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

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The Dissertation Committee for David Hay Jensen Certiﬁes 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

David Hay Jensen, B.A.

DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulﬁllment 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 speciﬁc 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 Deﬁnitions ...... 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 φ ...... 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 construct maps (with connected ﬁbers) 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 deﬁned.

A natural generalization is to consider the variety’s rational maps. More speciﬁcally, 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 effective divisors, up to numerical equivalence. The effective divisors on X, modulo numerical equivalence, form a cone inside RN . The cone of effective 1 1 divisors is denoted NE (X). In many cases, NE (X) is finitely generated, and thus provides us with a nice combinatorial parameterization of the variety’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 eﬀective 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 ﬁeld is that the edges of NE (M g,n) ought to be generated by geometrically meaningful divisors.

Among the ﬁrst 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.

Deﬁnition 1. Let g + 1 = (r + 1)(g − d + r). The Brill-Noether divisor

r r BNd ⊂ 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 eﬀective 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 deﬁne 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 eﬀective 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 ﬁrst introduced by Logan in [13].

Deﬁnition 2. Let g + 1 = (r + 1)(g − d + r) + α, and Z = (a0, . . . , ar) be an Pr increasing sequence of nonnegative integers with α = i=0 ai −i. The pointed r Brill-Noether divisor BNd,Z ⊂ M g,1 is the closure of the locus of pointed r curves (C, p) ∈ Mg,1 possessing a gd on C with ramiﬁcation 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 reﬁnement of earlier work on

M g.

One example of a pointed Brill-Noether divisor is the Weierstrass divisor, 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 ﬁrst 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 ﬁbration, 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 |BN4,(0,3)|.

The final map is also a fibration, but with a different associated divisor.

This is the divisor of pointed curves in M 6,1 such that the point is a “node of

2 a g6”:

Deﬁnition 3. Let D be the closure of the locus of pointed curves (C, p) ∈ M6,1

2 0 0 0 possessing a g6 L and a point p ∈ 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 corresponding complete linear series is |D + E| for some eﬀective 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. These nice coincidences allow us to construct explicit, very symmetric models of M g,1.

At the same time, showing that these maps correspond to geometrically meaningful divisors requires modern techniques from birational geometry and geometric invariant theory. There are many situations in algebraic geometry in which we would like to take the quotient of a variety X by a group G that does not act discretely on X. Unfortunately, because the orbits of X are not closed, a continuous quotient map X → X/G sending orbits to points cannot exist. The solution is to throw out some of the points, creating instead a quotient of a dense open subset U ⊆ X. This open set is not unique – it depends on a choice of a line bundle. By varying the line bundle on X, we obtain many diﬀerent quotients, each of them birational to the others. Most of the arguments for the claims above are made using this observation. By varying the line bundle on a given variety, we demonstrate that the desired rational map arises as a GIT quotient.

In what follows, we provide an outline of this paper. Chapter 2 provides the necessary background to understand our results. Aside from the end of section 2.2.2, it does not contain any new results. Section 2.1 provides an introduction to moduli of curves. Section 2.2 outlines the tools and techniques

5 from birational geometry that we use throughout. Section 2.3 covers geometric invariant theory, which serves as the main tool that we use to construct our rational maps. In order to study theses GIT quotients, we need to exhibit divisors that are invariant under the action of the group. A certain class of these, known as Hessians, are surveyed in Section 2.4.

In chapter 3 we construct each of our rational maps. This is done one moduli space at a time, with each section devoted to a diﬀerent genus.

6 Chapter 2

Background

In this chapter we provide the background necessary to understand our results. For the most part, this chapter contains nothing new and can safely be skipped by the knowledgeable reader. The exception to this occurs at the end of Section 2.2.2, where we provide proofs of several claims that we could not ﬁnd in the literature.

2.1 Moduli of Curves

The central objects of study in this paper are the moduli spaces of curves

M g,n. Heuristically, we may think of the moduli space Mg,n as the set of isomorphism classes (C, p1, . . . , pn), where C is a smooth curve of genus g and the pi’s are distinct points of C. This set can be endowed with a nautral geometric structure, in which a smoothly varying family of curves corresponds to a smoothly varying set of points in the moduli space. Equipped with this geometry, Mg,n is a quasiprojective variety, but it is not proper. Fortunately, by introducing curves with mild singularities, we may construct a natural compactiﬁcation M g,n.

7 More concretely, given a scheme S, we deﬁne a stable curve C of genus g over S to be a proper ﬂat morphism π : C → S satisfying:

1. The geometric ﬁbers Ct of π are reduced, connected, one dimensional schemes;

2. Ct has only nodes as singularities;

3. Every non-singular rational component of Ct meets the rest of the curve in at least 3 points;

4. Ct has arithmetic genus g.

A stable n-pointed curve over S consists of a stable curve over S, together with n sections σi : S → C such that:

1. σi(t) is a smooth point of Ct for all t;

2. The images of the σi’s are disjoint.

We can then deﬁne M g,n to be the variety that coarsely represents the functor which sends a scheme S to the set of stable n-pointed curves over S, up to isomorphism.

2.2 Birational Geometry 2.2.1 Deﬁnitions

In this section, we define the cone of effective divisors and other objects from birational geometry. Many of these definitions can be found in [17] and [6].

8 Let X be a variety and B1(X) the group of Weil divisors of X modulo algebraic equivalence. The Neron Severi Space of X is the vector space

1 1 NS(X) = BR(X) = B (X) ⊗Z R.

If X is smooth and projective, then B1(X) is a finitely generated abelian group, and thus NS(X) is a finite dimensional vector space. Moreover, if f : X → Y is a proper morphism, then the pushforward f∗ : Div(X) → Div(Y ) descends to algebraic equivalence (see [6]). It follows that, if f is surjective, then the map f∗ : NS(X) → NS(Y ) is a surjective linear map. By resolution of singularities this implies that NS(X) is finite dimensional even in the case where X is not smooth.

In later sections, we will often fail to distinguish between Weil divisors and Cartier divisors. This abuse of language is justiﬁed by the following: let

D1,D2 be Cartier divisors. We will say that D1 and D2 are numerically

1 equivalent if, for every curve C ⊂ X, C.D1 = C.D2. We let C (X) denote the group of Cartier divisors of X modulo numerical equivalence, and write

1 1 N (X) = C (X) ⊗Z R.

Lemma 2.2.1. (See [17]) Let X be a Q-factorial projective variety, and D1,D2

Weil divisors on X. If D1 and D2 are algebraically equivalent, then they are numerically equivalent as Cartier divisors.

Corollary 2.2.2. (See [17]) Let X be a projective variety. Then there is an injective linear map N 1(X) ,→ NS(X). If X is Q-factorial, this map is an isomorphism.

9 We now turn to the cone of eﬀective divisors, which is one of the main objects of study in this work. We let NE1(X) denote the positive linear span of irreducible eﬀective divisors in N 1(X). In other words,

X 1 ≥0 NE (X) = { aiDi|ai ∈ R ,Di an irreducible eﬀective divisor}.

1 The cone of eﬀective divisors NE (X) is the closure of NE1(X) in N 1(X).

1 Following [17], we will refer to elements of NE (X) as effective divisors. We will use the term “codimension one subvariety” where applicable to avoid confusion. Note that this cone does not in fact parameterize codimension one subvarieties, but limits of codimension one subvarieties. Indeed, 1 it is possible for NE (X) to contain elements that do not correspond to a non-negative real sum of codimension one subvarieties. The standard example of this is the cone of effective divisors of E × E, where E is an elliptic curve such that N 1(E × E) is 3-dimensional. In this case, it is known that 1 NE (E × E) is a round cone. Any point on the boundary of this cone cannot be expressed as a convex combination of other points in the cone, and if such a point lies at an irrational angle, it cannot be a positive real multiple of any 1 codimension one subvariety. For this reason, many authors refer to NE (X) as the “pseudoeffective” cone.

2.2.2 Rational Contractions

1 We are mainly interested in NE (X) because of the correspondence between eﬀective divisors and certain types of rational maps, called rational contractions. Heuristically, the location of a divisor within the cone of eﬀective

10 divisors determines properties of the corresponding rational map. In this section we discuss several results of this type. We begin with a few deﬁnitions, which can all be found in [11].

Deﬁnition 4. Let f : X 99K Y be a rational map between normal projective varieties. Let (p, q): W → X × Y be a resolution of f with W projective and p birational. We call f a birational contraction if it is birational and every p-exceptional divisor is q-exceptional. For a Q-Cartier divisor D ⊂ Y ,

∗ ∗ we deﬁne f (D) to be p∗(q (D)).

In [11], the authors extend this notion of rational contraction to the non-birational case.

Definition 5. Let X,Y, and W be as above. An effective divisor E on W is called q-fixed if no effective Cartier divisor whose support is contained in the support of E is q-moving. We call f a contraction if every p-exceptional divisor is q-fixed.

Examples of contractions include morphisms, birational contractions, and compositions of these. As mentioned above, our main interest in the cone of effective divisors stems from the correspondence between effective divisors and rational contractions. In particular, suppose that D is an effective divisor. We define the section ring

M 0 ⊗n R(X,D) = H (X, OX (D) ). n∈N

11 If D is eﬀective and R(X,D) is ﬁnitely generated, then there is an induced rational map

fD : X 99K P roj(R(X,D)).

Lemma 2.2.3. (See [11]) If f : X 99K Y is a contraction and D = f ∗(A) + E for A ample on Y and E ﬁxed by f, then D has a ﬁnitely generated section ring, and f = fmD for some m > 0.

We also need some results to help us describe the eﬀective cones of our varieties. The ﬁrst of these is known variously as the “Kodaira Lemma” or “Negativity of Contraction”.

Lemma 2.2.4. (See [12], [17]) Let f : X → Y be a proper birational mor-

n phism, with X regular in codimension one. Let {Ei}i=1 be a collection of f- 1 exceptional divisors, M an f-nef Q-Cartier divisor on X, and P ∈ NE (X) any element such that there is a sequence of Q-Cartier divisors {Dj} → P such that no Ei is contained in the support of any Dj. If L is a Q-Cartier divisor on Y such that Xn ∗ f (L) = M + P + eiEi i=1 then ei ≥ 0 ∀i.

Corollary 2.2.5. Let X be proper and Q-factorial. Then the complete set

{Ei} of exceptional codimension one subvarieties of a morphism f : X → Y as above is linearly independent in N 1(X).

Another major tool for describing cones is the following, from [17]:

12 Lemma 2.2.6. (See [17]) Let X be a projective variety and suppose that

Pn ≥0 1 i=1 R Ei is a subcone of NE (X) generated by codimension one subva- 1 rieties Ei. Then for any P ∈ NE (X), one can write Xn P = Q + ciEi i=1 where ci ≥ 0 ∀i and Q is the limit of eﬀective Q-Cartier divisors {Dj} such that no Ei is contained in the support of any Dj.

A consequence of this is the fact that exceptional divisors generate edges of the eﬀective cone. A proof of this can be found in [17].

Theorem 2.2.7. (See [17]) Let X and Y be normal, projective, Q-factorial varieties, f : X 99K Y a birational contraction. If {Ei} is the complete set of

P ≥0 exceptional codimension 1 subvarieties of f, then R Ei is a simplicial face 1 of NE (X).

We now turn our attention to a speciﬁc type of rational contraction – the composition of a birational contraction and a morphism. Throughout the remainder of this section, we let X,Y and Z be normal projective varieties with dimY < dim X, f : X 99K Z a birational contraction, c : Z → Y a proper morphism, and h : X 99K Y the composition. Let W be a resolution of f, giving us the following diagram:

W ppp 00 p ppp 0 pp q 00 ppp 0 xpp f 0 X / Z 00e AA 00 h AA 0 AA 00 c AA0 * Y

13 Note that, if X is a Mori dream space, then all rational contractions with lower dimensional image are of this type.

Because X is normal, the map h is deﬁned on divisors of X. There are essentially two types of these: those for which the restriction of h to the given divisor is dominant, and those for which it is not. We use this observation to 1 determine some faces of NE (X).

Lemma 2.2.8. Let D be a Q-Cartier divisor on Y such that c∗(D) generates

1 n an extremal ray of NE (Z). Let {Ei}i=1 be the set of all q-exceptional codi-

m mension one subvarieties on W such that e|Ei is not surjective, and {Fj}j=1 any subset of the remaining q-exceptional codimension one subvarieties. Fur- thermore, let

∗ 1 H = SpanR(e (D),E1,...,En) ∩ NE (W ).

Then Xm ≥0 H + R Fi i=1 1 is an extremal face of NE (W ).

k Proof. Let {Gi}i=1 be the remainder of the q-exceptional codimension one sub-

0 1 0 Pm ≥0 varieties. Suppose that P,P ∈ NE (W ) such that P + P ∈ H + i=1 R Fi. By the above, we may write

Xm Xk P = Q + aiFi + biGi i=1 i=1

14 where ai, bi ≥ 0 ∀i and Q is the limit of eﬀective Q-Cartier divisors {Dj} such that no Fi,Gi is contained in the support of any Dj. Similarly, we may write

Xm Xk 0 0 0 0 P = Q + aiFi + biGi i=1 i=1

∗ 0 Since c (D) generates an extremal ray, and since q∗(Q+Q ) is a multiple

∗ ∗ of c (D), we have that q∗(Q) is a positive multiple of c (D). It follows that

Xn Xm Xk ∗ Q = de (D) + eiEi + fiFi + giGi i=1 i=1 i=1 where d ≥ 0. By negativity of contraction, we see that fi, gi ≤ 0 ∀i. Thus, if we let Xm Xk R = − fiFi − giGi i=1 i=1 then Q + R ∈ H.

Now, let Fgen be a generic ﬁber of e and let r be the dimension of Fgen.

If r = 1, deﬁne C to be the class of Fgen in the cone of curves on W . Otherwise,

r−1 let A be an ample divisor on Fgen and deﬁne C to be A . Then, for any codimension one subvariety E in W , C · E ≥ 0, with equality if and only if e|E is not surjective. Thus, C · Fi > 0 and C · Gi > 0, but C · (Q + R) = 0.

This implies that fi = gi = 0 ∀i.

The same reasoning applies to P 0. By linear independence of q-exceptional

0 divisors, we see that bi + bi = 0. Since both are nonnegative, we have that

Pm ≥0 bi = 0. Thus, P ∈ H + i=1 R Fi. The result follows.

15 We now extend this result from a statement about morphisms to one about rational maps.

Corollary 2.2.9. Let D be a Q-Cartier divisor on Y such that c∗(D) generates

1 n an extremal ray of NE (Z). Let {Ei}i=1 be the set of all f-exceptional codi-

m mension one subvarieties on X such that h|Ei is not dominant, and {Fj}j=1 any subset of the remaining f-exceptional codimension one subvarieties. Fur- thermore, let

∗ 1 H = SpanR(f (D),E1,...,En) ∩ NE (X).

Then Xm ≥0 H + R Fi i=1 1 is an extremal face of NE (X).

˜ ˜ Proof. Let Ei, Fi be the strict transforms of Ei, Fi, respectively. Furthermore, let {Gi} be the complete set of p-exceptional codimension one subvarieties of W . Notice that the map h| is dominant if and only if the map e| is Ei E˜i surjective. By the above, if

˜ ∗ ˜ ˜ 1 H = SpanR(e (D), E1,..., En,G1,...,Gn) ∩ NE (W ) P ˜ m ≥0 ˜ 1 then H + i=1 R Fi is an extremal face of NE (W ).

0 1 0 Pm ≥0 Now, suppose that P,P ∈ NE (X) such that P +P ∈ H+ i=1 R Fi. P ∗ 0 ∗ ∗ 0 ˜ m ≥0 ˜ ∗ ˜ Then p (P + P ) = p (P ) + p (P ) ∈ H + i=1 R Fi. Thus, p (P ) ∈ H + P P m ≥0 ˜ ∗ m ≥0 i=1 R Fi. But this means that P = p∗p (P ) ∈ H + i=1 R Fi. The conclusion follows.

16 1 Having described an extremal face of NE (X), we now seek to establish some of its properties. The results below essentially show that the only possible irreducible divisor in the interior of this face is f ∗(D). In what follows, we let D1 be an irreducible codimension one subvariety of X, and suppose that

∗ D1 + D2 = h (D) for some D a codimension one subvariety of Y , D2 ∈ 1 NE (X). Suppose further that D1 is not numerically equivalent to a multiple of h∗(D). To prove our claim, we need a version of negativity of contraction that applies to this case. The argument below follows closely that of Lemma 2.19 in [12].

˜ 1 Lemma 2.2.10. Let D1 ∈ NE (W ) be the class of the strict transform of

n D1, and {Ei}i=1 the set of exceptional divisors of the map p : W → X. Let 1 P ∈ NE (W ) any element such that there exists a sequence of Q-eﬀective ˜ Cartier divisors {Pj} → P such that neither D1 nor any Ei is contained in the support of any Pj. If X ˜ P = e0D1 + eiEi

then ei = 0 ∀i.

Proof. If ei ≤ 0 ∀i, then, since 0 is extremal, we have that ei = 0. Thus, it suﬃces to show that all of the coeﬃcients are nonpositive. By cutting with dimY −1 general hypersurfaces in Y , we may assume that Y is a curve. Then, ˜ since the image of an irreducible divisor is irreducible, e(D1) is a point in Y . ˜ Note that anything in the span of D1 and the Ei’s pushes forward under p to

17 a multiple of D1. By assumption, D2 is not a multiple of D1, so there must be ˜ a component of the ﬁber of e(D1) not contained in the span of these divisors. Let G be such a component.

For each Ei, by cutting with dimqZ (Ei) general hypersurfaces in Z, we may assume that qZ (Ei) is zero dimensional. Now, let S ⊂ W be the intersection of dimW − 2 general hypersurfaces containing a general point of

˜ 0 ˜ 0 0 D1, G, and each of the Ei’s. Let D1 = D1 ∩ S, G = G ∩ S, and Ei = Ei ∩ S.

0 0 0 Each of these is an irreducible curve. Let P = P |S. By assumption, P ·D1 ≥ 0

0 0 and P · Ei ≥ 0 ∀i.

0 0 Note that the map q|S : S → Z contracts all of the Ei. Let F denote

0 the ﬁber of e|S(D1). From the theory of surfaces, we know that q|S factors ˜ 0 0 through a map q : S → S contracting those Ei that do not lie in F (as well

0 0 0 as other possible curves), but not contracting D ,G , or any of the Ei lying in

0 0 this ﬁber. In what follows, we set D1 = E0.

0 We claim that the intersection pairing on the span of the Ei is negative

0 deﬁnite. This pairing is known to be negative deﬁnite on the span of those Ei contracted by the map q : S → S˜. Since this subspace is orthogonal to the

0 0 span of those Ei contained in the fiber of e(D1), it suffices to show that the intersection pairing is negative definite on the latter subspace. We therefore

0 0 reduce to the case where all of the Ei are contained in a ﬁber. Let F denote

0 Pn 0 0 0 0 0 the class of a ﬁber, and E = i=0 Ei. Note that, for any Ei, Ei ·(F −E ) ≥ 0. 0 0 0 0 0 Since Ei is contained in a ﬁber, however, we have Ei · F = 0, so Ei · E ≤ 0. This in turn implies that

18 X 02 0 0 Ei ≤ − Ei · Ej j6=i

P 0 Now, let H = hiEi be any element in this subspace. Then X X Xn X 2 0 0 0 0 2 02 2 0 0 H = hihjEi·Ej = 2hihjEi·Ej+ hi Ei ≤ − (hi−hj) Ei·Ej ≤ 0 0≤i,j≤n 0≤i

0 where the ﬁrst inequality is an equality if and only if hi = 0 for every Ei that

0 0 intersects F − E , and the second is an equality if and only if hi = hj for

0 0 every Ei and Ej that intersect. Since the ﬁbers of this map are connected,

0 0 and E 6= F , this implies that equality holds throughout if and only if hi = 0 ∀i. Thus, the intersection pairing is negative deﬁnite.

We claim that this proves all the coeﬃcients to be nonpositive. Indeed, suppose that at least one of the coeﬃcients is positive and write

P 0 = R+ + R− P P where R+ = e E0 and R− = e E0. Since R+ 6= 0, we know ei≥0 i i ei≤0 i i +2 0 that R < 0. Therefore, there is some Ei with positive coeﬃcient such that

0 + 0 − 0 0 Ei · R < 0. Clearly, Ei · R ≤ 0, so Ei · P < 0, a contradiction. It follows that ei ≤ 0 ∀i, which in turn implies that ei = 0 ∀i.

P ≥0 ˜ ≥0 Corollary 2.2.11. Given the setup of the previous lemma, R D1 + R Ei 1 is an extremal face of NE (W ).

P 0 1 0 ≥0 ˜ ≥0 Proof. Suppose that P,P ∈ NE (W ) with P + P ∈ R D1 + R Ei. ˜ P Again, we can write P = Q + aD1 + biEi, where a, bi ≥ 0, and Q is the

19 ˜ limit of a sequence of Q-eﬀective divisors {Pj} such that neither D1 nor any of the Ei’s is contained in the support of any Pj. Similarly, we may replace

0 0 0 P 0 P with Q + a D + biEi. By rearranging these equations, we see that P 0 ˜ Q + Q = e0D1 + eiEi for some values of ei. By the proposition above, it

0 follows that ei = 0 ∀i, and thus Q + Q = 0. Since zero is extremal, we have P 0 0 ≥0 ˜ ≥0 Q = Q = 0. Thus, both P and P are elements of R D1 + R Ei.

1 Corollary 2.2.12. D1 generates an extremal ray of NE (X).

0 1 0 Proof. Suppose that P,P ∈ NE (X) with P + P = aD1 for some a. Then

∗ ∗ 0 ∗ ∗ ≥0 P ≥0 p P + p P = p D1. But p D1 ∈ R D + R Ei, which is extremal, so it follows that p∗P and p∗P 0 are elements of this extremal face as well. We may

∗ 0 therefore conclude that P = p∗p P is a multiple of D1, and similarly for P .

2.3 Geometric Invariant Theory

The rational contractions that we construct in the next chapter arise naturally as GIT quotients. In this section, we provide the necessary background to understand these constructions.

Let X be a smooth projective variety over C and G a reductive algebraic group acting on X. If we want to examine the orbits of this action, it is natural to ask for the existence of a quotient variety. In other words, we would like there to be a projective variety X//G such that, if X → Y is a G-equivariant

20 morphism of varieties, then it factors through a unique map X//G → Y . Unfortunately, such a quotient may not necessarily exist. It is often possible, however, to ﬁnd an open set U ⊆ X such that U//G exists. It has long been understood that this open subset is not unique; it depends on the choice of a G-ample line bundle. In particular, if L is such a line bundle, then we deﬁne the set of semistable points to be

Xss(L ) = {x ∈ X|∃s ∈ H0(X, L ⊗n)G for some n such that s(x) 6= 0} and the set of stable points to be

Xs(L ) = {x ∈ Xss(L )|G·x is closed in Xss(L ) and the stabilizer of x is ﬁnite}.

Xss(L )//G is a projective variety that satisﬁes the catgeorical deﬁni- tion of a quotient. Notice that, if X is irreducible, and if both Xss(L ) and Xss(L 0) are nonempty, then the quotients Xss(L )//G and Xss(L 0)//G are birational. We will make use of this observation to construct birational models for certain spaces by varying the choice of line bundle.

Let L be a line bundle on X. We consider G-linearizations of L , or actions of G on L that are linear on the fibers of L and commute with the projection to X. The action of G on L induces an action of G on H0(X, L ), which allows us to talk about the G-invariant sections of L . L is G-effective if some tensor power of it admits a G-invariant section. In other words, L is G-effective if Xss(L ) 6= ∅. An ample G-effective line bundle is called G- ample. If we let P icGX be the group of isomorphism classes of G-linearized line bundles on X, then there is a homomorphism P icGX → P icX whose

21 kernel is isomorphic to the group of characters on G. In the examples we will be considering, the group G admits no nontrivial characters, and so we will often refer to the line bundle without specifying its linearization.

The tensor product of two G-ample line bundles is also G-ample, so the G-ample line bundles form a cone inside P icGX. Since the quotient Xss(L )//G depends only on the algebraic equivalence class of L , we may restrict our attention to the cone spanned by the G-ample line bundles in

G NSQ . Following Dolgachev and Hu, we will call this the G-ample cone, denoted CG(X). The properties of this cone are studied extensively in [3] and [18]. The following theorem is a summary of some of the major results of those papers:

Theorem 2.3.1. [3] [18] The G-ample cone is divided into a ﬁnite number of convex cones, called chambers, by a ﬁnite number of codimension 1 homoge- neous spaces, called walls. These satisfy the following properties:

1. L lies on a wall if and only if Xss(L ) 6= Xs(L );

2. L lies on a boundary of CG(X) if and only if Xss(L ) 6= Xs(L ) = ∅;

3. L and L 0 lie in the same chamber if and only if Xs(L ) = Xss(L ) = Xss(L 0) = Xs(L 0);

4. If L (t) = L t ⊗ L 0(1−t), then Xss(L ) ∩ Xss(L 0) ⊆ Xss(L (t));

5. If L lies on a wall and L 0 lies is an adjacent chamber, then the inclusion Xss(L 0) ⊆ Xss(L ) induces a morphism Xss(L 0)//G → Xss(L )//G that is an isomorphism over Xs(L )//G.

22 The case of G = C∗ has been studied in detail. Let XG be the set of

G ﬁxed points of the action, and let X = X1 ∪X2 ∪...∪Xr be its decomposition into irreducible components. Let

+ Xi = {x ∈ X| lim t · x ∈ Xi} t→0

− Xi = {x ∈ X| lim t · x ∈ Xi}. t→∞

+ − We can think of Xi as the points of X that “ﬂow from” Xi, and Xi as those

+ + − − that “ﬂow to” Xi. The partition of X as X = X1 ∪· · ·∪Xr = X1 ∪· · ·∪Xr is known as the Bialynicki-Birula decomposition of X with respect to the action of C∗ [1]. Every point of X\XG is contained in exactly one subset of the following form:

+ − Cij = (Xi \Xi) ∩ (Xj \Xj)

Using the Cij’s, we can deﬁne a partial ordering on the Xi’s by saying that Xi is directly less than Xj if Cij 6= ∅, and Xi is less than Xj if there is a sequence

Xi = Xi0 ,Xi1 ,...,Xik = Xj such that Xin is directly less than Xin+1 for all n. (In other words, Xi is less than Xj if there are points that “ﬂow from” Xi to Xj.) An open subset U ⊆ X is then the semistable locus of some G-ample line bundle if and only if it is of the form

U = ∪i∈A−,j∈A+ Cij

− + − where A ,A is a partition of {1, . . . , r} such that, if i ∈ A and Xj is less

− than Xi, then j ∈ A .

23 In the case that G 6= C∗, less is known, but it is often possible to reduce to the case of a C∗ action. In [18], Thaddeus considers the following

t 1−t situation: let L +, L − be G-ample line bundles and deﬁne L (t) = L + ⊗ L − .

ss ss Suppose that there exists a t0 with X (L (t)) = X (L +) for all 1 > t > t0,

ss ss and X (L (t)) = X (L −) for all −1 < t < t0. This is the case, for example, when L + and L − are in adjacent chambers in the G-ample cone, and the line between them crosses a wall precisely at L (t0). In this case, we let Y =

∗ P (L + ⊕ L −), and Z = Y//G. There is then a family of C actions on Z such

ss ∗ ss −1 ss that X (L (t))//G = Z//Ct , and X (t) = q(p (Z (t))). Here, q : Y → X and p : Y → Z are the natural projection maps. Using this, we can deﬁne subsets of X analogous to the ﬂowlines of a C∗ action:

X± = Xss(0)\Xss(∓)

X0 = Xss(0)\(Xss(+) ∪ Xss(−)).

Thaddeus then proves:

Theorem 2.3.2. [18] Let x ∈ X0 be a smooth point of X. Suppose that G·x is

ss ∼ ∗ ss closed in X (0) and that Gx = C . Then the natural map X (L (±))//G → Xss(L )(0)//G is an isomorphism outside of X±//G(±). Furthermore, over a neighborhood of x in X0//G(0), X±//G(±) is a ﬁbration with weighted projective spaces as ﬁbers.

Note: This theorem requires the hypothesis that we are working in characteristic zero.

24 Given a G-ample line bundle L , it is often diﬃcult to describe the set Xss(L ). Mumford’s numerical criterion is a useful tool for determining whether or not a point is in this set. The action of G on the line bundle L determines a map G → GL(H0(L )∨). Now, deﬁne a 1-parameter subgroup to be a homomorphism λ : C∗ → G. Since the image of C∗ is an abelian subgroup

0 ∨ 0 ∨ of GL(H (L ) ), there exists a basis {b1, b2, . . . , bn} for H (L ) with respect to which λ(t) acts by the diagonal matrix   ta1 0 ··· 0  a2   0 t ··· 0  λ(t) =  . . . .  .  . . .. .  0 0 ··· tan

∗ We will refer to the ai’s as the weights of the C action. For a point x ∈ X,

0 ∨ 0 ∨ Pn consider its image in P H (L ) , lift it to H (L ) , and write x = i=1 xibi with respect to this basis. Deﬁne µλ(x) to be the smallest value of ai such that the corresponding xi is nonzero. Then x is semistable if and only if µλ(x) ≤ 0 for every 1-parameter subgroup λ of G, and x is stable if and only if µλ(x) < 0 for every nontrivial 1-parameter subgroup λ of G. We will use this criterion often to determine the (semi)stability of points in our space.

2.4 Hessians

We now turn our attention to families of curves on surfaces. We do this because, in the ﬁrst two cases that we will be considering, the general curve embeds canonically into a surface. In particular, the general genus 3 curve admits a canonical embedding into P 2. Similarly, the general genus 4 curve admits a canonical embedding into P 3 whose image is contained in a unique

25 quadric surface. This surface is generically isomorphic to P 1 × P 1.

Let Y be a smooth projective surface over C, L 0 an eﬀective line bundle on Y , and Z = P H0(Y, L 0). Let

X = {(C, p) ∈ Z × Y |C(p) = 0}.

We denote the various maps as in the following diagram:

π X i / Z × Y 1 / Y

f π2 Z id / Z

In other words, we will denote by π1 : Z × Y → Y and π2 : Z × Y → Z the natural projections, i the inclusion map i : X → Z × Y , and f the projection map f : X → Z. If L 0 is base-point free, then X is a projective space bundle over Y , so it is smooth, and if h0(L 0) ≥ 2, then P icX ∼= P icY × Z . There is a natural action of Aut(Y ) on X, given by

A(C, p) = (C ◦ A−1,A(p)).

We will later study the geometric invariant theory quotients of X by the action of this group. Our ﬁrst step is to identify a collection of G-eﬀective divisors on X, called Hessians.

If C is a curve on Y and L is another line bundle on Y , then for every

0 point p ∈ C there are n + 1 = h (C, L |C ) diﬀerent orders of vanishing of

0 sections s ∈ H (C, L |C ). When written in increasing order,

L L a0 (p) < ··· < an (p)

26 these integers are called the vanishing sequence of L at p. The weight of

L Pn L L at p is defined to be w (p) = i=0 ai (p) − i. A point p is said to be an L -flex if the weight of L at p is nonzero. In other words, p is an L -flex if the vanishing sequence of L at p is anything other than 0 < 1 < ··· < n. The divisor of L -flexes on C consists of all the L -flexes, each counted with

P L multiplicity equal to its weight. In other words, it is p∈C w (p)p. The divisor of L -ﬂexes corresponds to a section WL of a certain line bundle that is called the Wronskian of L . We say that a divisor H on Y is an L -Hessian if the restriction of H to C is precisely the divisor of L -ﬂexes.

Returning to our family of curves f : X → Z above, suppose that L

∗ is a line bundle on Y such that the pushforward f∗(i ◦ π1) L is locally free of rank n + 1. We deﬁne a relative L -Hessian to be a divisor H ⊆ X whose

∗ restriction to each ﬁber is the divisor of f∗(i ◦ π1) L -ﬂexes. In [2], Cukierman shows that the class of the relative L -Hessian is µ ¶ n + 1 (n + 1)c π∗L + c Ω1 − c f ∗f π∗L 1 1 2 1 X/Z 1 ∗ 1

For the remainder of this section, we will follow the proof in [2] to determine this class more explicitly in our case. We see that if I is the ideal sheaf of X in Z × Y , then we have the exact sequence

2 ∗ 1 1 0 → I/I → π1ΩY |X → ΩX/Z → 0 so we have

1 ∗ 1 2 c1ΩX/Z = c1π1ΩY |X − c1I/I .

27 Also, X is the scheme of zeros of a section of the line bundle E =

∗ 0 ∗ 2 ∼ ∗ ∗ π1L ⊗ π2O Z (1) on Z × Y . Note that I/I = E ⊗ O X = E |X . It follows that

1 ∗ 1 c1ΩX/Z = c1π1ΩY |X + c1E

∗ 1 ∗ 0 ∗ = c1π1ΩY |X + c1π1L + c1π2O Z (1).

Now, note that

∗ ∗ 0 0 0∗ π2∗(π1L ⊗ E ) = H (Y, L ) ⊕ H (Y, L ⊗ L ) ⊗ O Z (−1) so the class of the relative L -Hessian is µ ¶ n + 1 (n + 1)c π∗L + (c π∗Ω1 | + c π∗L 0 + c π∗O (1)) 1 1 2 1 1 Y X 1 1 1 2 Z

0 0 0∗ −c1(H (Y, L ) ⊕ H (Y, L ⊗ L ) ⊗ O Z (−1)).

These Hessians provide a large class of G-invariant sections to work with when we are trying to determine the GIT quotients of X.

2 0 In particular, suppose that Y = P , and L = O Y (d) for some d ≥ 4.

By the above, we see that for every m and d, a relative O Y (m)-Hessian Hm

1 exists. Since ΩY = O Y (−3), if m < d, Hm is cut out by a G-invariant section W of m µ ¶ µ ¶ n + 1 n + 1 O ((n + 1)m + (d − 3), ), X 2 2 ¡ ¢ 0 m+2 where n+1 = h (Y, L ) = 2 . In particular, H1 is cut out by a section W1 ∈ 0 H (O X (3(d−2), 3)). W1 vanishes at (C, p) if C is smooth at p and the tangent line to C at p intersects C with multiplicity at least 3, or if p is a singular

0 point of C. Similarly, H2 is deﬁned by a section of W2 ∈ H (O X (15d−33, 15)).

28 W2 vanishes at (C, p) if C is smooth at p and the osculating conic to C at p intersects C with multiplicity at least 6, or if p is a singular point of C.

Notice, however, that if

0 < a1(p) < a2(p) is a vanishing sequence for O C (1) at p, and a2(p) 6= 2a1(p), then

0 < a1(p) < min(2a1(p), a2(p)) < max(2a1(p), a2(p)) < a1(p) + a2(p) < 2a2(p)

O C (2) O C (1) is a vanishing sequence for O C (2) at p. Thus, w (p) ≥ 4w (p) − 3.

From this, we see that if (C, p) ∈ H1, then (C, p) ∈ H2 as well, and so H2 =

0 0 H1 + H2 is reducible. The points of H2 ∩ C are classically known as the

0 0 sextatic points of C, and H2 is cut out by a G-invariant section W2 of 9 O X (12(d − 4 ), 12). From the equation above, we see that if p ∈ H1 ∩ C, then

O C (1) H2 intersects C with the same multiplicity as H1 only if w (p) = 1. Thus,

0 O C (1) H2 ∩ C contains all the points where w (p) > 1. These include singular points and points where the tangent line to C is a hyperﬂex (a line that intersects C at p with multiplicity ≥ 4).

1 1 0 Similarly, suppose that Y = P ×P , and L = O Y (d, d). Note that, for

0 every (m1, m2, d) with mi < d, a relative O Y (m1, m2)-Hessian Hm1,m2 exists. ∗ In this case, our formulas show that the rank of f∗(i ◦ π1) O Y (m1, m2) is

0 n + 1 = h (O Y (m1, m2)) = (m1 + 1)(m2 + 1).

1 0 Also, since ΩY = O Y (−2, −2), we see that Hm1,m2 is cut out by a section 0 0 Wm1,m2 ∈ H (O X (a1, a2, b)) for µ ¶ n + 1 a = (n + 1)m + (d − 2) 1 1 2

29 µ ¶ n + 1 a = (n + 1)m + (d − 2) 2 2 2 µ ¶ n + 1 b = . 2

1 1 0 Since P × P has a natural involution, we know that Wm1,m2 cannot 0 0 be G-invariant if m1 6= m2. Notice, however, that Wm1,m2 ⊗ Wm2,m1 is a

G-invariant section of O X (a, a, b) for

n + 1 = (m1 + 1)(m2 + 1) µ ¶ n + 1 a = (n + 1)(m + m ) + 2 (d − 2) 1 2 2 µ ¶ n + 1 b = 2 . 2

We will use Wm1,m2 to denote the G-invariant section described here, and

Hm1,m2 to denote its zero locus.

0 In particular, the section W0,1 ∈ H (O X (2(d − 1), 2(d − 1), 2)) vanishes at a point (C, p) if C intersects one of the two lines through p with multiplicity at least 2 (or, equivalently, if the osculating (1, 1) curve is a pair of lines).

0 Similarly, the section W1,1 ∈ H (O X (2(3d−4), 2(3d−4), 6)) vanishes at a point (C, p) if there is a curve of bidegree (1, 1) that intersects C with multiplicity 4 or more at p.

30 Chapter 3

Results

3.1 Genus 3

In this section, we study the moduli space M 3,1. Our main result is the following:

Theorem 3.1.1. There is a birational contraction of M 3,1 contracting the

1 Weierstrass divisor BN3,(0,3).

Recall that the general genus 3 curve admits a canonical embedding into

2 P as a plane quartic. To construct a birational model for M 3,1, therefore, we consider GIT quotients of the universal family over the space of plane quartics.

The image of the Weierstrass divisor in this model is precisely the Hessian H1. Our goal therefore is to exhibit a GIT quotient in which this locus is contracted. Our methods work equally well if we consider more generally plane curves of any degree d.

Speciﬁcally, following the set-up of the previous section, let Y = P 2

0 0 2 and L = O Y (d) for d ≥ 4. Then π2 : X → P H (P , O (d)) is the family of all plane curves of degree d. By the above, we know that P icX ∼= Z ×Z . We will

∗ ∗ write O X (a, b) to denote (i ◦ π1) (O P 2 (a)) ⊗ f (O Z (b)). The automorphism

31 group of P 2 is G = PSL(3, C). We now study the GIT quotients of X by the action of this group.

G Proposition 3.1.2. O X (3(d − 2), 3) lies on a boundary of C (X).

ss s Proof. Let L = O X (3(d−2), 3). It suﬃces to show that X (L ) 6= X (L ) = ∅.

ss It is clear that X (L ) 6= ∅, since W1 is a G-invariant section of L .

To show that Xs(L ) = ∅, we invoke the numerical criterion. Let (C, p) ∈ X. By change of coordinates, we may assume that p = (0, 0, 1) and the tangent line to C at p is x0 = 0. So, if X i j k C = ai,j,kx0x1x2, i+j+k=d then a0,0,d = a0,1,d−1 = 0. Under the embedding determined by L , we write (C, p) in terms of the basis of monomials of the form

Y3 3(d−2) x2 aiα,jα,kα . α=1

Now consider the 1-parameter subgroup   t−1 0 0 λ(t) =  0 1 0  . 0 0 t

P3 Note that λ(t) acts on the monomial above with weight 3(d−2)+ α=1 iα −kα.

We want to know when this weight is negative. Let i, k be the values of iα, kα that minimize iα − kα. Then X3 3(d − 2) + iα − kα ≥ 3(d − 2) + 3(i − k) α=1

32 which is negative whenever i−k−2 < −d = −i−j −k, or 2i+j < 2. This only occurs when both i = 0 and j < 2, in other words, when either a0,0,d or a0,1,d−1 is nonzero. By assumption, however, this is not the case, so (C, p) ∈/ Xs(L ). Since (C, p) was arbitrary, it follows that Xs(L ) = ∅.

9 ss Proposition 3.1.3. Let L = O X (12(d − 4 ), 12). Then X (L ) = X\(H1 ∩ 0 s ss s H2), and X (L ) ⊆ X\H1. In particular, since X (L ) 6= X (L ), L lies on a wall in CG(X).

0 ss 0 Proof. First, notice that if (C, p) ∈/ H2, then (C, p) ∈ X (L ), since W2 is a

0 G-invariant section of L that does not vanish at (C, p). Thus, X\(H1 ∩ H2) ⊆ Xss(L ).

Now, suppose that (C, p) ∈ H1. As before, by change of coordinates, we may assume that p = (0, 0, 1) and the tangent line to C at p is x0 = 0. Since

(C, p) ∈ H1, either p is a singular point of C or this tangent line intersects C at p with multiplicity at least 3. Thus, if we write

X i j k C = ai,j,kx0x1x2, i+j+k=d then a0,0,d = a0,1,d−1 = 0, and either a1,0,d−1 = 0 (if p is singular) or a0,2,d−2 = 0 (if p is a ﬂex). Again, under the embedding determined by L , we write (C, p) in terms of the basis of monomials of the form

Y12 12d−27 x2 aiα,jα,kα . α=1

33 We ﬁrst examine the case where p is a ﬂex. In this case, consider the 1-parameter subgroup:   t−5 0 0 λ(t) =  0 t 0  . 0 0 t4

P12 λ(t) acts on the monomial above with weight 4(12d−27)+ α=1 5iα −jα −4kα. We want to know when this weight is negative. Let i, j, k be the values of iα, jα, kα that minimize 5iα − jα − 4kα. Then X12 4(12d − 27) + 5iα − jα − 4kα α=1 ≥ 4(12d − 27) + 12(5i − j − 4k) = 12(9i + 3j − 9) which is negative when 3i + j < 3. This happens when both i = 0 and j < 3, but, by assumption, this is not the case. Thus (C, p) ∈/ Xs(L ).

Notice furthermore that if the tangent line to C at p is a hyperﬂex, then a0,3,d−3 = 0 as well, and so the expression 3i + j − 3 above is strictly positive (rather than simply nonnegative), and thus (C, p) ∈/ Xss(L ).

Next we look at the case where p is a singular point. Consider the 1-parameter subgroup:   t−1 0 0 λ(t) =  0 t−1 0  . 0 0 t2

P12 λ(t) acts on the monomial above with weight 2(12d−27)+ α=1 iα +jα −2kα. We want to know when this weight is nonpositive. Let i, j, k be the values of iα, jα, kα that minimize iα + jα − 2kα. Then X12 9 2(12d − 27) + i + j − 2k ≥ 2(12d − 27) + 12(i + j − 2k) = 12(3i + 3j − ) α α α 2 α=1

34 3 which is nonpositive when i + j ≤ 2 . This happens when one of i, j is 0 and the other is at most 1. Since p is a singular point of C, however, this is not

ss s the case. Thus (C, p) ∈/ X (L ). It follows that X (L ) ⊆ X\H1.

0 If (C, p) ∈ H1 ∩H2, then either p is a singular point of C, or C is smooth at p and the tangent line to C at p is a hyperﬂex. From our observations

ss 0 above, we may therefore conclude that X (L ) ⊆ X\(H1 ∩ H2). It follows

ss 0 that X (L ) = X\(H1 ∩ H2).

a 9 Corollary 3.1.4. If L = O X (a, b) for any a, b such that d − 2 > b > d − 4 , ss then X (L ) = X\H1.

ss 9 ss Proof. If (C, p) ∈/ H1, then (C, p) ∈ X (O X (12(d − 4 ), 12)) ∩ X (O X (3(d − ss ss 2), 3)). It follows that (C, p) ∈ X (L ), so X\H1 ⊆ X (L ). It remains to

ss show that, if (C, p) ∈ H1, then (C, p) ∈/ X (L ). For this, mirror the proof above with the same 1-parameter subgroups.

We are left to consider the behavior of our quotient as we cross over the

9 wall deﬁned by O X (12(d − 4 ), 12) in the G-ample cone. For this, we will make 9 use of Theorem 2.3.2 and the notation from [18]. Let L (0) = O X (12(d− 4 ), 12), and L +, L − be line bundles lying in the chambers adjacent to L (0) in the

G-ample cone. In particular, L + = O X (a, b) for some a, b such that d − 2 > a 9 9 a b > d − 4 . Similarly, L − = O X (a, b) for some a, b such that d− 4 > b > d − 3. Our ﬁrst task is to determine X− and X0 in this situation.

35 − 0 0 Proposition 3.1.5. With the set-up above, X = H1\(H1 ∩ H2). X is the set of all pointed curves (C, p) with the following description: C is a reducible curve, consisting of a cuspidal cubic plus d − 3 copies of the “tangent line” to C at the cusp (this is the line in P 2 whose unique point of intersection with the cubic is the cusp). The point p is the (unique) smooth ﬂex point of the cuspidal cubic.

ss 0 ss Proof. We have already seen that X (0) = X\(H1 ∩H2) and X (+) = X\H1.

− 0 Thus, X = H1\(H1 ∩ H2).

To prove the statement about X0, let (C, p) ∈ X0. Notice that, since X0 ⊆ X−, p is a smooth point of C and the tangent line to C at p intersects C with multiplicity exactly 3. Since (C, p) ∈/ Xss(−), there must be a nontrivial 1-parameter subgroup λ : C → PSL(3, C) such that λ(t) acts on (C, p) with strictly positive weight. Choose a basis that diagonalizes λ:   tr0 0 0 λ(t) =  0 tr1 0  . 0 0 tr2

By change of basis, we may assume that r0 ≤ r1 ≤ r2. Since the image of λ is contained in PSL(3, C), we know that r0 + r1 + r2 = 0. Furthermore, since λ is nontrivial, we have that r0, r1, r2 are not all zero, which means that r0 < 0 and r2 > 0. Now, writing C in terms of this basis X i j k C = ai,j,kx0x1x2, i+j+k=d we can write (C, p) in terms of the basis consisting of monomials of the form Ya Yb

xlα aiβ ,jβ ,kβ . α=1 β=1

36 Again, we pick values that minimze everything, and reduce to the problem of ﬁguring out whether

arl − b(r0i + r1j + r2k) is negative. Notice that this is the same as asking whether a r < r i + r j + r k. b l 0 1 2 We divide this into cases, depending on p.

Case 1 – p = (0, 0, 1): In this case, rl = r2, so we are trying to

a a 9 determine when b r2 < r0i + r1j + r2k. Recall that b < d − 4 . Now, if a 9 5 b r2 ≥ r0 + (d − 1)r2, then (d − 4 )r2 > r0 + (d − 1)r2, so − 4 r2 > r0, or 1 a 9 r1 > 4 r2. This means that b r2 < (d − 4 )r2 < 3r1 + (d − 3)r2. It follows that a0,0,d = a0,1,d−1 = 0, and either a1,0,d−1 = 0 or a0,2,d−2 = a0,3,d−3 = 0. But we know that p is a smooth point of C and the tangent line to C at p intersects C with multiplicity exactly 3, so neither of these is a possibility.

Case 2 – p lies on the line x0 = 0, but not on the line x1 = 0: In

a this case, rl = r1, so we are trying to determine when b r1 < r0i + r1j + r2k. If a 9 r1 > 0, then since r1 ≤ r2, we have b r1 < (d − 4 )r1 ≤ dr1 ≤ r1j + r2(d − j), so we see that a0,0,d = a0,1,d−1 = ··· = a0,d,0 = 0. This means that p lies on a linear component of C, and therefore (C, p) ∈/ X−.

a On the other hand, if r1 ≤ 0, then since r2 ≥ −2r1, we see that b r1 ≤

(d−3)r1 ≤ (d−1)r1 +r2 ≤ r1j+(d−j)r2 +r2 for j ≤ d−1, so a0,0,d = a0,1,d−1 =

··· = a0,d−1,1 = 0. This means that either p lies on a linear component of C or the only point of C lying on the line x0 = 0 also lies on the line x1 = 0. Given our assumptions, neither of these is possible.

37 Case 3 – p does not lie on the line x0 = 0: In this case, rl = r0,

a so we are trying to determine when b r0 < r0i + r1j + r2k. Since r0 < 0 and a r1 < r0 < r2, we see that b r0 < (d − 3)r0 = (d − 2)r0 + r1 + r2 < r0i + r1j + r2k a for i ≤ d − 2, k 6= 0. Notice that, if b r0 < (d − 1)r0 + r2, then C is of the form X i j C = ai,j,0x0x1. i+j=d In other words, C is d distinct lines. In this case, the tangent line to every point of C is a component of C itself, so (C, p) ∈/ X− for any p. We therefore see that a 9 5 b r0 ≥ (d−1)r0 +r2, so (d− 4 )r0 ≥ (d−1)r0 +r2. But then 4 r0 ≤ −r2 = r0 +r1, a so r0 ≤ 4r1. It follows that b r0 < (d − 3)r0 ≤ (d − 4)r0 + 4r1 ≤ r0i + r1j for j ≥ 4.

We see that C is of the form

d−3 3 2 2 3 2 C = x0 (ad,0,0x0 + ad−1,1,0x0x1 + ad−2,2,0x0x1 + ad−3,3,0x1 + ad−1,0,1x0x2).

Thus, C consists of a cuspidal cubic together with d − 3 copies of the line whose unique point of intersection with the cubic is the cusp. The point p is the unique ﬂex point of the cuspidal cubic.

It is clear that this (C, p) ∈ X−, since the tangent line to C at p intersects C with multiplicity exactly 3. To see that (C, p) ∈/ Xss(−), consider the 1-parameter subgroup   t5 0 0 λ(t) =  0 t−1 0  . 0 0 t−4

Finally, notice that since all cuspidal plane cubics are projectively equivalent, X0 must be the set of all such curves.

38 Corollary 3.1.6. The map Xss(−)//G(−) → Xss(0)//G(0) contracts the lo-

0 cus H1\(H1 ∩H2) to a point. Outside of this locus, the map is an isomorphism.

d−3 2 Proof. We will use Theorem 2.3.2 to establish the claim. Let C = x2 (x0x2 −

3 0 x1), and p = (0, 0, 1). Then (C, p) ∈ X . Since all cuspidal plane cubics are projectively equivalent, G · (C, p) = X0, so G · (C, p) is closed in Xss(0) and X0/G(0) is a point. Notice that the stabilizer of (C, p) must ﬁx the following three points:

1. p = (0, 0, 1)

2. The cusp, which is (1, 0, 0)

3. The intersection of the linear component with the tangent line through p, which is (0, 1, 0).

Thus, the stabilizer of (C, p) must consist solely of diagonal matrices. A quick check shows that the stabilizer of (C, p) is the one-parameter subgroup   t5 0 0 λ(t) =  0 t−1 0  0 0 t−4 which is isomorphic to C∗. The conclusion follows by the theorem of Thaddeus above.

39 We are particularly interested in the case where d = 4, because in this

ss case X (−)//G(−) is a birational model for M 3,1. In particular, we have the following:

ss Proposition 3.1.7. There is a birational contraction β : M 3,1 99K X (−)//G(−).

−1 ss Proof. It suﬃces to exhibit a morphism β : V → M 3,1, where V ⊆ X (−)//G(−) is open with complement of codimension ≥ 2 and β−1 is an isomorphism onto its image. To see this, let U ⊆ Xss(−) be the set of all moduli stable pointed curves (C, p) ∈ Xss(−). Notice that the complement of U is strictly contained in the discriminant locus ∆, which is an irreducible hypersurface in Xss(−). Thus, in the quotient, we have the containment (Xss(−)\U)//G(−) ⊂ ∆//G(−) ⊂ Xss(−)//G(−), and ∆//G(−) is irreducible. Notice, furthermore, that both ∆ and Xss(−)\U are G-invariant, so if (C, p) ∈ Xs(−)\∆ (respectively, (C, p) ∈ Xs(−) ∩ ∆ ∩ U), then the orbit of (C, p) does not intersect ∆ (respectively, Xss(−)\U). Since this point is stable, this means that the image of (C, p) is not contained in ∆//G(−) (respectively, (Xss(−)\U)//G(−)). Thus, the containments (Xss(−)\U)//G(−) ⊂ ∆//G(−) and ∆//G(−) ⊂ Xss(−)//G(−) are strict. It follows that the complement of U has codimension ≥ 2.

By the universal property of the moduli space, since U → Z is a family of moduli stable curves, it admits a unique map U → Z → M 3,1. This map is certainly G-equivariant, so it factors uniquely through a map U//G(−) →

M 3,1. Since every degree 4 plane curve is canonical, two such curves are isomorphic if and only if they diﬀer by an automorphism of P 2. It follows that this map is an isomorphism onto its image.

40 Theorem 3.1.8. There is a birational contraction of M 3,1 contracting the

1 1 1 Weierstrass divisor BN3,(0,3). Furthermore, the divisors BN3,(0,3), BN2 , ∆1 1 and ∆2 span a simplicial face of NE (M 3,1).

ss ss Proof. The composition M 3,1 99K X (−)//G(−) → X (0)//G(0) is a birational contraction. By the above, the Weierstrass divisor is contracted by this

1 map, so it suﬃces to show that BN2 and the ∆i’s are contracted as well. It

1 is known that the image of BN2 is the set of double conics, which is just a point. Similarly, the image of ∆i is contained in the singular locus ∆, which is an irreducible hypersurface. Since the generic point of ∆ is an irreducible nodal curve, the image of ∆i is codimension 2 or greater for all i ≥ 1. The result follows from Theorem 2.2.7.

3.2 Genus 4

We now turn to the case of genus 4 curves. Our main result will be the following:

Theorem 3.2.1. There is a birational contraction of M 4,1 contracting the

1 pointed Brill-Noether divisor BN3,(0,2).

Recall that the general genus 4 curve admits a canonical embedding into P 1 × P 1 as a curve of bidegree (3, 3). In a similar way to the previous section, we will use this observation to construct a birational model for M 4,1.

41 Speciﬁcally, we consider GIT quotients of the universal family over the space of such curves. Here, the Hessian H0,1 is again the image of a pointed Brill- Noether divisor. As above, our goal is to ﬁnd a GIT quotient in which this locus is contracted.

1 1 0 Let Y = P × P and L = O Y (d, d) for d ≥ 3. Here, π2 : X → Z = P H0(P 1 × P 1, O (d, d)) is the family of all curves on P 1 × P 1 of bidegree

∼ 3 (d, d). By the above, we know that P icX = Z . We will write O X (a1, a2, b) to

∗ ∗ 1 1 denote (i◦π1) (O P 1×P 1 (a1, a2))⊗π2(O Z (b)). Recall that G = Aut(P ×P ) = (PSL(2, C) × PSL(2, C)) o Z /2Z ∼= PSO(4, C). Our goal, as before, is to study the GIT quotients of X by the action of this group.

G Proposition 3.2.2. O X (2(d − 1), 2(d − 1), 2) lies on a boundary of C (X).

ss Proof. Let L = O X (2(d − 1), 2(d − 1), 2). It suﬃces to show that X (L ) 6=

s ss X (L ) = ∅. It is clear that X (L ) 6= ∅, since W0,1 is a G-invariant section of L .

To show that Xs(L ) = ∅, we invoke the numerical criterion. Let (C, p) ∈ X. By change of coordinates, we may assume that p = (0, 1; 0, 1). So, if X i0 i1 j0 j1 C = ai0,i0,j0,j1 x0 x1 y0 y1 , i0+i1=j0+j1=d then a0,d,0,d = 0. Under the embedding determined by L , we write (C, p) in terms of the basis of monomials of the form

Y2 2(d−1) 2(d−1) x1 y1 ai0α,i1α,j0α,j1α . α=1

42 Now consider the 1-parameter subgroup µµ ¶ µ ¶¶ t−1 0 t−1 0 λ(t) = , ∈ PSL(2, C) × PSL(2, C). 0 t 0 t

P2 Note that λ(t) acts on the monomial above with weight 4(d − 1) + α=1 i0α − i1α + j0α − j1α. We want to know when this weight is negative. Again, ﬁnd values that minimize this expression. Then

X2 2(d−1) 2(d−1) 4(d−1)+ i0α −i1α +j0α −j1αx1 y1 ≥ 4(d−1)+2((i0 −i1)+(j0 −j1)) α=1 which is negative whenever (i0−i1)+(j0−j1) < −2(d−1) = −i0−i1−j0−j1+2, or i0 + j0 < 1. This only occurs when i0 = j0 = 0, in other words, when a0,d,0,d is nonzero. By assumption, however, this is not the case, so (C, p) ∈/ Xs(L ). Since (C, p) was arbitrary, it follows that Xs(L ) = ∅.

ss Proposition 3.2.3. Let L = O X (2(3d − 4), 2(3d − 4), 6). Then X (L ) =

s ss s X\(H0,1 ∩H1,1), and X (L ) ⊆ X\H0,1. In particular, since X (L ) 6= X (L ), L lies on a wall in CG(X).

ss Proof. First, notice that if (C, p) ∈/ H1,1, then (C, p) ∈ X (L ), since W1,1 is a G-invariant section of L that does not vanish at (C, p). Thus, X\(H0,1 ∩

ss H1,1) ⊆ X (L ).

Now, suppose that (C, p) ∈ H0,1. As before, by change of coordinates, we may assume that p = (0, 1 : 0, 1). Since (C, p) ∈ H0,1, C intersects one of the two lines through p with multiplicity at least 2. Without loss of generality,

43 we may assume that this line to be x0 = 0. Thus, if we write X i0 i1 j0 j1 C = ai0,i1,j0,j1 x0 x1 y0 y1 , i0+i1=j0+j1=d then a0,d,0,d = a0,d,1,d−1 = 0. Under the embedding determined by L , we write (C, p) in terms of the basis of monomials of the form

Y6 2(3d−4) 2(3d−4) x1 y1 ai0α,i1α,j0α,j1α . α=1

Now, consider the 1-parameter subgroup: µµ ¶ µ ¶¶ t−1 0 t−2 0 λ(t) = , ∈ PSL(2, C) × PSL(2, C). 0 t 0 t2

P6 λ(t) acts on the monomial above with weight 6(3d − 4) + α=1 i0α − i1α +

2j0α − 2j1α. We want to know when this weight is negative. Let i0, i1, j0, j1 be the values of i0α, i1α, j0α, j1α that minimize i0α − i1α + 2j0α − 2j1α. Then

X6 6(3d − 4) + i0 − i1 + 2j0 − 2j1 α=1

≥ 6(3d − 4) + 6(i0 − i1 + 2j0 − 2j1) = 12(i0 + 2j0 − 2) which is negative when i0 + 2j0 ≤ 2. This happens when both i0 ≤ 1 and j = 0, but, by assumption, this is not the case. Thus (C, p) ∈/ Xs(L ). We

s therefore see that X (L ) ⊆ X\H0,1.

Notice that if (C, p) ∈ H0,1 ∩ H1,1, this means that the osculating (1, 1) curve to C at p is the pair of lines through that point, and this curve intersects C with multiplicity at least 4. This means that either a0,d,1,d−1 = 0 or a2,d−2,0,d = 0, in addition to the conditions above, which would imply that

44 the expression i0 + 2j0 − 2 above is zero for at most one term, and strictly positive for all of the others. One can therefore slightly change the weights of the 1-parameter subgroup so that all of the weights become strictly positive.

ss ss Hence (C, p) ∈/ X (L ). It follows that X (L ) = X\(H0,1 ∩ H1,1).

a 4 Corollary 3.2.4. If L = O X (a, a, b) for any a, b such that d − 1 > b > d − 3 , ss then X (L ) = X\H0,1.

ss Proof. If (C, p) ∈/ H0,1, then (C, p) ∈ X (O X (2(3d − 4), 2(3d − 4), 6)) and

ss ss (C, p) ∈ X (O X (2(d − 1), 2(d − 1), 2)). It follows that (C, p) ∈ X (L ), so

ss X\H0,1 ⊆ X (L ). It remains to show that, if (C, p) ∈ H0,1, then (C, p) ∈/ Xss(L ). For this, mirror the proof above with the same 1-parameter subgroup.

Again, we want to use Thaddeus’ theorem to study the GIT quotients of the space X. We will let L (0) = O X (2(3d − 4), 2(3d − 4), 6). Let L + and

G L − be line bundles lying in the chambers adjacent to L (0) in the C (X).

a 4 In particular, L + = O X (a, a, b) for some a, b such that d − 1 > b > d − 3 . 4 a 3 Similarly, L − = O X (a, a, b) for some a, b such that d − 3 > b > d − 2 . Our ﬁrst task is to determine X− and X0 in this situation.

− 0 Proposition 3.2.5. With the set-up above, X = H0,1\(H0,1 ∩ H1,1). X is the set of all pointed curves (C, p) admitting the following description: C is a reducible curve consisting of a smooth curve of bidegree (1, 2) (or (2, 1)), together with d − 1 copies of the tangent line to this curve through a point that

45 is tangent to a line, and d − 2 copies of the other line through this same point. The marked point p is the unique other point on the smooth (1, 2) curve that is tangent to a line. (Here, the term “line” means a (0, 1) or (1, 0) curve).

ss ss Proof. We have already seen that X (0) = X\(H0,1 ∩ H1,1) and X (+) =

− X\H0,1. Thus, X = H0,1\(H0,1 ∩ H1,1).

To prove the statement about X0, let (C, p) ∈ X0. Notice that, since X0 ⊆ X−, exactly one of the two lines through p intersects C with multiplicity exactly 2. Since (C, p) ∈/ Xss(−), there must be a nontrivial 1-parameter subgroup λ : C → (PSL(2, C) × PSL(2, C)) o Z /2Z such that λ(t) acts on (C, p) with strictly positive weight. Choose a basis that diagonalizes λ: µµ ¶ µ ¶¶ t−r0 0 t−r1 0 λ(t) = , ∈ PSL(2, C) × PSL(2, C). 0 tr0 0 tr1

By change of basis, we may assume that r0 and r1 are nonnegative, and, using the involution, we may assume that r0 ≤ r1. Since λ is nontrivial, we have that r0 and r1 are not both zero, so r1 > 0. Now, writing C in terms of this basis X i0 i1 j0 j1 C = ai0,i0,j0,j1 x0 x1 y0 y1 , i0+i1=j0+j1=d we can write (C, p) in terms of the basis consisting of monomials of the form Ya Yb

xlα ymα ai0β ,i1β ,j0β ,j1β . α=1 β=1 Again, we pick values that minimze everything, and reduce to the problem of ﬁguring out whether

l m b(r0(i0 − i1) + r1(j0 − j2)) − a((−1) r0 + (−1) r1)

46 is negative. Notice that this is the same as asking whether

a ((−1)lr + (−1)mr ) > r (i − i ) + r (j − j ). b 0 1 0 0 1 1 0 1

We divide this into cases, depending on p.

Case 1 – p = (0, 1 : 0, 1): In this case, l = m = 1, so we are trying to

a a 4 determine when b (−r0 − r1) > r0(i0 − i1) + r1(j0 − j1). Recall that b < d − 3 , a 4 so b (r0 + r1) < (d − 3 )(r0 + r1) < (d − 1)(r0 + r1) ≤ (d − 2)r0 + dr1. Now, if a 4 b (r0 +r1) ≥ dr0 +(d−2)r1, then (d− 3 )(r0 +r1) > dr0 +(d−2)r1, so r1 ≥ 2r0. a 4 This means that b (r0 + r1) < (d − 3 )(r0 + r1) < (d − 4)r0 + dr1. It follows that a0,d,0,d = a1,d−1,0,d = 0, and either a0,d,1,d−1 = 0 or a2,d−2,0,d = 0. But we know that exactly one of the two lines through p intersects C with multiplicity exactly 2, so neither of these is a possibility.

Case 2 – p lies on the line y0 = 0, but not the line x0 = 0: In

a this case, l = 1 and m = 0, so we are trying to determine when b (r0 − r1) > a 4 r0(i0−i1)+r1(j0−j1). Here, b (r0−r1) > (d− 3 )(r0−r1) ≥ d(r0−r1) ≥ kr0−dr1 for all k ≤ d. It follows that ak,d−k,0,d = 0 for all values of k, which means that y0 = 0 is a linear component of C. Thus p lies on a linear component of C, so (C, p) ∈/ X−, which is impossible.

Case 3 – p lies on the line x0 = 0, but not on the line y0 = 0: In

a this case, l = 0 and m = 1, so we are trying to determine when b (−r0 + r1) > a r0(i0 −i1)+r1(j0 −j1). Note that b (−r0 +r1) > (d−2)(−r0 +r1) ≥ −dr0 +kr1 for all k ≤ d − 2. We therefore see that a0,d,k,d−k = 0 for all k ≤ d − 2. If a0,d,d,0 6= 0, then every point of C that lies on the line x0 = 0 also lies on the

47 line y0 = 0, a contradiction. We therefore see that a0,d,d,0 = 0 as well, but this means that p lies on a linear component of C, and therefore (C, p) ∈/ X−.

Case 4 – p does not lie on either of the lines x0 = 0 or y0 = 0:

a In this case, l = m = 0, so we are trying to determine when b (r0 + r1) > a r0(i0 − i1) + r1(j0 − j1). Now note that b (r0 + r1) > (d − 2)(r0 + r1), so ak0,d−k0,k1,d−k1 = 0 if k0 and k1 are both less than d. Furthermore, since r0 < r1,(d − 2)(r0 + r1) > dr0 + (d − 4)r1, so ad,0,k,d−k = 0 for k < d − 1.

3 a Now, if (d − 2 )(r0 + r1) ≤ (d − 6)r0 + dr1, then 3r0 ≤ r1, so b (r0 + r1) > 3 (d − 2 )(r0 + r1) ≥ dr0 + (d − 2)r0. It follows that either ad,0,d−1,1 = 0, in which case C is a product of 2d lines, which is impossible, or ak,d−k,d,0 = 0 for all k < d − 2. We therefore see that C is of the form

d−2 d−1 2 2 2 C = x0 y0 (ad,0,d,0x0y0 + ad,0,d−1,1x0y1 + ad−1,1,d,0x0x1y0 + ad−2,2,d,0x1y0).

Thus, C consists of three components. One is a curve of bidegree (2, 1). The other two components consist of multiple lines through one of the points on this curve that has a tangent line. The point p is forced to be the unique other such point.

It is clear that this (C, p) ∈ X−, since by deﬁnition, one of the lines through p intersects C with multiplicity greater than 1, and it is impossible for it to intersect a smooth curve of bidegree (2, 1) with higher multiplicity than 2, or for the other line through p to intersect the curve with multiplicity at all. To see that (C, p) ∈/ Xss(−), consider the 1-parameter subgroup µµ ¶ µ ¶¶ t−1 0 t−2 0 λ(t) = , . 0 t 0 t2

48 Finally, notice that all such curves are in the same orbit of the action of G, so X0 must be the set of all such curves. To see this, note that if we ﬁx the two points that have tangent lines to be (1, 0 : 1, 0) and (0, 1 : 0, 1), then the curve is determined uniquely by the third point of intersection of the curve with the diagonal. Since PSL(2, C) acts 3-transitively on points of P 1, we obtain the desired result.

Corollary 3.2.6. The map Xss(−)//G(−) → Xss(0)//G(0) contracts the locus H0,1\(H0,1 ∩ H1,1) to a point. Outside of this locus, the map is an isomorphism.

d−2 d−1 2 Proof. Again, we prove this using Theorem 2.3.2. Let C = x1 y1 (x0y1 +

2 0 0 x1y0), and p = (0, 1 : 0, 1). Then (C, p) ∈ X . As we have seen, X is the orbit of (C, p), so G · (C, p) is closed in Xss(0) and X0/G(0) is a point. Notice that the stabilizer of (C, p) must ﬁx p = (0, 1 : 0, 1), and the other ramiﬁcation point, which is (1, 0 : 1, 0). Thus, the stabilizer of (C, p) must consist solely of pairs of diagonal matrices. A quick check shows that the stabilizer of (C, p) is the one-parameter subgroup µµ ¶ µ ¶¶ t−1 0 t−2 0 λ(t) = , 0 t 0 t2 which is isomorphic to C∗. Again, the conclusion follows from the theorem of Thaddeus above.

49 Our main interest is the case where d = 3. As above, this is because in

ss this case X (−)//G(−) is a birational model for M 4,1. In particular, we have the following:

ss Proposition 3.2.7. There is a birational contraction β : M 4,1 99K X (−)//G(−).

−1 Proof. As above, it suﬃces to exhibit a morphism β : V → M 4,1, where V ⊆ Xss(−)/G(−) is open with complement of codimension ≥ 2 and β−1 is an isomorphism onto its image. Again, we let U ⊆ Xss(−) be the set of all moduli stable pointed curves (C, p) ∈ Xss(−). The proof in this case is exactly like that in the case of P 2, as the discriminant locus ∆ ⊆ X is again an irreducible G-invariant hypersurface.

By the universal property of the moduli space, since U → Z is a family of moduli stable curves, it admits a unique map U → Z → M 4,1. This map is certainly G-equivariant, so it factors uniquely through a map U/G(−) → M 4,1. Since every curve of bidegree (3, 3) on P 1 × P 1 is canonical, two such curves are isomorphic if and only if they diﬀer by an automorphism of P 1 × P 1. It follows that this map is an isomorphism onto its image.

Theorem 3.2.8. There is a birational contraction of M 4,1 contracting the

1 pointed Brill-Noether divisor BN3,(0,2). Moreover, if P is the Petri divisor, then 1 1 the divisors BN3,(0,2), P , ∆1, ∆2, and ∆3 span a simplicial face of NE (M 4,1).

ss ss Proof. The composition M 4,1 99K X (−)//G(−) → X (0)//G(0) is a birational contraction. By the above, the given pointed Brill-Noether divisor is

50 contracted by this map, so it suﬃces to show that P and the ∆i’s are contracted as well. The image of P is known to be the set of triple (1, 1) curves, which is just a point. As above, the image of ∆i is contained in the singular locus ∆, which is an irreducible hypersurface. Since the generic point of ∆ is an irreducible nodal curve, the image of ∆i is codimension 2 or greater for all i ≥ 1. The result follows from the contraction theorem (Theorem 2.2.7).

3.3 Genus 5

We now study the case of genus 5 curves. On M 5,1, there are four pointed Brill-Noether divisors. These are:

1 1. BN3 – the pullback of the (non-pointed) Brill Noether divisor on M 5, consisting of limits of trigonal curves;

1 2. BN5,(0,5) – the Weierstrass divisor, consisting of pointed curves (C, p) such that p is a limit of Weierstrass points;

1 1 3. BN4,(0,3) – the divisor of pointed curves (C, p) possessing a limit g4 with ramiﬁcation sequence (0, 3) at p;

2 2 4. BN6,(0,2,4) – the divisor of pointed curves (C, p) possessing a limit g6 with ramiﬁcation sequence (0, 2, 4) at p.

Unlike the previous two sections, in this section we will not construct a birational contraction of M 5,1. Instead, we will construct a rational ﬁbration

51 Figure 3.1: The map φ corresponding to a pointed Brill-Noether divisor. We provide a geometric description of this map here: the general genus 5 curve C admits an embedding into P 4 as the complete intersection of 3 quadrics. For any point p ∈ C, the set of quadrics containing both C and the tangent line to C at p forms a 2-dimensional vector space. Let S be the intersection of the quadrics in this space, which is a del Pezzo surface of degree 4. Now, let H ⊆ P 4 be the osculating hyperplane to C at p, and consider the intersection H ∩ S. Since

TpC ⊂ H ∩ S, we may write H ∩ S as the union of two components TpC ∪ R, where R is generically a twisted cubic in H.

Notice that, since C is a complete intersection of quadrics, it has no trisecants, and thus the intersection multiplicity of C and TpC at p on S must be 2. Since H intersects C at p with order of vanishing 4 or more, we see that

R must be tangent to C. The three curves C, R, and TpC all therefore have the same tangent direction at p. If we blow up S at p, the strict transforms of all three curves will pass through the same point on the exceptional divisor E. If we then blow up again at this point, the new exceptional divisor will be a P 1 with 4 marked points on it – namely, the points of intersection of this P 1

52 with the strict transforms of C, R, E, and TpC (see Figure 3.1).

In this way, we obtain a rational map

φ : M 5,1 99K M 0,4

. Our main result relates this to the natural map

1 f 1 : M 5,1 99K P rojR(M 5,1, mBN ). mBN4,(0,3) 4,(0,3)

In particular:

Theorem 3.3.1. φ = f 1 for some m > 0. mBN4,(0,3)

We divide this into two subsections. The ﬁrst, much like the previous two sections, focuses on semistability arguments and variation of GIT quotients. In the second, we consider how this GIT picture relates to the geometry of M 5,1.

3.3.1 Semistability Arguments

4 Let Y = P and Z = Gr(3, O Y (2)) be the Grassmmannian of 3-dimensional subspaces of the space of quadrics in Y . Since a general genus 5 canonical curve is the complete intersection of 3 quadrics in P 4, the general point in Z corresponds to a genus 5 curve. Now, let

X = {(I C (2), p) ∈ Z × Y |Q(p) = 0 ∀Q ∈ I C (2)}.

As before, we denote the various maps as in the following diagram:

π X i / Z × Y 1 / Y

f π2 Z id / Z

53 Since X is a Grassmmannian bundle over P 4, it is smooth, and P icX ∼= Z ×Z .

∗ ∗ We will write O X (a, b) to denote (i ◦ π1) (O P 4 (a)) ⊗ f (O Z (b)). There is a natural action of Aut(Y ) = PSL(5, C) on X. Our goal is to study the GIT quotients of X by the action of this group.

Notice that, since the general point of X corresponds to a pointed genus

5 curve, there is a rational map X 99K M 5,1. Our ﬁrst step is to calculate the pullback of the pointed Brill-Noether divisors under this map.

Proposition 3.3.2. For every pointed Brill-Noether divisor, there is some a, b

a 3 with b = 2 , such that the pullback of this divisor under the map X 99K M 5,1 is a section of O X (a, b).

Proof. To prove this, we introduce two test curves on X. Let F1 be a ﬁber of a general point in Z under the map f : X → Z. In other words, F1 is obtained by ﬁxing a general genus 5 curve C and varying the marked point. Notice that a section of O X (0, 1) has intersection number 0 with F1, and a section of

O X (1, 0) has intersection number deg(C) = 8 with F1.

Let S be the complete intersection of two general quadric hypersurfaces in Y . Then S is a smooth del Pezzo surface of degree 4. Fix a point p ∈ S and let F2 be a Lefschetz pencil of curves in | − 2KS| through p with marked point p. Since F2 lies entirely inside a ﬁber of the map i ◦ π1 : X → Y , a section of

O X (1, 0) has intersection number 0 with F2. Moreover, since the image of F2 under the map f : X → Z is a line in Z, a section of O X (0, 1) has intersection number 1 with F2. It is clear that the class of a divisor on X is determined uniquely by its intersection with F1 and F2.

54 Notice that the pullback of the (non-pointed) Brill-Noether divisor has intersection number 0 with both F1 and F2, and therefore pulls back to zero under this rational map. Because every pointed Brill-Noether divisor is an eﬀective combination of this divisor and the Weierstrass divisor, we therefore see that the pullbacks of all the pointed Brill-Noether divisors lie on a single ray in NS(X). It thus suﬃces to compute the intersection numbers of the pullback of a single such divisor with F1 and F2. Here we examine the Weierstrass

1 divisor, BN5,(0,5).

The general genus 5 curve possesses 4 · 5 · 6 = 120 Weierstrass points,

1 1 so BN5,(0,5) · F1 = 120. To compute BN5,(0,5) · F2, we use the class of the

Weierstrass divisor in M 5,1:

1 BN5,(0,5) = 15ω − λ + 10δ1 + 6δ2 + 3δ3 + δ4

Since F2 is a Lefschetz pencil, δi = 0 ∀i > 0. The total space of this pencil is the blow-up of S at 16 points, with p being one of these points. It follows

tot gen that the Euler characteristic of the total space F2 is 24. If F is a generic ﬁber of this pencil, then

tot 1 gen χ(F2 ) = χ(P )χ(F2 ) + #{singular ﬁbers}

2 It follows that δ0 = 24 + 2 · 8 = 40. Moreover, K tot = −12, and so κ = F2 −12 + 2 · 2 · 8 = 20. Since λ = 12(κ + δ), we have λ = 5. Finally, ω is the negative self-intersection of the section corresponding to the ﬁxed point p,

tot which is just the exceptional divisor lying over p in F2 . It follows that ω = 1,

1 and so BN5,(0,5) · F2 = 15 − 5 = 10.

55 1 Together, these intersection numbers show that the pullback of BN5,(0,5) is cut out by a section of O X (15, 10). This concludes the proof.

Our next result shows that this ray generates an edge of the G-eﬀective cone.

G Proposition 3.3.3. O X (3, 2) lies on a boundary of C (X).

ss s Proof. Let L = O X (3, 2). It suﬃces to show that X (L ) 6= X (L ) = ∅. It is clear that Xss(L ) 6= ∅, since the pullbacks of all of the pointed Brill-Noether divisors are G-invariant sections of L ⊗n for some n.

To show that Xs(L ) = ∅, we invoke the numerical criterion. Let (C, p) ∈ X. By change of coordinates, we may assume that p = (0, 0, 0, 0, 1). Furthermore, we may assume that the tangent line to C at p is the line x0 = x1 = x2 = 0. Note that the map

0 1 I C (2) → H (P , O P 1 (2) − 2p) given by restricting the quadric to the tangent line TpC is a linear map. Since the codomain has dimension 1, there are at least two linearly independent quadrics in C containing TpC. Again, by change of coordinates, we may assume that the tangent spaces to these two quadrics at p both contain the plane x0 = x1 = 0. In addition, let V ⊆ I C (2) be the subspace of quadrics containing

TpC. Notice that, if Q ∈ V , then the restriction of Q to this plane is the union of TpC and a second line through p. Consider the map

0 2 V → H (P , O P 1 (1) − p)

56 given by restricting this second line to TpC. As above, since the codomain has dimension 1, there must be a quadric in V whose restriction to this plane is a double line. By change of coordinates, we may assume that the tangent space to this quadric at p is the hyperplane x0 = 0. So, if X

C = (ai0,j0,i1,j1,i2,j2 )xi0 xj0 ∧ xi1 xj1 ∧ xi2 xj2 ,

0≤iα≤jα≤4

then ai0,j0,i1,j1,4,4 = ai0,j0,i1,j1,3,4 = ai0,j0,3,3,2,4 = a1,4,2,3,2,4 = 0. In particular, notice that ai0,j0,i1,j1,i2,j2 = 0 if

i0 + j0 + i1 + j1 + i2 + j2 > 15.

Under the embedding determined by L , we write (C, p) in terms of the basis of monomials of the form

Y2 3 x4 ai0α,j0α,i1α,j1α,i2α,j2α . α=1 Now, consider the 1-parameter subgroup   t−2 0 0 0 0  −1   0 t 0 0 0    λ(t) =  0 0 1 0 0  ∈ PSL(5, C).  0 0 0 t 0  0 0 0 0 t2

We see that λ(t) acts on the monomial above with weight 6 + 2(12 − (i0 + j0 + i1 + j1 + i2 + j2)), which is negative when i0 + j0 + i1 + j1 + i2 + j2 > 15. By assumption, this is not the case, so (C, p) ∈/ Xs(L ). Since (C, p) was arbitrary, it follows that Xs(L ) = ∅.

57 We now let L (0) = O X (3, 2),and L − be a line bundle lying in the chamber of CG(X) adjacent to L (0). By Theorem 2.3.1 we know that there is

ss ss a morphism X (L −)//G → X (L (0))//G. We show that φ is equal to the

ss composition of the natural map M 5,1 99K X (L −)//G with this map.

ss ss Proposition 3.3.4. The map X (L −)//G → X (L (0))//G is the natural

ss map X (L −)//G → M 0,4 described above.

ss Proof. It suﬃces to show that, if two points in X (L −) have the same image under the map X 99K M 0,4, then they have the same image under the map

ss ss X (L −)//G → X (L (0))//G. To prove this, we show that both points lie in the same orbit closure. As above, we assume that p = (0, 0, 0, 0, 1), and that C = Q1 ∩ Q2 ∩ Q3, where X Qα = aα,i,jxixj. 0≤i≤j≤4

Also, as above, we assume that aα,i,j = 0 if i+j > 3+α. Note that, by acting on this curve by a diagonal matrix, we may further assume that a3,3,3 = −a3,2,4.

ss We also note that, since (C, p) ∈ X (L −), the terms a1,0,4, a2,1,4, a3,2,4 are all nonzero. This can be veriﬁed using one-parameter subgroups. We can therefore scale all of the quadrics so that a1,0,4 = a2,1,4 = a3,2,4 = 1.

We now determine the image of (C, p) under the map to M 0,4. Notice that the del Pezzo surface S is the intersection Q1 ∩ Q2, and the osculating hyperplane to C at p is cut out by x0 = 0. Taking the intersection of S with this hyperplane and projecting into the the tangent plane x0 = x1 = 0, x4 = 1,

58 we obtain a curve in A2 cut out by the following equation:

2 x2(a1,1,1a1,2,2x2 − a1,1,1a2,2,3a1,1,3x3 + higher order terms)

Simplifying this, we see two components. TpC is cut out by x2 = 0, and R is cut out by

2 a1,2,2x2 − a2,2,3a1,1,3x3 + higher order terms

We obtain the inverse image of R in the blow-up by considering this equation

2 1 along with tx2 = ux3 in A × P . If t 6= 0, we can set t = 1 and substitute x2 = ux3 to obtain:

2 a1,2,2ux3 − a2,2,3a1,1,3x3 + higher order terms

= x3(a1,2,2u − a2,2,3a1,1,3x3 + higher order terms) which intersects the exceptional divisor at the point u = 0. To blow up the

0 resulting surface at this point, we repeat this process and substitute u = u x3 to obtain:

0 x3(a1,2,2u − a2,2,3a1,1,3 + higher order terms) which intersects the exceptional divisor at the point u0 = a2,2,3a1,1,3 . We there- a1,2,2 fore see that, in the coordinates (t0, u0), the points of intersection of this P 1 with the strict transforms of R, E, and the tangent line to C at p are the points (a2,2,3a1,1,3, a1,2,2), (0, 1), and (1, 0) respectively. A similar calculation shows that the strict transform of C intersects the exceptional divisor at the point (1, 1). We now show that every curve with the same ratio a2,2,3a1,1,3 lies a1,2,2 in the same orbit closure.

59 Consider the 1-parameter subgroup from the argument above:   t−2 0 0 0 0  −1   0 t 0 0 0    λ(t) =  0 0 1 0 0  ∈ PSL(5, C).  0 0 0 t 0  0 0 0 0 t2

As t approaches zero, we see that the curve (C, p) limits toward the curve cut out by the following three quadrics:

2 Q1 = x0x4 + a1,1,3x1x3 + a1,2,2x2

Q2 = x1x4 + a2,2,3x2x3

2 Q3 = x2x4 − x3

Now, consider the image of these quadrics under the action of the matrix:   α−1 0 0 0 0    0 α 0 0 0    λ(t) =  0 0 β 0 0  ∈ PSL(5, C).  0 0 0 1 0  0 0 0 0 β−1

The image is

2 3 2 Q1 = x0x4 + α βa1,1,3x1x3 + αβ a1,2,2x2 β2 Q = x x + a x x 2 1 4 α 2,2,3 2 3

2 Q3 = x2x4 − x3

Notice that, if a1,1,3 and a1,2,2 are nonzero, then you can choose values for α

2 2 3 β a2,2,3a1,1,3 and β that make α βa1,1,3 = αβ a1,2,2 = 1. This forces a2,2,3 = . α a1,2,2

60 We therefore see that our curve lies in the same orbit closure as the one cut out by the three quadrics:

2 Q1 = x0x4 + x1x3 + x2

a2,2,3a1,1,3 Q2 = x1x4 + x2x3 a1,2,2 2 Q3 = x2x4 − x3

ss So all curves with the same such ratio are identiﬁed under the map X (L −) →

ss X (L (0))//G. (A similar argument shows this result to hold if either a1,1,3 =

0 or a1,2,2 = 0.) This completes the proof.

ss Proposition 3.3.5. There is a birational contraction p : M 5,1 99K X (−)//G.

−1 ss Proof. It suﬃces to exhibit a morphism p : V → M 5,1, where V ⊆ X (−)/G(−) is open with complement of codimension ≥ 2 and p−1 is an isomorphism onto its image. Let U ⊆ Xss(−) be the set of all moduli stable pointed curves (C, p) ∈ Xss(−). Notice that the complement of U is strictly contained in the locus ∆ of singular curves. Since ∆ is a determinantal variety of expected codimension, it is irreducible. Thus, in the quotient, we have the containment (Xss(−)\U)// ⊂ ∆//G ⊂ Xss(−)//G, and ∆//G is irreducible. Notice, furthermore, that both ∆ and Xss(−)\U are G-invariant, so if (C, p) ∈ Xs(−)\∆ (respectively, (C, p) ∈ Xs(−) ∩ ∆ ∩ U), then the orbit of (C, p) does not intersect ∆ (respectively, Xss(−)\U). Since this point is stable, this means that the image of (C, p) is not contained in ∆//G (respectively, (Xss(−)\U)//G).

61 Thus, the containments (Xss(−)\U)//G ⊂ ∆//G and ∆//G ⊂ Xss(−)//G are strict. It follows that the complement of U has codimension ≥ 2.

By the universal property of the moduli space, since U → Z is a family of moduli stable curves, it admits a unique map U → Z → M 5,1. This map is certainly G-equivariant, so it factors uniquely through a map U//G → M 5,1. Since every smooth complete intersection of 3 quadrics in P 4 is a canonical genus 5 curve, two such curves are isomorphic if and only if they diﬀer by an automorphism of P 4. It follows that this map is an isomorphism onto its image.

This proposition places us in a setting where we may use the results on birational geometry from Section 2.2.2. In particular, we see that the map φ can be expressed as the composition of a birational contraction and a morphism.

3.3.2 Consequences

1 We now consider the three boundary divisors on M 0,4. If p ∈ BN5,(0,5), then the osculating hyperplane to C at p vanishes to order 5 at p. In terms of the intersection product on S, this means that (C · (TpC + R))p = 5. If C is not trigonal, then no line intersects C in three points, so (C · TpC)p = 2. This means that (C · R)p = 3. In other words, the strict transforms of C and R pass through the same point of the exceptional divisor. It follows that the

Weierstrass divisor is contained in the pullback of the point on M 0,4 pictured

62 in Figure 3.2.

1 Suppose that 2p + q + r is a g4 on C, ramiﬁed at p. By Riemmann-

4 Roch, there is a plane in P containing q, r, and the line TpC. This means that the line through q and r intersects the line TpC. Since S contains three points on this line, and S is a complete intersection of quadrics, this line lies

1 on S. It follows that the 5 g4’s on C that are ramiﬁed at p are cut out by divisors on S of the form TpC + L, where L is a −1 curve on S that intersects

1 TpC. If p ∈ BN4,(0,3), then there must be a −1 curve L on S, other than

TpC, that passes through p. We see from this description that R is the union of L and another rational curve passing through p. Since R is singular at p, its inverse image under the ﬁrst blow-up contains the exceptional divisor with multiplicity 2. It follows that this pointed Brill-Noether divisor is contained in the pullback of the point on M 0,4 pictured in Figure 3.3.

2 2 Now, let p ∈ BN6,(0,2,4). By deﬁnition, there exists a g6 D on C with ramiﬁcation sequence (0, 2, 4) at p. It follows that D − 2p and K − D + 2p are

1 both g4’s on C that are ramiﬁed at p. From the description above, we see that

0 0 there are two −1 curves L, L on S such that L + L + 2TpC is a hyperplane

0 section of S. We therefore see that R is the union of L, L , and TpC. Since R contains TpC as a component, this divisor is contained in the pullback of the point pictured in Figure 3.4.

These descriptions of pointed Brill-Noether divisors determine fundamental properties of the map φ. In particular,

Theorem 3.3.6. φ = f 1 for some m > 0. mBN4,(0,3)

63 1 Figure 3.2: Image of the Weierstrass divisor BN5,(0,5)

1 Figure 3.3: Image of the Pointed Brill-Noether Divisor BN4,(0,3)

2 Figure 3.4: Image of the Pointed Brill-Noether Divisor BN6,(0,2,4)

64 1 Proof. By the argument above, we know that BN4,(0,3) is contained in the pullback by φ of a boundary divisor on M 0,4. Let A be the class of a point

∗ 1 on M 0,4, so h (A) = BN4,(0,3) + D, where D is an eﬀective divisor on M 5,1. 1 1 1 1 By [13], we know that BN4,(0,3) = BN5,(0,5) + BN3 , and thus BN4,(0,3) clearly 1 does not lie on an extremal ray of NE (M 5,1). By Corollary 2.2.12, therefore,

1 ∗ we know that BN4,(0,3) is numerically equivalent to a multiple of h (A). The result follows.

1 1 1 Corollary 3.3.7. BN5,(0,5) and BN3 generate extremal rays of NE (M 5,1).

Proof. Note that both divisors are irreducible. Since their sum is numerically equivalent to a multiple of h∗(A), but neither divisor is itself a multiple, we see that both divisors generate extremal rays by Corollary 2.2.12.

2 1 Corollary 3.3.8. BN6,(0,2,4) is numerically equivalent to a multiple of BN4,(0,3).

2 Proof. By [13], we know that BN6,(0,2,4) is numerically equivalent to an eﬀec- 1 1 tive combination of BN5,(0,5) and BN3 . On the other hand, if it is not numer- 1 ically equivalent to a multiple of BN4,(0,3), then by the preceding argument it 2 generates an extremal ray. This would imply that BN6,(0,2,4) is a multiple of 1 1 either BN5,(0,5) or BN3 . But our earlier argument shows that neither of these divisors move, so this is impossible.

3.4 Genus 6

Finally, we consider the case of genus 6 curves. As in the previous section, we will construct a rational ﬁbration of M 6,1 corresponding to a divisor of

65 interest. The map is simple to describe: the general genus 6 curve admits an embedding into a smooth quintic del Pezzo surface Y as a section of | − 2KY |. This embedding is unique up to an automorphism of the surface. By forgetting the curve and simply remembering the marked point, we obtain a rational map

∼ ψ : M 6,1 99K Y/S5 = M 0,5/S5.

Recall that we are interested in the divisor D ⊂ M 6,1 consisting of pointed curves (C, p) that have a degree 6 map C → P 2 that maps p to a singular point. Our main result is that D lies on the boundary of the eﬀective cone of M 6,1. Furthermore, given some assumptions we can relate ψ to the natural map

fmD : M 6,1 99K P rojR(M 6,1, mD).

Theorem 3.4.1. Let P ⊂ M 6,1 be the Petri divisor, and suppose that ψ|P and

ψ|∆i are dominant ∀i ≥ 1. Then ψ = fmD for some m > 0.

In order to establish our results about the map ψ, we must show that it arises as the composition of a birational map with a morphism. If we let

0 H = {(C, p) ∈ P (H (Y, −2KY )) × Y |p ∈ C}, there is then a rational map

f / M 6,1 H/S5 ψ c ' ∼ Y/S5 = M 0,5/S5

66 Proposition 3.4.2. The map f is a birational contraction. The exceptional divisors of f are the Petri divisor P , and the boundary divisors ∆i for i ≥ 1.

Proof. This is probably well known. For the ﬁrst part, it suﬃces to exhibit

−1 a morphism f : V → M 6,1, where V ⊂ H/S5 is open with complement of codimension ≥ 2 and f −1 is an isomorphism onto its image. To see this, let

U ⊂ H/S5 be the set of all moduli stable pointed curves (C, p) ∈ H/S5. Notice that the complement of U is strictly contained in the locus T of singular curves, which is an irreducible hypersurface in H/S5. It follows that the complement of U has codimension ≥ 2.

0 By the universal property of the moduli space, since U → P H (Y, −2KY )/S5 is a family of moduli stable curves, it admits a unique map U → M 6,1. Since the embedding of a genus 6 curve in Y is unique up to an automorphism of Y , two such curves are isomorphic if and only if they diﬀer by an element of

S5. Since the general genus 6 curve possesses no non-trivial automorphisms, it follows that this map is generically injective.

Now, notice that the general element of T has only nodes as singularities. Furthermore, none of these nodes are disconnecting – if they were, then the blow-up of Y at the node would have a disconnected member of | − 2K|, which is impossible. Thus, the image of ∆i for i ≥ 1 has codimension ≥ 2.

2 Furthermore, any element of U admits 5 limit g6’s, given by the 5 blow-downs

2 Y → P . Thus, the Petri divisor is f-exceptional as well. Since H/S5 is a projective bundle over M 0,5/S5, it has Picard number 2, and thus the relative

Picard number of the map f is 6. Since both M 6,1 and H/S5 are Q-factorial,

67 this implies that there are exactly 6 f-exceptional divisors. It follows that these are all of them.

2 The general genus 6 curve has 5 g6’s, corresponding to the 5 blow-downs Y → P 2. From this description, it is clear that a point on a genus 6 curve will

2 map to a node under some g6 if and only if it is contained in a −1 curve on Y .

Thus, if ∆ ⊂ M 0,5/S5 is the boundary divisor, there exists an eﬀective divisor E such that D + E is equivalent to a multiple of ψ∗(∆). Since ψ∗(∆) is not big, we obtain:

1 Proposition 3.4.3. D lies on the boundary of NE (M 6,1).

Ideally, we would like to have the stronger results that D generates an 1 extremal ray of NE (M 6,1) and that ψ is the corresponding rational map. At present, we do not know how to show this. Note, however, that this would follow if we knew that the map ψ is dominant when restricted to the Petri divisor and the boundary divisors ∆i.

Proposition 3.4.4. Suppose that ψ|P and ψ|∆i are dominant ∀i ≥ 1. Then

≥0 ∗ ≥0 P5 ≥0 1 R ψ (∆) + R P + i=1 R ∆i is a simplicial face of NE (M 6,1).

Proof. Since c is a ﬁbration, c∗(∆) is not big. Thus, it lies on the boundary

1 ∗ of NE (H/S5). Since H/S5 has Picard number 2, this means that c (∆) 1 generates an extremal ray of NE (H/S5). By Corollary 2.2.9, therefore, we are done.

68 Theorem 3.4.5. Suppose that ψ|P and ψ|∆i are dominant ∀i ≥ 1. Then D

∗ is equivalent to a positive multiple of ψ (∆). Furthermore, ψ = fmD for some m > 0.

Proof. Since D + E is equivalent to a multiple of ψ∗(∆), and ψ∗(∆) generates an extremal ray, D is equivalent to a positive multiple of ψ∗(∆). Furthermore, since ∆ is ample on M 0,5/S5, we have ψ = fmD for some m > 0.

Corollary 3.4.6. Suppose that ψ|P and ψ|∆i are dominant ∀i ≥ 1. Then

≥0 ≥0 P5 ≥0 1 R D + R P + i=1 R ∆i is a simplicial face of NE (M 6,1).

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71 Vita

David Hay Jensen is originally from Bedford, NH, a small town just outside of Manchester. While there, he attended Derryﬁeld School. He then went to Williams College, where he majored in math and philosophy, and wrote an honors thesis under the supervision of Susan Loepp. He is currently a graduate student in mathematics at the University of Texas at Austin, where he studies algebraic geometry under the direction of Sean Keel.

Permanent address: 620 W 51st St, Apt No. 201, Austin, TX 78751

This dissertation was typeset with LATEX‡ by the author.

‡LATEX is a document preparation system developed by Leslie Lamport as a special version of Donald Knuth’s TEX Program.