AN ABSTRACT OF THE DISSERTATION OF. Tatsuhiko Hatase for the degree of Doctor of Philosophy in Mathematics presented on

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2 AN ABSTRACT OF THE DISSERTATION OF Tatsuhiko Hatase for the degree of Doctor of Philosophy in Mathematics presented on August 24, Title: Algebraic Pappus Curves Abstract approved: Thomas A. Schmidt We show that Pappus Curves, introduced by R. Schwartz to study his dynamical system in the real projective plane generated by iterated applications of the classical Pappus Theorem, are algebraic exactly in the linear case. Our approach is to use properties of projective curves such as singular points, genus, number of automorphisms and to apply elementary invariant theory. As a complement, we study fixed points of projective transformations of order four.

3 c Copyright by Tatsuhiko Hatase August 24, 2011 All Rights Reserved

4 Algebraic Pappus Curves by Tatsuhiko Hatase A DISSERTATION submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented August 24, 2011 Commencement June 2012

5 Doctor of Philosophy dissertation of Tatsuhiko Hatase presented on August 24, 2011 APPROVED: Major Professor, representing Mathematics Chair of the Department of Mathematics Dean of the Graduate School I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request. Tatsuhiko Hatase, Author

6 ACKNOWLEDGEMENTS Academic I owe my deepest gratitude to my thesis advisor Thomas A. Schmidt for keeping me motivated throughout the whole process for years. I would like to thank my thesis committee members. I am grateful for all that I have learned and for years of financial support that Mathematics Department of Oregon State University has provided me. Many thanks to David Wing for TeX support and various other advices. Personal I would like to thank my friends for all the moral support over the years. Their constant encouragement kept me going. Special thanks to my continuing source of inspiration without whom I probably would not have come up with most of my best ideas. exist. Also, I would like to thank my parents. Without them, I almost surely would not

7 TABLE OF CONTENTS Page 1. INTRODUCTION Motivation Statement of the Main Problem Organization of This Thesis BACKGROUND INFORMATION Projective Geometry Projective Space Projective Transformations The Modular Group Algebraic Curves Algebraic Curves Singularities Polarities Riemann Surfaces Invariant Theory SCHWARTZ S PAPPUS CURVES Marked Boxes Box Operations Orbit Ω and the Incidence Graph Γ Return of The Modular Group ALGEBRAIC PAPPUS CURVES ARE LINEAR Linear Case Excluding Higher Degrees Cubic Case

8 TABLE OF CONTENTS (Continued) 4.4. Conic Case Page 5. GEOMETRICAL SIGNIFICANCE OF FIXED POINTS OF ORDER FOUR PROJECTIVE TRANSFORMATIONS CONCLUSION BIBLIOGRAPHY APPENDICES A APPENDIX An Additional Proposition By Schwartz

9 LIST OF FIGURES Figure Page 1.1 Pappus Theorem: A pair of collinear triple points gives us new collinear triple points An Overmarked Box with vertices p, q, r, and s, distinguished points t and b, and distinguished edges T and B Box Operations: τ 1, τ 2, and ι. The images of τ 1 and τ 2 are nested inside the original marked box Incidence Graph Γ: Each edge represents a marked box and its vertices represent top and bottom of the box Normalization R3; τ 1 ι is realized by an order three rotation Normalization P2; ι is a polarity with respect to the conic x 2 + y 2 + z 2 = Maps θ, Proj, and ν are PSL 2 (Z)-equivariant. G, M, and M define group actions of PSL 2 (Z) on their respective spaces A Linear Pappus Curve: The marked box is symmetric with respect to the Pappus Curve Conics Invariant Under T 0 : They are concentric circles centered at [0 : 0 : 1] At a fixed point of an order four transformation, conics are tangent Pappus Curve Λ and Its Transverse Linefield L

10 ALGEBRAIC PAPPUS CURVES

11 2 1. INTRODUCTION 1.1. Motivation Pappus Theorem is as old as the hills. R. Schwartz Given a pair of triples of collinear points in the projective plane P 2 k for field k of characteristic neither two nor three, Pappus Theorem determines the location of a new triple of collinear points. We denote the line joining two distinct points a and b by ab and the intersection of two distinct lines A and B by AB. Furthermore, we denote the intersection of lines ab and cd by (ab)(cd) and the line joining AB and CD by (AB)(CD). Theorem (Pappus Theorem) Suppose the points a, b and c are collinear and the points a, b and c are collinear in the projective plane P 2 k for field k of characteristic neither two nor three. Then the points a = (ab )(a b), b = (ac )(a c) and c = (bc )(b c) are collinear. a b c a b c a b c FIGURE 1.1: Pappus Theorem: A pair of collinear triple points gives us new collinear triple points. Details for the rest of this section will be given in Chapter 3.

12 Richard Schwartz, in his paper Pappus s Theorem and the Modular Group [11], defines an object called a marked box which is a certain finite configuration of points 3 and lines in the projective plane P 2 R. Besides vertices at intersections of lines, each marked box contains two distinguished points; similarly, it has two distinguished edges, its top and bottom. By iterating Pappus Theorem on marked boxes, Schwartz generates collections of distinguished points and edges. Schwartz defines box operations that map marked boxes to marked boxes. Composition gives a group multiplication for the box operations. The group of operations is generated by three basic operations; using these Schwartz proved that the box operation group is isomorphic to the classical modular group. We give more detail about the modular group in Section See Theorem Naturally enough, the group of box operations acts on the set of all marked boxes. Schwartz proved that the set of distinguished points from any orbit of convex marked boxes is dense in a homeomorphic image of S 1 in the projective plane. He calls this S 1 in the projective plane a Pappus Curve. The technical definition of Pappus Curve is Definition Statement of the Main Problem By Schwartz s definition, a Pappus Curve is a topological circle. Furthermore, he shows that a Pappus Curve, in fact, is an analytic curve. So, one naturally may ask whether a Pappus Curve is an algebraic curve as well. In this thesis, to answer this question, we prove the following theorem. Main Theorem A Pappus Curve is algebraic if and only if it is linear. Algebraic here means that the points of the given Pappus Curve satisfy an irreducible

13 4 polynomial equation. Such an equation defines an algebraic curve (See Section 2.2.1); note that, on this algebraic curve, there may be other points on this algebraic curve C Λ not on the Pappus Curve Λ (and in the complex projective plane, there always are such points). We define Pappus Curve and algebraic Pappus Curve in Section 3.4. In this thesis, to prove the main theorem, we break down the problem into smaller cases and prove a series of propositions. First, by an example, we show that a linear Pappus Curve exists. Then by applying projective transformations, we show that any line in P 2 R contains a Pappus Curve. Then, we show that any algebraic Pappus Curve must be smooth. By using this fact along with Plücker s Formula and Hurwitz s Automorphism Theorem, we see that any algebraic curve of degree greater than or equal to four cannot be a Pappus Curve. To rule out cubics, we show that a cubic Pappus Curve must have infinitely many flexes which contradicts the fact that a smooth cubic has at most nine flexes. We define flexes in Section Then, finally, we see that no conic can be a Pappus Curve by using invariant theory and computation Organization of This Thesis In Chapter 2, we review the basic concepts of projective geometry, algebraic curves, and invariant theory. We define key concepts that are crucial to understand the problem clearly. Here, we sketch a proof that smooth complex algebraic curves are compact Riemann surfaces then state some well known theorems. In Chapter 3, we explore Schwartz s paper in detail to clearly define convex marked boxes and their distinguished points and edges. In particular, we review the fact that the

14 group of box operations is generated by a projective transformation of order three and a polarity. 5 In Chapter 4, we prove our Main Theorem as outlined in Section 1.2. In Chapter 5, we investigate projective transformations of order four. The inspiration for this section comes from one of the box operations defined by Schwartz. We study the significance of the fixed points of projective transformations of order four. In Chapter 6, we give a conclusion and discuss possible extensions of this thesis. In an appendix, we discuss another aspect of Schwartz s paper. An orbit of (convex) marked boxes gives a topological circle in P 2 R (P 2 R) that satisfies certain geometric property. We fill in some details to the proof of Theorem A0.9.

15 6 2. BACKGROUND INFORMATION 2.1. Projective Geometry In this section, we discuss projective spaces and their properties. Most of the material in this section can be found in Projective Geometry by H. S. M. Coxeter [1] and Undergraduate Algebraic Geometry by Miles Reid [10] Projective Space Definition Let k be a field. Then the projective n-space over k is defined to be the set of all one dimensional linear subspaces of k n+1. We denote the projective n-space over k by P n k. In the vector space k n+1 two nonzero vectors (x 1,..., x n+1 ) and λ(x 1,..., x n+1 ) span the same subspace of dimension one for λ 0 in k. We denote the point p P n k which corresponds to the subspace spanned by (x 1,..., x n+1 ) by p = [x 1 :... : x n+1 ]. We call this coordinate system for points in P n k the homogeneous coordinate system where [x 1 :... : x n+1 ] = [λx 1 :... : λx n+1 ] for all λ k. The set of points p = [x 1 :... : x n+1 ] with x n+1 0 in P n k is called (the standard) affine space and the set of points with x n+1 = 0 is the hyperplane at infinity. A projective n-space can be seen as the union of the affine part from above, which is k n embedded into P n k and the hyperplane at infinity. Here, in an affine space, we ignore the vector space structure of k n. Now, we define more general affine space of a projective space.

16 Definition An affine space in P n k is a subspace that is isomorphic to k m for m n. 7 When n = 2, a projective 2-space is called a projective plane. Let p, q P n k be points, and v p, v q k n+1 be vectors corresponding to the respective points. The line through p and q in P n k is the collection of points that are represented by vectors of the form av p + bv q where a, b k. Each such point corresponds to a one dimensional subspace in the two dimensional subspace of k n+1 spanned by the vectors v p and v q. In k 3, any two distinct subspaces of dimension two must have a common subspace of dimension one. Suppose they do not intersect, then we can get four linearly independent vectors, two from each subspace, and this is not possible in a vector space of dimension three. Thus we have that, in P 2 k, any two distinct lines must intersect. Because of this, we have that given a pair of distinct lines, there is a unique point where they intersect. Analogous to the fact that a pair of distinct points determines a unique line joining them, each pair of distinct lines meets in a unique point. The uniqueness of the intersection point is guaranteed by the distinctness of the lines; if there are two points of intersection, then the two subspaces of dimension two corresponding to our lines share two linearly independent vectors, and that contradicts the subspaces being distinct. This idea gives us the duality of a projective plane. Any geometric statement for P 2 k remains true when the roles of points and lines are switched Projective Transformations The group GL n+1 (k), the set of (n + 1) (n + 1) matrices in k with nonzero determinants, represents the set of invertible linear transformations of the vector space

17 8 k n+1. Invertible linear transformations map one dimensional subspaces to one dimensional subspaces, and, in general, m dimensional subspaces to m dimensional subspaces. For each linear transformation in k n+1, there is a corresponding map in P n k that takes points to points, lines to lines, and, in general, dimension d linear subspaces to dimension d linear subspaces while preserving intersections and joins. We call this map a projective transformation on P n k. It is easy to see that the set of projective transformations forms a group. If S, T GL n+1 (k) are such that, for some nonzero λ k, S = λt, then for any x k n+1, Sx = λt x. Since [x 1 :... : x n+1 ] = [λx 1 :... : λx n+1 ] in P n k, S and T induce the same map on P n k. So, we have that PGL n+1 (k), the quotient group GL n+1 (k)/k is the group of projective transformations on P n k. Given a pair of m dimensional subspaces of k n+1, there is a linear transformation in GL n+1 (k) that maps one to the other; given a pair of points or lines in P n k, there is a projective transformation that maps one to the other. Lemma A projective transformation maps an affine plane to an affine plane [5]. Proof. A projective transformation on P 2 R maps a line to a line. Hence it maps the complement of a line to the complement of a line. Therefore, a projective transformation maps an affine plane to an affine plane. PGL 3 (k). In particular, the group of the projective transformations on P 2 k is (isomorphic to)

18 Definition We say that a quadruple of points in the projective plane are in general position if no triple of them are collinear. 9 Remark A group of projective transformations act sharply four transitively on the projective plane; given a pair of quadruples of points in general position, there exists a unique projective transformation that maps one to the other. To see this, given a pair of four points in general position, we find the unique projective transformation that maps one to the other by solving for a matrix in PGL 3 (k). Due to its utility, we state an obvious implication of the sharp four transitivity. Lemma A projective transformation on the projective plane that fixes four points in general position is the identity map. We sketch a proof of the following well-known result. Lemma A projective transformation that fixes three points on a line fixes the entire line pointwise. Proof. Suppose that T fixes collinear the points u, v, and w. Then there exists a projective transformation S (not unique) such that u [1 : 0 : 1] v [0 : 0 : 1] w [ 1 : 0 : 1]

19 10 We first show that any projective transformation that fixes [1 : 0 : 1], [0 : 0 : 1], and [ 1 : 0 : 1] also fixes all the other points on the line y = 0. Then by conjugacy, we have that T fixes the line containing u, v, and w pointwise as well. Thus, it suffices to show the claim holds for this specific case. Suppose that T is a projective transformation that fixes the three points shown above on the line y = 0. Then computation shows that T = α β 0 0 γ 0, 0 δ α and, with this, one easily shows that T : [x : 0 : z] [αx : 0 : αz]. Thus, T fixes all the points on the line y = 0. We are most interested in the case where k = R. In some cases, we will study P 2 R as a subset of P 2 C. On P 2 R, the collection of projective transformations is represented by the group PSL 3 (R) = SL 3 (R)/{±I} where SL 3 (R) is the group of 2 2 matrices in R with determinant one and T is the identity matrix. This is because PSL 3 (R) and PGL 3 (R) are isomorphic as groups; for any M SL 3 (R), there is a matrix M GL 3 (R) and λ R such that M = λm The Modular Group The material in this subsection comes from Fuchsian Groups by Svetlana Katok [7].

20 The modular group, PSL 2 (Z) is a subset of the collection of all fractional linear transformations. For each element of PSL 2 (R), a fractional linear transformation on C is 11 defined by z az + b cz + d where a, b, c, d R and ad bc = 1. For the modular group, we have that a, b, c, d Z. Fractional linear transformations are isometries for Poincaré s upper half-plane model of hyperbolic geometry. The modular group is generated by two transformations S and ST where S : z 1 z and T : z z + 1. We have that S 2 = (ST ) 3 = I. A fractional linear transformation is said to be hyperbolic if and only if it fixes two real points at the boundary of H = {z C : Im(z) > 0}. A hyperbolic map attracts toward one of the fixed points, and expands from the other with respect to the hyperbolic metric defined on H. A matrix representing an element of PSL 2 (R) is hyperbolic if and only if its trace is greater than two in absolute value. So, there are infinitely many hyperbolic elements in PSL 2 (Z) since it is very easy to have a + d > 2 and ad bc = 1 by letting b = 1 and c = ad 1 for any choice of a and d. Lemma The action of PSL 2 (Z) is transitive on ˆQ = Q [7]. Proof. For any a c Q, with gcd(a, c) = 1, there exists b, d Z such that ad bc = 1. Therefore, a transformation T (z) = az+b cz+d sends to a c. Thus, any two points in ˆQ are congruent under the action of PSL 2 (Z).

21 12 Corollary For any z ˆR = R { }, the PSL 2 (Z) orbit of z is dense in ˆR. By the previous lemma, for any q ˆQ, its images under the action of PSL 2 (Z) is ˆQ. Since ˆQ is dense in ˆR, it follows that the images of q are dense in ˆR. For z / ˆQ, we can show the same by taking the conjugate of PSL 2 (Z) by an element of PSL 2 (R) such that T (z) ˆQ. We use a single SL 2 (Z) orbit to show the following. Corollary The hyperbolic fixed points of PSL 2 (Z) are dense in ˆR. The map T (z) = 2z+1 z+1 is hyperbolic and fixes z = 1± 5 2. By the previous lemma, the images of either fixed point under the action of PSL 2 (Z) are dense in ˆR. Thus, by conjugation, one easily shows that the hyperbolic fixed points of PSL 2 (Z) are dense in ˆR Algebraic Curves In this section, we discuss some basic ideas of algebraic curves. Most of this section is from Algebraic Curves by William Fulton [4] Algebraic Curves Definition Let R be a ring. Then we let R[x] denote the ring of polynomials in x with coefficients in R. The degree of a polynomial a i x i is the largest d such that a d 0. We say that a polynomial of degree d is monic if a d = 1. We let R[x 1,..., x n ] denote the ring of polynomials in n variables over R. For our ease, when n = 3, we will write R[x, y, z].

22 Definition The monomials in R[x 1,..., x n ] are polynomials of the form x i 1 1 x in n with non-negative integers i j. Here, the degree of the monomial is i i n. A polynomial F is homogeneous or a form of degree d if all the terms in F are monomials of degree d with coefficients in R. 13 Now that we have a solid definition of the ring of polynomials, we discuss the algebraic curves defined by the polynomials. Definition A point p = [x 1 :... : x n+1 ] P n k is said to be a zero of a homogeneous polynomial f k[x 1,..., x n+1 ] if f(x 1,..., x n+1 ) = 0. Furthermore, the set of all the zeros of a homogeneous polynomial is called an algebraic hypersurface. If the degree of the homogeneous polynomial is d, then we say that the degree of the algebraic hypersurface is d. When n = 2, an algebraic hypersurface is called an algebraic curve. Note that zeros of homogeneous polynomials and algebraic curves are well-defined. Since f is homogeneous, if for p = [x 1 :... : x n+1 ], f(x 1,..., x n+1 ) = 0, then for p = [λx 1 :... : λx n+1 ] we have that f(λx 1,..., λx n+1 ) = λ d f(x 1,..., x n+1 ) = 0 where d is the degree of f. Algebraic curves in P 2 k are also referred to as projective plane curves. Projective transformations send algebraic curves to algebraic curves and preserve the degree. For any set S of polynomials in k[x 1,..., x n+1 ], we define V(S) = {p P n k : p is a zero of each f S}. If I is the ideal generated by the set S, then V(I) = V(S). We call V(I) an algebraic set.

23 Definition We say that an algebraic set V P n k is irreducible if it is not the 14 union of two or more distinct algebraic sets. Otherwise, an algebraic set is said to be reducible. Each algebraic set in a reducible algebraic set is called a component. We have that V( f ) is irreducible if and only if f is an irreducible polynomial where f denotes the ideal in k[x 1,..., x n+1 ] generated by f. Let X P n k, then we define I(X) = {f k[x 1,..., x n+1 ] : every p X is a zero of f}. This set I(X) is an ideal in k[x 1,..., x n+1 ] since for any f I(X) and g k[x 1,..., x n+1 ], for every p X, fg(p) = f(p)g(p) = 0 g(p) = 0, and so fg I(X). We call I(X) the ideal of X. Given an ideal I in a ring R, the radial of I is the set rad(i) = { r R : r n I for some n Z + }. We call an ideal I a radical ideal if I = rad(i) [2]. There is a one-to-one correspondence between the set of radical ideals of k[x 1,..., x n+1 ] and the set of algebraic sets in P n k. Definition Given f 1,..., f m k[x 1,..., x n ], if there is no nonzero polynomial F k[x 1,..., X m ] such that F (f 1,..., f m ) 0, then the set of polynomials {f 1,..., f m } is said to be algebraically independent. Algebraic curves of degrees one, two, and three are called: lines, conics, and cubics respectively. Thus our Main Theorem states that only lines can be algebraic Pappus Curves.

24 Given a pair of algebraic curves, then for each point in the projective plane, there is a non-negative integer called the intersection number which describes the multiplicity 15 of their intersection at this point. See Fulton s Algebraic Curves [4] for details of the definition. The following theorem is very important while studying algebraic curves. Theorem Bézout s Theorem Let f and g be projective plane curves of degrees d f and d g respectively over an algebraically closed field. Assume that f and g have no common component. Then f and g intersect d f d g times counted with multiplicities [4] Singularities Definition Given an algebraic hypersurface f(x 1,..., x n+1 ) = 0, p P n k is a singular point of f if f(p) = f x 1 (p) = = f x n+1 (p) = 0. A hypersurface with no singular points is said to be smooth or non-singular. With this definition, we have the following corollaries of Bézout s Theorem. Corollary A reducible algebraic curve has singularities. Proof. Suppose that C is a reducible algebraic curve defined by a reducible homogeneous polynomial equation fg = 0. Let C f and C g be the algebraic curves given by the polynomial equations f = 0 and g = 0 of degrees d f and d g respectively. Then, by Bézout s Theorem, f and g intersect d f d g times counted with multiplicities. One easily shows by computation that those intersection points are singular points of C.

25 Corollary An irreducible algebraic curve over a field of characteristic 0 has finitely many singular points. 16 Proof. By Bézout s Theorem, unless f, f the system of equations x, f y, and f z f(p) = f f f (p) = (p) = x y z (p) = 0 have a common component, has only finitely many solutions. Now, if f is irreducible, then it has only one component. Since deg( f x ) = deg( f y f ) = deg( z ) = deg(f) 1, they cannot have a common component with f. Hence, an irreducible curve has only finitely many singular points. Remark Let T be a projective transformation. Suppose that C is an algebraic curve defined by a homogeneous polynomial f. Then T (C) is defined by the homogeneous polynomial g = f T 1. Then, for p C, g(t (p)) = f T 1 (T (p)) = f(p) = 0. Now, suppose that p is a singular point of C. Then by applying the chain rule g x i (T (p)) = a i f x i (T 1 (T (p))) = a i f x i (p) = 0 for some a i = a i (T ) k. Thus, we have that T (p) is a singular point of T (C). Hence, a projective transformation T maps singular points of a curve C to singular points of the curve T (C). Definition In P 2 k, the multiplicity of a point p on a curve C = V(f) is the smallest integer m such that for some i+j = m, of the point p. m f x i y j 0. Let m p denote the multiplicity

26 Definition A line L is said to be tangent to a curve C at p if the intersection number of L and C at p is greater than m p [14]. 17 When the multiplicity of a point is one, we call the point a simple point of the curve. Note that a simple point is non-singular. Also, C has exactly one tangent line at any simple point. A singular point of C at which there are more than one distinct tangent line is called a multiple point of C. Definition A simple point p on a curve C is said to be a flex of the curve C if its tangent line intersects C at p with multiplicity greater than or equal to three. Flexes can also be defined in terms of the polynomial that defines the curve. For a given homogeneous polynomial f of degree d, define its Hessian to be f xx f xy f xz H = det f yx f yy f yz. f zx f zy f zz We have that H is a form of degree 3(d 2). Suppose that the curve C and its Hessian intersect at a point p. Then p is either a multiple point or a flex of C. Lemma Projective transformations map flexes to flexes. Proof. Suppose that C = V(f). By Remark , given a projective transformation T, we have that T (f xx )(T (p)) = (f T 1 ) xx (T (p)) = f xx (p). Similar equations hold for other second partial derivatives. Therefore, it follows that T (H)(T (p)) = H(p) for any p P 2 R. Thus, we have that if p is a flex of C, then T (p) is a flex of T (C).

27 18 Corollary (of Bézout s Theorem) nine flexes. A smooth cubic over a field of characteristic 0 has at least one but no more than Proof. Given a smooth cubic algebraic curve C defined by a cubic polynomial f, its Hessian is of degree three. By Bézout s Theorem, we have that f and H intersect nine times counting with multiplicities. Since C is smooth, none of these intersections are multiple points. Hence, f has at least one but no more than nine flexes. In fact, one can show that there are nine distinct flexes in a smooth cubic. Another important characteristic an algebraic curve is the genus which we define later in this section. This value g can be computed using the following proposition. Proposition Plücker s Formula Let C be a smooth projective plane curve over C. Then the genus g of C is g = (d 1)(d 2) 2 where d is the degree of C [8]. The following is a handy fact about conics. Proposition If p 1,..., p 5 P 2 k are distinct points such that no four are collinear, then there exists exactly one conic through p 1,..., p 5 [10]. Proof. On P 2 k, the equation of a conic is given by Ax 2 + Bxy + Cy 2 + Dxz + Exy + F z 2 = 0.

28 Suppose C is a conic and p i = [x i : y i : z i ] C for i = 1,..., 5 where those points are such that no quartet of them are collinear. Then we get a system of five linear equations in k 6 19 Ax 2 i + Bx i y i + Cy 2 i + Dx i z i + Ex i y i + F z 2 i = 0 where i = 1,..., 5. Since we have five equations, this is enough to solve for five of the six coefficients as linear terms in the six variables. Now, for any λ 0, the equation λ(ax 2 + Bxy + Cy 2 + Dxz + Exy + F z 2 ) = 0 defines the same conic as the previous equation. Thus, we have that p 1,..., p 5 define a unique conic. As can be seen in the above proof, any five linearly independent equations are enough to be able to uniquely solve for the coefficients. For example, if we have three non-collinear points and the equations of tangent lines for two of those points, we can determine a unique conic that satisfies those conditions. This gives a proof to the following proposition. Proposition Five linearly independent conditions define a unique conic. Corollary For each d, an irreducible algebraic curve of degree d is uniquely determined by finitely many points it passes through. Proof. For any d, a polynomial of degree d has only finitely many coefficients. Thus, given enough points in general position, we have sufficeuntly many equations to solve for the coefficients uniquely Polarities The material in this subsection is from Algebraic Curves by R. Walker [14] and Plane Algebraic Curves by H. Hilton [6].

29 20 Definition Suppose that we are given a line l that intersects the conic C at the points p and q. Then the pole of l is defined to be the point at which the tangent lines of C at p and q intersect. If p = q, then we have that l is tangent to C and its pole is p. Now, suppose that we are given a point p. Then we have that the polar of p is the collection of poles of all the lines through p. One can show that the polar of p is a line. Let the polarity with respect to a given base conic to be the operation which takes a point to its polar and a line to its pole with respect to the conic. A polarity is an involution on a projective plane. A polarity preserves incidences. Suppose that P is the polar of p. If p P, then we say that p and P are self-conjugate. Example Let the conic f(x, y, z) = x 2 +y 2 +z 2 = 0 be given. Let p = [A : B : C] and q = [A : B : C ] be the points at which the conic and the line ax + by + cz = 0 intersect. Since f x = 2x, f y = 2y, and f z = 2z, the equations of the tangent lines at p and q are Ax + By + Cz = 0 and A x + B y + C z = 0 respectively. Because p and q are on the line ax + by + cz = 0, we have that Aa + Bb + Cc = A a + B b + C c = 0. This equation also indicates that the point [a : b : c] is on both Ax + By + Cz = 0 and A x + B y + C z = 0. Therefore, [a : b : c] is the point where the two lines intersect. So we have that the polar of the point [a : b : c] with respect to x 2 + y 2 + z 2 = 0 is the line ax + by + cz = 0. Similarly, the pole of the line ax + by + cz = 0 is the point [a : b : c]. The set of projective transformations and polarities together generate the group of projective symmetries. The non-identity elements of projective transformations, polarities,

30 21 and projective symmetries are also known as collineations, correlations, and projectivities, respectively [1]. More details about projectivities can be found in Projective Geometry by H. S. M. Coxeter [1] Riemann Surfaces Algebraic curves over C can be viewed as Riemann surfaces. Here, we review some basic concepts of Riemann surfaces. The material in this section is mostly from Algebraic Curves and Riemann Surfaces by R. Miranda [8]. Basically, Riemann surfaces are spaces that locally look like an open set in the complex plane. We use complex charts, defined below, to show how a space can look like an open set in the complex plane. Definition Suppose that f is a complex valued function on C. Then for z C, we can express it as z = x + iy where x, y R. Also, we can write the function f(z) as f(x + iy) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) are real valued functions. Then we say that f is differentiable at z if the Cauchy-Riemann equations u x = v y and u y = v x are satisfied. A function f is holomorphic if it is differentiable at all points in its domain [3]. A holomorphic map whose inverse is also holomorphic is said to be biholomorphic [8].

31 22 Definition A complex chart, or simply a chart, on a topological space X, is a homeomorphism φ : U V, where U X is an open set in X, and V C is an open set in the complex plane. The chart φ is said to be centered at p U if φ(p) = 0. Furthermore, let φ 1 : U 1 V 1 and φ 2 : U 2 V 2 be two complex charts on X. We say that φ 1 and φ 2 are compatible if either U 1 U 2 =, or is biholomorphic. φ 2 φ 1 1 : φ 1 (U 1 U 2 ) φ 2 (U 1 U 2 ) With charts, we give the space X local complex coordinates. Definition A complex atlas A on X is a collection A = {φ α : U α V α } of pairwise compatible complex charts where {U α } covers X. Two complex atlases are said to be equivalent if every chart of one is compatible with every chart of the other. If two atlases are equivalent, then their union is also an atlas. We call a maximal atlas, by inclusion, a complex structure of X. Definition If a space X has a countable basis for its topology, then X is said to be second countable [9]. Definition A topological space X is called a Hausdorff space is for each pair x 1, x 2 of distinct points of X, there exist neighborhoods U 1 and U 2 of x 1 and x 2, respectively, that are disjoint [9].

32 Definition A Riemann surface is a second countable connected Hausdorff topological space X together with a complex structure. 23 If a Riemann surface X is a compact space, then we say that X is a compact Riemann surface. The genus of a Riemann surface X, is the non-negative integer g such that H 1 (X; Z) = Z 2g where H 1 (X; Z) is the first homology group of X with integer coefficients [3]. Informally, the genus of a Riemann surface is the largest number of closed curves such that removing them keeps the surface connected. Definition A mapping F : X Y is holomorphic at p X if and only if there exists charts φ 1 : U 1 V 1 on X with p U 1 and φ 2 : U 2 V 2 on Y with F (p) U 2 such that the composition φ 2 F φ 1 1 is holomorphic at φ 1 (p). We say that F is a holomorphic map if and only if F is holomorphic on all of X. Definition An isomorphism between Riemann surfaces is a holomorphic map F : X Y which is bijective, and whose inverse F 1 : Y X is holomorphic. A self-isomorphism F : X X is called an automorphism of X. For a compact Riemann surface X with genus g, there is an important result. Theorem Hurwitz s Automorphism Theorem Let G be a group of automorphisms of a compact Riemann surface X of genus g 2. Then G 84(g 1).

33 Now, in order to apply Hurwitz s Automorphism Theorem, we show that smooth algebraic curves are compact Riemann surfaces. 24 Theorem The Implicit Function Theorem Let f(u, v) C[u, v] and X = {(u, v) C 2 f(u, v) = 0}. Let p = (u 0, v 0 ) be a point on X. Suppose that f v (p) 0. Then there exists a function g(u) defined and holomorphic in a neighborhood of u 0 such that, near p, X is equal to the graph v = g(u). Moreover g = f u / f v near u 0. Theorem Any smooth algebraic curve over C is a compact Riemann surface. in [8]. Here, we sketch a variant of the proof provided by Miranda in sections 1.2 and 1.3 Proof. Suppose that C is a smooth algebraic curve given by a homogeneous polynomial F. Let A z denote the affine plane {[x : y : z] P 2 C : z 0}. Then let C z be the affine curve defined by the polynomial f(y, z) = F (x, y, 1). We have that C z = C A z. Now, suppose that T is a projective transformation represented by a matrix a b c T = d e f, g h i then let A T denote the affine plane to which T maps A z. Now, let C T be the affine curve defined by the polynomial f T (u, v) = F (u, v, 1) where u = Then, we have that C T = C A T. ax + by + cz gx + hy + iz and v = dx + ey + fz gx + hy + iz.

34 25 We have that a smooth algebraic plane curve is irreducible since otherwise we would have singularities at the intersection of the components. Thus, it follows that C T, is also smooth and irreducible. On a smooth affine plane curve, we can obtain complex charts by the Implicit Function Theorem. Thus we have that C T is a Riemann surface. Now, we have that {C T } are open subsets of C by subspace topology since each of them is an intersection of C and an affine space which is an open subset of P 2 C. Also, the complex charts defined by the Implicit Function Theorem can be shown to be compatible with each other. Thus, we have that C is also a Riemann surface. Furthermore, since C is a closed subset of P 2 C, it is compact. Thus, any smooth algebraic plane curve is a compact Riemann surface Invariant Theory In this section, we discuss invariant theory. This material can be found in Bernd Sturmfels Algorithms in Invariant Theory [12]. Even though invariant theory is about algebraic curves, it is not used commonly or as well known as other concepts we see in this thesis. So we dedicate this section for it. We are interested in the polynomials that remain invariant under the action of a finite group of projective transformations that is represented by a group G PGL n (C). Let C[x] denote the ring of polynomials in n variables x = (x 1,..., x n ). Given a finite group G, let C[x] G be the ring of invariant polynomials under the action of G in C[x]. Our aim is to determine a set {I 1,..., I m } that generates the invariant subring C[x] G. When none of the polynomial in the set can be expressed in terms of the others, we call each polynomial in the set a fundamental invariant.

35 26 The following operator, called the Reynolds operator, takes a polynomial in C[x] and gives us a polynomial that is invariant under the action of a finite matrix group G. The Reynolds operator is defined by : C[x] C[x] G, f f := 1 f π G π G where f π is defined to be the polynomial f(πx) with πx being the transpose of the matrix multiplication of π and the column vector x T. The Reynolds operator has the following properties. The Reynolds operator is a C-linear map. The Reynolds operator acts as the identity map on C[x] G. Since πx has linear terms in each coordinate, deg(f) = deg(f ). Elementary calculation with the Reynolds operator can be easily performed. Example Let G 3 = {g 1, g 2, g 0 } PGL 3 (C) where g 1 = , 0 0 1

36 g 2 = g 0 = , 27 Then, by Reynolds operator, we have that ( x = 1 3 ( 1 2 x 3 2 y) + ( 1 2 x + ) 3 2 y) + x = 0, y = 1 3 ( ( 3 2 x 1 ) 3 y) + ( 2 2 x 1 2 y) + y = 0, z = 1 (z + z + z) = z, 3 (x 2 ) = 1 3 (x 3 ) = 1 3 (y 3 ) = 1 3 ( ( ( 1 2 x 3 2 y)2 + ( 1 2 x + ( 1 2 x 3 2 y)3 + ( 1 2 x + ( ( ) 3 2 y)2 + x 2 = 1 2 (x2 + y 2 ), ) 3 2 y)3 + x 3 = 1 4 (x3 3xy 2 ), 3 2 x 1 ) 3 2 y)3 + ( 2 x 1 2 y)3 + y 3 = 1 4 (y3 3x 2 y). The following proposition shows that every finite subgroup of PGL n (C) has at least n invariants.

37 Proposition Every finite matrix group G PGL n (C) has n algebraically independent invariants, i.e., the ring C[x] G has transcendence degree n over C. 28 Proof. Let, for each i {1,..., n}, P i := π G(x i π t) C[x][t] and consider P i (t) as a monic polynomial in t with coefficients from C[x]. Since P i is invariant under the action of G on the x-variables, its coefficients are also invariant, so P i C[x] G [t]. Now, we have that t = x i is a root for P i (t), because one of the elements in G is the identity map. This means that the variables x 1,..., x n are algebraically dependent on some invariant polynomials in C[x][t]. Hence the invariant subring C[x] G and C[x] have the same transcendence degree n over C. The following theorem of Hilbert guarantees that any invariant ring C[x] G with G a finite matrix group in PGL n (C) is finitely generated. Theorem Hilbert s Finiteness Theorem The invariant ring C[x] G of a finite matrix group G PGL n (C) is finitely generated. Proof. Let I G = C[x] G + be the ideal in C[x] that is generated by all homogeneous invariants of positive degree. Then, it follows that I G is generated by the polynomials (x e 1 1 xen n ) where (e 1,..., e n ) ranges over all nonzero, nonnegative integers. By Hilbert s basis theorem, we have that every ideal in C[x] is finitely generated, so there are finitely many homogeneous invariants I 1,..., I m such that I G = I 1,..., I m. Furthermore, we have that this set of invariants generates C[x] G.

38 We continue our discussion of finding finitely many polynomials to generate the invariant ring. 29 Let C[x] G d denote the set of all homogeneous invariants of degree d. We have that the invariant ring C[x] G is the direct sum of the finite-dimensional C-vector spaces C[x] G d. Definition The Hilbert series of C[x] G is the generating function Φ G (t) = dim C ( C[ x ] G d ) td. d=0 of C[x] G. The following theorem by Molien in 1897 provides a formula for the Hilbert series Theorem Molien [12] The Hilbert series of the invariant ring C[x] G equals Φ G (t) = 1 1 G det(id tπ). π G The following lemma helps us determine when we have found a set of invariants that generate the invariant subring C[x] G. Lemma Let p 1,..., p m be algebraically independent elements of C[x] which are homogeneous of degrees d 1,..., d m respectively. Then the Hilbert series of R := C[p 1,..., p m ] is H(R, t) := (dim C R d ) t d = n=0 1 (1 t d 1 ) (1 t d m).

39 Proof. Let R d be the set of polynomials of degree d in R. Since the p i are algebraically independent, the set 30 {p i 1 1 p im m i 1,..., i m N and i 1 d i m d m = d} forms a C-basis for the R d as a vector space. Thus the dimension of R d is equal to the cardinality of the set A d = {(i 1,..., i m ) N m i 1 d i m d m = d}. Then 1 1 (1 t d 1 ) (1 t d m) = 1 t d t dm ( ) ( ) = t i 1d 1 t imdm = and this proves the lemma. i 1 =0 d=0 (i 1,...,i m) A d t d i m=0 = A d t d d=0 Now, with an example, we see how to find a set of invariants that generate C[x] G by use of its Hilbert series. Example Let G 3 be as in Example Then the Hilbert series of C[x, y, z] G 3 is Φ G3 (t) = 1 [ ] 1 3 det(id tg 1 ) + 1 det(id tg 1 ) + 1 det(id tg 1 ) = 1 [ ] 1 3 (1 t)(t 2 + t + 1) + 1 (1 t)(t 2 + t + 1) + 1 (1 t) 3 = 1 + t 3 (1 t)(1 t 2 )(1 t 3 )

40 31 = 1 + t + 2t 2 + 4t 3 + 5t 4 + 7t 5 + The coefficients of t, t 2, and t 3 of this Hilbert series tells us that the invariant subring has one polynomial of degree one, one polynomial of degree two, and two polynomials of degree three in its generating set. In Example , using the Reynolds operator, we found such invariant polynomials I 1 := z I 2 := x 2 + y 2 I 3 := x 3 3xy 2 I 4 := y 3 3x 2 y. We know that there are at most three algebraically independent invariants here and we have four invariant polynomials, so we know that at least one can be eliminated by an algebraic relation. We find that the equation I 2 4 = I 3 2 I 2 2 is satisfied. One easily shows that there is no algebraic relation between just I 1, I 2, and I 3. Thus we have that C[I 1, I 2, I 3, I 4 ] = C[I 1, I 2, I 3 ] I 4 C[I 1, I 2, I 3 ]. We have that C[I 1, I 2, I 3 ] is a subring generated by algebraically independent homogeneous polynomials, so by the lemma, the Hilbert series of C[I 1, I 2, I 3 ] is 1 (1 t)(1 t 2 )(1 t 3 ). Now, since the elements of degree d in C[I 1, I 2, I 3 ] are in one-to-one correspondence with the elements of degree d + 3 in I 4 C[I 1, I 2, I 3 ], the Hilbert series of I 4 C[I 1, I 2, I 3 ] is equal to t 3 (1 t)(1 t 2 )(1 t 3 ). Hence, we have that the Hilbert series of C[I 1+t 1, I 2, I 3, I 4 ] is 3 (1 t)(1 t 2 )(1+t 3 ) which is the Hilbert series of C[x, y, z] G 3. Hence we have that C[x, y, z] G 3 I 1, I 2, I 3, and I 4. is generated by

41 Corollary The only conic invariant under the standard rotation of order three that fixes the point o = [0 : 0 : 1] and passes through t = [1 : 0 : 1] is x 2 + y 2 z 2 = As we saw in Example , any invariant conic under the rotation is of the form a(x 2 + y 2 ) + bz 2 = 0. This equation is satisfied when the equation is a(x 2 + y 2 z 2 ) = 0. In the previous example, we saw a case where all group elements are in R C, and the invariant polynomials we found have real coefficients. Since invariant theory thus far has been defined using C, it may not be not obvious that when G PGL n (R), the set of invariants with real coefficients that generates C[x] G also generates R[x] G. The following proposition guarantees that when G PGL n (R), the set of invariants with real coefficients that generates C[x] G also generates R[x] G. Proposition When G PGL n (R), the set of invariants with real coefficients that generates C[x] G also generates R[x] G. Proof. Let G PGL n (R) be a finite subgroup. Since G is a finite subgroup of PGL n (C), by hypothesis there exist I 1,..., I m with real coefficients such that they generate C[x] G and {I 1,..., I n } is algebraically independent. Thus, we have that m C[x] G = I k C[I 1,..., I n ]. k=n+1 Since I 1,..., I m R[x] and they are invariant under G, we have that m R[x] G I k R[I 1,..., I n ]. k=n+1 The coefficients are real since we can apply the Reynolds operator to a term with real coefficients and since our G has matrices with real entries, the output of the operator also has real coefficients.

42 33 Now, suppose that I R[x] G. Then, we can express I as m I = (a k,j + ib k,j )I j n=n+1 j where I j is a monomial in {I 1,..., I n } and a k,j and b k,j are real. Then, we can express I as I = p + iq where p, q R[x] with m p = a k,j I j k=n+1 j and m q = b k,j I j. k=n+1 j Since I is a polynomial with real coefficients, we have that q = 0. But then it follows that m I k b k,j = 0 k=n+1 for all j since {I 1,..., I n } was chosen so that the set is algebraically independent. Hence m m I = a k,j I j I k R[I 1,..., I n ] k=n+1 j k=n+1 and m R[x] G I k R[I 1,..., I n ]. k=n+1 Therefore, we have shown that m R[x] G = I k R[I 1,..., I n ]. k=n+1 Hence, we may apply the invariant theory with base field R.

43 34 3. SCHWARTZ S PAPPUS CURVES In this section, we summarize pertinent material from Richard Schwartz s paper Pappus s Theorem and the Modular Group [11]. This section, including the figures, is adapted from Schwartz s paper Marked Boxes Schwartz uses Pappus Theorem (Theorem ) to create a collection of marked boxes (defined below); he iterates the theorem by applying it to a new pair of three collinear points a, b and c and a, b and c (or a, b and c and a, b and c ). Schwartz defines objects called marked boxes and derives curves of points and lines from Pappus Theorem. Definition An overmarked box is a pair of 6-tuples of points and lines in P 2 R ((p, q, r, s; t, b), (P, Q, R, S; T, B)) where p, q, r, and s are the vertices of the box, and P = ts, Q = tr, R = bq, S = bp, T = pq, and B = rs are lines with t T and b B. See Figure 3.1. Remark The collection of distinct points (p, q, r, s; t, b) uniquely determines an box. There is an involution on the set of overmarked boxes defined by φ((p, q, r, s; t, b), (P, Q, R, S; T, B)) ((q, p, s, r; t, b), (Q, P, S, R; T, B)).

44 Since the lines pq and qp are the same, we have that this involution maps overmarked boxes to themselves while exchanging the labeling of the vertices. 35 Definition A marked box is an equivalence class of overmarked boxes under φ. Let Θ = ((p, q, r, s; t, b), (P, Q, R, S; T, B)), then the pair (t, T ) is the top of Θ and (b, B) is the bottom of Θ where T and B are distinguished edges and t and b are distinguished points of Θ. Definition A marked box is said to be convex if p and q separate t and T B on the line T, and r and s separate b and T B on the line B as shown in Figure 3.1. T p t q P Q S R s B FIGURE 3.1: An Overmarked Box with vertices p, q, r, and s, distinguished points t and b, and distinguished edges T and B. b r This definition of convexity is the reason why we are working specifically with R and not just any field; we need the field to have an ordering to define the convexity in the way that works for us. Informally, a convex marked box is a convex quadrilateral in P 2 R with a distinguished top edge, a distinguished bottom edge, a distinguished top point, and a distinguished bottom point.

45 Definition The interior of a convex marked box ((p, q, r, s; t, b), (P, Q, R, S; T, B)) is the convex region bounded by t, B, ps, and qr that contains the intersection pr qs Box Operations There are three natural box operations one can perform on marked boxes defined using Pappus Theorem. We let ι be the operation where the top vertices, edge, and the distinguished point and the respective bottom ones in a given marked box are exchanged so that the resulting box is convex. Let τ 1 be the operation where the top vertices, edge, and the distinguished point are fixed and the bottom ones go to the points and the line defined by Pappus Theorem such that the resulting box is convex. Let τ 2 be the operation where the bottom ones are fixed and the top ones go to the points and the line defined by Pappus Theorem so that the resulting box is convex. Explicitly, those operations are defined by ι(θ) = ((s, r, p, q; b, t), (R, S, Q, P ; B, T )), τ 1 (Θ) = ((p, q, QR, P S; t, (qs)(pr)), (P, Q, qs, pr; T, (QR)(P S))) τ 2 (Θ) = ((QR, P S, s, r; (qs)(pr), b), (pr, qs, S, R; (QR)(P S), B)). We also let I denote the identity map. Figure 3.2 indicates how the operations work. With these explicit expressions of the box operations, one easily computes the following relations.