Pezzo which are of the latter type. This work is joint with Ulrich Derenthal, and has been published in [DL10]. We also include a previously unknown example of a a singular del Pezzo surface of degree four that is an equivariant compactification of G a G m, to highlight the fact that a proof of Manin s conjecture for varieties of this type will even apply to some del Pezzo surfaces. In fact since [DL10] has appeared, a proof for Manin s conjecture for such varieties has been announced [TT11]. The layout of this thesis is as follows. In Chapter 2 we recall various important facts on the geometry of del Pezzo surfaces and related surfaces. 5

13 Chapter 1. Introduction Chapter 3 contains the classification of those singular del Pezzo surfaces that are equivariant compactifications of G 2 a. In Chapter 4, we prove the above mentioned result on an asymptotic formula for a restricted divisor problem for four linear forms. In Chapter 5 we give a brief overview on counting rational points on projective varieties, and in particular a precise formulation of Manin s conjecture for singular del Pezzo surfaces. Chapters 6 and 7 then contain the proofs of Manin s conjecture for the singular sextic and singular quartic del Pezzo surface respectively. The logical dependence of the chapters is summarised in figure Figure 1.1: Logical dependence between chapters. 6

14 Chapter 2 The geometry of del Pezzo surfaces In this chapter we begin with recalling various facts about divisors and blowups, many of which can be found in [Har77] and [Lar07]. Drawing on these, we define del Pezzo surfaces and study some of their simple properties, a good reference for these being [Man86]. We then study the closely related notions of singular and generalised del Pezzo surfaces, which will in fact be the main focus of study for the rest of this thesis. A good resource for these surfaces is [CT88]. We then finish by discussing split surfaces and also give a criterion for a geometrically rational smooth projective surface to be split. In this chapter we work in the category of algebraic varieties over a perfect field k, with fixed algebraic closure k. 2.1 Divisors on algebraic varieties Throughout this section, X is a normal variety. Such varieties are well-behaved from a geometrical point of view since any normal variety is non-singular in codimension one, which leads to a working theory of Weil divisors. The following definitions and facts can be found in [Har77, II.6]. 7

15 Chapter 2. The geometry of del Pezzo surfaces A Weil divisor on X is defined to be a formal sum of irreducible closed subvarieties of codimension one defined over k. A Weil divisor is called Cartier if it is locally principal. That is to say, locally it is given as the zeros and poles of some rational function defined over k. To any Cartier divisor D on X, we can associate a line bundle O X (D) equipped with a section corresponding to D. Conversely, any line bundle is associated to some Cartier divisor in this way. If X is smooth, then Weil divisors and Cartier divisors coincide, but this is not necessarily the case when X is singular. If f : X Y is a morphism of varieties and D is a Cartier divisor on Y such that Im f D, then we may pull D back to get a Cartier divisor f D on X. Given a Cartier divisor D, we can associate to it a rational map X P n. We say that D is very ample if this map is in fact a closed immersion, and we say that D is ample if some multiple of it is very ample. Every variety X comes equipped with an important divisor, called the canonical divisor, which encodes a lot of the geometry of X (See [Har77, Sec. II.8]). If X is smooth, then the highest exterior power of its cotangent bundle is a line bundle, the canonical line bundle. A canonical divisor K X is any divisor associated to this line bundle, and we often abuse notation and speak of the canonical divisor. We also call K X the anticanonical divisor. If X is singular normal, then normality implies its singular locus has codimension two. We may then define its canonical divisor to be the closure of the canonical divisor of the smooth locus. Note that this is only a Weil divisor in general, and not Cartier. 8

16 2.2. Blow-ups The group of all divisors is quite big, so it is useful to quotient out by a suitable equivalence relation to get the Picard group (See [Har77, Sec. II.6]). The Picard group Pic(X) of X is defined to be the collection of all line bundles on X up to isomorphism, with group operation being tensor product. This group is naturally isomorphic to the group of all Cartier divisors up to linear equivalence. If X is also projective, then given a collection D 1,..., D n of Cartier divisors which intersect transversely at smooth points of X, we define their intersection number to be D 1... D n = #(D 1... D n ). This extends to give a unique symmetric multilinear form on Pic(X) (see [Lar07, Sec. 1.1.C] for the complex case or [Ful98, Ch. 2] for the general case). We denote the self-intersection of a divisor with itself by D n. 2.2 Blow-ups Blow-ups are fundamental in algebraic geometry, and can be thought of as a way of turning subvarieties into divisors. For example, blowing up a smooth point on a surface can be thought of as replacing that point by all of its possible tangent directions, which is a copy of P 1. Blow-ups can also be used to resolve the singularities of a variety. For example Hironaka [Hir64] has shown that any variety X over a field of characteristic zero admits a resolution of singularities, which is achieved by repeatedly blowing up the singularities of X. We define blow-ups through their universal property. Definition Let X be a variety, and Z X a closed subscheme over k (e.g. a rational point, or a Galois orbit of points over k). Then the blow-up of X at Z is a variety X (also denoted Bl Z X) together with a morphism π : X X, such that the inverse image of Z is a Cartier divisor, called the exceptional divisor of the blow-up. Moreover it is the universal such object, namely if 9

17 Chapter 2. The geometry of del Pezzo surfaces φ : Y X is any morphism such that the inverse image of Z is a Cartier divisor, then φ factors uniquely through π. Blow-ups always exist, and moreover the morphism π is an isomorphism away from the exceptional divisor. Their construction and basic properties can be found in [Har77, Ch. II.7]. In the case of blowing-up points on varieties, we shall use the following terminology. In what follows x X(k) is a rational point, and π : X X is the blow-up of X at x. We say that a point x X(k) is infinitely near x if it lies in the support of the exceptional divisor of X. Given an effective divisor D on X, we define its strict transform D to be the closure of π 1 (D \ {x}) in X. Most of the properties of blow-ups that we shall need are summarised in Theorem Before we state the result, recall that given a normal projective surface X and n N, a ( n)-curve (or simply a negative curve) on X is a smooth geometrically integral curve E of genus zero on X such that E 2 = n. Such curves are important for us since they are created through blowing-up points on surfaces. Theorem Let X be a smooth projective surface, and let π : X X be the blow-up of X at a rational point, with exceptional divisor E. Then X is also a smooth projective surface and the canonical divisor on X satisfies K X = π K X + E. Moreover the natural map Pic(X) Z Pic( X) (D, n) π (D) + ne, is an isomorphism. The following facts completely determine the intersection behaviour of divisors on X. (a) π (D 1 ) π (D 2 ) = D 1 D 2 for any two divisors D 1, D 2 on X. 10

18 2.3. Del Pezzo surfaces (b) π (D) E = 0 for any divisor D on X. (c) E is a ( 1)-curve. (d) π (D) = D + re, where D is any effective divisor on X with multiplicity r through x. In particular D 2 = D 2 r. In fact, any ( 1)-curve on X arises as the exceptional divisor of the blow-up of some surface at a smooth rational point. Proof. The facts on the Picard group can be found in [Har77, Prop. V.3.2]. The proof of the expression for the canonical divisor is given in [Har77, Prop. V.3.3], and (d) is [Har77, Prop. V.3.6]. A proof of the fact that any ( 1)- curve is the exceptional curve of some blow-up (this is called Castelnuovo s contraction criterion) can be found in [Har77, Thm. V.5.7]. We will see in the next section that any del Pezzo surface over an algebraically closed field is the blow-up of P 2 at 9 points in general position. Thus Theorem will allow us to calculate the geometric invariants of such surfaces, on recalling that Pic(P 2 ) = Z, with a generator being the class of a line L, and K P 2 = 3L (see [Har77, Prop.II.6.4] and [Har77, Exa. II ]). 2.3 Del Pezzo surfaces We now come to the definition of del Pezzo surfaces. Definition A del Pezzo surface is a smooth projective surface X whose anticanonical divisor K X is ample. We define the degree of X to be KX 2. It follows immediately from this definition and the adjunction formula [Har77, Ex. II.8.4] that any non-singular cubic surface in P 3, and any nonsingular intersection of two quadrics in P 4 is a del Pezzo surface. The best way to visualise del Pezzo surfaces of arbitrary degree is through the following theorem. 11

19 Chapter 2. The geometry of del Pezzo surfaces Theorem Let X be a del Pezzo surface of degree d over an algebraically closed field. Then 1 d 9, and either X = P 1 P 1 or X is the blow-up of P 2 in 9 d points in general position. Where for d 3, general position means 1. No three points lie on a line. 2. No six points lie on a conic. Conversely, every del Pezzo surface arises in this way. Proof. See [Man86, Thm. 24.4] We remark that there is a similar (although slightly more technical) meaning of general position for d = 1, 2. However, in this thesis we shall mostly be concerned with del Pezzo surfaces of degree 3, so we have omitted it. See [Man86, Thm. 26.2] for a precise statement. Morally however, general position means never blow-up a point which lies on a ( 1)-curve. Thus by Theorem 2.2.2, we only allow ( n)-curves with n 1 on del Pezzo surfaces. Del Pezzo surfaces of degree one and two are special for another reason. If X is a del Pezzo surface of degree d, then K X always defines a rational map X P d. If d 3, then this map is in fact a closed immersion, i.e. K X is very ample, and moreover the negative curves on X k are exactly the lines under this embedding. However, for obvious reasons of dimension this is not the case for d = 1, 2 (see [Man86, Thm. 24.5] and [Man86, Rem ]). Thus for simplicity of exposition, sometimes in this thesis we omit the cases d = 1, 2, or deal with them separately. If X is a del Pezzo surface over a non-algebraically closed field, then Theorem can be slightly misleading. Since it is quite possible for X to not be rational over the ground field, even if it contains rational points. Examples of such surfaces are minimal cubic surfaces [Man86, Thm. 33.1]. However one can use Theorem to good effect, as much of the arithmetic of del Pezzo surfaces over non-algebraically closed fields is governed by action of Gal(k/k) on the exceptional curves of X k. 12

20 2.4. Singular del Pezzo surfaces Theorem together with Theorem allow us to deduce many of the geometric properties of del Pezzo surfaces. For example we see that if X is a del Pezzo surface of degree d, then Pic(X k ) = Z 10 d, with a set of generators given by the pull-back of a line in P 2 and the exceptional curves created through blowing-up points in P 2 (at least if X k = P 1 P 1 ). 2.4 Singular del Pezzo surfaces There are two natural ways to generalise the definition of del Pezzo surfaces. Firstly, we can retain smoothness but weaken the condition that K X is ample, which leads to generalised del Pezzo surfaces. One can instead retain the condition that K X is ample but allow mild singularities, leading to singular del Pezzo surfaces. It turns out (see Theorem 2.4.4) that these two approaches are essentially equivalent, as any generalised del Pezzo surface is the minimal desingularisation of a singular del Pezzo surface. Most of our work in this section is drawn from [CT88] First recall that a Cartier divisor D on a projective variety X is called nef (numerically effective) if D C 0 for every irreducible curve C X, and is called big if for large enough m the rational map associated to md is birational onto its image. There is a useful numerical criterion for testing whether a nef divisor D is big, namely D is big if and only if D dim X > 0 (See [Lar07, Thm ]). Definition A generalised del Pezzo surface X is a smooth projective surface whose anticanonical divisor K X is nef and big. We define the degree of X to be KX 2. We also define the type of a generalised del Pezzo surface X of degree d to be the data (d, G), where G is the extended Dynkin diagram of X k, that is, the dual graph of the configuration of negative curves on X k. We briefly discuss our definition of generalised del Pezzo surfaces and its relation to the literature, as there are a few competing definitions which can 13

21 Chapter 2. The geometry of del Pezzo surfaces lead the unwary astray. First note that a del Pezzo surface is clearly a generalised del Pezzo surface, since ample divisors are big and nef by the Nakai- Moishezon criterion for ampleness [Har77, Thm. V.1.10]. Next, what we call a generalised del Pezzo surface is also sometimes called a weak del Pezzo surface (see e.g. [Sch07]). Finally, some authors define them to be surfaces for which K X is semi-ample and of positive degree, that is to say, some multiple of K X is base-point free and ( K X ) 2 > 0, (see e.g. [CGL10], where in fact they abuse notation and simply call such surfaces del Pezzo surfaces). This definition is in fact equivalent to ours, as such a divisor is necessarily nef and big since some multiple of any semi-ample divisor must be effective and hence K X C 0 for all curves C on X. Moreover, we shall see in Theorem that K X being nef and big actually implies that K X is semi-ample. Next, we show that generalised del Pezzo surfaces are classified in a manner analogous to Theorem In the statement of the theorem F 2 denotes a Hirzebruch surface. We say more on such surfaces in Section 2.5. Theorem Let X be a generalised del Pezzo surface of degree d over an algebraically closed field. Then 1 d 9, and either X = P 1 P 1, X = F 2, or X is the blow-up of P 2 in 9 d (possibly infinitely near) points in almost general position. Where by almost general position, we mean that we never blow-up a point lying on a ( 2)-curve. Conversely, every generalised del Pezzo surface arises in this way. Proof. See [CT88, Prop. 0.4] Our next aim is to define singular del Pezzo surfaces, but to do this we first need to do some singularity theory. Let X be a singular surface. We say that π : X X is a resolution of singularities if X is a smooth surface and π is a birational morphism. We say that the resolution is minimal (or a minimal desingularisation) if every other resolution factors through it. 14

22 2.4. Singular del Pezzo surfaces Minimal resolutions of surfaces always exist and are unique up to isomorphism. Such resolutions can be constructed by blowing-up the singular points on the surface. This is a theorem of Zariski [Zar39] in characteristic zero, and Abhyankar [Abh56] in non-zero characteristic. The singularities that we are primarily interested in are du val singularities (also called rational double points). For our purposes it will be sufficient to know that over an algebraically closed field of characteristic zero, a singularity is du Val if and only if local analytically it is isomorphic to one of the forms in Table 2.1 (see [Has09, Prop. 4.10] or [KM98, Thm. 4.22]). Here, the resolution graph has one vertex for each exceptional curve on the minimal desingularisation above the singularity, and an edge joining two vertices if and only if the corresponding curves intersect. The type of each singularity is then defined to be the Dynkin diagram given by the resolution graph. Singularity type Equation Resolution graph A n x 2 + y 2 + z n+1 D n x 2 + y 2 + z n 1 E 6 x 2 + y 3 + z 4 E 7 x 2 + y 3 + yz 3 E 8 x 2 + y 3 + z 5 Table 2.1: Du Val Singularities Although it is not perhaps a priori clear from this definition, these singularities are quite natural from our perspective. For example, suppose that X is a cubic surface or the intersection of two quadrics in P 4 over C with only isolated singularities, and that X is not a cone over an elliptic curve. Then X has only du Val singularities [CT88, Prop. 0.2, Prop. 0.3]. It is these surfaces which are a natural generalisation of del Pezzo surfaces, indeed, cones over elliptic curves are not even rational varieties. 15

23 Chapter 2. The geometry of del Pezzo surfaces Definition A singular Del Pezzo surface is a singular normal projective surface X with only du Val singularities, whose anticanonical divisor K X is ample. Its degree is defined to be KX 2. We also define the type of a singular del Pezzo surface X of degree d to be the data (d, G), where G is the extended Dynkin diagram of X k, that is, the dual graph of the configuration of negative curves on the minimal desingulariation X of X over k. We note that from Section 2.1, the canonical divisor of a singular algebraic variety is only a Weil divisor in general, and we only defined ampleness and intersection numbers for Cartier divisors. However it turns out that in fact K X is automatically Cartier as long as X has only du Val singularities [KM98, p. 123]. Also, what we have called singular del Pezzo surfaces are sometimes referred to as Gorenstein log del Pezzo surfaces or alternatively log del Pezzo surfaces of index one in the literature, see e.g. [Ye02] or [AN06]. As in the case of del Pezzo surfaces, K X is very ample for a singular del Pezzo surface X of degree d if and only if d 3 [CT88, Prop. 0.6]. In [CT88, p. 29], they define singular del Pezzo surfaces to be singular normal surfaces whose minimal desingularisation is a generalised del Pezzo surface. The following theorem shows that in fact this definition is equivalent to ours. Theorem Let X be a singular del Pezzo surface and π : X X a minimal desingularisation. Then X is a generalised del Pezzo surface with K X = π ( K X ), and the inverse image of the singularities of X is exactly the collection of ( 2)-curves on X k. Conversely, any generalised del Pezzo surface which is not a del Pezzo surface arises in this way. Proof. First let π : X X be the minimal desingularisation of a singular del Pezzo surface X. Since du Val singularities are canonical (see e.g. [KM98, Thm. 4.20]), it follows that K X = π ( K X ). Hence K X is nef since the pull-back 16

24 2.4. Singular del Pezzo surfaces of a nef divisor is nef [Lar07, Exa.1.4.4]. Moreover we have K 2 X = π (K X ) 2 > 0 and so K X is big. Indeed, we may find two divisors K 1, K 2 linearly equivalent to K X whose support does not contain the singular locus of X. Then since π is an isomorphism outside this locus, we have ( K X ) 2 = ( K 1 ) ( K 2 ) = π ( K 1 ) π ( K 2 ) = ( K X) 2 as required. So X is a generalised del Pezzo surface. Now, every curve created through resolving the singularities of X is a ( 2)- curve by [Has09, Prop. 4.2], so it suffices to show that every ( 2)-curve E on X k arises in this way. The push-forward π (E) of E is either a non-zero effective divisor on X, or else π (E) = 0. In the latter case E is sent to a single point on X, so it must lie above the singularities of X. So suppose that π (E) is a non-zero effective divisor on X. Then the projection formula [Ful98, Prop. 2.3(c)] implies that E π ( K X ) = π (E) ( K X ). Since K X is ample, π (E) ( K X ) > 0 by the Nakai-Moishezon criterion for ampleness [Har77, Thm. V.1.10]. However, the adjunction formula [Har77, Prop. V.1.5] implies that E ( K X) = 0, which gives a contradiction since K X = π ( K X ). Thus π (E) = 0 as required. The converse result follows from [Has09, Cor. 4.6]. This theorem implies that the two notions of type defined for generalised del Pezzo surfaces and singular del Pezzo surfaces coincide. It is also possible to classify which types may occur, at least over algebraically closed fields of characteristic zero. Bruce and Wall [BW79] achieved this classification for singular cubic surfaces. For singular del Pezzo surfaces of higher degree, this can be found in [CT88, Prop. 6.1] and [CT88, Sec. 8]. The case of singular del Pezzo surfaces of degree 1 or 2 is handled in [AN06]. Often, the type of a singular del Pezzo surface X is determined by its degree d and its singularity type alone. For example, there is unique singular del Pezzo surface over k of degree 8 and singularity type A 1. Indeed, the minimal desingularisation X for such a surface is a generalised del Pezzo surface of 17

25 Chapter 2. The geometry of del Pezzo surfaces degree 8, so is either P 1 P 1, F 2 or the blow-up of P 2 at a single point, by Theorem However of these surfaces, only F 2 contains a ( 2)-curve, hence X k = F2 and the type is uniquely determined. There are however singular del Pezzo surfaces with the same degree and singularity type, but with different types. For example a singular del Pezzo surface of degree six and singularity type A 1 may have two types [CT88, Prop. 8.3]. In order to differentiate between these two types it suffices to note that one type has three lines (a slight abuse of notation for the ( 1)-curves appearing on the minimal desingularisation) and the other has four lines. Hence throughout this thesis when we label the type of a surface, we abuse notation and give its singularity type and its degree. If this does not uniquely determine the type, then we also state the number of lines it has. 2.5 Split surfaces We finish this chapter with a discussion on split surfaces. Such surfaces generally have simpler arithmetic than non-split ones, and the two surfaces that we consider in Chapter 6 and Chapter 7 are split. Definition We say that a smooth projective geometrically rational surface X is split if the natural map Pic(X) Pic(X k ) is an isomorphism, i.e. every divisor over the algebraic closure is linearly equivalent to one defined over the ground field. We say that a singular projective geometrically rational surface is split if its minimal desingularisation is split. Naturally any smooth projective geometrically rational surface X is split after some finite field extension. Simply choose a finite set of divisors over k generating Pic(X k ) and let L be their field of definition. Then X L is a split surface. In this section we give a criteria to determine whether or not X is in fact split over k, by appealing to the classification of rational surfaces over algebraically closed fields. Moreover we shall show that our definition of 18

26 2.5. Split surfaces split is equivalent to the usual definition of split (see e.g. [DL10]), namely that a generalised del Pezzo surface is split if and only if it is isomorphic to P 2, P 1 P 1, F 2 or the blow-up of P 2 at collection of points all of which are rational. First recall that we say that a smooth projective surface over an algebraically closed field is minimal if it contains no ( 1)-curves. Such surfaces arise naturally in the birational classification of surfaces, since given any ( 1)- curve we may always contract it by Theorem Running this process iteratively, one arrives at a (possibly non-unique) minimal surface. See [Har77, Thm. 5.8] for further details. Minimal rational surfaces over an algebraically closed field admit a particulary simple description. Namely, such a surface is either P 2, P 1 P 1 or a Hirzebruch surface F n with n 2 (see e.g. [Bea96, Thm. V.10]). The Hirzebruch surfaces F n for n 0 are the rational ruled surfaces, that is, the surfaces X equipped with a morphism X P 1, such that every fibre is isomorphic to P 1. We have the exceptional isomorphism F 0 = P 1 P 1, and F 1 is isomorphic to P 2 blown-up at a single point. For our purposes it suffices to know that Pic F n = Z 2, generated by the class of a fibre F and the class of the zero section B. These divisors satisfy F 2 = 0, B 2 = n, F B = 1, and B is the unique irreducible curve on F n with negative self-intersection if n 1. These facts can be found in [Bea96, Prop. IV.1]. We begin the classification of split surfaces with a well-known result on the arithmetic of negative curves. Lemma Let X be a smooth projective surface. Then any negative curve on X is the unique effective curve in its divisor class. In particular, if E is a negative curve on X k whose divisor class is invariant under the action of Gal(k/k), then E is in fact defined over k. Proof. Let E be a negative curve on X and suppose that E is an effective 19

27 Chapter 2. The geometry of del Pezzo surfaces divisor on X linearly equivalent, but not equal to, E. In particular E E and hence E E 0. However since E is linearly equivalent to E, we see that E E < 0, a contradiction. To prove the second part of the Lemma, it suffices to note that the action of Gal(k/k) sends effective divisors to effective divisors. Next recall that we say a variety X is a form of a variety Y if X k = Yk. The forms of X are classified by the cohomology set H 1 (k, Aut X k ) [HS00, Exa. C.5.1]. Lemma Let X be a smooth projective geometrically rational surface. If X k is minimal and X k = P 1 P 1, then X is split if and only if X(k). If X k is not minimal, then X is split if and only if each negative curve on X k is defined over k. Moreover if X has a form which is split, then such a form is unique up to isomorphism. Proof. First suppose that X k = P 2. Clearly P 2 itself is split, and moreover it is a classical fact that any form of P n containing a rational point is in fact isomorphic to P n, so we have uniqueness in this case. For a proof see [Jah08, Prop. I.5.9]. Next, suppose that X k = P 1 P 1. If X is split, then there are curves L 1 and L 2 on X corresponding to the two rulings of P 1 P 1. The rational maps associated to these divisors give us two morphisms π i : X P 1, for i = 1, 2. Putting these together gives a morphism X P 1 P 1, which is clearly an isomorphism. This also proves uniqueness in this case. If on the other hand X k = Fn for some n 1, then since B is the unique curve on X k with negative self intersection, we see that B is in fact defined over k by Lemma Now suppose that X(k) and let X X be the blow-up of X at a rational point x X(k) with exceptional curve E. Then by Theorem 2.2.2, the only negative curves on X k are E and the strict transforms B, F of the curve B and the fibre F on X k passing through x. However, E and 20

28 2.5. Split surfaces B are both defined over k, and hence so it F by Lemma Contracting E, we see that F is also defined over k, and thus X is split. Similarly, if X is split it is clear that X(k). Indeed the intersection of a fibre with B yields a rational point. We now show that if X is a split form of F n for some n 1, then X = F n. The rational map associated to a fibre F yields a morphism to a curve C of genus zero. However, X(k) thus C(k), so in fact C = P 1. Thus X is a ruled surface over P 1, and it is also has a section corresponding to the curve B. Ruled surfaces with a section are classified by vector bundles of rank two on the base curve [Har77, Prop. 2.2], and any vector bundle of rank two on P 1 is a direct sum of two line bundles [Har77, Cor. 2.14]. The surfaces corresponding to these vector bundles are exactly the Hirzebruch surfaces, by definition. Thus X = F n. We now suppose that X k is not minimal. If X is split, then certainly each negative curve on X k is defined over k by Lemma Conversely we may contract each ( 1)-curve iteratively to get a morphism to a geometrically minimal surface Y. To prove that X is split it therefore suffices to show that Y is split. First note that Y (k) as the image of any negative curve is a rational point. As we have seen, this shows Y is split if Y k = P 1 P 1. If Y k = P 1 P 1, then we may blow-up the rational point corresponding to the last contracted ( 1)-curve, to get a surface Ỹ with an exceptional curve E. However, Ỹ k contains three ( 1) curves corresponding to E and the two rulings on P 1 P 1. By assumption, each of these is defined over k. Hence contracting E we see that Y is also split in this case. Since Y has a unique split form, it follows that the same is true for X. We finish this section by remarking that not every surface has a split form. Indeed, if X is the blow-up P 2 in sufficiently many conjugate points then Aut X k is trivial, so H 1 (k, Aut X k ) = 0 and X has no non-trivial forms. How- 21

29 Chapter 2. The geometry of del Pezzo surfaces ever obviously X is not split. 22

30 Chapter 3 Singular del Pezzo surfaces and equivariant compactifications The work in this chapter first appeared in [DL10], and is joint with Ulrich Derenthal. 3.1 Introduction In this chapter we classify those generalised and singular del Pezzo surfaces that are equivariant compactifications of G 2 a. This result is motivated with arithmetic applications in mind. Indeed, Manin s conjecture (see Section 5.2) is already known for such surfaces by [CLT02]. So given a generalised or singular del Pezzo surface, it is useful to know whether or not this surface is already covered by [CLT02]. Manin s conjecture is also known for all toric varieties [BT98a], i.e. varieties which are equivariant compactifications of an algebraic torus. Toric del Pezzo surfaces have been classified: del Pezzo surfaces are toric precisely in degree 6. In lower degrees, there are some toric singular del Pezzo surfaces, for example a cubic surface with 3A 2 singularities, for which Manin s conjecture was proved not only by the general results of [BT98a], [Sal98], but also by more direct methods in [Fou98], [Bre98], [HBM99]. The classification of all 23

31 Chapter 3. Singular del Pezzo surfaces and equivariant compactifications toric singular del Pezzo surfaces is known and can be found in [Der06], for example. The main result of this chapter is the following. Theorem Let S be a (possibly singular or generalised) del Pezzo surface of degree d, defined over a field k of characteristic 0. Then S is an equivariant compactification of G 2 a over k if and only if one of the following holds: S has a non-singular k-rational point and is a form of P 2, P 1 P 1, the Hirzebruch surface F 2 or the corresponding singular del Pezzo surface, S is a form of Bl 1 P 2 or Bl 2 P 2, d = 7 and S is of type A 1, d = 6 and S is of type A 1 (with 3 lines), 2A 1, A 2 or A 2 + A 1, d = 5 and S is of type A 3 or A 4, d = 4 and S is of type D 5. In Lemma 3.2.5, we will give a criterion that will reduce the number of candidates of generalised del Pezzo surfaces that might be equivariant compactifications of G 2 a, to a short list of surfaces that are connected by blow-ups and blow-downs as presented in Figure 3.1. Using a strategy described in Section 3.3, we will show explicitly that the surfaces of type A 1 in degree 6, type A 3 in degree 5 and type D 5 in degree 4 are equivariant compactifications of G 2 a, while type D 4 in degree 4 and type E 6 in degree 3 cannot have this structure. From these borderline cases, some general considerations will allow us to complete the classification over algebraically closed fields. Over non-closed fields, some additional work will be necessary. In Section 3.5, we will give an example of a del Pezzo surface that is neither toric nor an equivariant compactification of G 2 a, but an equivariant compactification of a semidirect product G a G m. This shows that it could be worth- 24

32 3.2. Preliminaries while even for del Pezzo surfaces to extend the harmonic analysis approach to Manin s conjecture for equivariant compactifications of more general algebraic groups than tori and vector spaces. Indeed, since the publication of [DL10], there have been further developments in this direction [TT11]. d = /2/1 A 2 + A 1 A 4 D 5 E 6 F 2 A 1 A 2 E 7 P 2 Bl 1 P 2 Bl 2 P 2 2A 1 A D 3 4 E 8 P 1 P 1 A 1 Figure 3.1: Generalised del Pezzo surfaces S defined over k that satisfy #{negative curves on S} rk Pic(S). The boxed ones are equivariant compactifications of G 2 a. Arrows denote blow-up maps. 3.2 Preliminaries In this section we gather some elementary results on equivariant compactifications of G 2 a, and in particular their behaviour under blow-ups. We work over a field k of characteristic 0 with algebraic closure k. If G is a geometrically connected linear algebraic group defined over k, then we say that a proper variety X defined over k is an equivariant compactification of G over k or alternatively a G-variety over k, if G acts on X, with the action being defined over k, and there exists an open subset U X which is equivariantly isomorphic to G over k. By an equivariant morphism, we mean a morphism commuting with the action of G. We note that any algebraic group over k which is isomorphic to G n a over k, is also isomorphic to G n a over k. 25

33 Chapter 3. Singular del Pezzo surfaces and equivariant compactifications An equivalence between G-varieties X 1, X 2 is a commutative diagram G X 1 X 1 (α,j) G X2 j X 2 (3.2.1) where α : G G is an automorphism and j : X 1 X 2 is an isomorphism. Lemma Up to equivalence, there are exactly two distinct G 2 a-structures on P 2 over k. They are given by the following representations of G 2 a: τ(a, b) = a 1 0 b 0 1, ρ(a, b) = a 1 0 b a2 a 1. Proof. See [HT99, Proposition 3.2]. Lemma Let S be a non-singular G 2 a-variety over k, and E S k a ( 1)-curve which is invariant under the action of the Galois group Gal(k/k). Then there exists a G 2 a-equivariant k-morphism that blows down E. Proof. See [HT99, Proposition 5.1] for the corresponding statement over k. It is clear that if E is invariant under the action of the Galois group Gal(k/k), then the corresponding morphism is defined over k. Lemma Let G be an algebraic group over k, and let S be a projective surface which is a G-variety over k. Let π : S S be the blow-up of S at a closed subscheme Z S over k, that is invariant under the action of G. Then S can be equipped with a G-structure over k in such a way that π : S S is a G-equivariant k-morphism. Proof. By the universal property of blow-ups (see Definition 2.2.1), we see that there exists a morphism f making a commutative diagram G X f X (id,π) G X f X. 26 π

34 3.2. Preliminaries Here we use that fact that G S is isomorphic to the blow-up of G S at G Z. A priori, we only know that the map f satisfies the identities ex = x and (gh) 1 g(h(x)) = x for all g, h G and x S \ E, where E be the exceptional divisor of the blow-up. However any morphism which is equal to the identity on an open dense subset must also be equal to the identity on the whole space. That is, these identities do in fact hold on all of S and we get an action of G on S over k. Lemma Let S be a singular del Pezzo surface over k, and S its minimal desingularisation. Then S is a G 2 a-variety over k if and only if S is. Proof. Suppose S is a G 2 a-variety over k. Since G 2 a is connected, the orbit of a singularity under this action is connected as well. Furthermore, every point in the orbit is a singularity as well (since translation by an element of G 2 a is an isomorphism). But there is only a finite number of (isolated) singularities. Therefore, the orbit is just one point, so that each singularity is fixed under the G 2 a-action. By a similar argument, we see that the Galois group Gal(k/k) at worst swaps any singularities. Hence we can resolve the singularities via blow-ups and applying Lemma 3.2.3, we see that S is also a G 2 a-variety over k. Next, suppose that S is a G 2 a-variety over k. The anticanonical class is defined over k, and hence the anticanonical map (or a multiple of it in degrees 1 and 2) is defined over k and contracts precisely the ( 2)-curves, so that its image is the corresponding singular del Pezzo surface S (see Theorem and [CT88, Prop. 0.6]). This map is G 2 a-equivariant by [HT99, Proposition 2.3] and [HT99, Corollary 2.4]. Lemma If a generalised del Pezzo surface S is an equivariant compactification of G 2 a over k, then the number of negative curves contained in S k is at most the rank of Pic( S k ). Proof. As explained in [HT99, Section 2.1], the complement of the open G 2 a- orbit on S k is a divisor, called the boundary divisor. By [HT99, Proposi- 27