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Birational Geometry of the Moduli Spaces of Curves with One Marked Point

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Birational Geometry of the Moduli Spaces of Curves with One Marked Point The Dissertation Committee for David Hay Jensen Certi¯es that this is the approved version of the following dissertation: BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT Committee: Sean Keel, Supervisor Daniel Allcock David Ben-Zvi Brendan Hassett David Helm BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT by David Hay Jensen, B.A. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Ful¯llment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN May 2010 To Mom, Dad, and Mike Acknowledgments First and foremost, I would like to thank my advisor, Sean Keel. His sug- gestions, perspective, and ideas have served as a constant source of support during my years in Texas. I would also like to thank Gavril Farkas, Joe Harris, Brendan Hassett, David Helm and Eric Katz for several helpful conversations. iv BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT Publication No. David Hay Jensen, Ph.D. The University of Texas at Austin, 2010 Supervisor: Sean Keel We construct several rational maps from M g;1 to other varieties for 3 · g · 6. These can be thought of as pointed analogues of known maps admitted by M g. In particular, they contract pointed versions of the much- studied Brill-Noether divisors. As a consequence, we show that a pointed 1 Brill-Noether divisor generates an extremal ray of the cone NE (M g;1) for these speci¯c values of g. v Table of Contents Acknowledgments iv Abstract v List of Figures vii Chapter 1. Introduction 1 Chapter 2. Background 7 2.1 Moduli of Curves . 7 2.2 Birational Geometry . 8 2.2.1 De¯nitions . 8 2.2.2 Rational Contractions . 10 2.3 Geometric Invariant Theory . 20 2.4 Hessians . 25 Chapter 3. Results 31 3.1 Genus 3 . 31 3.2 Genus 4 . 41 3.3 Genus 5 . 51 3.3.1 Semistability Arguments . 53 3.3.2 Consequences . 62 3.4 Genus 6 . 65 Bibliography 70 Vita 72 vi List of Figures 3.1 The map Á ............................. 52 1 3.2 Image of the Weierstrass divisor BN5;(0;5) . 64 1 3.3 Image of the Pointed Brill-Noether Divisor BN4;(0;3) . 64 2 3.4 Image of the Pointed Brill-Noether Divisor BN6;(0;2;4) . 64 vii Chapter 1 Introduction The moduli spaces of curves M g;n are among the most fundamental and heav- ily studied objects in algebraic geometry, yet their birational geometry remains somewhat mysterious. A possible approach to studying M g;n would be to con- struct maps (with connected ¯bers) from the moduli space to simpler objects that we could analyze explicitly. One of the main results of [7] is that every morphism from M g;n to a variety of lower dimension factors through the map M g;n ! M g;k given by forgetting all but k of the marked points. In addition, every morphism from M g;n onto a variety of equal dimension restricts to an isomorphism on the interior Mg;n. We cannot expect, therefore, to learn much about the geometry of M g;n by studying such maps. We discover a much richer picture, however, if we do not require our maps to be everywhere de¯ned. A natural generalization is to consider the variety's rational maps. More speci¯cally, we consider those rational maps that are determined by complete linear series, often called rational contractions. If the rational contraction f : X 99K Y is birational, we refer to Y as a birational model for X. Using this language, we can restate our original goal by saying that we would like to construct birational models for M g;n. 1 The rational maps admitted by a variety X are in correspondence with its e®ective divisors, up to numerical equivalence. The e®ective divisors on X, modulo numerical equivalence, form a cone inside RN . The cone of e®ective 1 1 divisors is denoted NE (X). In many cases, NE (X) is ¯nitely generated, and thus provides us with a nice combinatorial parameterization of the va- riety's rational maps. If a divisor lies in the interior of the cone, then the corresponding rational map is birational. If, on the other hand, there is a birational model for X in which a given divisor is contracted, then the divisor lies on an edge of the cone. There are many examples of e®ective divisors on M g;n, but only some of them are geometrically meaningful. These are the loci of pointed curves possessing some interesting geometric property. A motivating principle in this 1 ¯eld is that the edges of NE (M g;n) ought to be generated by geometrically meaningful divisors. Among the ¯rst to study the birational geometry of the moduli spaces of curves were Harris and Mumford. A key element of their proof that M g is of general type for genus g ¸ 24 is the computation of the class of certain divisors on M g, known as Brill-Noether divisors. De¯nition 1. Let g + 1 = (r + 1)(g ¡ d + r). The Brill-Noether divisor r r BNd ½ M g is the closure of the locus of curves in Mg possessing a gd. In their work, Harris and Mumford discovered that all of the Brill- 1 Noether divisors lie on a single ray in NE (M g); this later led to Harris and Morrison's conjecture that this ray is an edge of the e®ective cone [9]. The 2 Harris-Morrison Slope Conjecture has recently been proven false in [4] and subsequently in [5], but the statement is known to hold for certain small values of g. In particular, it holds for g · 11, g + 1 composite. Our main goal here is to carry out a pointed analogue of the discussion above in some low genus cases. Here, we can de¯ne pointed versions of the Brill-Noether divisors and formulate a pointed analogue of the Slope Conjec- ture. In particular, we can ask whether these pointed Brill-Noether divisors generate edges of the e®ective cone. Much as was done for M g, we prove this conjecture for certain values of g by explicitly constructing rational maps. r The gd's on a curve exhibit unusual behavior at certain special points of the curve. In particular, some of the points have unexpected higher-order tangencies. The notion of pointed Brill-Noether divisors was ¯rst introduced by Logan in [13]. De¯nition 2. Let g + 1 = (r + 1)(g ¡ d + r) + ®, and Z = (a0; : : : ; ar) be an Pr increasing sequence of nonnegative integers with ® = i=0 ai ¡i. The pointed r Brill-Noether divisor BNd;Z ½ M g;1 is the closure of the locus of pointed r curves (C; p) 2 Mg;1 possessing a gd on C with rami¯cation sequence Z at p. Notice that, unlike the Brill-Noether divisors on M g, which only exist when g + 1 is composite, there are multiple pointed Brill-Noether divisors on M g;1 in every genus. Studying them is a natural re¯nement of earlier work on M g. One example of a pointed Brill-Noether divisor is the Weierstrass divi- sor, consisting of pointed curves (C; p) such that p is a Weierstrass point of C. 3 1 In [17], Rulla proves that this divisor generates an edge of NE (M 2;1). Our main result is the construction of four maps that extend this to higher genera. The ¯rst two maps contract a pointed-Brill Noether divisor: Theorem 1.0.1. There is a birational contraction of M 3;1 contracting the 1 Weierstrass divisor BN3;(0;3). Similarly, there is a birational contraction of 1 M 4;1 contracting the pointed Brill-Noether divisor BN3;(0;2). The third map is a ¯bration, given by the complete linear series of a pointed Brill-Noether divisor: Theorem 1.0.2. There is a natural map M 5;1 99K M 0;4. The corresponding 1 complete linear series is jBN4;(0;3)j. The ¯nal map is also a ¯bration, but with a di®erent associated divisor. This is the divisor of pointed curves in M 6;1 such that the point is a \node of 2 a g6": De¯nition 3. Let D be the closure of the locus of pointed curves (C; p) 2 M6;1 2 0 0 0 possessing a g6 L and a point p 2 C such that h (C; L ¡ p ¡ p ) ¸ 2. Theorem 1.0.3. There is a natural map M 6;1 99K M 0;5=S5. The correspond- ing complete linear series is jD + Ej for some e®ective divisor E. Among the consequences of these constructions is the following: Corollary 1.0.4. For 3 · g · 5, there is a pointed Brill-Noether divisor that 1 generates an extremal ray of NE (M g;1). The divisor D lies on the boundary 1 of NE (M 6;1). 4 In some sense, these constructions are very classical. The map M 5;1 99K M 0;4, for example, can be described entirely in terms of the fact that the general genus 5 curve is the intersection of 3 quadrics in P4 { an observation familiar to 19th century algebraic geometers. Similarly, the map M 6;1 99K M 0;5=S5 arises as a direct consequence of the fact that the general genus 6 curve can be embedded in a quintic del Pezzo surface.
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