Towards a General Theory of Weierstrass Points on Curves Sami Douba Department of Mathematics & Statistics McGill University, Montr´eal June 2017 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Masters of Science c Sami Douba 2017 2 Abr´eg´e Nous pr´esentons plusieurs g´en´eralisationsalg´ebro-g´eom´etriquesde la notion classique d'un point de Weierstrass sur une surface de Riemann. Nous montrons qu'une large part de l'intuition acquise par la th´eorie classique est vraie en caract´eristique 0 et en grande caract´eristique.Finalement, nous calculons et examinons les points de Weierstrass de la quartique de Klein en caract´eristiques2 et 3 pour d´emontrer que ces points peuvent faire preuve d'un comportement int´eressant en petite caract´eristique. Abstract We present various algebro-geometric generalizations of the classical notion of a Weierstrass point on a Riemann surface. We show that much of the intuition gained from the classical theory holds in characteristic 0 and in high characteristic. Finally, we compute and examine the Weierstrass points of the Klein quartic in characteristics 2 and 3 to demonstrate that these points may exhibit interesting behaviour in low characteristic. 3 4 Preface The definitions and the proofs in Chapter 1 are modelled very closely after Hershel Farkas and Irwin Kra's exposition of Weierstrass points in [FK80], but the discussion is slightly more general. The theory in Chapter 2 and Section 3.1 is taken entirely from the papers [Lak81], [Lak84] by Dan Laksov, and [LT94] by Laksov and Anders Thorup. Section 3.2 makes extensive use of Noam Elkies' paper [Elk99] on the Klein quartic, but the results in that section are, to the author's knowledge, original. Acknowledgements I would like to express my immense gratitude to my supervisor, Prof. Eyal Goren, for his support, his guidance, and his patience. I would also like to thank Prof. Jacques Hurtubise for agreeing to examine my thesis. Last but not least, I am indebted to my fellow students, with and from whom I gained much of the background knowledge necessary to complete this thesis. 5 6 Contents 1 A na¨ıve approach to Weierstrass points 13 1.1 The gap theorems ............................... 13 1.2 Weierstrass points and wronskians ...................... 17 1.3 Weierstrass points on hyperelliptic curves .................. 23 1.4 Higher Weierstrass points .......................... 26 2 Laksov's Weierstrass points on curves 29 2.1 The bundle of principal parts ........................ 30 2.2 Weierstrass points ............................... 35 2.3 The wronskian ................................ 36 2.4 Local computations .............................. 40 2.5 The classical situation ............................ 43 3 Weierstrass points in families 45 3.1 Weierstrass points on schemes ........................ 45 3.2 Reductions of the Klein quartic ....................... 50 7 8 Introduction Since elliptic curves have non-trivial automorphisms, one cannot hope to construct a fine moduli space parametrizing isomorphism classes of these objects. However, an elliptic curve E is endowed with distinguished points | the group E[N] of its N-torsion points. We can use this group to endow an elliptic curve with additional structure, namely, a full level N structure, and then stipulate that isomorphisms between elliptic curves preserve this structure. The resulting moduli problem admits a fine moduli space for large enough N; it is the modular curve Y (N). Now what about curves of genus greater than 1? Is there a \non-abelian" analogue of the set E[N]? In the early 1860s, Weierstrass proved his so-called L¨uckensatz, or \gap theorem", which can be stated as follows: For any point on a compact Riemann surface of genus g, there are exactly g integers m, called \gaps", such that there is no meromorphic function on that surface having a pole of order m at that point as its only singularity. For all but a finite number of special points, the surface's so-called Weierstrass points, these gaps are the integers 1 ≤ m ≤ g. Furthermore, it is possible to attach weights to these points so that the sum of their weights is a constant that depends only on g. From the perspective of classifying curves of a fixed genus, this seems promising. Since compact Riemann surfaces can be thought of as algebraic curves over C, it is natural to ask if one can talk about Weierstrass points on more general algebraic curves, and if one can use these points to \rigidify" spaces of such curves and construct moduli spaces, as is done with torsion points on elliptic curves. In the first chapter, we provide a na¨ıve theory of Weierstrass points on smooth pro- jective curves over algebraically closed fields. This approach is completely analogous to 9 10 the classical theory of Weierstrass points on compact Riemann surfaces. However, this theory breaks down when the characteristic of the ground field is low relative to the genus. Indeed, in low enough characteristic, there are examples of curves C of genus g with the property that each point on C has a non-classical gap sequence, that is, a gap sequence that differs from 1; : : : ; g (see Example 2.0.1). In the second chapter, we resolve this issue by introducing Laksov's theory of Weier- strass points on curves. Instead of defining a Weierstrass point as a point whose gap sequence is non-classical, Laksov associates to each curve a generic gap sequence, and defines a point on that curve to be Weierstrass if the gap sequence at that point differs from the generic gap sequence. Under this approach, a curve has finitely many Weier- strass points regardless of the characteristic of the ground field. However, the weighted number of Weierstrass points may vary between curves of a fixed genus (see Section 3.2). This is already an indication that Weierstrass points may not be satisfactory candidates for higher-genus analogues of torsion points on elliptic curves. In the final chapter, we expose Laksov and Thorup's notion of Weierstrass points on families of curves over arbitrary base schemes. This generalizes the theory of the previous chapter. We end with a thorough examination of the Klein quartic. We show that none of the points obtained as reductions mod 3 of Weierstrass points of the classical Klein quartic is a Weierstrass point of the reduction in the sense of Laksov. This shows that Laksov's Weierstrass points have another drawback: their behaviour in algebraic families appears to be subtle. Literature Review For the following discussion, let g ≥ 2 be an integer, and denote by X a compact Riemann surface of genus g. Furthermore, let Mg be the moduli space of Riemann surfaces of genus g, and let Mg be the Mumford-Deligne compactification of Mg ([DM69]). The study of Weierstrass points dates back to the nineteenth century, when Weier- strass stated and proved the L¨uckensatz (discussed above). The theorem first appeared in the dissertation of Weierstrass's student Schottky in 1875 (later published as [Sch77]). Through their work on adjoints of plane curves, Brill and Noether implicitly proved in 11 1873 that the set of Weierstrass points on X is nonempty and has size at most g3 − g ([BN74]). In 1882, Noether proved the L¨uckensatz for any sequence of points P1;P2;::: on X, generalizing Weierstrass's result ([Noe84]; see 1.1.4 for a modern | and more general | version of this theorem). He also proved the finiteness of Aut(X) using Weierstrass points ([Noe82]), but the first proof of this fact is probably due to Klein ([dC08, p. 42]). In his landmark 1893 paper [Hur92], H¨urwitzintroduced the wronskian and the notion of \weights" to the study of Weierstrass points. He even introduced the concept of \higher" Weierstrass points, although the first time Weierstrass points were thus labelled was in a paper by Haure from 1896 ([Hau96]). Segre adopted this label when he discussed \punti di Weierstrass" in his short 1899 publication [Seg99], in which he proved that the maximal weight of a Weierstrass point on a nonhyperelliptic curve of genus g is (g − 1)(g − 2)=2 + 1. The first half of the twentieth century is considered a period of relative dormancy in the study of Weierstrass points, but it was in this period that a paper of great relevance to this thesis was published. In his 1939 paper [Sch39], Schmidt used the wronskian method to extend the notion of Weierstrass points to curves defined over fields of positive characteristic. The gap sequences and Weierstrass points defined by Laksov in [Lak81], whose modern approach to the topic is the focus of this thesis, are the same as those given by Schmidt. Rauch ushered in a period of renewed interest in Weierstrass points with the publica- tion of his paper [Rau59] in 1959. Ahlfors had already endowed the set of isomorphism classes Mg of Riemann surfaces of genus g with the structure of a (3g − 3)-dimensional complex analytic space ([Ahl60]). In his 1959 paper, Rauch proved that the Riemann surfaces of genus g carrying a Weierstrass point with a prescribed first non-gap constitute a complex-analytic subvariety of Mg. Some years later, expanding on the work of Rauch, Arbarello examined the closure W n;g in Mg of the set of Riemann surfaces of genus g possessing a Weierstrass point x such that h0(C; nx) ≥ 2, where 2 ≤ n ≤ g (so, for example, W 2;g is the closure of the hyperelliptic locus, and in general W n;g contains the space of Riemann surfaces of genus g possessing a Weierstrass point whose first non-gap 12 is n). Among other things, Arbarello proved that W 2;g is an irreducible analytic variety of dimension 2g + n − 3 ([Arb74]).