Geometric invariant theory and moduli spaces of pointed curves · 2014-11-06 · Geometric invariant theory and moduli spaces of pointed curves David Swinarski Ph.D dissertation Columbia

Geometric invariant theoryand moduli spacesof pointed curves

David Swinarski

Ph.D dissertationColumbia University, 2008

Advisors: Ian Morrisonand Michael Thaddeus

Abstract

The main result of this dissertation is that Hilbert points parametrizing smoothcurves with marked points are GIT-stable with respect to a wide range of linearizations.This is used to construct the coarse moduli spaces of stable weighted pointed curvesMg,A, including the moduli spaces Mg,n of Deligne-Mumford stable pointed curves, aswell as ample line bundles on these spaces.

Contents

Contents i

Acknowledgements iii

1 Preliminaries 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Geometric Invariant Theory (GIT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Weighted pointed curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 An outline of this dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Stability of smooth pointed curves 82.1 Introduction to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The GIT setup for pointed curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 A review of Gieseker’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Why Gieseker’s proof doesn’t cover marked points . . . . . . . . . . . . . . . . . 242.5 The filtration X• and its profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6 The discrepancy between the profile and virtual profile for X• . . . . . . . . . . 392.7 Bounding the weight of the virtual profile T vir . . . . . . . . . . . . . . . . . . . . 472.8 GIT stability of smooth pointed curves . . . . . . . . . . . . . . . . . . . . . . . . 572.9 Additional remarks on the GIT-Stability Theorem . . . . . . . . . . . . . . . . . . 59

3 Constructing moduli spaces 673.1 Statement of the Construction Theorem . . . . . . . . . . . . . . . . . . . . . . . . 673.2 Three lemmas used in the construction . . . . . . . . . . . . . . . . . . . . . . . . 683.3 Smoothness, I, and J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.4 Proof of the Construction Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Polarizations on Mg 794.1 The Polarization Formula for Mg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 Interpreting the Polarization Formula . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 87

A The potential stability theorem 89A.1 Strategy and setup for the Potential Stability Theorem . . . . . . . . . . . . . . . 90A.2 First properties of GIT-semistable curves . . . . . . . . . . . . . . . . . . . . . . . 94A.3 The only singularities allowed are nodes . . . . . . . . . . . . . . . . . . . . . . . 105A.4 GIT-semistable curves are reduced . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.5 The behavior of weighted marked points on a GIT-stable curve . . . . . . . . . 119

B Linearizations for which Jss is closed in Jss124

Index 128

i

Acknowledgements

It is a great pleasure to thank many people for the incredible support they have given

me during my five years at Columbia University.

To my advisors, Ian Morrison and Michael Thaddeus, for giving me many research ideas,

including this project; for your uncountably many suggestions for improving the writing of

this dissertation; and for guiding me through the academic job search;

To Elizabeth Baldwin, for getting me interested in these problems, sharing a great deal

of unpublished work with me, and for our ongoing collaboration, one fruit of which is the

Potential Stability Theorem in the appendix here;

To Brendan Hassett, David Hyeon, Dawei Chen, Frances Kirwan, Julius Ross, Richard

Thomas, Diane Maclagan, Angela Gibney, and Karen Smith, for listening and asking questions

and giving me the encouragement of knowing that people are interested;

To Johan de Jong, whose door seemed always open, for technical advice;

To Dave Bayer, for many discussions on Hilbert schemes, computing, and typography;

To the faculty and staff of the Columbia Math Department, the library staff, and my

defense committee members. In particular, to Robert Friedman, John Morgan, Enrico

Arbarello, Brian Conrad, Johan de Jong, and Jason Starr for teaching excellent geometry and

algebraic geometry classes at Columbia;

To my algebraic geometry classmates at Columbia, Sonja Mapes, Danny Gillam, Matt

Deland, Joe Ross, Thibaut Pugin, Ming-Lun Hsieh, and Xander Faber, for mathematical

assistance and comraderie;

To my family, for your love and support during my years and years as a student;

To my teammates at Front Runners, my fellow chorines at the NYC Gay Men’s Chorus,

and my fellow parishioners at Ascension. The math department kept my mind in shape

these past five years; you, my body and soul;

And to my role models, accomplices, and friends: Matt, Mike, Paul & Greg, Phil & Shawn,

Mark & Rob, Mike, Brendan, Sonja, Mike, and Marc:

Thank you with all my heart.

iii

Chapter 1

Preliminaries1.1 Introduction

The main goal of this dissertation is to give geometric invariant theory constructions of

moduli spaces of Deligne-Mumford stable weighted pointed curves.

The notion that a compact orientable topological surface of genus g ≥ 1 may be endowed

with infinitely many different complex analytic structures goes back to the nineteenth

century. In the first edition of Geometric Invariant Theory (1965), Mumford showed that the

set of isomorphism classes of smooth curves of genus g has the structure of an algebraic

variety. (In fancier language, there is a quasiprojective orbifold Mg which coarsely represents

the moduli functor of smooth curves of genus g.)

The idea behind Mumford’s construction is easy to describe: Let C be a smooth curve of

genus g ≥ 1, and let K be its canonical divisor. Then νK is very ample for any integer ν ≥ 3,

and a choice of basis for H0(C, νK) determines an embedding C ⊂ PN . It is relatively easy

to find an algebraic variety J parametrizing all ν-canonically embedded smooth curves of

genus g in PN . Then, by identifying points which represent the same curve with different

embeddings (that is, by forming the quotient J/PGL(N + 1)), one expects to obtain Mg.

Making this rigorous requires a great deal of work; constructing the moduli space of smooth

curves was a primary motivation for the development of geometric invariant theory (GIT).

In the 1960s Mumford, Mayer, and Deligne discovered an especially nice compactification

of the moduli space of smooth curves. Denoted Mg, this moduli space also includes

nodal curves of genus g with finite automorphism groups. Notably, they did not find this

compactification by GIT; another ten years passed before Mumford and Gieseker gave GIT

constructions of Mg .

The past thirty years have seen the introduction of many more moduli spaces as well as

new techniques for constructing them. Some of the most important new spaces include Mg,n,

1

2 CHAPTER 1. PRELIMINARIES

the space of Deligne-Mumford stable pointed curves; Mg,n(X, β), the Kontsevich spaces

of stable maps; and versions of these spaces, due to Hassett and many others, where the

marked points are allowed to collide in certain configurations. As regards techniques, stacks

(already employed by Deligne and Mumford) have gained more common usage in algebraic

geometry, and Kollár pioneered a powerful new approach by which one constructs the

moduli space first as an algebraic space and then constructs an ample line bundle on it (by

semipositivity results, for instance), showing that the moduli space is actually a projective

scheme.

With all this activity, it seems surprising that no GIT construction of Mg,n appeared

in the literature until Elizabeth Baldwin’s 2006 Oxford D.Phil dissertation. (She gave a

GIT construction of Mg,n(Pr , d); this includes Mg,n as the special case of maps to a point.)

The novel part of her construction is proving that pointed smooth curves (or maps whose

domains are pointed smooth curves) are GIT-stable. Her argument is a delicate induction on

g and the number of marked points n; elliptic tails are glued to the marked points one by

one, ultimately relating GIT-stability of an n-pointed genus g curve to Gieseker’s result for

genus g + n unpointed curves.

We hoped to improve upon Mumford’s, Gieseker’s, and Baldwin’s results in three ways.

First, we wanted to construct moduli spaces of weighted pointed curves; it appears that

Baldwin’s proof can accommodate some, but not all, sets of weights. Second, we want to

study quotients in situations where Gieseker’s and Baldwin’s proofs don’t apply. (Specifically,

in terminology introduced in the next section, we are interested in linearizations not covered

by Gieseker’s and Baldwin’s proofs.) Third, we are also interested in 2- and 3-canonical

linear systems. The Log Minimal Model Program for Mg, begun by Hassett and Hyeon in

[HH], has generated interest in these quotients and their pointed analogues. Gieseker’s

proof works in these degrees for smooth unpointed curves, but due to its use of elliptic

tails, Baldwin’s proof cannot be used to study the analogous pointed curves, as elliptic tails

are known to be GIT-unstable in these cases.

In the next two sections we give some background on GIT and weighted pointed curves.

Section 1.4 is an outline of this dissertation.

1.2. GEOMETRIC INVARIANT THEORY (GIT) 3

1.2 Geometric Invariant Theory (GIT)

Forming quotients in algebraic geometry requires more machinery than in the topological

or smooth categories. If a scheme X is acted on by an algebraic group G, one must take

care to ensure that the quotient X/G is also a scheme and that the quotient map X → G is

a morphism. Mumford’s geometric invariant theory provides conditions under which one

obtains a quotient morphism.

Here is a brief summary of GIT. Details and references will be given in later chapters as

needed:

In GIT, two ingredients are needed to form a quotient: a parameter space with group

action as well as a linearization of the group action (a lifting of the group action to sections of

a line bundle). In this dissertation, the parameter space X will always be a projective scheme,

and the line bundle L we linearize will always be ample. Then the quotient scheme, denoted

X//LG, is just Proj(R), where R :=⊕

n∈N H0(X, L⊗n)G is the ring of invariant sections of

positive powers of L. There is a rational map from X to X//LG. Points x ∈ X for which

there exists a nonvanishing invariant section are called GIT-semistable; the quotient map is

actually a morphism at such points. If the orbit of x is closed and the stabilizer Gx is finite,

then x is called GIT-stable. Mumford gave a numerical criterion which equates GIT-stability

with respect to the whole group G to GIT-stability for every one parameter subgroup (1-PS)

in G. He also defined a function µL(x, λ) which measures GIT-stability for 1-PS λ, in the

sense that

µL(x, λ) ≤ 0 a x is λ-semistable

µL(x, λ) < 0 a x is λ-stable

µL(x, λ) > 0 a x is λ-unstable.

Thaddeus [Thaddeus] and Dolgachev and Hu [DH] studied the effect of changing the

linearization for a fixed parameter space X. This theory is often abbreviated VGIT, for

variation of GIT. Assume X is normal and irreducible. The space of linearizations is divided

into chambers. If L1 and L2 lie within the same chamber, then the quotients X//L1 G and

X//L2 G are isomorphic. If L1 and L2 lie in adjacent chambers, then the quotients X//L1 G


and X//L2 G are birational to each other. Specifically, they are related by a flip (a blowup

followed by a blowdown).

1.3 Weighted pointed curves

We now give the definition of DM-stable weighted pointed curves, the main objects of

interest in this dissertation.

Definition 1.3.1. A genus g weighted pointed curve (C, P1, . . . , Pn,A) over an algebraically

closed field k consists of the following:

• a reduced connected projective algebraic curve C over k of genus g with at worst nodes

as singularities;

• closed points P1, . . . , Pn, which lie on C and are ordered (note we do not require that

they be distinct, nor that they be smooth points of C);

• an ordered n-tuple A= (a1, . . . , an) of rational numbers ai with 0 ≤ ai ≤ 1 for all i.

We say a weighted pointed curve (C, P1, . . . , Pn,A) is DM-stable if

• ai = 0 if Pi is a node;

• if a subset of the marked points Pi : i ∈ I ⊂ [1..n] coincide, then∑

I ai ≤ 1;

• the Q-line bundle ω(∑

aiPi) is ample on C.

Remark. The adjective “DM-stable” is not historically accurate when applied to weighted

pointed curves (Deligne and Mumford studied unpointed curves; Mumford and Knudsen

studied pointed curves, but did not consider weights), but the descriptive power of this

terminology seems to justify the abuse. We will similarly refer to DM-stable maps (instead

of Kontsevich stable maps, or stable maps) for uniformity of language.

Hassett introduced DM-stable weighted pointed curves in [Hass]; the theory has been

extended to maps by several people ([BM], [AG], [MM]). Note that Alexeev and Guy in [AG]

allow ai ∈ R, but this makes no important difference.

1.3. WEIGHTED POINTED CURVES 5

In 21st century algebraic geometry, moduli problems are described by functors. That

machinery is not needed for the results of this dissertation, and we shall therefore avoid it,

except in recalling the important results summarized below:

Theorem 1.3.2 ([Hass] Theorem 2.1, [Hass] §2.1.1, [Hass] Theorem 4.1).

1. For any g and set of weights A with 2g − 2+ a > 0, there is a connected stack Mg,A,

smooth and projective over Spec Z which represents this moduli problem and has a

projective coarse moduli space Mg,A.

2. If ai > 0 for all i, then the moduli stack Mg,A is smooth, its boundary is a normal

crossings divisor, and its coarse moduli space has only finite quotient singularities.

3. If some of the weights are zero, then the moduli space is a fibered power of simpler

ones: Say a1, . . . , ak > 0 while ak+1, . . . , an = 0. Write ` = n− k and write A′ for the set

of weights (a1, . . . , ak), and write Cg,A′ for the universal family over Mg,A′ . Then

Mg,A = Cg,A′ ×Mg,A′· · · ×Mg,A′

Cg,A′︸︷︷︸` times

.

In general, when some of the weights in A are zero, the moduli space Mg,A is singular.

4. (“Wall crossing.”) Fix g and n and suppose A= (a1, . . . , an) and B = (b1, . . . , bn) satisfy

bi ≤ ai for each i = 1, . . . , n. Then there exists a natural birational morphism

Mg,A →Mg,B.

The image of [C, P1, . . . , Pn,A] under this map is the isomorphism class of the weighted

pointed curve obtained by contracting any components of C on which ωC(∑

biPi) is

not ample.

These results will not be used in any proofs in this dissertation; they are stated here

because they provide our motivation for studying the spaces Mg,A. Property 4 above tells us

that the spaces Mg,A are all birational to Mg,n, and Properties 3 and 4 together tell us that

we can think of this collection of spaces as interpolating between Mg,n and the fibered power

Cg ×Mg · · · ×Mg Cg . Hence, a major motivation to study the spaces Mg,A is for applications

to the birational geometry of Mg,n.


Hassett’s results raise many interesting questions: Is it possible to construct the moduli

spaces by GIT? (Hassett’s construction of these spaces follows Kollár’s program, not GIT.)

If they can be constructed by GIT, what spaces will appear as the linearization varies? Is

there a way to get the entire collection of moduli spaces by variation of GIT? Which ample

line bundles on these spaces arise from GIT, and how do they relate to the semipositivity

line bundle? In this dissertation we answer the first question affirmatively; I am actively

researching the others.

As mentioned above, it is also possible to define moduli spaces of DM-stable weighted

pointed maps, and using these one may define weighted Gromov-Witten invariants. It would

be interesting to know what new information, if any, these may contain beyond the usual

Gromov-Witten invariants.

1.4 An outline of this dissertation

We seek to construct the moduli spaces of DM-stable weighted pointed curves Mg,A via

GIT. To do this, we must describe parameter spaces J whose points parametrize embedded

DM-stable weighted pointed curves (C ⊂ PN , P1, . . . , Pn,A), and then describe linearizations

for which these points coincide with the GIT-stable locus.

In the existing GIT constructions of moduli spaces of DM-stable curves and maps

([Mum3], [Gies], [BS]), no one has ever yet shown directly that nodal DM-stable curves or

maps are GIT-stable. Instead, all proofs to date proceed indirectly via the following strategy,

which we outline informally as follows: First, one shows that the smooth objects of the

moduli problem are GIT-stable. Second, one shows that anything in the GIT-semistable locus

is DM-stable, or nearly so. Finally, to prove that the nodal objects of the moduli problem

are GIT-stable, one relates their GIT-stability to the GIT-stability of the smooth objects via a

deformation argument.

The construction of Mg,A given here also follows this strategy, and this is reflected in

the structure of this dissertation.

In Chapter 2, we prove that smooth pointed curves are GIT-stable, for a wide range of

parameter spaces and linearizations. I refer to the main theorem of this chapter (Theorem 2)

as the GIT-Stability Theorem. Note that throughout Chapter 2 we study smooth embedded

1.4. AN OUTLINE OF THIS DISSERTATION 7

pointed curves (C ⊂ PN , P1, . . . , Pn) and ignore the weights; this may seem strange, but we

will see later that the main theorem is sufficiently flexible to accommodate weights and

meets our needs in constructing Mg,A. I consider the proof of the GIT-Stability Theorem

the major achievement of this dissertation, and for this reason I have included a lengthy

introduction in Chapter 2 as well as a section of additional remarks at the end of the chapter.

The second step of the strategy outlined above, that GIT-semistable points of the

parameter space are nearly DM-stable, is covered by a result known as the Potential Stability

Theorem. The version needed for Mg,A is derived from the paper [BS], which was written

jointly by Elizabeth Baldwin and me; for this reason, I have included this material as an

appendix rather than in the main body. The second appendix chapter includes the proof

(also derived from [BS]) that Jss , the GIT-semistable locus of the quasiprojective parameter

space used to construct Mg,A, is closed in Iss , the GIT-semistable locus of a larger projective

scheme I containing J.

In Chapter 3 we construct the moduli spaces Mg,A. I refer to Theorem 3.1.2, the main

result of this chapter, as the Construction Theorem: Given g, n, and A, there exists a

parameter space J and a linearization on it such that the GIT quotient J//SL(N + 1) is

isomorphic to Mg,A. This chapter also includes three lemmas used in the construction;

they are in the literature, so I have given outlines of their proofs. We also prove that the

parameter space J is reduced and smooth, and that more generally the Hilbert scheme is

smooth at points parametrizing nonspecial l.c.i. curves. Although these smoothness results

are not used anywhere in the dissertation, it is reassuring to know that the parameter spaces

we use are well-behaved.

As discussed above, a GIT quotient of a projective scheme carries a natural polarization.

In Chapter 4 we present a formula in terms of standard classes on Mg for the polarizations

on which arise from GIT constructions for different parameter spaces and linearizations.

Finally, we explore the implications of this formula for the role of GIT in studying the ample

cones of moduli spaces.

Chapter 2

Stability of smooth pointed curves2.1 Introduction to Chapter 2

In the 1960s Mumford established Chow stability of smooth unpointed genus g curves

embedded by complete linear systems of degree d ≥ 3g. In the late 1970s Gieseker

established asymptotic Hilbert stability of smooth curves under complete linear systems

of degree d ≥ 2g + 1, and Mumford used Gieseker’s ideas to improve his Chow result to

d ≥ 2g + 1. (Here, the word “asymptotic” describes the linearizations under consideration;

these will be described in detail in Section numerical criterion.) Both of them then use

an indirect argument to show that nodal DM-stable curves are GIT-stable. The first GIT

construction of Mg,n was in 2006, due to Baldwin. Her argument is inductive; elliptic tails

are glued to the marked points one by one, ultimately relating GIT-stability of an n-pointed

genus g curve to Gieseker’s result for genus g + n unpointed curves.

In this chapter I give a direct proof that smooth curves with marked points are GIT-stable

with respect to a wide range of parameter spaces and linearizations. In later chapters we

will show that some of these yield the coarse moduli spaces of DM-stable weighted pointed

curves Mg,n and Mg,A; the GIT-stability result of this chapter is the key new result needed

for the construction. The proof presented here applies to 2- or 3-canonical linear systems.

It is logically independent of Gieseker’s, though many of the ideas used are inspired by his

work, and in the unpointed case it works for approximately the same linearizations.

My approach follows Gieseker’s in reducing the GIT problem to a combinatorial problem,

though the solution is very different from his. Here is a description of my approach:

Let x be a point parametrizing an embedded smooth pointed curve (C ⊂ PN , P1, . . . , Pn).

We reformulate the numerical criterion in a way that permits a more combinatorial approach.

A 1-PS λ of SL(N+1) induces a weighted filtration of H0(C,O(1)) and a weighted filtration of

H0(C,O(m)). The value of Mumford’s function µL(x, λ) may be interpreted as the “minimum

8

2.1. INTRODUCTION TO CHAPTER 2 9

weight of a basis of H0(C,O(m)) compatible with this filtration plus a contribution from

the marked points.” (From now on, whenever we refer to a basis of H0(C,O(m)), we always

implicitly mean one that is compatible with the weighted filtration.) The numerical criterion

says that if µL(x, λ) is sufficiently small, then x is GIT-stable with respect to λ. Any basis

therefore gives an upper bound for µL(x, λ), so the goal becomes: find a basis of sufficiently

small weight.

Our main tool for computing (a bound for) the weight of a basis is something I call a

profile. This is a graph which may be associated to any filtration of a vector space such that

the weight decreases at each stage. Suppose F• is such a filtration of H0(C,O(m)). (I use

tildes for filtrations of H0(C,O(m)); no tilde indicates a filtration of H0(C,O(1)).) Suppose

the weight on the kth stage of F• is rk. Then the profile associated to F• is just the decreasing

step function in the first quadrant of the (codimension×weight)-plane whose value is rk

over the interval [codim Fk, codim Fk+1). Given any profile, it is possible to choose a basis

whose weight is less than or equal to the area under the profile.

There is a notion of an absolute weight filtration on H0(C,O(m)) (see Section 2.2); the

area under its profile is the minimum weight of a basis, and hence computes Mumford’s

function µL(x, λ). This is perhaps the most natural filtration to consider, but it is too

difficult to compute. So, like Gieseker, we study other filtrations.

The action of a 1-PS λ induces a filtration V• of H0(C,O(1)). By considering specific

spaces of degree m monomials in elements of V diagonalizing the λ-action, Gieseker

produces a very straightforward filtration V• of H0(C,O(m)) as well as a second, slightly

fancier filtration G•. Gieseker is able to show that the weight (or area) associated to G• is

sufficiently small to establish λ-stability of smooth unpointed curves. Unfortunately, as we

show with a concrete example, the analogue of G• is not sufficient to establish λ-stability

when there are marked points.

One could try to improve G•, but it is too difficult (at least for me) to show that the

sum of its area and the marked points contribution is sufficiently small. Therefore I use

V• as a starting point to build a new filtration, X•, which is obtained by taking spans of

carefully chosen spaces of monomials. In a convenient (though not rigorous) soundbite:

where Gieseker uses monomials, my proof uses polynomials. The recipe for X• is given in

10 CHAPTER 2. STABILITY OF SMOOTH POINTED CURVES

terms of the combinatorics of the base loci of the stages of the filtration V•. Although X•

is rather tedious to define, it has the virtue that we can bound the sum of its area and the

marked points contribution sufficiently well to show that smooth curves with marked points

are GIT-stable. The key new ingredients in my proof are the definition/choice of X•; an

easy but important lemma (Lemma 2.4.2) which allows us to compute spans of spaces of

monomials in the Vj ’s using multiplicities of points in the base loci; and the combinatorial

argument (see the proof of Lemma 2.7.1) which allows us to effectively bound the sum of

the marked points contribution and the area of the profile associated to X•.

Gieseker’s proof establishes stability for smooth unpointed curves embedded by com-

plete linear systems of degree d ≥ 2g + 1. (There are some misleadingly placed hypotheses

in [Gies], but one can check that everything works with the hypotheses just mentioned.) At

the present time it is necessary for me to make the hypotheses:

• If n = 0, the parameter space satisfies N ≥ 2g − 2, or, equivalently, d ≥ 3g − 3.

• If n ≥ 1, then either the parameter space satisfies N ≥ 2g − 1, or else the linearization

satisfies the following condition (the notation is explained in Section 2.2): b > g−1N .

Here is an outline of the chapter: in Section 2.2 I describe the GIT problem carefully,

specifying the parameter spaces and linearizations we will consider, and reformulate the

numerical criterion in the form in which we shall use it. Profiles are also defined here. In

Section 2.3 I review Gieseker’s proof, with a few enhancements, to fix notation; a reader

familiar with Gieseker’s proof should be able to read it very quickly. In Section 2.4 I give an

example showing why his proof does not suffice for marked points, and a hint illustrating

how we will go about fixing it.

Throughout Sections 2.2–2.4 we steadily extract combinatorial data from the algebro-

geometric action of a 1-PS λ acting on the Hilbert point of a smooth pointed curve. The last

result of this type is Lemma 2.4.2, which allows us to compute codimensions of spans of

monomial-type sublinear series of H0(C,O(m)) using only the multiplicities of points in

the base loci. After this, the problem becomes almost entirely combinatorial.

In Section 2.5, I produce the filtration X• on H0(C,O(m)) which is built using the

filtration V• as scaffolding. The goal is now to show that the area under the profile for

2.1. INTRODUCTION TO CHAPTER 2 11

X• plus the contribution from the marked points is less than the bound specified by the

numerical criterion.

This is established in two steps: first, I describe a second, simpler graph called the

virtual profile which is bounded above by the profile for X•. Basically it is the graph of the

piecewise linear function connecting the left endpoints of the steps in the weight profile.

(I’m oversimplifying things a little here—I’m glossing over some rounding errors.) The

virtual profile is not really the profile of any filtration, nor does it compute or bound the

weight of a basis; the most rigorous interpretation I have for it is on the level of graphs.

Again, while it is easy to compute the area of the profile (it’s a step function, after all!), when

it is time to add the contribution from the marked points, it is easier to do this with the

virtual profile than with the profile. In Section 2.6 I bound the discrepancy between the

areas of the two graphs and show that this is relatively small when m is large. Then in

Section 2.7 I bound the sum of the area under the virtual profile for X• and the weight from

the marked points. Everything comes together in Section 2.8 to show that smooth pointed

curves with marked points have GIT stable Hilbert points.

Finally, this chapter concludes with a section of additional remarks giving more de-

tailed comparisons to Gieseker’s and Baldwin’s proofs, as well as discussing two proposed

strategies for improving the main result.

Here is a picture illustrating the profile and virtual profile associated to an example

that is explained in detail in Section 2.5. Note that I will always fill in the graphs of step

functions to obtain staircase figures.

Figure 2.1: Example profile and virtual profile

Codimension

Weight

In summary:


action of one 1-PS λ on a smooth pointed curve⇓

a filtration V• of H0(C,O(1)) and a filtration V• of H0(C,O(m))⇓

another filtration X• of H0(C,O(m))and two graphs associated to X• (a profile and a virtual profile)

⇓a basis of H0(C,O(m)) of small weight

⇓a bound for µL(x, λ)

⇓stability of the smooth pointed curve with respect to λ

Two remarks on notation here may prevent alarm for those readers skimming the proof:

Note that from Section 2.5 onward it may appear at times as though we are using

rational numbers as exponents of monomials. Although the resulting “virtual” spaces are

usually nonsensical, in cases where they do make sense they are useful in motivating some

definitions and calculations. However, such spaces are never used to produce basis elements

in H0(C,O(m)); to get basis elements, we always round exponents.

Also, we work with two-dimensional arrays of integers cj,i . That is, j indexes the row,

and i indexes the column, opposite the usual alphabetic convention. The reason for this

is that we always use the index i for the marked points Pi , and these correspond to the

columns.

2.2 The GIT setup for pointed curves

The parameter spaces and linearizations we use

In this chapter we investigate GIT-stability for the following general setup. Let P(t) :=

dt − g + 1 be a degree one polynomial. We form the incidence locus

I ⊂ Hilb(PN , P(t))×∏n

i=1 PN where the points in the projective space factors lie on the

curve in PN parametrized by the point in the first factor. We study the GIT stability of

points of I. Note two things: no sets of weights A appear in this paragraph; we will see

in Section 3.1 that considering weighted marked points influences the choice of d, but

weights play no direct role in the GIT-stability proof. Also, we do not assume that C ⊂ PN is

2.2. THE GIT SETUP FOR POINTED CURVES 13

pluricanonically embedded, or even that the degree of C ⊂ PN matches the degree of the

pluricanonical embedding— we can investigate GIT-stability for more general setups than

just those which are obviously useful for constructing moduli spaces of curves. All we need

is that the embedding C ⊂ PN is by a complete linear system, and (possibly) some precise

degree/dimension bounds in terms of the genus, which will be carefully stated at the end in

Theorem 2.8.1.

To do GIT, one must specify a linearization on the G-space (here, I). Although not neces-

sary, perhaps the easiest way to do this is to embed Hilb(PN , P(t))×∏n

i=1 PN equivariantly

in a high-dimensional projective space and use its O(1).

Let C ⊂ PN be a subscheme with Hilbert polynomial P(t). For sufficiently large m, m′i ,

the mapsevm

C : H0(PN ,O(m)) → H0(C,OC(m))

evm′

iPi

: H0(PN ,O(m′i )) → H0(Pi ,OPi (m′

i )) C

are surjective. The first map gives rise to an embedding of the Hilbert scheme in a Grass-

mannian, which in turn embeds in a projective space by the Plücker embedding. The maps

in the second line correspond to m′i -uple embeddings of PN . Finally, a Segre embedding of

all these projective spaces yields an embedding of Hilb(PN , P(t))×∏n

i=1 PN into a very large

projective space, as desired.

Now, to specify a linearization on I ⊂ Hilb(PN , P(t))×∏n

i=1 PN , it suffices to specify the

ratios between m and each m′i . I will do this as follows: let B = (b1, . . . , bn) ∈ Qn ∩ [0, 1]n

be a set of weights, which I call the linearizing weights. Then set m′i = bim2. Finally, write

b :=∑n

i=1 bi .

The numerical criterion for our setup

By being a little more explicit, we obtain a useful reformulation of the numerical criterion.

In Gieseker’s paper and this paper we use Grothendieck’s convention that if V is a vector

space, then P(V ) is the collection of equivalence classes under scalar action of the nonzero

elements of the dual space V∨. One consequence of this convention is that the numerical

criterion takes the opposite sign from that in [GIT].

Let X be a projective algebraic scheme with the action of a group G linearized on a very

ample line bundle L. Let λ : Gm → G be a 1-PS of G. Choose a basis e0, . . . , eN of H0(X, L)


diagonalizing the λ action and ordered so that the weights r0 ≤ · · · ≤ rN ∈ Z increase. The

weights on the dual basis then have the opposite signs: −r0, . . . ,−rN .

A point x ∈ X is represented by some nonzero x =∑N

i=0 xie∨i ∈ H0(X, L)∨. Define

µL(x, λ) := minri|xi ≠ 0.

Then, with our sign conventions, we have the following characterization of GIT-stability:

Theorem 2.2.1 (cf. [GIT] Theorem 2.1).

x ∈ Xss(L) ⇐⇒ µL(x, λ) ≤ 0 for all 1-PS λ ≠ 0

x ∈ Xs(L) ⇐⇒ µL(x, λ) < 0 for all 1-PS λ ≠ 0.

In our situation X is the incidence scheme I, the point x ∈ X parametrizes an embedded

pointed curve (C ⊂ PN , P1, . . . , Pn), the scheme I is embedded in

P(∧P(m) Symm V ⊗

⊗ni=1 Symm′

i V ) where V = H0(PN ,O(1)), and L is the O(1) on this very

large projective space. Let λ be a 1-PS of SL(V ). One particularly nice basis of∧P(m) Symm V ⊗⊗n

i=1 Symm′i V is given by elements of the form

(M1 ∧ · · · ∧MP(m))⊗ (M′1)⊗ · · · ⊗ (M′

n), (2.1)

where each Mj is a monomial of degree m and each M′i is a monomial of degree m′

i in the

basis elements of V diagonalizing λ.

The numerical criterion may be translated as follows: a point of I is stable with respect

to λ if and only if there is a basis element of the form (2.1) such that

1. the images of the M` under the evalution map form a basis of H0(C,OC(m)),

2. M′i does not vanish at Pi ,

3. the SL(N + 1) weights satisfy

P(m)∑`=1

wtλ(M`)+n∑

i=1wtλ(M′

i ) < 0

In fact, it will be convenient to normalize the λ weights so that they decrease to 0 and

sum to 1. If sN , . . . , s0 are the original weights, (so sN ≥ · · · ≥ s0 and∑

sj = 0), then the


desired transformation is rj = (sN−j − s0)/((N + 1)|s0|). Also, we write

A :=P(m)∑`=1

wtλ(M`)

T :=P(m)∑`=1

wtλ(M`)+n∑

i=1wtλ(M′

i )

for parts of the left hand side of condition 3. above. We may rewrite condition 3. as follows.

Lemma 2.2.2. Condition 3. above with the unnormalized weights sj is equivalent to the

following condition:

3.′ With the normalized weights rj , the following inequality is satisfied:

T :=P(m)∑`=1

wtλ(M`)+n∑

i=1wtλ(M′

i ) <(

1+ g − 1N + 1

)m2 + 1

N + 1

n∑i=1

m′i −

g − 1N + 1

m

=(

1+ g − 1+ bN + 1

)m2 − g − 1

N + 1m. (2.2)

Proof. Suppose that we have the required collection of monomials satisfying

P(m)∑`=1

wtλ(M`)+n∑

i=1wtλ(M′

i ) < 0

with the weights sj . Let w0, . . . , wN be a basis of H0(C,O(1)) diagonalizing the λ action. If

M` = wf`,00 · · ·w

f`,NN , then wtλ(M`) =

∑Nj=0 f`,j sj .

Let j(i) be the function whose value for each i = 1, . . . , n is the largest index (hence giving

the smallest weight) such that the section wj(i) does not vanish at Pi . Then wtλ(M′i ) = m′

i sj(i).

Thus condition 3. may be rewritten

P(m)∑`=1

N∑j=0

f`,j sj +n∑

i=1m′

i sj(i) < 0

aP(m)∑`=1

N∑j=0

f`,N−j ((N + 1)|s0|rj + s0)+n∑

i=1m′

i ((N + 1)|s0|rN−j(i) + s0) < 0.

We proceed to divide by |s0|. Note that our conventions imply that s0 < 0:

P(m)∑`=1

N∑j=0

f`,N−j ((N + 1)rj − 1)+n∑

i=1m′

i ((N + 1)rN−j(i) − 1) < 0

a (N + 1)P(m)∑`=1

N∑j=0

f`,N−j rj −P(m)∑`=1

N∑j=0

f`,N−j + (N + 1)n∑

i=1m′

i rN−j(i) −n∑

i=1m′

i < 0

aP(m)∑`=1

N∑j=0

f`,N−j rj + (N + 1)n∑

i=1m′

i rN−j(i) <1

N + 1(

P(m)∑`=1

N∑j=0

f`,N−j +n∑

i=1m′

i )


But we have∑N

j=0 f`,N−j = m since each M` is a monomial of degree m. Hence we obtain

P(m)∑`=1

N∑j=0

f`,N−j rj + (N + 1)n∑

i=1m′

i rN−j(i) <dm− g + 1+

∑ni=1 m′

iN + 1

Finally, we apply the relation mi = bim2 associated to the linearization and use b =∑

bi :

P(m)∑`=1

N∑j=0

f`,N−j rj + (N + 1)n∑

i=1m′

i rN−j(i) <dm− g + 1+ bm2

N + 1(2.3)

Now, if we let vj = wN−j , then the term∑N

j=0 f`,N−j rj is the weight of the monomial

vf`,00 · · ·v

f`,NN . Also, vN−j(i) is the smallest weight section among the vj ’s which does not

vanish at Pi . Thus we may interpret the left hand side of (2.3) as: the r -weight of a collection

of monomials restricting to the basis of H0(C,O(m)) plus the r -weight of a collection of

degree m′i monomials which do not vanish at Pi .

This argument can be run in reverse, so given a collection of monomials satisfying 3.′

we can produce a collection of monomials satisfying 3.

Note that property 1. above requires a set of monomials in H0(PN ,O(m)) which map

to a basis of H0(C,O(m)) of small weight. We want to turn things around, and instead

start on the curve in H0(C,O(m)) and work our way back to H0(PN ,O(m)). The action

of a 1-PS λ of SL(V ) on the Hilbert point of a curve induces a weights on elements of

H0(C,OC(m)) (cf. [HM] p. 208). Briefly, take a basis of H0(PN ,O(1)) diagonalizing the λ

action. There is an obvious way to define the weight of any degree m monomial, the weight

of any degree m homogeneous polynomial is defined to be the maximum weight of its

constituent monomials, and the weight of an element of H0(C,OC(m)) is the minimum of

the weights of its preimages in H0(PN ,O(m)).

The next proposition says that to establish GIT stability, it is enough to show that there

exists any basis of H0(C,O(m)) of small weight.

Lemma 2.2.3. If there exist a basis of H0(C,O(m)) of λ-weight W , and monomials M′1, . . . , M′

n

satisfying condition (2) above, and together these satisfy

W +n∑

i=1wtλ M′

i ≤(

1+ g − 1+ bN + 1

)m2 − g − 1

N + 1m,

then there are monomials M1, . . . , MP(m) which together with M′1, . . . , M′

n satisfy conditions 1,

2, and 3’ of the numerical criterion.


Proof. Let q1, . . . , qP(m) be a basis of H0(C,O(m))satisfying

W +n∑

i=1wtλ M′

i ≤(

1+ g − 1+ bN + 1

)m2 − g − 1

N + 1m.

We may assume that the q’s are in order of decreasing weight. Let p1, . . . , pP(m) be a set of

preimages of the q’s of minimal weight (that is, wt pi = wt qi for each i). Let Mi,j be the

monomials constituting pi , so that pi =∑ji

j=1 αi,j Mi,j .

Write the list of monomials Mi,j in order of decreasing weight. If there are ties, choose

any order on the tied entries. Write y = #Mi,j. Form the (P(m)×y)-matrix whose entry in

row i and the column labelled by Mi,j is the coefficient of Mi,j in pi . Each row has a leading

monomial (the monomial corresponding to the leftmost column with a nonzero entry in

that row). Row reduce this matrix to upper triangular form; this can only lower the leading

weight in each row. Now choose the leading monomials in each row. Either these map to a

basis of H0(C,O(m)) having weight less than or equal to the weight of the basis given by

q1, . . . , qP(m), or else there is a relation between these terms after restriction to the curve. If

this happens, delete the column corresponding to the leftmost monomial appearing in the

relation, and begin again (row reduce to upper triangular form, check whether the leading

terms in each row give a basis...). Eventually we must arrive at a set of monomials which give

a basis for H0(C,O(m)) (since ρ(pi) is a basis of H0(C,O(m))) and the weight of this set

of monomials is less than or equal to the weight of the basis given by q1, . . . , qP(m).

Generalities on profiles

As mentioned in the introduction, the main tool for computing the weight of a basis is

something I call a profile. (Gieseker uses profiles in his proof, but he doesn’t use the word

“profile.”) We define this abstractly now.

Let V be a vector space such that every element of V has a weight associated to it. Let F•

be a decreasing weighted filtration on W . That is, V = F0 ⊃ F1 ⊃ · · · ⊃ FN = 0, and there is

a (finite) decreasing sequence of weights r0 > r1 > · · · > rN = 0 such that all the elements

of Fh have weight less than or equal to rh.

Definition 2.2.4. The profile of a decreasing weighted filtration F• as described above is the

graph of the decreasing step function in the (codimension×weight)-plane whose value is rh


over the interval [codim Fh, codim Fh+1).

This is like a distribution function bounding how many linearly independent elements

have at most a given weight. Indeed, given a profile, it is possible to choose a basis whose

weight is no greater than the area under the profile. We will sometimes speak of the “weight

of a filtration” or “weight of a profile”; of course what we mean by this is the area underneath

the profile, which is a bound for the weight of a basis adapted to this filtration. We use this

to bound Mumford’s µL(x, λ), since the weight of any basis gives a bound for the minimal

weight of a monomial basis.

Now, there is a notion of an absolute weight filtration. It may be described as follows:

For each possible weight rh, form

Ω(rh) := Spanv : v ∈ V , wt(v) ≤ rh.

Then the profile associated to Ω• can be used to choose a basis of minimum weight, as it

tells exactly how many elements of high weight must be added to the basis before elements

of lower weight may be added.

In this paper, we will encounter filtrations of H0(C,O(1)) and H0(C,O(m)). To help

keep track of the ambient vector space of the filtration, we will use tildes for filtrations of

H0(C,O(m)). The filtration of greatest importance for us, X• (to be defined in Section 2.5),

is of this type.

2.3 A review of Gieseker’s proof

Let us quickly review Gieseker’s proof from [Gies], viewing it as the n = 0 case of the

above setup. We have recast the numerical criterion to say: the m-th Hilbert point of a

smooth curve is GIT-stable if and only if there exists a basis of H0(C,OC(m)) such that the

sum of its weights is less than (1+ ε)m2.

As discussed before Lemma 2.2.3, the action of a 1-PS λ of SL(N + 1) on the Hilbert

point of a curve induces weights on elements of H0(C,OC(m)) (cf. [HM] p. 208). Now, it

is probably most natural to consider the absolute weight filtration on H0(C,O(m)). If one

could compute its profile, then one could compute Mumford’s function µL(x, λ) on the nose.

However, this is too difficult to compute, so Gieseker considers another filtration instead.

2.3. A REVIEW OF GIESEKER’S PROOF 19

Here is a brief and slightly simplified description of the weighted filtration G• Gieseker

uses and its profile. Given: a curve and a 1-PS λ. As before, renormalize the λ-weights so

that they are decreasing and sum to 1. Let wi be a basis of H0(C,OC(1)) H0(PN ,O(1))

diagonalizing the λ action (and compatible with the order of the ri). Let

Vi := Span(wj |j ≥ i) ⊆ V . The normalization ensures that all the points (im, rim) lie in

the first quadrant. Form the lower envelope of these points, and let 0 = i0, i1, . . . , index the

subsequence of points lying on the lower envelope. Then in H0(PN ,OPN (m)) Symm V we

have the following filtration:

Symm V = V mi0

V 0i1

⊃ V m−1i0

V 1i1

⊃ · · · ⊃ V m−pi0

V pi1

⊃ · · · ⊃ V 0i0

V mi1

V mi1

V 0i2

⊃ V m−1i1

V 1i2

⊃ · · · ⊃ V m−pi1

V pi2

⊃ · · · ⊃ V 0i1

V mi2

etc.(2.4)

The image of this filtration under restriction to the curve gives a filtration G• of

H0(C,OC(m)). We can compute the dimension of each stage of the filtration in

H0(C,OC(m)), and we know the weight of each stage, so this is the data of a profile. The

profile is the graph of a step function; its left endpoints lie on the lower envelope of the

set of points (im, rim). A picture is given on the next page. Looking ahead, the lower

envelope here is the inspiration for what I will later call the virtual profile.

Any basis adapted to this filtration will establish stability, as the area A under the profile

is very close to the area under the lower envelope, and the area under the lower envelope is

less than 1m2, by a combinatorial lemma due to Morrison ([Morr], Section 4).

Figure 2.2: Lower envelope for the filtration G•

Codimension

Weight


The weighted filtration on H0(C,O(1))

For brevity, many important details of Gieseker’s proof were left out of the previous

subsection. We now take the opportunity to begin building up the definitions and notation

we need; I have grouped these in this section with his proof, because most of the ideas here

are extracted from his proof or follow easily from it.

As we have observed already, the action of the 1-PS λ induces most fundamentally a

weighted filtration on H0(C,O(1)), but to establish stability we need to find a basis of

H0(C,O(m)) of small weight. We will be going back and forth between these two vector

spaces for the rest of the proof. We begin with H0(C,O(1)), and see what our knowledge

of this filtration tells us about filtrations on H0(C,O(m)). Once we find formulas for the

area under the profile for a certain filtration on H0(C,O(m)), we will ultimately bound the

weight of the basis by relating quantities back to their counterparts in H0(C,O(1)).

Let V• be the weighted filtration on H0(C,O(1)) induced by the action of the 1-PS λ.

That is, the stages of the filtration are distinguished by decreasing weight. Let zj be the

size of the j th stage of the filtration, so zj = codim Vj+1− codim Vj , and let rj be the weight.

Assume that the weights rj have been normalized so that they are decreasing to zero and

sum to 1 (that is, rN = 0 and∑

zj rj = 1). Let Dj be the base locus of the sublinear series Vj ,

and let dj = deg Dj . Let Q1, ..., Qq be the points in SuppDN . (It will be convenient to order

these points, but the choice of order does not matter.) The marked points Pi may or may

not show up among the Q’s; set

Bi = ∑

Pk=Qi bk,0, Qi ≠ Pk for any k.

(2.5)

Let cj,i be the multiplicity of Qi in Dj . (Note that the indices are not in alphabetic order,

opposite the usual convention.) In general Vj is contained in but not equal to

H0(C,O(1)(−Dj )). My experience with this problem leads me to conjecture that the maxi-

mum of Mumford’s µL(x, λ) function occurs for 1-PS where equality holds at every stage.

Relating codegrees and codimensions in H0(C,O(1))

We have one obvious bound on the weights:∑

zj rj = 1. We will need to relate codegrees

dj =∑n

i=1 cj,i and codimensions∑j−1

τ=0 zτ .


Near the top of the weighted filtrations, the base loci have low degree, so O(1)(−Dj )

has high degree, and the dimension/codimension of H0(C,O(1)(−Dj )) may be computed

using Riemann-Roch. More precisely: if deg Dj > d − 2g + 1, then codim Vj > N − g. So if

codim Vj ≤ N−g, then deg Dj ≤ d−2g+1, so degO(1)(−Dj ) > 2g−2, so h1(O(1)(−Dj )) = 0.

Since Vj ⊆ H0(C,O(1)(−Dj )), we get a bound: the codegree of O(1)(−Dj ) cannot exceed

the codimension of Vj . Recall from the definition of the zj ’s that codim Vj =∑j−1

τ=0 zτ .

Writing Dj =∑q

i=1 cj,iQi , we have: deg Dj =∑q

i=1 cj,i . We thus obtain:

if∑j−1

τ=0 zτ ≤ N − g, then∑q

i=1 cj,i ≤∑j−1

τ=0 zτ . (2.6)

I call this the Riemann-Roch region of the filtration. Write jRR for the largest index j which

satisfies∑j−1

τ=0 zτ ≤ N − g.

On the other hand, if O(1) itself is special, or for stages of the filtration of high

codimension (that is, near the bottom), the line bundles O(−Dj ) have low degree, and we

might have h1(O(−Dj )) ≠ 0. Here we can use Clifford’s Theorem to get the following bound:

if∑j−1

τ=0 zτ > N − g, then∑q

i=1 cj,i ≤∑j−1

τ=0 zτ +(∑j−1

τ=0 zτ − (N − g))− h1(C,O(1)). (2.7)

I call this the Clifford region of the filtration and write jCliff for the smallest index j which

satisfies∑j−1

τ=0 zτ > N − g. (So of course jCliff = jRR + 1.)

Note that in the case of principal interest (when d = ν(2g− 2+ a) and ν is large, so that

N is also large), the Riemann-Roch region accounts for the lion’s share of the filtration.

Passing to H0(C,O(m))

We want to use the base loci Dj to control how multiples of the Vj intersect, and

this would work best if Vj = H0(C,O(1)(−Dj )). Gieseker observed that if we pass from

H0(C,O(1)) to H0(C,O(m)) (which is where we ultimately need to produce a basis anyway),

then we will be able to treat an arbitrary 1-PS λ as if it were of this form. Most of the

proof of Lemma 2.3.1 below comes from pages 54–55 of [G2]. However, I want to add a few

comments to Gieseker’s proof, so I will run through the argument here.

Let (V u−ws V w

t V0)v denote the subspace of H0(C,O((u+ 1)v)) generated by expressions

of the form x1 · · ·xv(u−w)y1 · · ·yvw z1 · · ·zv where the x’s come from Vs , the y ’s come from

Vt , and the z’s come from V0.


Lemma 2.3.1 (“Gieseker’s Multiplication Lemma.”). Let u, v, w be nonnegative integers with

0 ≤ w ≤ u and v ≥ 1. Suppose C is an arbitrary subscheme of PN with Hilbert polynomial

dt − g + 1 and

v ≥ d2(u+ 1)2 − d(u+ 1)2

− g + 1.

Then

(V u−ws V w

t V0)v = H0(C,O((u+ 1)v)(−(u−w)Ds −wDt ))

Remark. Note that the bound on v depends on u and the Hilbert polynomial P(z) =

dz − g + 1, but not on the curve C or the line bundle OC(1) embedding C into PN .

Proof. Let Ls and Lt be the line bundles generated by the sections in Vs and Vt . Here is the

first comment to add to Gieseker’s proof: then Ls = OC(1)(−Ds). We have

(V u−ws V w

t V0)v ⊂ H0(C, (Lu−ws Lw

t L0)v ) = H0(C,O((u+ 1)v)(−(u−w)Ds −wDt )).

Now, since sections in V u−ws V w

t generate Lu−ws Lw

t , and V0 is very ample, we have that

V u−ws V w

t V0 is very ample, and hence determines an embedding C PM . We have a short

exact sequence

0 → I(v) → OPM (v) → OC(v) → 0.

(We now have two OC(1)’s in this proof, corresponding to the embeddings in PN and PM , but

it is not difficult to tell them apart.) Write ds = deg Ds , respectively for t ; then deg Ls = d−ds

and deg Lt = d − dt . Then the Hilbert polynomial for C ⊂ PM is

P(z) = ((d − ds)(u−w)+ (d − dt )(w)+ d)z − g + 1.

The Gotzmann number for this Hilbert polynomial is

m0 =((d − ds)(u−w)+ (d − dt )(w)+ d)2 − ((d − ds)(u−w)+ (d − dt )(w)+ d)

2− g + 1;

recall that the Gotzmann number for a Hilbert polynomial has the property that it is the

maximum regularity for any sheaf with that Hilbert polynomial ([Gotz] Lemma 2.9). Hence,

H1(I(v)) = 0 since v is larger than the Gotzmann number. But then

H0(PM ,O(v)) → H0(C, (Lu−ws Lw

t L0)v


is surjective.

Comparing this to the definition of (V u−ws V w

t V0)v , this says that

(V u−ws V w

t V0)v = H0(C, (Lu−ws Lw

t L0)v ) = H0(C,O((u+ 1)v)(−(u−w)Ds −wDt ))

as desired.

Finally note that d − ds and d − dt are no larger than d; hence taking

v ≥ d2(u+ 1)2 − d(u+ 1)2

− g + 1.

ensures that v is greater than or equal to the Gotzmann number for any Vs and Vt .

Remark. We will be applying this result when C is a smooth curve in PN ; for this

application, the Gotzmann number is much larger than what is needed. I hope to improve

this result significantly, which should be helpful (if not necessary) when studying stability

for small values of m.

Let m = (u+ 1)v . Then H0(C,O(m)) is filtered by the subspaces (V uj V0)v .

Note that if there are two successive stages in V• where the base locus does not increase,

the images of these two stages are the same upon passing to H0(C,O(m)). Thus, we will

want to record the subsequence of the j ’s where the degree of the base locus increases. I

will subindex these by the letter k, and I will simply write k rather than writing jk.

In fact the space H0(C,O(m)) is filtered by spaces of the form (V u−wk V w

k+1V0)v . This

defines a very important filtration, which we denote V•, and it is indexed by pairs (k, w) in

lexicographic order.

I will use tildes for quantities associated to V•. We have Vk = H0(C,O(m)(−Dk)), where

Dk = uvDjk . We write dk := uvdjk and ck,i := uvcjk,i . Then

Vk = H0(C,O(m)(−ck,1Q1 − · · · − ck,qQq))

and elements of this space have weight ≤ rk := uvrjk + vr0.


Define N to be the smallest index giving the vr0-weight space. We have:

Space Weight

V0 = H0(C,O(m)) r0

V1 = H0(C,O(m)(−c1,1Q1 − · · · − c1,qQq)) r1

V2 = H0(C,O(m)(−c2,1Q1 − · · · − c2,qQq)) r2

......

VN = H0(C,O(m)(−cN,1Q1 − · · · − cN,qQq)) rN = vr0

(2.8)

We may extract the multiplicities of the points in the base loci in the weighted filtration

V• and the weights to obtain an (N + 1)× (q + 1) array:

c0,1 · · · c0,q r0

c1,1 · · · c1,q r1...

......

...cN,1 · · · cN,q rN = vr0

(2.9)

This array has the following properties: the ck,i ’s are all nonnegative integers; the

ri ’s are rational numbers weakly decreasing to vr0; and in the first row the c0,i ’s are all

zero. Furthermore we see that the sum of the entries in row k is governed by either a

Riemann-Roch bound (2.6) or a Clifford bound (2.7).

2.4 Why Gieseker’s proof doesn’t cover marked points

To my knowledge, Elizabeth Baldwin first wrote down the straightforward generalization

of Gieseker’s result to Mg,n (unpublished), and it is not difficult to see that the analogue of

Gieseker’s filtration does not suffice to establish stability in cases where bi is more than a

little larger than 0. Here is a counterexample:

Example 1

This example shows that the profile associated to G• (which equals V• in this example)

does not suffice to establish asymptotic Hilbert stability when there are marked points.

Suppose n ≥ 3. Consider the 1-PS λ which acts with linearly decreasing weights on the

marked points. That is, λ induces the following weighted filtration:

2.4. WHY GIESEKER’S PROOF DOESN’T COVER MARKED POINTS 25

Space Weight

V0 = H0(C,O(1)) 12

V1 = H0(C,O(1)(−P1)) 13

V2 = H0(C,O(1)(−P1 − P2)) 16

V3 = H0(C,O(1)(−P1 − P2 − P3)) 0

The points (im, rim) all lie on their lower envelope. Also, we have r0 + r1 + r2 = 1. Using

bi = 1/2, we have T ≈ 1m2 − 14 m2 + bm2 = 5/4m2 > (1+ ε)m2.

So the straightforward adaptation of Gieseker’s proof is not enough to establish the

stability of smooth pointed curves with respect to the linearizations we have specified.

The key observation

In fact it is not difficult to show that the 1-PS of Example 1 is not destabilizing.

We use the following easy linear algebra lemma:

Lemma 2.4.1. Let V1, . . . , Vn be subspaces of a vector space V . Write Vij := Vi ∩ Vj , Vijk :=

Vi ∩ Vj ∩ Vk, etc. Then

codim SpanV1, . . . , Vn (2.10)

=∑

codim Vi −∑i<j

codim Vij +∑

i<j<kcodim Vijk − · · · + (−1)n−1 codim V123···n.

Gieseker’s proof proceeds as follows: H0(C,O(m)) contains the following spaces with

the following codimensions and weights:

Wt Codim Space12 m 0 H0(C,O(m))

12 m− 1

6 1 H0(C,O(m)(−P1))12 m− 2

6 2 H0(C,O(m)(−2P1))12 m− 3

6 3 H0(C,O(m)(−3P1))12 m− 4

6 4 H0(C,O(m)(−4P1))...

......

As discussed above, if one basis element is chosen from each of these spaces, then A is

approximately (1− 1n+1 )m2 = 3

4 m2.

The key observation is that we know more subspaces corresponding to each weight in

the left column. For instance, elements of the spaces H0(C,O(m)(−2P1)) and


H0(C,O(m)(−P1 − P2)) each have weight 12 m − 2

6 . These spaces each have codimension

2, and their intersection H0(C,O(m)(−2P1 − P2)) has codimension 3, so their span has

codimension 2+ 2− 3 = 1.

Now we try again to choose a basis of H0(C,O(m)) of lowest weight. For the first basis

element, we may be obliged to choose one element of top weight 12 m. But for the second

basis element, we now know that we may bypass the elements of weight 12 m− 1

6 and instead

choose an element of weight 12 m− 2

6 . My proof repeatedly uses this trick, suggested by Ian

Morrison, to establish a filtration and profile giving a basis of lower weight than Gieseker’s.

Minimizing multiplicities

Soon we are going to put a lot of effort into minimizing multiplicities. The following

lemma shows that this makes easy work of computing spans of spaces of the form we have

encountered.

Lemma 2.4.2 (“Span Lemma”). Suppose we are given q subspaces E1, . . . , Eq of H0(C,O(m))

of the form:

E1 = H0(C,O(m)(−d1,1Q1 − · · · − d1,qQq)E2 = H0(C,O(m)(−d2,1Q1 − · · · − d2,qQq))

...Eq = H0(C,O(m)(−dq,1Q1 − · · · − dq,qQq))

The Ei need not be distinct, and though the notation looks a little similar to that of filtrations

above, we do not mean in any way to imply that the Ei form a filtration—in the applications

we have in mind, they do not.

Suppose that Ei minimizes the multiplicity of Qi—that is, the minimum in each column

appears along the diagonal. Suppose also that

q∑i=1

maxj

dj,i < dm− 2g.

Then

Span(E1, . . . , Eq) = H0(C,O(m)(−q∑

i=1di,iQi))

and

codim Span(E1, . . . , Eq) = d1,1 + d2,2 + · · · + dq,q.

2.5. THE FILTRATION X• AND ITS PROFILE 27

Proof. The conditionq∑

i=1max

jdj,iQi < dm− 2g.

ensures that the codimension of the intersection of any subset of these q spaces may be

computed using Riemann-Roch. Thus, for each subset I ⊆ 1, . . . , q, say I = i1, . . . , ik we

have

codim Ei1···ik = max(di1,1, . . . , dik,1)+max(di1,2, . . . , dik,2)+ · · · +max(di1,q, . . . , dik,q).

Suppose j 6∈ I. Then I claim that in (2.10), the term max(di1,j , . . . , dik,j ) is cancelled by

a term coming from I ∪ j. Being a subset of cardinality one greater, the codimension of

the intersection indexed by I ∪ j gets opposite sign from that indexed by I. And since

by hypothesis dj,j is the smallest term in column j , it drops out of max(di1,j , . . . , dik,j , dj,j ),

giving us exactly the cancellation we claimed. Given I, every j ∈ 1, . . . , q is either in I or

not in I, so it is clear whether the term max(di1,j , . . . , dik,j ) is cancelling or being cancelled.

The only terms surviving are the di,i since there are no double intersections of the form Eii

in our setup to cancel them.

Finally, the base locus of Span(E1, . . . , Eq) must be∑q

i=1 di,iQi (since we can find sections

that vanish to each Qi to exactly order di,i). This gives

Span(E1, . . . , Eq) ⊂ H0(C,O(m)(−q∑

i=1di,iQi)). (2.11)

But the codimensions of the two spaces in line (2.11) are the same, so we must actually have

equality.

2.5 The filtration X• and its profile

Subscript conventions

In the course of the proof we will need to keep track of a set of subsequences of a

subsequence of a sequence. I use the following notation and conventions.

Tildes

Tildes are when we are working in H0(C,O(m)). Recall from Section 2.3 that quantities

associated to V• (such as the multiplicities c•,i and the weights r•) have no tildes and are


fundamentally indexed by j ’s. Quantities associated to V• (like c•,i and weights r•) are

written with tildes and indexed by k’s, where k indexes the subsequence of the rows j of

the original filtration V• where the base locus increases.

Avoiding nested subscripts

When I want to refer to the subsequence of c•,i or r• corresponding to stages of V• where

the base locus increases, rather than using nested subscripts and writing for instance rjk I

will simply write rk.

Cases I-IV and the functions s(k, i) and t(k, i)

It is useful to define two functions s and t . We will take the time now to define four

cases, which will be referred to in this section and in Section 2.6.

I. We have ck,i < ck+1,i < ck+2,i . That is, the multiplicity of the point Qi jumps at row k

and again at row k+ 1. In this case we define s(k, i) = k and t(k, i) = k+ 1.

II. We have ck,i = ck+1,i = ck+2,i . That is, the multiplicity of Qi does not jump at row k or

at row k+ 1. Define s(k, i) to be the last row where this multiplicity jumped, and let

t(k, i) be the next row where it jumps, or else N if ck,i = cN,i . In symbols, in Case II,

s(k, i) is the largest index strictly less than k such that cs(k,i),i < cs(k,i)+1,i , and t(k, i)

is the smallest index strictly greater than k such that ct(k,i),i < ct(k,i)+1,i if this exists,

or else N.

III. We have ck,i = ck+1,i < ck+2,i . That is, the multiplicity of Qi does not jump at row k

but jumps at row k+ 1. Then as in Case II we define s(k, i) to be the last row where

this multiplicity jumped, and we define t(k, i) = k+ 1.

IV. We have ck,i < ck+1,i = ck+2,i . That is, the multiplicity of Qi jumps at row k but not at

row k+ 1. We define s(k, i) = k, and as in Case II let t(k, i) be the next row where this

multiplicity jumps, or else N if ck,i = cN,i .

It follows from the definitions of Cases II, III, and IV that if row k is in Case II, then we

have s(k+ 1, i) = s(k, i) and t(k+ 1, i) = t(k, i).


Defining s and t differently in Cases I–IV as we have done permits us to treat these cases

simultaneously in Section 2.6, which more than makes up for the extra work involved here.

Eliminating redundancies

Rather than printing i redundantly in subscripts, whenever I can I will leave it off the

second time. For example I will simply write cs(k,i) for cs(k,i),i .

The functions j(i, `) and k(i, `)

We will also want to keep track of the subset of j ’s or k’s where the multiplicity of the

point Qi in the base locus increases. I will do this as follows:

Say the multiplicity of Qi jumps Ki times between the top of the filtration and the

bottom. We start counting from zero, so these stages of the filtration are the 0th jump up

through the (Ki − 1)th jump. As a convention, we append N (the index of the last row of the

filtration V•) or N (the index of the last row of the filtration V•) as the Kthi element of this

sequence. We write two increasing set functions

j(i,•) : 0, . . . , Ki → 0, . . . , N

and

k(i,•) : 0, . . . , Ki → 0, . . . , N

and use these to index the rows where the multiplicity of the point Qi in the base locus

increases. That is, the function j(i,•) takes values in the j ’s, and similarly k(i,•) takes

values in the k’s. Here is an example to give a little practice with this notation: j(i, 0) means

the index j where the multiplicity of Qi jumps for the 0th time. This is the lowest row of

the filtration where Qi is not in the base locus, so rj(i,0) is the least weight of a section not

vanishing at Qi .

As before, when i appears more than once in a subscript, we will leave it off the second

time. Thus cj(i,0),i becomes cj(i,0) and we have cj(i,0) = 0 while cj(i,0)+1 = cj(i,1) > 0.

A consequence of these conventions

As a consequence, note that previously when going between the filtrations V• and V• we

had ck,i = uvcjk,i . But now with our new notation we can write ck(i,`) = uvcj(i,`). In this sense


the definitions of j(i, `) and k(i, `) have eliminated some of the need for nested subscripts.

Finally we note that although the notations are similar in format, j and k are somewhat

different in character from s and t . We might describe j and k as “lookup” functions,

whereas s and t are “previous” and “next” functions.

The filtration X• and its profile

Here we describe the filtration X• of H0(C,O(m)) and its weight profile. X• is obtained

from the filtration V• by taking spans of the stages of V• with other cleverly chosen spaces.

Like V•, the filtration X• has (N × u)+ 1 stages and is indexed by pairs (k, w), ordered

lexicographically.

For each k = 0, . . . , N − 1, and for each w = 0, . . . , u − 1 we want to describe the space

Xk,w . Our starting point is the space (V u−wk V w

k+1V0)v . Elements of this space have weight

less than or equal to v(u−w)rk + vwrk+1 + vr0.

Our goal: for each i from 1 to q, find a subspace of H0(C,O(m)) such that

1. the weight is less than or equal to the weight of (V u−wk V w

k+1V0)v

2. the multiplicity of Qi in its base locus is less than the multiplicity of Qi in the base

locus of (V u−wk V w

k+1V0)v .

We do this as described in the following definition. Also, it is convenient to define certain

quantities x(k, w, i) at this time; their role will be explained soon.

Definition 2.5.1 (The filtration X• and its profile).

First, X0,0 = H0(C,O(m)).

For the remaining triples (k, w, i) with (k, w) ≠ (0, 0), where k = 0, . . . , N − 1, w =

0, . . . , u− 1, we begin with (V u−wk V w

k+1V0)v . For each i = 1, . . . , q, there may be an additional

contribution to the profile, as follows:

• If the multiplicity of Qi is zero in row k+ 1 (and hence zero in row k also), there is no

contribution to Xk,w , and x(k, w, i) = 0.

• If the multiplicity of Qi is nonzero in row k + 1 and we are in Case I as defined in

Section 2.5, so the multiplicity of Qi jumps at row k and row k+ 1, there is no further


contribution to Xk,w beyond (V u−wk V w

k+1V0)v , and x(k, w, i) is the multiplicity of Qi in

(V u−wk V w

k+1V0)v ;

• If the multiplicity of Qi is nonzero in row k + 1 and we are in Case II, III, or IV as

defined in Section 2.5, so the multiplicity of Qi jumps at no more than one of the

rows k and k + 1, let s(k, i) and t(k, i) be as defined there. For each w we find the

smallest integer W = W (u, v; k, w, i) such that (V u−Ws(k,i)V W

t(k,i)V0)v has weight less than

v(u − w)rk + vwrk+1 + vr0. Then (V u−Ws(k,i)V W

t(k,i)V0)v is added to Xk,w , and x(k, w, i) is

the multiplicity of Qi in the base locus of (V u−Ws(k,i)V W

t(k,i)V0)v .

Then

Xk,w = Span(V u−wk V w

k+1V0)v , spaces of type (V u−Ws(k,i)V W

t(k,i)V0)v if there are any,

and let x(k, w) be the codimension of Xk,w .

Note Xk,w is the span of between 1 and q + 1 distinct spaces; there may be fewer than

q + 1 distinct spaces in the span, as there may be points Qi , which make no contribution,

and/or repeats may occur among the spaces of the form (V u−Ws(k,i)V W

t(k,i)V0)v .

Finally, for the last stage of the filtration, define XN := VN .

Thus, the profile associated to X• is the graph of decreasing step function whose value

over the intervals [x(k, w), x(k, w + 1)) is v(u−w)rk + vwrk+1 + vr0, and whose value over

the interval [codim XN , dim H0(C,O(m))] is vr0.

Illustration: The filtration X• and its profile for Example 1

Recall that Example 1 concerns the 1-PS with n = q = 3 which induces the following

weight filtration V•:

Space Weight

V0 = H0(C,O(1)) 12

V1 = H0(C,O(1)(−P1)) 13

V2 = H0(C,O(1)(−P1 − P2)) 16

V3 = H0(C,O(1)(−P1 − P2 − P3)) 0

After passing to H0(C,O(m)) we obtain the filtration V•:


Space Weight

V0 = H0(C,O(m)) 12 uv + 1

2 v

V1 = H0(C,O(m)(−uvP1)) 13 uv + 1

2 v

V2 = H0(C,O(m)(−uvP1 − uvP2)) 16 uv + 1

2 v

V3 = H0(C,O(m)(−uvP1 − uvP2 − uvP3)) 12 v

We compute the filtration X• and its profile for Example 1. For this, we ought to specify

u, v first. We choose u = 3 and v = 5. Of course, this value of v is really too small to use

with Lemma 2.3.1, but let us ignore this in the interest of presenting a reasonably sized

example. Also, in this example, we will always have x(k, w) =∑q

i=1 x(k, w, i). (How do I

know this? See Section 2.9 for a hint.)

The filtration X• has ten stages. The first and the last are easy to compute—we have

X0,0 = H0(C,O(m)) and X3 = (V 33 V0)5. Let’s compute one of the middle stages, X1,1, as

an example: The multiplicity of P1 does not increase from row 1 to row 2 to row 3, so we

are in Case II, and s(1, 1) = 0 and t(1, 1) = 3. We find W = 2. (Here W may be computed

from its defining properties, or by skipping ahead and using Formula (2.20) derived in

Section 2.6.) Thus the contribution to X1,1 from P1 is (V 10 V 2

3 V0)5, and x(1, 1, 1) = 10. The

multiplicity of P2 increases from row 1 to row 2, but not from row 2 to row 3, so we are in

Case IV, and s(1, 2) = 1 and t(1, 2) = 3. Here W = 1, and the contribution from P2 to X1,1 is

(V 21 V 1

3 V0)5, and x(1, 1, 2) = 5. The multiplicity of P3 is zero in both row 1 and row 2, so P3

does not contribute to X1,1. We have: X1,1 = Span(V 21 V 1

2 V0)5, (V 10 V 2

3 V0)5, (V 21 V 1

3 V0)5, and

x(1, 1) = 15.

Here is the filtration X•. I have left the spans unsimplified.

Stage Space Codim WtX0,0 = H0(C,O(m)) 0 10X0,1 = Span(V 2

0 V 11 V0)5, (V 2

0 V 13 V0)5 5 55/6

X0,2 = Span(V 10 V 2

1 V0)5, (V 20 V 1

3 V0)5 5 50/6X1,0 = Span(V 3

1 V 02 V0)5, (V 2

0 V 13 V0)5, (V 3

1 V 03 V0)5 5 45/6

X1,1 = Span(V 21 V 1

2 V0)5, (V 10 V 2

3 V0)5, (V 21 V 1

3 V0)5 15 40/6X1,2 = Span(V 1

1 V 22 V0)5, (V 1

0 V 23 V0)5, (V 2

1 V 13 V0)5 15 35/6

X2,0 = Span(V 32 V 0

3 V0)5, (V 10 V 2

3 V0)5, (V 11 V 2

3 V0)5, (V 32 V 0

3 V0)5 20 5X2,1 = Span(V 2

2 V 13 V0)5, (V 0

0 V 33 V0)5, (V 1

1 V 23 V0)5, (V 2

2 V 13 V0)5 30 25/6

X2,2 = Span(V 12 V 2

3 V0)5, (V 00 V 3

3 V0)5, (V 01 V 3

3 V0)5, (V 12 V 2

3 V0)5 40 20/6X3 = (V 3

3 V0)5 45 15/6


Notice that X0,1 = X0,2 = X1,0, and X1,1 = X1,2. Nothing in our definitions prevents this,

and it does not harm us either—all it means is that when we compute the area under the

profile between these stages of the filtration, we will obtain a complicated expression for

zero.

Here is an illustration of the profile for X• in Example 1 with u = 3, v = 5.

Figure 2.3: Profile for X• in Example 1 with u = 3, v = 5

5

2.5

Codimension

Weight

Using the profile of X•

Note that the spaces used to construct each Xk,w in Definition 2.5.1 satisfy the de-

gree hypothesis of Lemma 2.4.2: every space going into the span is either of the form

(V u−wk V w

k+1V0)v or (V u−W (k,w,i)s(k,i) V W (k,w,i)

t(k,i) V0)v . But the base locus of any space of this form is

bounded by the base locus of (V uNV0)v , which is uvcN,1+· · ·+uvcN,q. That is, maxjdj,i ≤

uvcN,i , so we have

q∑i=1

maxjdj,i ≤

q∑i=1

uvcN,i ≤ uvd < uvd + ud − 2g = dm− 2g.

However, it is not true that (V u−wk V w

k+1V0)v or (V u−Ws(k,i)V W

t(k,i)V0)v always minimizes the

multiplicity of Qi among these q spaces. (It is possible to find the minimum, but we will

not do this now. See Section 2.9 for a little more discussion.) Therefore, we cannot apply

Lemma 2.4.2 to conclude that x(k, w) =∑q

i=1 x(k, w, i). However, we may use Lemma 2.4.2

to conclude that x(k, w) ≤∑q

i=1 x(k, w, i), since the minimum multiplicity for the point Qi

must be smaller than x(k, w, i). Of course, this is not enough to bound x(k, w + 1)− x(k, w).

But since the rk’s are decreasing, the weight A of this profile will only decrease if some

x(k, w) <∑q

i=1 x(k, w, i). So computing using equality at every stage gives the following


upper bound for A:

A ≤N−1∑k=0

u−1∑w=0

(v(u−w)rk + vwrk+1 + vr0)(x(k, w + 1)− x(k, w))+ (dim XN)vr0. (2.12)

We have XN = H0(C,O(m)(−uvDN)), and so we may compute

dim XN = dm− uvdN − g + 1 = (d − dN)uv + dv − g + 1.

Substituting this into (2.12), we obtain

A ≤N−1∑k=0

u−1∑w=0

(v(u−w)rk+vwrk+1+vr0)(x(k, w+1)− x(k, w))+((d−dN)uv+dv−g+1)vr0.

(2.13)

Rather than trying to bound the right hand side of (2.13), we will follow a different

approach. We will define a “virtual” profile whose graph has area Avir nearly the same as the

area of the graph A of the actual profile, but which is computationally a little easier to work

with. Let ∆ = A−Avir be the discrepancy. Also, for each i between 1 and q, recall that rj(i,0)

is the rj such that cj,i = 0 and cj+1,i > 0. Then

T ≤ Avir +∆+n∑

i=1Birj(i,0)(u+ 1)2v2. (2.14)

We use the rest of this section to define the virtual profile. In the next section we bound

∆, and in Section 2.7 we bound Avir +∑n

i=1 Birj(i,0)(u+ 1)2v2. Putting this all together with

(2.14), we will get a bound for T .

The virtual profile

The virtual profile simplifies the graph of the profile in three ways:

• In the profile, we form a span of q spaces for all k and for all w , so the step function is

defined over (N × u)+ 1 intervals; in the virtual profile, we only partition the domain

(the codimension axis) into N + 1 intervals.

• In the profile, we round so that W = W (u, v ; k, w, i) is always an integer, so exponents,

multiplicities, and codimensions are integers; in the virtual profile, their counterparts

are rational numbers. (This is the origin of the djective “virtual.”)


• In particular the quantity f (k) (defined below) is the virtual counterpart to x(k, 0). The

profile is a step function, so the two points (x(k, 0), uvrk + vr0) and

(x(k+ 1, 0), uvrk+1 + vr0) are connected by a staircase; but in the virtual profile, we

connect the two points (f (k), rk) and (f (k+ 1), rk+1) by straight line segments.

We will call the figure so obtained the virtual profile and use Avir, the area under the virtual

profile, to approximate A.

Definition 2.5.2 (The virtual profile). For each k = 0, . . . , N −1, we define f (k) as follows. We

begin by defining fi(k) for each i. Fix i. Graph the set of points (rk(i,`), ck(i,`)) : ` = 0, . . . , Ki

and connect these by straight line segments. Then fi(k) is the piecewise linear function whose

value at k is the second coordinate of the point on this graph lying over rk.

The picture described above translates into the following rules. We refer to Cases I-IV as

defined in 2.5:

• If ck+1,i = 0, then fi(k) = 0.

• Otherwise, let s(k, i) and t(k, i) be as defined at the beginning of Section 2.5. Then

fi(k) =(

rk − rt(k,i)

rs(k,i) − rt(k,i)cs(k,i) + (1− rk − rt(k,i)

rs(k,i) − rt(k,i))ct(k,i)

).

Note that in Case I and Case IV the formula above just gives fi(k) = ck,i , since s(k, i) = k

in these cases.

Finally,

f (k) :=q∑

i=1fi(k).

The virtual profile is the graph of the piecewise linear function connecting the points

(f (k), rk).

Note the switch in the order of the coordinates that takes place: fi(k) is defined by a

graph in the (weight×multiplicity of Qi)-plane, whereas the virtual profile is graphed along

with the profile in the (codimension×weight)-plane.


The quantity f (k) is an approximate upper bound for the codimension of the rk-weight

space in H0(C,OC(m)). We have:

Avir =N−1∑k=0

12

(f (k+ 1)− f (k))(rk+1 + rk)+ (dim VN)vr0

=N−1∑k=0

12

(f (k+ 1)− f (k))(rk+1 + rk)+ (d − dN)uv + dv − g + 1. (2.15)

Also, for each i between 1 and q, recall that rj(i,0) is the rj such that cj,i = 0 and

cj+1,i > 0. Let T vir = Avir + (u + 1)2v2∑qi=1 Birj(i,0) denote the approximation to T obtained

by approximating A by Avir. We have the following upper bound for T vir:

T vir ≤N−1∑k=0

12

(f (k+1)− f (k))(rk+1+ rk)+((d−dN)uv+dv−g+1)vr0+(u+1)2v2n∑

i=1Birj(i,0).

(2.16)

Illustration: the virtual profile for X• in Example 1

Let us compute the virtual profile for Example 1. We can compute the virtual profile for

an arbitrary u, v :

For k = 0 there is nothing to compute.

For k = 1, the multiplicity of P1 does not jump from row 1 to row 2. We are in Case II.

Looking at where the multiplicity P1 jumps, we have s(1, 1) = 0 and t(1, 1) = 3, and we find

that f1(1) = 13 uv . The multiplicity of P2 jumps between row 1 and row 2; we are in Case IV,

and we have f2(1) = c1,2 = 0. Finally, since the multiplicity of P3 is zero in both row 1 and

row 2, f3(1) = 0. Then f (1) = 13 uv . Also, r1 = 1

3 uv + 12 v .

For k = 2, the multiplicity of P1 does not jump from row 2 to row 3. We are in Case II,

s(2, 1) = 0 and t(2, 1) = 3, and f1(2) = 23 uv. The multiplicity of P2 does not jump between

row 2 and row 3; we are in Case II, and s(2, 2) = 1 and t(2, 2) = 3, giving f2(2) = 12 uv.

Finally, the multiplicity of P3 jumps at row 2; we are in Case IV, so f3(2) = c2,3 = 0. Then

f (2) = 76 uv . Also, r2 = 1

6 uv + 12 v .

Finally, for k = N = 3 there is also nothing to compute.

The area of the region under the graph connecting the points (0uv, 12 uv + 1

2 v),

( 13 uv , 1

3 uv+ 12 v), ( 7

6 uv, 16 uv+ 1

2 v) and (3uv, 12 v) is 1

2 u2v2+ 32 uv2. To this we add the weight

of the vr0 region, which is (dim VN)vr0 = ((d − 3)uv + dv − g + 1)( 12 v). We have:


Avir = 12

u2v2 + 12

duv2 + 12

dv2 − 12

(g − 1)v.

Using Bi = 12 , the contribution from the marked points is 1

2 (u2v2 + 2uv2 + v2). We have:

T vir = 1u2v2 + (12

d + 1)uv2 + (12

d + 1)v2 − 12

(g − 1)v.

Interpreting the vertices of the virtual profile

If we suppose that the integer uv is divisible by 6, we can give a little more meaning to

the calculations above.

For k = 1 we can begin with the space V1, which gives us the point (1uv, 13 uv + 1

2 v). To

this we add the space V23 uv

0 V13 uv

3 V v0 to minimize the multiplicity of P1. Similarly we add

V uv1 V v

0 to minimize the multiplicity of P2. The multiplicity of P3 is zero in all the spaces of

this weight. The codimension of V uv1 V v

0 is uv , and the codimension of V23 uv

0 V13 uv

3 V v0 is also

uv . However, using Lemma 2.4.2, the codimension of their span is 13 uv . In other words, the

point (1uv, 13 uv + 1

2 v) in the profile of V• slides left to ( 13 uv, 1

3 uv + 12 v) in the virtual profile

for X•.

A similar analysis for k = 2 yields the list of spaces V13 uv

0 V23 uv

3 V v0 , V

12 uv

1 V12 uv

3 V v0 , and

V uv2 V v

0 minimizing the multiplicities of P1, P2, and P3 respectively. The codimension of their

span is 76 uv , so the point (2uv, 1

6 uv + 12 v) in the profile of V• slides left to ( 7

6 uv, 16 uv + 1

2 v)

in the virtual profile for X•.

It seems that for any fixed 1-PS λ we could choose uv sufficiently divisible to clear any

denominators which may arise. However, we cannot do this across all 1-PS, so we will

consider this interpretation of the vertices of the virtual profile as motivational, not part of

the rigorous proof. Also, even when we have such divisibility, so that the virtual profile’s

vertices have this interpretation, I see no rigorous way to interpret the straight line segments

connecting the vertices. So, it seems best to regard the virtual profile merely as a graph and

not an algebro-geometric object of any kind.

The profile and virtual profile for Example 1 together

Here are the profile and virtual profile for X• in Example 1 with u = 3, v = 5.


Figure 2.4: Example 1. Profile and virtual profile for X• with u = 3, v = 5

5

2.5

Codimension

Weight

In this picture the area under the profile looks significantly larger than the area under

the virtual profile, but for larger values of u these areas become relatively closer. This is

made rigorous in the next section, but as an example, here is an approximate picture of

the profile and virtual profile for X• in Example 1 with u = 20, v = 5. (A duplicate of this

illustration is displayed in the introduction to this chapter.)

Figure 2.5: Example 1. Profile and virtual profile for X• with u = 20, v = 5

10

5

Codimension

Weight

Progress report

We have at last defined all the key ingredients mentioned in the introduction:

one filtration V• of H0(C,O(1)),two filtrations V• and X• of H0(C,O(m)),

and two graphs associated to X•

In Sections 2.6, 2.7, and 2.8 it remains to study these filtrations and graphs more closely

and show that they have the properties claimed.

2.6. THE DISCREPANCY BETWEEN THE PROFILE AND VIRTUAL PROFILE FOR X• 39

2.6 The discrepancy between the profile and virtual profile forX•

This section is devoted to showing that the areas of the profile and virtual profile are

very close when m is large. That is, we bound the discrepancy ∆ := A−Avir. The approach

used in this section is extremely straightforward, though tedious.

We will bound ∆ by computing bounds for several terms which contribute to it. Roughly

speaking, we will compute the discrepancy ∆k,i for each k and i, but it takes a little care to

say exactly what we mean by this, as the regions of the graph may be offset a little bit. For

instance, in the picture corresponding to Example 1 with u = 3, v = 5, we would partition

the virtual profile at codimension 17.5 (a breakpoint of the piecewise linear function) but

the corresponding partition for the profile occurs at codimension 20.

For the virtual profile this is straightforward. The area under the graph of the virtual

profile may be divided in an obvious way into N trapezoids and one final rectangle. Let us

focus on the area Avirk of the kth trapezoid:

Avirk = 1

2(f (k+ 1)− f (k))(rk+1 + rk)

= 12

q∑i=1

fi(k+ 1)−q∑

i=1fi(k)

(rk+1 + rk)

=q∑

i=1

12

(fi(k+ 1)− fi(k))(rk+1 + rk).

I will write Avirk,i for the ith summand:

Avirk,i =

12

(fi(k+ 1)− fi(k))(rk+1 + rk).

We compute Avirk,i now.

Computing Avirk,i

Avirk,i is the area of the trapezoid whose vertices are (fi(k), 0), (fi(k+1), 0), (fi(k+1), rk+1),

and (fi(k), rk). To compute fi(k+ 1)− fi(k), recall the definition of fi(k) given in Definition

2.5.2. We use the four cases defined in Section 2.5.

I. The multiplicity c•,i jumps at row k and again at row k+1. Then the spaces contributing

to the profile are V uvk V v

0 at the kth vertex and V uvk+1V v

0 at the (k+ 1)th vertex, and in


between, spaces of the form V (u−w)vk V wv

k+1V v0 are used. Thus in the virtual profile we

are calculating as if spaces of the form V αuvk V (1−α)uv

k+1 V v0 were being used between

these two vertices with α ranging from 0 to 1.

II. The multiplicity c•,i does not jump at row k or at row k + 1. Recall that we have

(s(k, i), t(k, i)) = (s(k+ 1, i), t(k+ 1, i)). In the profile, spaces of the form

V (u−W )vs(k,i) V W v

t(k,i)V v0 are being used between these two vertices. In the virtual profile, we

are calculating as if spaces of the form V αuvs(k,i)V (1−α)v

t(k,i) V v0 were being used between these

two vertices (though here the range of α is a subinterval strictly in the interior of

[0, 1]).

III. The multiplicity c•,i does not jump at row k but jumps at row k + 1. Recall that

t(k, i) = k+1. Once again, in the profile, spaces of the form V (u−W )vs(k,i) V W v

t(k,i)V v0 are being

used in this region. For this reason Case III is very similar to Case II. In the virtual

profile, we are calculating as if spaces of the form V αuvs(k,i)V (1−α)v

t(k,i) V v0 were being used in

this region, with α beginning at a value strictly smaller than 1 and decreasing to 0.

IV. The multiplicity c•,i jumps at row k but not at row k+1. By the definition of s we have

s(k, i) = k, and in the profile spaces of the form V (u−W (k,w,i))vs(k,i) V W (k,w,i)v

t(k,i) V v0 are being

used in this region. In the virtual profile, we are calculating as if spaces of the form

V αuvs(k,i)V (1−α)v

t(k,i) V v0 were being used in this region, with α starting at 1 and ending at a

value strictly greater than 0.

We do the calculation first for Case II. We have:

fi(k+ 1) = rk+1 − rt(k+1,i)

rs(k+1,i) − rt(k+1,i)cs(k+1,i) +

(1− rk+1 − rt(k+1,i)

rs(k+1,i) − rt(k+1,i)

)ct(k+1,i)

and

fi(k) = rk − rt(k,i)

rs(k,i) − rt(k,i)cs(k,i) +

(1− rk − rt(k,i)

rs(k,i) − rt(k,i)

)ct(k,i),


and (s(k, i), t(k, i)) = (s(k+ 1, i), t(k+ 1, i)). Thus

Avirk,i = 1

2(rk+1 + rk)(fi(k+ 1)− fi(k))

= 12

(rk+1 + rk)(

rk − rk+1

rs(k,i) − rt(k,i)(ct(k,i) − cs(k,i))

)

= 12

(uvrk+1 + uvrk + 2vr0)(

uvrk − rk+1

rs(k,i) − rt(k,i)(ct(k,i) − cs(k,i))

)

= u2v2

(12

(rk+1 + rk)(ct(k,i) − cs(k,i))rk − rk+1

rs(k,i) − rt(k,i)

)

+uv2

(r0(ct(k,i) − cs(k,i))

rk − rk+1

rs(k,i) − rt(k,i)

)(2.17)

By similar calculations, and using some of the information presented in paragraphs I, III

and IV above, we derive the same formula in Cases I, III and IV.

Computing bounds for Ak,i

We have defined Avirk,i but have not yet defined a corresponding quantity Ak,i . We do this

now. Let Ak,i denote the following sum:

Ak,i :=u−1∑w=0

((u−w)rk +wrk+1 + r0)(x(k, w + 1, i)− x(k, w, i)). (2.18)

In pictures,∑q

i=1 Ak,i is the area under the profile between x(k, 0) and x(k+ 1, 0).

We wish to bound Ak,i . We split into Cases I-IV as in Section 2.6.

Case I. Case I is covered by the general calculation below. However, in Case I we can give

a sharper estimate via an easier calculation than in general. We present this now.

Using Definition 2.5.1 we have x(k, w + 1, i) = v(u − (w + 1))ck,i + v(w + 1)ck+1,i and

x(k, w, i) = v(u−w)ck,i + vwck+1,i , so x(k, w + 1, i)− x(k, w, i) = ck+1,i − ck,i . We have:

Ak,i =u−1∑w=0

((u−w)rk +wrk+1 + r0)(x(k, w + 1, i)− x(k, w, i))

=u−1∑w=0

((u−w)rk +wrk+1 + r0)(ck+1,i − ck,i)

= u2v2(

12

(rk+1 + rk)(ck+1,i − ck,i))+ uv2

((r0 +

12

(rk+1 + rk))(ck+1,i − ck,i))

.

(2.19)

The general calculation. From here to the end of Section 2.6, we suppress the sub-

scripts k,i as much as possible, as they do not change. We reintroduce them at the end

of this subsection in line (2.37).


Recall that in Definition 2.5.1, for each w , we defined W = W (k, w, i) to be the smallest

integer such that the space (V u−Ws V W

t V0)v has weight less than or equal to

v(u−w)rk + vwrk+1 + vr0. We use this property to get an expression for W :

v(u−W )rs + vW rt + vr0 ≤ v(u−w)rk + vwrk+1 + vr0

a W ≥ u(rs − rk)+w(rk − rk+1)rs − rt

⇒ W (w) = W (k, w, i) =⌈

u(rs − rk)+w(rk − rk+1)rs − rt

⌉(2.20)

It is useful to write

ζ = ζk,i := rk − rk+1

rs − rt(2.21)

ξ = ξk,i := rs − rk

rs − rt. (2.22)

Then

W = duξ +wζe. (2.23)

Since s ≤ k and k ≤ t and s < t , we have 0 ≤ ζ ≤ 1 and 0 ≤ ξ ≤ 1.

Proceeding, we have:

x(k, w, i) = v(u−W (w))cs + vW (w)ct

x(k, w + 1, i) = v(u−W (w + 1))cs + vW (w + 1)ct

⇒ x(k, w + 1, i)− x(k, w, i) = v(ct − cs)(W (w + 1)−W (w)).

Putting this into (2.18) we have:

Ak,i =u−1∑w=0

v ((u−w)rk +wrk+1 + r0) v ((ct − cs)(W (w + 1)−W (w)))

= v2(ct − cs)

u−1∑w=0

(urk + r0 −w(rk − rk+1))(W (w + 1)−W (w))

= v2(ct − cs)

(urk + r0)u−1∑w=0

(W (w + 1)−W (w))

−(rk − rk+1)u−1∑w=0

w(W (w + 1)−W (w))

. (2.24)


Calculating pieces of (2.24)

Before we continue computing Ak,i it is helpful to work out the sums appearing in (2.24).

This is accomplished in the next lemma. We use the notation

〈y〉 := y − byc, (2.25)

so that 〈y〉 denotes the fractional part of y . We write

Λ := duζ + 〈uξ〉 − 1e. (2.26)

Lemma 2.6.1.

u−1∑w=0

w(W (w + 1)−W (w)) =

Λ∑`=1

⌊` − 〈uξ〉

ζ

⌋, ζ ≠ 0

0, ζ = 0

(2.27)

andu−1∑w=0

(W (w + 1)−W (w)) =

Λ, 〈uξ〉 ≠ 0Λ+ 1, 〈uξ〉 = 0

(2.28)

Proof. For the first of these formulas,

u−1∑w=0

w(W (w + 1)−W (w)) =u−1∑w=0

w(duξ + (w + 1)ζe − duξ +wζe)

=u−1∑w=0

w(d〈uξ〉 + (w + 1)ζe − d〈uξ〉 +wζe). (2.29)

The factor (d〈uξ〉 + (w + 1)ζe − d〈uξ〉 + wζe) is 0 except when the first term has

just passed an integer and the second summand has not caught up yet, and then this

factor is 1. We can describe the values of w which are multiplied by nonzero coefficient:

d〈uξ〉 + ζ(w + 1)e > d〈uξ〉 + ζwe if and only if there exists an integer ` such that

〈uξ〉 + ζw ≤ ` < 〈uξ〉 + ζ(w + 1)

a w ≤ ` − 〈uξ〉ζ

< w + 1

⇒ w = b `−〈uξ〉ζ c.

(2.30)

Next we identify the range of summation for the index `. For the lower bound, if 〈uξ〉 ≠ 0,

then ` must begin at 1, giving the smallest possible w (since w must be nonnegative). If

〈uξ〉 = 0, then ` begins at 0, giving w = 0. But since we are multiplying by w , we may start

with ` = 1 in this case, too.

For the upper bound: Λ is the correct upper bound if


1. Λ is a positive integer

2. bΛ−〈uξ〉ζ c ≤ u− 1

3. bΛ+1−〈uξ〉ζ c ≥ u.

The inequalities of the preceding two lines imply the following:

Λ− 〈uξ〉ζ

< u ≤ Λ+ 1− 〈uξ〉ζ

a uζ + 〈uξ〉 > Λ ≥ uζ + 〈uξ〉 − 1.

By the integrality of Λ, we must have Λ = duζ + 〈uξ〉 − 1e. This gives the first formula of

the lemma.

Now we consider∑u−1

w=0(W (w + 1)−W (w)). Simplifying as above,

u−1∑w=0

(W (w + 1)−W (w)) =u−1∑w=0

(d〈uξ〉 + (w + 1)ζe − d〈uξ〉 +wζe). (2.31)

The sum in line (2.31) increases by one every time the first term passes an integer and the

second term hasn’t caught up yet. By the discussion above on the range of `, we see that

this happens Λ times if 〈uξ〉 ≠ 0 or Λ+ 1 times if 〈uξ〉 = 0.

It is helpful to decompose Λ according to u-degrees. We write

Λ = duζ + 〈uξ〉 − 1e

=

uζ + 〈uξ〉 − 1, if 〈uζ + 〈uξ〉 − 1〉 = 0uζ + 〈uξ〉 − 1− 〈uζ + 〈uξ〉 − 1〉 + 1, if 〈uζ + 〈uξ〉 − 1〉 ≠ 0

We define

η :=〈uξ〉 − 1, if 〈uζ + 〈uξ〉 − 1〉 = 0〈uξ〉 − 〈uζ + 〈uξ〉 − 1〉, if 〈uζ + 〈uξ〉 − 1〉 ≠ 0.

(2.32)

Note that −1 ≤ η < 1. The definition of η allows us to write

Λ = uζ + η. (2.33)

We also rewrite the formulas of Lemma 2.6.1 using indicators 1ζ≠0 = 1ζk,i≠0 and

1〈uξ〉=0 = 1〈uξk,i〉=0. There are two reasons for this. First, it helps us to avoid writing separate

cases many times. Also, ζ appears in the denominator of the first formula of Lemma 2.6.1,

and ζ can take the value 0. It is easy to forget that these two things do not happen at


the same time, causing concern that this summand (or later quantities) is undefined. The

indicator helps to remind us that when ζ = 0, we add 0.

Thus the formulas of Lemma 2.6.1 become

u−1∑w=0

w(W (w + 1)−W (w)) = 1ζ≠0

uζ+η∑`=1

⌊` − 〈uξ〉

ζ

⌋(2.34)

andu−1∑w=0

(W (w + 1)−W (w)) = uζ + η+ 1〈uξ〉=0. (2.35)

The main calculation resumed

We now resume the main calculation. We reprint line (2.24) and then substitute in (2.35)

and (2.34):

Ak,i = v2(ct − cs)

(urk + r0)u−1∑w=0

(W (w + 1)−W (w))

−(rk − rk+1)u−1∑w=0

w(W (w + 1)−W (w))

= v2(ct − cs)

(urk + r0)(uζ + η+ 1〈uξ〉=0)

−(rk − rk+1)1ζ≠0

uζ+η∑`=1

⌊` − 〈uξ〉

ζ

⌋

= v2(ct − cs)

(urk + r0)(uζ + η+ 1〈uξ〉=0)

−(rk − rk+1)1ζ≠0

uζ+η∑`=1

(`ζ− 〈uξ〉

ζ−⟨

` − 〈uξ〉ζ

⟩)

= v2(ct − cs)

u2rkζ + ((η+ 1〈uξ〉=0)rk + ζr0)u+ r0(η+ 1〈uξ〉=0)

−1ζ≠0(rk − rk+1)

ζ

(12

(uζ + η)(uζ + η+ 1))

+1ζ≠0(rk − rk+1)

ζ

uζ+η∑`=1

(〈uξ〉 + ζ

⟨` − 〈uξ〉

ζ

⟩) (2.36)

In the last line, we have 0 ≤ 〈`−〈uξ〉ζ 〉 < 1. We apply this bound to obtain an upper bound

for Ak,i , and we also begin grouping terms by their u-degree:


Ak,i ≤ v2(ct − cs)

(rkζ − 1ζ≠0(rk − rk+1)

ζ12

ζ2

)u2

+(

(η+ 1〈uξ〉=0)rk + ζr0 + 1ζ≠0(rk − rk+1)(〈uξ〉 + ζ − η− 1

2

))u

+(

r0(η+ 1〈uξ〉=0)+ 1ζ≠0(rk − rk+1)

ζ

(η〈uξ〉 + ζη− 1

2η(η+ 1)

))1

= v2(ct − cs)

(12

(rk + rk+1)ζ)

u2

+(

(η+ 1〈uξ〉=0)rk + ζr0 + 1ζ≠0(rk − rk+1)(〈uξ〉 + ζ − η− 1

2

))u

+(

r0(η+ 1〈uξ〉=0)+ 1ζ≠0(rk − rk+1)

ζ

(η〈uξ〉 + ζη− 1

2η(η+ 1)

))1

.

Finally, we restore the k, i symbols which have been suppressed throughout this subsec-

tion, yielding:

Ak,i ≤ v2(ct(k,i) − cs(k,i))

(12

(rk + rk+1)ζk,i

)u2 (2.37)

+(

(ηk,i + 1〈uξk,i〉=0)rk + ζk,ir0 + 1ζk,i≠0(rk − rk+1)(〈uξk,i〉 + ζk,i − ηk,i −

12

))u

+(

r0(ηk,i + 1〈uξk,i〉=0)+ 1ζk,i≠0(rk − rk+1)

ζk,i

(ηk,i〈uξk,i〉 + ζk,iηk,i −

12

ηk,i(ηk,i + 1)))

1

.

This completes our calculation of Ak,i in the general case.

Bounding the discrepancy

We now have all the ingredients we need to bound ∆.

∆ := A−Avir ≤N−1∑k=0

q∑i=1

(Ak,i −Avirk,i ).

By comparing (2.17) and (2.37), and using the definition of ζk,i at (2.21), we see that

∆k,i := Ak,i −Avirk,i ≤ (2.38)

v2(ct(k,i) − cs(k,i))

((ηk,i + 1〈uξk,i〉=0)rk + 1ζk,i≠0(rk − rk+1)(〈uξk,i〉 + ζk,i − ηk,i −

12

))u

+(

(ηk,i + 1〈uξk,i〉=0)r0 + 1ζk,i≠0(rs(k,i) − rt(k,i))(

ηk,i〈uξk,i〉 + ζk,iηk,i −12

η2k,i −

12

ηk,i

))1

.

2.7. BOUNDING THE WEIGHT OF THE VIRTUAL PROFILE T vir 47

Recall that the weights rj and the fractional parts of any quantity must be between 0

and 1, and −1 ≤ ηk,i < 1. Therefore we may make various coarse estimates:

(ηk,i + 1〈uξk,i〉=0)rk < 2;

〈uξk,i〉 + ζk,i − ηk,i −12

< 1+ 1+ 1− 12

⇒ 1ζk,i≠0(rk − rk+1)(〈uξk,i〉 + ζk,i − ηk,i −

12

)<

52

;

(ηk,i + 1〈uξk,i〉=0)r0 < 2;


η2k,i −

12

ηk,i < 1+ 1− 0+ 12

⇒ 1ζk,i≠0(rs − rt )(


η2k,i −

12

ηk,i

)<

52

. (2.39)

Combining these inequalities with (2.39) we obtain:

∆k,i ≤ uv2(ct(k,i) − cs(k,i))(

92

)+ v2(ct(k,i) − cs(k,i))

(92

). (2.40)

Next, I claim that the estimates given for each k in (2.40) together yield

N−1∑k=0

∆k,i ≤ uv2(

92

cN,i

)+ v2

(92

cN,i

). (2.41)

Refer back to the definition of s and t in Section 2.5. Equation (2.41) follows because the

pairs (s, t) fit together in such a way that when the estimates from (2.40) are summed over

k, the sum telescopes.

Finally, using the estimates obtained in (2.41), we obtain

∆ ≤q∑

i=1

N−1∑k=0

∆k,i ≤ uv2(

92

d)+ v2

(92

d)

. (2.42)

Observe that ∆ is of order uv2 and not of order u2v2.

2.7 Bounding the weight of the virtual profile T vir

The reader is strongly encouraged to review the subscript notations introduced in Section

2.5, especially the definitions of j(i, `) and k(i, `), before proceeding.

Setting up a comparison

Recall that in line (2.16) we obtained the following bound on T vir:


T vir ≤N−1∑k=0

12

(f (k+ 1)− f (k))(rk+1 + rk)+ ((d − dN)uv + dv − g + 1)vr0

+(u+ 1)2v2q∑

i=1Birj(i,0)

=N−1∑k=0

12

(f (k+ 1)− f (k))(rk+1 + rk)+ ((d − dN)uv + dv − g + 1)vr0

+u2v2q∑

i=1Birj(i,0) +

q∑i=1

Birj(i,0)(2uv2 + v2) (2.43)

Everything in this sum is in terms of k (it is, after all, the weight of a basis of

H0(C,O(m))). Almost the only bound available is that the weights sum to 1:∑N

j=0 zj rj = 1.

Our goal in this subsection is to rewrite (2.43) in a form that makes it easy to compare to∑zj rj .

We focus on the first term of (2.43):

N−1∑k=0

12

(f (k+ 1)− f (k))(rk+1 + rk) =N−1∑k=0

12

(q∑

i=1fi(k+ 1)−

q∑i=1

fi(k))(rk+1 + rk)

=N−1∑k=0

q∑i=1

12

(fi(k+ 1)− fi(k))(rk+1 + rk) (2.44)

Let Avirk,i denote the area of the region described in Definition 2.5.2. Then we have:

N−1∑k=0

q∑i=1

12

(fi(k+ 1)− fi(k))(rk+1 + rk) =N−1∑k=0

q∑i=1

Avirk,i

=q∑

i=1

N−1∑k=0

Avirk,i (2.45)

where in the last line we have changed the order of summation. Let Aviri =

∑N−1k=0 Avir

k,i .

Observe that, for a fixed i, it may not be necessary to partition this region into N vertical

trapezoids to compute the area Aviri ; a partition corresponding to the domains of definition

of the piecewise linear function fi , which may be coarser than that given by the full set of

k’s, will do.

Recall that k(i,•) indexes the rows k where the multiplicity c•,i jumps. Then we may


compute:

Aviri =

N−1∑k=0

12

(fi(k+ 1)− fi(k))(rk+1 + rk)

=Ki−1∑`=0

12

(ck(i,`+1) − ck(i,`))(rk(i,`+1) + rk(i,`))

= u2v2

Ki−1∑`=0

12

(cj(i,`+1) − cj(i,`))(rj(i,`+1) + rj(i,`))

+ uv2(cN,ir0) (2.46)

We develop the coefficient of the u2v2 term of (2.46):Ki−1∑`=0

12

(cj(i,`+1) − cj(i,`))(rj(i,`+1) + rj(i,`))

=

Ki∑`=1

12

(cj(i,`) − cj(i,`−1))rj(i,`) +Ki−1∑`=0

12

(cj(i,`+1) − cj(i,`))rj(i,`)

=

Ki−1∑`=1

12

(cj(i,`+1) − cj(i,`−1))rj(i,`) +12

cj(i,1)rj(i,0)

. (2.47)

Once again, cj(i,1) is the first nonzero multiplicity of Qi in a base locus in V•, and rj(i,0) is

the least weight of a section not vanishing at Qi . Putting (2.47), (2.46), and (2.45) into (2.44),

we have:

T vir ≤ u2v2q∑

i=1

Ki−1∑`=1

12

(cj(i,`+1) − cj(i,`−1))rj(i,`) +(

12

cj(i,1) + Bi

)rj(i,0)

(2.48)

+ q∑

i=1Birj(i,0)

(2uv2 + v2)+ ((d − dN)uv + dv − g + 1)vr0 + uv2

q∑i=1

cN,ir0

It is convenient to define Ij to be the set of i’s where the multiplicity jumps at row j ,

and not for the first or last time:

Ij := i | ∃` ≠ 0, Ki s.t. j = j(i, `). (2.49)

We switch the order of summations in (2.48) to obtain:

T vir = u2v2N∑

j=0

∑Ij

12

(cj(i,`+1) − cj(i,`−1))+∑

i: j=j(i,0)

(12

cj(i,1) + Bi

) rj (2.50)

+ q∑

i=1Birj(i,0)

(2uv2 + v2)+ (duv + dv − g + 1)vr0

which is of the form we desired.


Comparing

The next lemma gives a bound for the coefficient of u2v2 in (2.50).

Lemma 2.7.1.

N∑j=0

∑Ij

12

(cj(i,`+1) − cj(i,`−1))+∑

i: j=j(i,0)

(12

cj(i,1) + Bi

) rj ≤N∑

j=0Zj rj ,

where

Zj :=

zj , j < jRR

zj + j∑

τ=0zτ − (N − g)

, j = jRR

2zj , j ≥ jCliff

Idea of proof (Wall Street version). Think of j as being time in days, the Zj ’s as daily

income, and the coefficient of rj on the left hand side as daily losses. We will show that

every time you have a losing day, you have enough in the bank to see you through.

Idea of proof (algebraic geometry version). The Zj ’s defined above bound the change in

degree of the base loci from Vj to Vj+1. The only way there can be a jump larger than this is

if dj lags behind the maximum allowable degree for this codimension. In this case, we are

using more small weights and fewer large weights than we conceivably could, so the weight

of the resulting basis will not be maximal.

Proof. We may rewrite the desired inequality as

N∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rj ≥ 0.

We work successively on each index j where

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1)) < 0.

If there are no such j , we are done. So suppose there is at least one such index, and let

the set of these be indexed je beginning with e = 1. By the definition of j1 we have

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1)) > 0

for all j < j1, soj1−1∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rj

≥j1−1∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rj1


andj1−1∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

≥ 0.

We wish to establish that

j1∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rj

≥j1∑

j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rj1

(which is easy) and that

j1∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

≥ 0.

We rewrite this last inequality as j1∑j=0

Zj

− j1∑j=0

∑i: j=j(i,0)

(12

cj(i,1) + Bi

)+∑Ij

12

(cj(i,`+1) − cj(i,`−1))

≥ 0. (2.51)

We study the second sum in (2.51) above. Each i falls into exactly one of the following

cases:

Case 0. If c•,i does not jump before or at j1—that is, j(i, 0) > j1—then this i does not

contribute.

Case 1. If c•,i jumps exactly once before or at j1—that is, j(i, 0) ≤ j1 < j(i, 1)—then this

i contributes

12

cj(i,1) + Bi ≤12

cj1+1,i +12≤ cj1+1,i ,

since cj(i,1) = cj1+1,i and Bi ≤ 12 and cj1+1,i ≥ 1.

Case 2. If c•,i jumps exactly twice before or at j1—that is, j(i, 1) ≤ j1 < j(i, 2)—then the

contribution to the second term is

12

cj(i,1) + Bi +12

cj(i,2) ≤ cj1+1,i .

This follows because cj(i,2) = cj1+1,i and cj(i,2) ≥ cj(i,1) + 1.

Case 3. If c•,i jumps three or more times before or at j1, then some telescoping occurs,

and the contribution is

12

cj(i,1) + Bi +12

ct(j1,i) +12

cs(j1,i) −12

cj(i,1) ≤ cj1+1,i .


Here I am abusing notation a little (according to Section 2.5 the first argument of s(•, i) or

t(•, i) is supposed to be a k, not a j). Here s(j1, i) denotes the largest index less than or

equal to j1 where c•,i jumps, and t(j1, i) denotes the smallest index strictly greater than j1

index where c•,i jumps. Thus, ct(j1,i) = cj1+1,i and cs(j1,i) ≤ cj1,i .

To summarize, in each case, we see that the contribution is no more than cj1+1,i .

If j1 < jRR, so that j1 + 1 is in the Riemann-Roch region, then by (2.6) we have

q∑i=1

cj1+1,i ≤j1∑

j=0zj ,

so the left hand side of (2.51) is indeed nonnegative: j1∑j=0

Zj

− j1∑j=0

∑i: j=j(i,0)

(12

cj(i,1) + Bi

)+∑Ij

12

(cj(i,`+1) − cj(i,`−1))

≥ j1∑

j=0zj

− j1∑

j=0zj

= 0.

We have thus dealt with the first index, if it falls inside the Riemann-Roch region. We

may repeat the argument at each je in the Riemann-Roch successively, stopping when either

the je’s are exhausted or we reach the Clifford region. At each step we need to show two

things in order to proceed to the next step: first,

je∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rj

≥je∑

j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rj1

(which is always easy to check), and second,

je∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

≥ 0.

Next suppose that je = jRR, so je + 1 = jCliff . Then by (2.7) we have

q∑i=1

cje+1,i ≤je∑

j=0zj +

je∑j=0

zj − (N − g),

and je∑j=0

Zj

− je∑j=0

∑i: j=j(i,0)

(12

cj(i,1) + Bi

)+∑Ij

12

(cj(i,`+1) − cj(i,`−1))

≥

je∑j=0

zj +je∑

j=0zj − (N − g)

− je∑

j=0zj +

je∑j=0

zj − (N − g)

= 0.


Finally suppose that some je + 1 falls within the Clifford region. Then by (2.7) we have

q∑i=1

cje+1,i ≤je∑

j=0zj +

je∑j=0

zj − (N − g).

Using the definitions given in the statement of the lemma, we compute

je∑j=0

Zj =jRR−1∑

j=0Zj + ZjRR +

je∑j=jCliff

Zj

=jRR−1∑

j=0zj +

zjRR +jRR∑j=0

zj + (N − g)

+ 2zjCliff + · · · + 2zje

= 2je∑

j=0zj − (N − g),

and once again the left hand side of (2.51) is nonnegative: je∑j=0

Zj

− je∑j=0

∑i: j=j(i,0)

(12

cj(i,1) + Bi

)+∑Ij

12

(cj(i,`+1) − cj(i,`−1))

≥

2je∑

j=0zj − (N − g)

− je∑

j=0zj +

je∑j=0

zj − (N − g)

= 0.

Again, proceed to the next je until the set of these has been exhausted.

Ideally, we would now show that the bound obtained in Lemma 2.7.1 is smaller than

what is required in the numerical criterion. Unfortunately, this is not always true. Lemma

2.7.1 is sufficient for most, but not all, sets of linearizing weights B. Below I have listed five

cases which exhaust all possibilities. This partitioning may look strange, but it is in order of

difficulty of proof. In Cases A-C, I can prove asymptotic stability of smooth curves. In Cases

D and E, I cannot prove stability, so I will ultimately impose hypotheses to ensure that these

cannot occur.

Choose any sufficiently small value ε > 0. (The size of ε allowed will be made clear in

Cases B and C below, and the role of ε will become clear in the proof of Theorem 2.8.1.)

Then we consider the following five cases:

Case A. n ≥ 1 and b ≥ g−1N + ε(N + 1).

Case B. n ≥ 1 and b < g−1N + ε(N + 1) < 1

2Case C. n = 0 and N ≥ 2g − 2Case D. n = 0 and N < 2g − 2Case E. n ≥ 1 and b < g−1

N + ε(N + 1) ≥ 12

(2.52)


Let us proceed first with Case A: To apply Lemma 2.7.1 to our problem, we need to

bound∑N

j=0 Zj rj . Let rN−g+1, . . . , rN−1, rN = 0 be the last g weights (that is, ignore the index

j and list the smallest weights as many times as indicated by their multiplicities). Then we

have

N∑j=0

Zj rj ≤∑

zj rj + rN−g+1 + · · · + rN

≤ 1+ rN−g+1 + · · · + rN

The next lemma gives a bound for rN−g+1 + · · · + rN .

Lemma 2.7.2. rN−g+1 + · · · + rN ≤ g−1N .

Proof. Recall that rN = 0, so we may omit it from all the following sums. We argue similarly

to [Morr] Theorem 4.1. We wish to maximize rN−g+1 + · · · + rN−1, which is linear in the

r ’s, subject to the constraints∑N−1

j=0 zj rj = 1 and that the r ’s are decreasing. In the affine

hyperplane in (N − 1)-dimensional r -space determined by the equation∑N−1

j=0 zj rj = 1, the

condition that the r ’s are decreasing defines an (N−1)-simplex. The vertices of this simplex

correspond to sequences of the following form:

r0 = · · · = rh > rh+1 = · · · = rN−1 = 0.

The function must take its maximum at (at least) one of these vertices, and it is easy to

check that the maximum occurs when

r0 = · · · = rN−1 > 0,

or rj = 1N for all j , yielding a maximum value of g−1

N .

Also, the defining hypothesis of Case A at line (2.52) may be written as follows.

b ≥ g − 1N

+ ε(N + 1)

ag − 1

N≤ g − 1+ b

N + 1− ε

Therefore, as a trivial extension of Lemma 2.7.2, we have:

rN−g+1 + · · · + rN ≤g − 1+ b

N + 1− ε. (2.53)


We combine (2.53) with the bound found in (2.50) to obtain:

T vir ≤(

1+ g − 1+ bN + 1

− ε)

u2v2+ q∑

i=1Birj(i,0)

(2uv2+v2)+ (duv+dv−g+1)vr0. (2.54)

Note that the leading coefficient 1+ g−1+bN+1 − ε is less than the leading coefficient 1+ g−1+b

N+1

of the numerical criterion (2.2) by ε. This completes our discussion of Case A.

Next we turn to Cases B and C, defined in line (2.52). In these cases, the bound given in

Lemma 2.7.2 is too large to use with the numerical criterion. Fortunately, if we examine the

proof of Lemma 2.7.1 closely, we can improve the bound there a little bit.

Lemma 2.7.3. 1. Suppose a sufficiently small ε > 0 has been chosen and n ≥ 1 and

b < g−1N + ε(N + 1) < 1

2 , so that we are in Case B. Then

N∑j=0

∑Ij

12

(cj(i,`+1) − cj(i,`−1))+∑

i: j=j(i,0)

(12

cj(i,1) + Bi

) rj ≤N∑

j=0Zj rj −

(12− b

)rN−1,

where the Zj are as in Lemma 2.7.1, and

rN−1 =

0, zN > 1rN−1, zN = 1.

2. Suppose n = 0. Then

N∑j=0

∑Ij

12

(cj(i,`+1) − cj(i,`−1))+∑

i: j=j(i,0)

(12

cj(i,1) + Bi

) rj ≤N∑

j=0Zj rj −

12

rN−1,

where the Zj are as in Lemma 2.7.1, and

rN−1 =

0, zN > 1rN−1, zN = 1.

Proof. Note this is a trivial extension of Lemma 2.7.1 if zN > 1, as then rN−1 = 0. So suppose

zN = 1; then∑N−1

j=0 zj = N − 1. By the proof of Lemma 2.7.1 we know that

N−1∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rj

≥N−1∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

rN−1

andN−1∑j=0

Zj −∑

i: j=j(i,0)

(12

cj(i,1) + Bi

)−∑Ij

12

(cj(i,`+1) − cj(i,`−1))

≥ 0.


So ifN−1∑j=0

Zj −q∑

i=1cN,i ≥

12− b, (2.55)

then we are done. Note the left hand side of (2.55) is a nonnegative integer. So suppose the

left hand side of (2.55) is zero; we will explain how to improve the estimates used in the

proof of Lemma 2.7.1 by at least 12 − b.

First, if n = 0, there are no marked points, and Bi = 0 for all i. Since we estimated Bi ≤ 12 ,

we have the improvement we need.

So suppose n ≥ 1. If there is at least one point Qi appearing in a base locus in V• which

is not one of the marked points Pi , then similarly since Bi = 0 and we always estimated

Bi ≤ 12 , we have the improvement we need. So we may suppose that every Qi is a Pj (hence

q < n).

If there are no points Qi—that is, the base locus of VN is empty—then the weight vr0

space has codimension 0 in H0(C,O(m)), and we can easily show T vir is smaller than what

is required by the numerical criterion.

So suppose there is at least one point Q1 in the base locus of VN . But now, on the one

hand we have by hypothesis that Bi ≤ b < g−1N+1 + ε(N + 1) ≤ 1

2 ; but in the proof of Lemma

2.7.1 we only estimated Bi ≤ 12 ; so we see that we may improve our estimate by at least the

desired amount.

We proceed with Case B. We may argue just as we did in Lemma 2.7.2 to get

rN−g+1 + · · · + rN −(

12− b

)rN−1 ≤

(g − 1−

(12− b

))1N

(2.56)

Combining (2.56) with (2.50), we obtain:

T vir ≤1+

g − 32 + bN

u2v2 + q∑

i=1Birj(i,0)

(2uv2 + v2)+ (duv + dv − g + 1)vr0. (2.57)

We desire that the leading coefficient should be smaller than what is required by the

numerical criterion by ε. That is, we want:

g − 32 + bN

≤ g − 1+ bN + 1

− ε

a ε ≤ 12N(N + 1)

(N − 2g + 3− 2b). (2.58)

2.8. GIT STABILITY OF SMOOTH POINTED CURVES 57

The right hand side of (2.58) is positive because the hypotheses of Case B imply that

N ≥ 2g − 1, and we also have b < 12 . Thus, when ε is sufficiently small (depending on N, ν ,

and B) then (2.58) is satisfied.

Next we consider Case C. Lemma 2.7.3.2 covers this situation, and we may argue just as

we did in Lemma 2.7.2 to get

rN−g+1 + · · · + rN −12

rN−1 ≤(

g − 32

)1N

(2.59)

Then, we want to arrange that

g − 32

N≤ g − 1

N + 1− ε

a ε ≤ 12N(N + 1)

(N − 2g + 3). (2.60)

Since N ≥ 2g − 2 in Case C, (2.60) is satisfied for all ε sufficiently small.

This completes our discussion of Cases B and C.

Unfortunately, in Cases D and E, I know of no way to improve the bound of Lemma

2.7.1 in order to get the leading coefficient of T vir small enough to use with the numerical

criterion in this case! Therefore, in the GIT-Stability Theorem of the next section, we will

include hypotheses to ensure that only Cases A, B, C may occur.

2.8 GIT stability of smooth pointed curves

The GIT-Stability Theorem

We are ready to prove the main result of this chapter:

Theorem 2.8.1 (GIT-Stability Theorem). Suppose g, n, d, B, and ε are given, and that they

satisfy one of the following three sets of hypotheses:

A. n ≥ 1, b ≥ g−1N + ε(N + 1).

B. n ≥ 1, b < g−1N + ε(N + 1) < 1

2 , ε ≤ 12N(N+1) (N − 2g + 3− 2b).

C. n = 0, N ≥ 2g − 2, ε ≤ 12N(N+1) (N − 2g + 3).

Consider a point in the incidence locus I parametrizing a genus g smooth pointed curve

(C ⊂ PN , P1, . . . , Pn) embedded in PN by a complete linear system of degree d. We do not

assume that the points Pi are distinct, nor that C is pluricanonically embedded.


If n ≥ 1, suppose that B satisfies the following condition: for any subset I ⊆ [1..n] for

which the marked points Pi coincide,∑

i∈I bi < 12 . (In particular, applying this to the singletons

I = i, we have that bi < 1/2 for i = 1, . . . , n.)

Let m = (u + 1)v. Then for certain large values of m, the point of I parametrizing

(C ⊂ PN , P1, . . . , Pn) is GIT-stable for the SL(N + 1)-action with the linearization specified by

m′i = bim2 for each i. More precisely, there exist:

1. a positive integer u0 depending on g, n, d, B and ε, but not on the curve C, the points

Pi , or the embedding C ⊂ PN ; and

2. a function v0(u) whose domain is all integers greater than u0, and which depends on g,

n, d, B, ε and u, but not on the curve C, the points Pi , or the embedding C ⊂ PN ;

such that for any integers u ≥ u0 and v ≥ v0(u), the point of I parametrizing

(C ⊂ PN , P1, . . . , Pn) is GIT-stable for the SL(N + 1)-action with the linearization specified by

m′i = bim2 for each i.

Proof. By the numerical criterion, it is enough to show λ-stability for an arbitrary 1-PS λ

of SL(N + 1). So suppose λ is given. Apply all the constructions of this chapter to λ; in

particular, associated to λ we have the filtration X• and its profile and virtual profile.

By (2.54) and (2.42) we have

T = T vir +∆

≤(

1+g−1+bN + 1

−ε)

u2v2 + q∑

i=1birj(i,0)

(2uv2 + v2)

+(duv + dv − g + 1)vr0 +92

duv2 + 92

dv2

≤(

1+g−1+bN + 1

−ε)

u2v2 +2

n∑i=1

birj(i,0) +112

d

uv2 +2

n∑i=1

birj(i,0) +112

d

v2 + v

≤(

1+g−1+bN + 1

−ε)

u2v2 +(

2b + 112

d)

uv2 +(

2b + 112

d)

v2 + v (2.61)

Note that this bound depends on g, n, d, B, and ε. But we emphasize that in every case,

this bound does not depend on the particular curve C, the points Pi , the embedding C ⊂ PN ,

or the 1-PS λ.

2.9. ADDITIONAL REMARKS ON THE GIT-STABILITY THEOREM 59

Recall the bound required in the numerical criterion:

(1+ g − 1+ b

N + 1

)m2− g − 1

N + 1m =

(1+ g − 1+ b

N + 1

)(u2v2+2uv2+v2)− g − 1

N + 1(uv+v). (2.62)

We want to show that (2.61) is less than (2.62), or equivalently that

0 ≤

(g − 1+ bN + 1

−(

g − 1+ bN + 1

− ε))

u2 +(

2+ 2g − 2+ 2bN + 1

− 2b − 112

d)

u

+(

1+ g − 1+ bN + 1

− 2b − 112

d)v2 −

(g − 1N + 1

(u+ 1)+ 1)

v. (2.63)

But the coefficient of u2 in the coefficient of v2 is ε > 0. So there exists a u0 such that

for all u ≥ u0, the polynomial

εu2 +(

2+ 2g − 2+ 2bN + 1

− 2b − 112

d)

u+(

1+ g − 1+ bN + 1

− 2b − 112

d)

is positive; but then for all sufficiently large v (that is, for all v ≥ v0(u)), the polynomialεu2 +(

2+ 2g − 2+ 2bN + 1

− 2b − 112

d)

u+(

1+ g − 1+ bN + 1

− 2b − 112

d)v2

−(

g − 1N + 1

(u+ 1)+ 1)

v

is positive, too. Once again, we emphasize that the size of u required depends on g, n, d, B,

and ε, but not on the particular curve C, the points Pi , the embedding C ⊂ PN , or the 1-PS λ.

Similarly the size of v required depends on g, n, d, B, ε, and u, but not on the particular

curve C, the points Pi , the embedding C ⊂ PN , or the 1-PS λ.

Remark. Theorem 2.8.1 as stated does not establish stability for all large values of m,

only for some large values of m. Similarly, Gieseker’s stability proof ([Gies], Theorem 1.0.0)

only establishes stability for some, not all, large values of m. In both cases it seems possible

that one may be able to use VGIT arguments to conclude stability for all sufficiently large

values of m, but I have not checked this.

2.9 Additional remarks on the GIT-Stability Theorem

I hinted in Section 2.4 at how we first got the idea for the proof of the GIT-Stability

Theorem, but the definition of the filtration X• in Section 2.5 may still feel unmotivated.


In between these two sections we posed and solved the combinatorial problem below. It

appears nowhere else, so I include it here for completeness.

This provides an opportunity to discuss a suggestion for improving the GIT-Stability

Theorem, relate my proof to Gieseker’s in the unpointed case, and discuss a conjecture

about the worst 1-PS. Finally, I discuss a second suggestion for improving the GIT-Stability

Theorem, and relate Baldwin’s results and mine.

A related combinatorial problem

The following combinatorial problem is an idealized version of what arises in my proof

of GIT stability of weighted pointed curves.

Let N, n, q ∈ N with q ≥ n. Suppose we are given an (N + 1) × (q + 1) matrix of the

following form: c0,0 · · · c0,q r0

c1,0 · · · c1,q r1...

......

...cN,0 · · · cN,q rN = 0

(2.64)

Following the convenction of this chapter, let i index the columns, and let j index the

rows. Suppose this matrix has the following properties: the cj,i ’s are all nonnegative integers,

and the rj ’s are distinct rational numbers between 0 and 1. In the first row the c0,i ’s are all

zero. The c portions of any two consecutive rows j and j + 1 are the same except in one

column only, where cj+1,i = cj,i + 1. We may use this to partition the r ’s: we will say rj is

associated to column i if we have cj+1,i = cj,i + 1. In the far right column of the matrix, the

rj ’s are decreasing and sum to 1. Furthermore rN = 0 and rj > 0 if j < N.

(This corresponds to taking m = 1 in the stability proof and a case where the 1-PS λ

induces a filtration for which each stage is a complete sublinear series of codimension 1 in

the previous stage, and making the idealized assumption that Riemann-Roch always holds.)

Define

xj,i = minσ≤j−1,τ≥j

( rj − rτ

rσ − rτcσ ,i +

(1− rj − rτ

rσ − rτ

)cτ,i

).

Also for each j, i record the pairs (σj,i , τj,i) giving the minimum value xj,i . (It will be

clear in the solution what to do if more than one pair (σ , τ) works.)


Let

xj =q∑

i=1xj,i .

Also, for each i between 1 and n, define rPi to be the rj such that cj−1,i = 0 and cj,i = 1.

Finally, define

T :=N−1∑j=0

12

(xj+1 − xj )(rj+1 + rj )+ 12

n∑i=1

rPi

Theorem 2.9.1. For a matrix (c | r) as above, T ≤ 1.

We study the minimizing pairs (σj,i , τj,i) en route to proving the theorem. At a first

glance at the definition of the xj,i , it may seem that the pairs (σj,i , τj,i) could jump all over,

but the next series of results show that they do not.

Lemma 2.9.2. The sequences σj,i and τj,i may be found as follows: For a given i, find the

subsequence jk where c jumps in column i. That is, cjk+1 = cjk + 1. Form the lower envelope

of the set of points (rjk , cjk,i) and index these by a further subscript l. For any j , then, σj,i is

the largest index such that jkl is a point on the lower envelope with rjkl> rj , and τj,i is jkl+1 .

We use two additional lemmas in the proof of Lemma 2.9.2:

Lemma 2.9.3. If j ≠ jk for any k, then xj,i < cj,i .

Proof. Take σ to be the largest jk which is smaller than j , and take τ = jk+1. The hypothesis

j ≠ jk ensures τ > j . We have cjk,i = cj,i − 1 and cjk+1,i = cj,i . But then

rj − rjk+1

rjk − rjk+1

(cj,i − 1)+(

1− rj − rjk+1

rjk − rjk+1

)cj,i < cj,i ,

since the left hand side is a convex combination of cj,i with something smaller.

The next lemma says it is always advantageous to take σ and τ as large as possible until

cσ ,i or cτ,i jumps:

Lemma 2.9.4. Suppose σ1 < σ2 and τ1 < τ2 and cσ1,i = cσ2,i < cτ1,i = cτ2,i . Then

rj − rτ2

rσ2 − rτ2

cσ2,i +(

1− rj − rτ2

rσ2 − rτ2

)cτ2,i <

rj − rτ1

rσ1 − rτ1

cσ1,i +(

1− rj − rτ1

rσ1 − rτ1

)cτ1,i


Proof. We may write cσ ,i for cσ1,i and cσ2,i and write cτ,i for cτ1,i and cτ2,i . Since σ1 < σ2, we

have rσ1 > rσ2 . Also, rj − rτ1 ≥ 0, so

rj − rτ1

rσ1 − rτ1

cσ ,i <rj − rτ1

rσ2 − rτ1

cσ ,i .

But then

rj − rτ1

rσ2 − rτ1

cσ ,i +(

1− rj − rτ1

rσ2 − rτ1

)cτ,i <

rj − rτ1

rσ1 − rτ1

cσ ,i +(

1− rj − rτ1

rσ1 − rτ1

)cτ,i .

Similarly, starting with τ1 < τ2 we get(1− rj − rτ1

rσ2 − rτ1

)= rσ2 − rj

rσ2 − rτ1

>rσ2 − rj

rσ2 − rτ2

=(

1− rj − rτ2

rσ2 − rτ2

)

⇒ rj − rτ2

rσ2 − rτ2

cσ ,i +(

1− rj − rτ2

rσ2 − rτ2

)cτ,i <

rj − rτ1

rσ2 − rτ1

cσ ,i +(

1− rj − rτ1

rσ2 − rτ1

)cτ,i .

Proof of Lemma 2.9.2. If (rj , cj,i) is on the lower envelope of the points (rjk , cjk,i), there

is nothing to prove. So suppose it is not, that is, j ≠ jkl for any l. If we also have j ≠ jk

for any k, then it follows from Lemma 2.9.3 and Lemma 2.9.4 that we will do better with a

pair (σ , τ) where σ and τ are jk’s. Plot the points (rjk , cjk,i) for cj0,i = 0 to cN,i , and plot

(rj , cj,i) as well. Now, for this given j and any pair (σ , τ) with σ < j and τ ≥ j ,

rj − rτ

rσ − rτcσ ,i +

(1− rj − rτ

rσ − rτ

)cτ,i

is the y-coordinate on the line segment between (rσ , cσ ,i) and (rτ , cτ,i) lying over rj . So the

canonical pair minimizing this quantity consists of the nearest two points on the lower

envelope.

Proof of Theorem 2.9.1. Recall that T was defined as follows:

T =N−1∑j=0

12

(xj+1 − xj )(rj+1 + rj )+ 12

n∑i=1

rPi

where

xj =n+q∑i=1

xj,i .

If we also declare rPi = 0 for n+ 1 ≤ i ≤ q (that is, for the Qi ’s), then we may rewrite T as

T =n+q∑i=1

N−1∑j=0

12

(xj+1,i − xj,i)(rj+1 + rj )+ 12

rPi

.


We focus on the inner sums

Ti :=N−1∑j=0

12

(xj+1,i − xj,i)(rj+1 + rj )+ 12

rPi .

In the proof of Lemma 2.9.2 above, we found that the xj,i may be computed using the lower

envelope of the points (rjk , cjk,i). Then the term

N−1∑j=0

12

(xj+1,i − xj,i)(rj+1 + rj )

in Ti is just the area of the region in the first quadrant and underneath the lower envelope.

This area is bounded by the area of the region obtained by connecting all the points

(rjk , cjk,i) without taking the lower envelope. However, recall that the jk have exactly

the property that cjk+1 = cjk + 1. So we may usefully recompute the area underneath all

the points by doing a “Riemann sum” along the y-axis instead. Put another way, we are

switching from vertical trapezoids to horizontal trapezoids. The area is

cjk,i=cN,i∑cj0=0

12

(rjk+1 + rjk )

Note that if 1 ≤ i ≤ n then what is labelled rj0 in this sum is what we have called rPi

previously. It is the r value corresponding to the largest index for which cj,i = 0.

Then,

T =n+q∑i=1

Ti ≤N∑

j=0rj = 1.

The middle inequality follows since in every row j there is exactly one i such that

cj+1,i = cj,i + 1. Recall that the rj ’s are partioned into the sets rj,i for i = 1 to q; each

rj appears in exactly one of the Ti ’s. If it is not the first index, it appears twice with a

coefficient of 1/2 each time. If j = j0 for that i, then it only appears once in the sum, but

then we add on an extra 12 rPi if i ≤ n. If i > n then we will not add in the extra 1

2 rPi so we

will get a strict inequality.

From this analysis it is clear that the examples for which T = 1 will be those for which

each point rj lies on the ith lower envelope and for which q = 0, as predicted.

Lower convex envelopes might give better bounds for T

Recall from Section 2.5 that in the definition of X•, we did not minimize the multiplicity

of each Qi . It is not hard to find the minima; in light of the combinatorial problem above,


we see that instead of using the functions s(k, i) and t(k, i), defined as “ ‘previous’ and

‘next’ among values where the multiplicity of Qi jumps,” we should use σ (k, i) and τ(k, i),

defined as “ ‘previous’ and ‘next’ among values where the multiplicity of Qi jumps which lie

on the lower envelope of these.” That is, there are q lower envelopes to keep track of.

It is possible that if one defines a filtration Y• using lower envelopes like this, one might

be able to prove stability under weaker hypotheses than those given in the GIT-Stability

Theorem. In particular I believe that this might yield a proof of asymptotic stability of

canonically embedded smooth nonhyperelliptic curves. The obstacle is the proof of the

analogue of Lemma 2.7.1. I don’t know how to make this work using lower envelopes instead

of just the next value; instead of relating everything to cj1+1 one would need to work with

much later c’s, and I don’t see how to do this.

Comparison to Gieseker and Morrison’s results, and the worst 1-PS

We may interpret Gieseker’s proof ([Gies], Theorem 1.0.0) as the n = 0, q = 1 case of

Theorem 2.8.1. This easily leads to a coarse upper bound for T . The bound so obtained

is not quite as good as the bound given in [Morr] Section 4 and used in Gieseker’s proof.

However, after running the proof here, one can perform their analysis on top of that, and

the resulting bounds for the leading coefficient would then agree.

Kempf and Rousseau showed that when x is GIT-unstable, there is a “worst 1-PS” desta-

bilizing x. This suggests the following strategy for proving stability: suppose for purposes

of contradiction that x is GIT-unstable, then find the worst 1-PS, then show that it is actually

not destabilizing. Morrison and I have never gotten this strategy to work in our situation

(we can’t find the worst 1-PS, for the same reason that we can’t compute the absolute weight

filtration discussed in Section 2.2).

However, we can describe the 1-PS for which it is most difficult to prove stability using

our methods: it is the 1-PS for which there is only one point Q1 = Pi in the base locus of VN ,

where bi is the largest value in B, every stage of the filtration is a complete sublinear series

of H0(C,O(1)), and the weights are linearly decreasing (hence, uniquely determined by the

conditions that they decrease to zero and sum to 1).

Of course, just because it is hard for us to show that this 1-PS is GIT-stable does not


mean it is actually the worst 1-PS, but it certainly is a candidate. I believe it would be

interesting either to show that this is the worst 1-PS, or exhibit another 1-PS which is worse.

In the meantime, I mention this 1-PS for its value as an heuristic test for GIT stability

for parameter spaces and linearizations where this is currently unknown, and for testing

putative stability proofs.

Can we improve these results if we use a more complicated filtration than V• asscaffolding?

In the definition of the filtration X• we only take the span of “three-layer” spaces

V αuvs V (1−α)uv

t V v0 . It is natural to ask whether we might get better results by defining a

filtration using spaces of the form V αuvs V βuv

t V (1−α−β)uvw V v

0 .

When m is large, the following heuristic argument suggests that adding more layers will

not change anything. We never really asked what is the best way to produce a basis. We

always began with a space of the form V αuvk V (1−α)uvV v

0k+1 having weight

αrkuv + (1−α)rk+1uv + vr0 and asked the question: for what choice of βj for j = 0 to N

will codim Span(V αuvk V (1−α)uv

k+1 , V β0uv0 V β1uv

1 · · ·V βN uvN V v

0 ) be minimized?

There are constraints. First,∑N

j=0 βk = 1. Also, the weight of the second space in the

span should be less than or equal to that of the first, so

β0r0 + · · · + βNrN ≤ αrk + (1−α)rk+1.

These conditions give a polytope in β-space. Minimizing the multiplicity of each Pi

means minimizing the linear function

f (β0, . . . , βN) = c1,iβ1 + c2,iβ2 + · · · + cN,1βN

over this polytope. The minimum must occur on the boundary, specifically at one (or more)

of the vertices of the polytope, and these are precisely the “three-layer” spaces.

The argument just given should be approximately true when m is very large and divisible

(so that all the exponents are integers), but it could break down badly for small m. So, for

small m stability, we might want to consider filtrations which are much more complicated

than those used in this dissertation.


Comparison to Baldwin’s results

As noted in the introduction to this chapter, Elizabeth Baldwin gave the first GIT

construction of Mg,n in her 2006 Oxford D.Phil dissertation. In fact, she constructed

Mg,n(Pr , d), the Kontsevich-Manin moduli space of stable pointed maps, and Mg,n is the

special case of maps to a point. It may appear that a major difference between Baldwin’s

proof and the proof presented in this chapter is that Baldwin considers maps, not just

curves, but this is misleading; her stability proof is just as original and difficult when applied

to curves as it is for maps. My master’s thesis and Baldwin’s D.Phil work taught us that

going from curves to maps is not difficult (clever modifications to Gieseker’s proof suffice),

but adding marked points in either place is very difficult (genuinely new ideas are needed).

It seems highly likely that if one modified the stability proof given in this chapter in exactly

the way that I modified Gieseker’s stability proof in my master’s thesis, then one would

obtain a new proof of stability of stable pointed maps when the domain curve is smooth.

On the other hand, Baldwin’s proof does not easily accommodate weighted marked

points. When glueing elliptic tails to marked points, it seems very helpful to know that

the marked point has weight one. Perhaps this obstacle may be circumvented, but neither

Baldwin nor I have made an effort to do so.

To me, the major difference between the two results (that is, leaving aside all differences

in techniques) is in the range of linearizations we use. They are closely related but disjoint.

Baldwin’s proof works for a parameter space and range of linearizations where mine does

not (specifically ai = 1 and bi ∈ [ 12 , 1

2 +1

4ν−1 ]), and my proof works in a range where hers

does not (specifically, ai < 1 and bi < 12 ).

Chapter 3

Constructing moduli spaces3.1 Statement of the Construction Theorem

My motivation for studying the stability problem of Chapter 2 was to give GIT construc-

tions of moduli spaces of DM-stable weighted pointed curves. We describe the parameter

spaces and linearizations for this application now.

Let (C, P1, . . . , Pn,A) be a DM-stable weighted pointed curve with n marked points. Write

a :=∑

ai , and assume that 2g − 2 + a > 0. Then for ν sufficiently large and divisible,

(ωC(∑

aiPi))⊗ν =: OC(1) is a very ample line bundle. Write

V = H0(C, (ωC(∑

aiPi))⊗ν ) = H0(C,OC(1))

d = degOC(1) = ν(2g − 2+ a)

N + 1 = dim V = ν(2g − 2+ a)− g + 1

P(t) = h0(C,OC(t)) = dt − g + 1.

Definition 3.1.1 (Definition of the loci I, U , and J.).

• Write I ⊂ Hilb(PN , P(t))×∏n

i=1 PN for the incidence locus where the points in the second

factor land on the curve C ⊂ PN parametrized by a point of the first factor.

• Write U for the maximal open subscheme in I where

1. the curve Ci is connected, where Ci is the curve parametrized by the point i ∈ I

2. the curve Ci ⊂ PN is nondegenerately embedded

3. a relative dualizing sheaf for the morphism C |U → U exists.

• Write J for the locally closed subscheme of U parametrizing weighted pointed curves

embedded by (ωC(∑

aiPi))⊗ν .

67

68 CHAPTER 3. CONSTRUCTING MODULI SPACES

Remark. This description specifies J set-theoretically. We could specify a scheme

structure on J using Fitting ideals, but we will see that J is reduced (see Proposition 3.3.1

below) so I have omitted this.

The DM-stable weighted pointed curve (C, P1, . . . , Pn,A) is represented by a PGL(N +1)-

orbit in the locus J.

See Section 2.2 for a description of the linearizations on I under consideration.

I have chosen not to add any further decorations to I and J, but it is important to

remember that d, N, P(t) all depend on g, n, A and ν. Hence, I and J also depend on

these parameters as well. So, even if g and n are held constant, if A or ν varies, one is

moving between loci in different Hilbert schemes—that is, one is using different parameter

spaces—and this is not variation of GIT in the sense of Thaddeus and Dolgachev and Hu.

On the other hand, if g, n, A, and ν are held constant and only B varies, this is VGIT in the

sense of Thaddeus and Dolgachev and Hu.

The following theorem is a main theorem of this dissertation:

Theorem 3.1.2 (Construction Theorem). Given g, n, and a set of embedding weights A with

0 ≤ ai < 1 for i = 1, . . . , n, and satisfying 2g − 2+ a > 0, it is possible to choose ν, m, and a

set of linearizing weights B such that the GIT quotient J//m,BSL(N + 1) Mg,A.

The proof of Theorem 3.1.2 is postponed to Section 3.4.

3.2 Three lemmas used in the construction

Three lemmas will be needed in the proof of Theorem 3.1.2.

Smoothing weighted pointed stable curves

Lemma 3.2.1 (Smoothability). Let (C, P1, . . . , Pn,A) be a DM-stable weighted pointed curve

over an algebraically closed field k. Then there exists a (possibly singular) family C →

S := Spec k[[t]] such that C0 C and the generic fiber CK is smooth, plus sections σi :

Spec k[[t]] → C satisfying σi(0) = Pi for i = 1, . . . , n.

Proof. Probably the best approach is to consider the pointed curve as a map Σ → C, where Σ

is the disjoint union of n reduced points, and study the deformations of this map. Hassett

3.2. THREE LEMMAS USED IN THE CONSTRUCTION 69

carries this out, at least when ai > 0 ([Hass] Proposition 3.8). However, I will sketch my own

argument below, which avoids the additional homological algebra machinery needed for

deformation theory of maps as detailed in [Ran].

The result is trivial if C is smooth (just take the trivial family over S), so we may assume

that C is nodal in the rest of the proof. It is well-known (though quite involved) that a

Deligne-Mumford stable curve C can be smoothed. Our strategy will be to smooth the curve

and then find the desired sections. A slight nuisance is that the underlying nodal curve

(without the marked points) could have infinitesimal automorphisms. So, we keep track of

enough marked points to kill infinitesimal automorphisms, then find the sections afterward.

Step 1: Reduce to distinct marked points. Weighted marked points may collide. If Pi = Pj

with i < j , then for the remaining discussion we forget Pj and at the end simply take σj = σi

once σi has been found.

Step 2: Split up the smooth marked points and singular marked points. Suppose after Step

1 that there are k distinct marked points on C. We may assume without loss of generality

that the marked points P1, . . . , P` lie at nonsingular points of C while P`+1, . . . , Pk lie at nodes

of C. The cases ` = 0 (all the marked points are nodes) and ` = k (all the marked points are

nonsingular) are possible. We know that a`+1, . . . , ak = 0.

The result of Steps 1 and 2 is that now (C, P1, . . . , P`) is a Deligne-Mumford stable

`-pointed curve, and hence has finitely many automorphisms.

Step 3: Smooth the Deligne-Mumford stable pointed curve. Now we deform the Deligne-

Mumford stable pointed curve (C, P1, . . . , P`) obtained from Steps 1 and 2. We follow [DM]

§2 for this.

First, for dimension reasons and homological dimension reasons (using that nodes

are l.c.i.) one shows that T2 = Ext2C(ΩC(P1 + · · · + P`),OC) vanishes. Thus, infinitesimal

deformations are unobstructed. Next, one proves that T0 = 0; this comes from the fact that

Deligne-Mumford stable pointed curves have no infinitesimal automorphisms. Thus, there is

a universal deformation space for (C, P1, . . . , P`). A happy accident in dimension one is that

this deformation is actually algebraizable. (I am treating this result as a black box, but two

relevant sources are [Vistoli] Prop. 6.3 and [SGA1] Prop. 7.2). Finally, by carefully tracking

what happens to the nodes in the deformation space, we can see that the locus of nodal


curves in the deformation space is a closed subvariety. Thus, [DM] yields a family of curves

π : C → Spec R deforming C whose generic fiber is smooth, where R = k[[t]] if char k = 0,

or else R = W [[t]] where W is the ring of Witt vectors of k ([DM] page 79, [Des] page 4).

Step 5: Finding sections. For this we apply the following result from EGA IV, with

Y = Spec R and X = C and f = π .

Proposition 3.2.2 ([EGA IV] Corollaire 14.5.4). Let Y = Spec A where A is a complete local

noetherian domain. Let f : X → Y be a morphism locally of finite type, and let x be a closed

point of the central fiber of f . Suppose f is equidimensional at x. Then there exists a local

domain A′ containing A, which is finite as an A-algebra, with the property that there is a

section Y ′ → X′ (where Y ′ = Spec A′ and X′ = X ×Y Y ′), and the composition Y ′ → X′ → X is

an immersion whose image contains x.

(Note that since k is algebraically closed, hence perfect, we may characterize W as the

unique complete local domain with maximal ideal (p) and residue field k. Cf. [E], [Mum2],

[Serre]. Since W is complete and local and its maximal ideal is finitely generated, then by

[Mat] Theorem 29.4, W is Noetherian. Thus W [[t]] satisfies the hypotheses above.)

Thus, possibly after making a finite base change to the family obtained in Step 4, we will

be able to find the desired sections.

Finally, if char p ≠ 0, since k is the residue field of W , we have a homomorphism

W [[t]] → k[[t]], which induces a morphism Spec k[[t]] → Spec W [[t]]. By pulling back the

family C and its sections, we get the desired family over k[[t]].

Remark 1. Note that in general the analogous statement for weighted pointed stable

maps fails—there are maps which are not smoothable. In general this reflects the fact that

for curves, the locus J and the moduli space are irreducible, but for maps, the parameter

space and moduli space are reducible.

Remark 2. By working harder, we could get the points to be distinct on the generic fiber,

but we do not need this for the proof of Theorem 3.1.2.


GIT semistable replacement

We need the following general GIT result. The notation of this subsection is local in the

sense that the symbols m, n, X, B, N, etc. do not have the meaning they have in the rest of

this dissertation.

Lemma 3.2.3 (“GIT semistable replacement”). Suppose a reductive group G acts on a projec-

tive scheme X with a linearization given by L. Let S be a smooth, one-dimensional scheme,

s0 a closed point of S; write K := S − s0. (If S is Spec R for a valuation ring R, K is the open

point corresponding to Spec K.) Suppose a morphism f : K → Xss is given. Then there exists

• a ramified cover S′ h→ S

• a morphism f ′ : S′ → Xss

• a point s′0 ∈ S′ with h(s′0) = s0

• a morphism g : S′ − s′0 → G

such that for every s′ ∈ S′ − s′0, we have f (h(s′)) = g(s′) · f ′(s′).

Versions of this result appear in several places. This is the content of [Seshadri] Theorem

4.1.i. His argument consists of several reduction steps; his proof and remarks together

occupy five pages. The result is also stated in [Mum3] Lemma 5.3 in the important special

case X = P(V ), with a one paragraph proof. Finally, Caporaso gives it this wonderfully

memorable name in [Cap] 1.2.1. I will outline Seshadri’s proof below, referring to [Seshadri]

for full details.

Proof. Write XK for X ×S K and GK for G ×S K. Define ZK be the closure in XK of the GK

orbit of the image of the map f : K → Xss , and write ZS for the unique closed subscheme of

XS faithfully flat over S such that the generic fiber of ZS → S agrees with ZK (cf. [Seshadri]

Remark 4.4).

Reduction 1: It is enough to show that q : ZssS → S is faithfully flat. For, then by [EGA IV]

§14.17, we know that there exists a surjective finite morphism S′ h→ S such that the base

change of q by h has a section θ. Then θK′ is in the open GS′ -orbit of ZssS′ corresponding to

K′.


Reduction 2: It is enough to show that q : ZssS → S is surjective. Flat and surjective implies

faithfully flat. We argue that q is flat: ZssS → ZS is an open immersion, hence flat. ZS → S is

flat by construction. Thus q is flat, since it is the composition of two flat morphisms.

For the next steps, write NSν→ ZS for the normalization of Z, and write q′ := q ν.

Similarly we will write N0 for the preimage of s0 in NS , and NK for the preimage of K in NS ,

etc. Finally, we abuse notation and continue write L for ν∗L. (Note: we do not use normality

until Reduction 6 below, but the sensible place to set it up is here.)

Reduction 3: It is enough to show that the fiber Nss0 is nonempty. By hypothesis, our

original f satisfied Im(f ) ⊂ Xss , so the only point where we don’t know surjectivity is s0.

Reduction 4: It is enough to show that for some m > 0, we have H0(NssS , L⊗m

S )GS ≠ 0.

Suppose H0(NssS , L⊗m

S )GS ≠ 0. Then, by dividing by powers of the uniformizer at s0 if

necessary, we can find a τ ∈ H0(NssS , L⊗mn

S )GS such that τ not identically zero on N0 red. We

have τ|N0 is G-invariant, so this shows Nss0 ≠ .

Reduction 5: It is enough to show that H0(NssK , L⊗n

K )GK ≠ 0. We have

H0(NssS , L⊗m

S )GS ≠ 0 ⇐⇒ H0(NssK , L⊗n

K )GK ≠ 0.

One direction is restriction, the other direction is multiplying by powers of the uniformizer

at s0 ∈ S to clear denominators.

Reduction 6: It is enough to show that H0(U, L⊗nK )GK ≠ 0, where U is an open orbit in NK .

This uses the nontrivial fact that H0(NssK , L⊗n

K )GK H0(U, L⊗nK )GK , for which I will simply

cite [Seshadri], Theorem 4.2.ii, page 529. (Note he proves part ii. before proving part i. so

we do not have any problems with circular logic here.) This step requires normality.

Reduction 7: It is enough to show that L⊗nK |U is trivial. Then we clearly have

H0(U, L⊗nK )GK ≠ 0.

To show that L⊗nK |U is trivial, we apply Proposition 3.2.4 below, and this completes the

sketch of the proof of Lemma 3.2.3.

Proposition 3.2.4 ([Seshadri] Remarks 4.8, 4.9). Let x ∈ Xss , and let U be the G-orbit through

x with its reduced subscheme structure. Then there exists n > 0 such that Ln|U is trivial.

Proof. Let H be the isotropy subgroup scheme of G at x. Then U = H \G. Let χ : H → Gm

be the rational character of H defining L|U . Then we want to show that χn is the trivial


character for some n. Suppose not. Then we can find a 1-PS λ : Gm → H such that χ λ

is surjective. But then µL(x, λ) ≠ 0, so either λ or λ−1 is destabilizing, which contradicts

x ∈ Xss .

Uniqueness of stable reduction

We also need the uniqueness part of the stable reduction theorem. Versions of this for

DM-stable weighted pointed curves are given in [Hass] Proposition 3.7 and [AG] Theorem

1.11. Here we prove a slightly weaker version, which is what we use in the proof of Theorem

3.1.2.

Lemma 3.2.5 (Uniqueness of stable reduction). Let S be a smooth, one-dimensional scheme,

and let s0 a closed point of S. Then a family of smooth weighted pointed curves over S − s0

has at most one extension to a family of DM-stable weighted pointed curves over S.

Proof. We temporarily ignore any sections for which ai = 0 and return to these at the end.

Suppose there are two families C and C ′ of DM-stable weighted pointed curves over S

extending the original family. The minimal resolutions of each of these families are two

smooth families of curves over S which agree generically. If g ≥ 1, then by the uniqueness

of minimal models of surfaces (cf. [Licht] Theorem 4.4 and [Shaf]) there is a canonical

isomorphism between the blowups, and we thus obtain in a canonical way a regular surface

C dominating the two original families. If g = 0 we can still obtain a regular surface C

dominating the two original families via two sequences of monoidal transformations (cf.

[Licht] Theorem 1.15).

We can describe the maps C → C and C → C ′ very precisely. The families C and C ′ are

regular away from the singularities of the central fibers, and at nodes of the central fiber

they are formally isomorphic to k[[x, y, t]]/(xy − tm) for some m ([DM] page 81, [Des] page

4). Hence the monoidal transformations giving C introduce chains of P1’s in the central

fiber, and every component of the exceptional divisor is connected to the rest of the central

fiber at exactly two nodes.

Note that the proper transforms of the divisors Di ⊂ C and D′i ⊂ C ′ agree. Write Di for

these lifts.


Then the families C and C ′ are each obtained from C by contracting the (−2)-curves on

which KC +∑

aiDi is not ample. This shows that C and C ′ are isomorphic over S.

Finally, if any of the ai are 0, the corresponding sections σi : S → C and σ ′i : S → C ′

agree on C0 C ′0 also, since C C ′ is separated.

3.3 Smoothness, I, and J

The next two results are not logically required for the proof of Theorem 3.1.2. Neverthe-

less it is certainly reassuring to know that the parameter spaces we consider are smooth,

and this may be useful or necessary for future applications.

Proposition 3.3.1 (cf. Lemma 3.35 in [HM]). The locus J is reduced and smooth of dimension

(3g − 3+ n)+ (N2 + 2N).

Proof. The following proof is adapted from Lemma 3.35 in [HM] with only minor modifica-

tions. The analogous proposition in [Gies] is Theorem 2.0.1.

We have a naive dimension count for J: the curve depends on 3g−3 moduli, the marked

points add another n, and choosing a basis (up to scalars) of H0(C, ων (∑

νPi)) contributes

another (N + 1)2 − 1.

Next we introduce the locus Y in I where (C ⊂ PN , P1, . . . , Pn) is abstractly isomorphic to

(C, P1, . . . , Pn). That is, the embedding can run over all line bundles L on C, and all choices

of bases of H0(C, L). A naive dimension count for Y is g + (N + 1)2 − 1.

We will show that the tangent space to J ∩ Y has dimension (N + 1)2 − 1. This can only

happen if the tangent spaces to J and Y are each as small as possible, showing that both J

and Y are smooth and reduced.

Consider a tangent vector Spec k[ε]ξ→ I. (Here we follow the standard practice of writing

Spec k[ε] for Spec k[ε]/(ε2).) We obtain a family of curves D by pulling back the universal

family C over I:

Dξ //

C

Spec k[ε]

ξ // I

Now, ξ is tangent to J if and only if ξ∗OC (1) ωνD/ Spec k[ε](

∑νaiDi).

3.3. SMOOTHNESS, I, AND J 75

On the other hand, ξ is tangent to Y iff D is a first order isotrivial deformation of C;

then D C × Spec k[ε]. Putting this together, ξ is tangent to J ∩ Y iff

ξ∗OC (1) OSpec k[ε] ⊗ωνC(∑

νaiPi). Hence we see that a choice of basis (up to scalars) of

H0(C, ωνC(∑

νaiPi) determines a tangent vector ξ : Spec k[ε] → J ∩ Y , and vice versa, and

this establishes the dimension count (N + 1)2 − 1.

In the next proposition we show that a much larger locus in I is smooth. Again, this

result is not used anywhere in this dissertation, but could be very useful for future VGIT

studies. I have drawn heavily from [HM] page 103 and [Cap] Lemma 2.2; I noticed that their

argument proves a more general statement than just what they claimed, but all I have really

added to their argument is details and references.

Proposition 3.3.2. Let [C, P1, . . . , Pn] be a point of the incidence locus

I ⊂ Hilb(PN , P(t))×∏n

i=1 PN , where C is a one-dimensional local complete intersection in PN

with H1(C,OC(1)) = 0. Then I is smooth and reduced at [C, P1, . . . , Pn].

Proof. There is only one Hilbert scheme in the picture, so we will abbreviate our notation

and simply write Hilb. Also, note that the result will follow for I if we prove that Hilb is

smooth at [C].

In the language of tangent-obstruction theory (cf. [FG]), the tangent space to Hilb is

HomOC (ΩC ,OC) and the obstructions lie in Ext1OC

(ΩC ,OC). We define the normal sheaf of

C ⊂ PN to be

N :=NC/PN :=H omOC (I/I2,OC).

We can use the normal sheaf to get cohomological interpretations of the tangent space and

obstruction space:

T[C]Hilb = HomOC (ΩC ,OC) H0(C,N )

Ext1OC

(ΩC ,OC) H1(C,N )

Thus, to show that Hilb is smooth at [C], it suffices to show that H1(C,N ) = 0.

To see that H1(C,N ) = 0, we use the familiar exact sequence for the normal sheaf:

0 // TC // TPN |Cf // N // T1

C// 0. (3.1)


Here TC is the tangent sheaf of C; TPN is the tangent bundle of PN ; N is the normal sheaf

as explained above; and T1C = Ext1

OC(ΩC ,OC). The exactness of the sequence (3.1) follows

from [LS] 2.4.2, 3.1.1, and 3.1.2.

Break (3.1) into two short exact sequences:

0 // TC // TPN |C // Im(f ) // 0

0 // Im(f ) // N // T1C

// 0

(3.2)

From the long exact sequence associated to the first line of (3.2), we see that H1(C,TPN |C)

surjects onto H1(C, Im(f )). Observe that T1C is supported on the singular locus of C, which

is a zero-dimensional scheme, so H1(C, T1C) = 0. Thus we get a surjection of H1(C, Im(f ))

onto H1(C,N ). Composing these maps, we get a surjection of

H1(C,TPN |C) % H1(C,N ). (3.3)

But the Euler sequence for PN tells us that TPN is a quotient of OPN (1)⊕N+1. Restricting

this to C, we get a surjection

H1(C,OC(1)⊕N+1) % H1(C,TPN |C) (3.4)

since H2(C, Ker(OC(1)⊕N+1 → TPN |C)) = 0 because C is one-dimensional.

Composing the surjections (3.3) and (3.4), we obtain

H1(C,OC(1)⊕N+1) % H1(C,N ). (3.5)

Also H1(C,OC(1)⊕N+1) =N+1⊕

H1(C,OC(1)). But by hypothesis we have H1(C,OC(1)) =

0. Hence H1(C,N ) = 0, and the Hilbert scheme has the expected dimension at [C].

Remark. This result may seem surprising to those familiar with “Murphy’s Law in

algebraic geometry” (cf. [Vakil]), but there is no contradiction. Mumford’s original example

is for genus 24 curves of degree 14 in P3, which are special (at least, if the embedding is

complete).

3.4 Proof of the Construction Theorem

In this section we give the proof of the Construction Theorem (Theorem 3.1.2).

3.4. PROOF OF THE CONSTRUCTION THEOREM 77

Proof of the Construction Theorem (Theorem 3.1.2). We are given g, n, and A. We need to

choose ν , m, and B and an ε, of course taking care that the choices are made in a noncircular

fashion.

Step 1. We begin with the choice of ν. Choose ν sufficiently large and divisible that all

the following conditions are satisfied. (There is a specific motivation for imposing each

condition; for conditions 1.–4. this will be explained later in the proof.)

1. ν > (9g − 7)/(2g − 2+ a)

2. For every subset I ⊆ [1..n] such that∑

I ai < 1, choose ν sufficiently large that

ν > 1/(2− 2∑

I ai).

3. For all i = 1, . . . , n, if ai ≠ 0, then ν > ga(2g−2+a)ai

4. For every subset I ⊆ [1..n] such that∑

I ai > 1, choose ν sufficiently large that

ν2(2g−2+a)

∑I

ai − 1

+ν

−g∑

Iai −

∑Ic

ai +12

(2g − 2+ a)−g−1

+12

(g+1) > 0.

(Note that since 2g − 2+ a > 0 and the condition only applies when∑

I ai > 1 we can

always arrange for this by taking ν sufficiently large.)

5. νai ∈ Z for all i = 1, . . . , n

Since g and A were given, the choice of ν determines all the following:

d := ν(2g − 2+ a)N := d − g + 1

the Hilbert polynomial P(t) = dt − g + 1the loci J ⊂ I ⊂ Hilb(PN , P(t))×

∏ni=1 PN

Step 2. Now that ν is chosen, we take bi = ν2ν−1 ai for all i = 1, . . . , n. Condition 2. in the

choice of ν ensures that∑

I bi < 1/2 whenever this is required in Theorem 2.8.1. Next, we

choose an ε (to be used in Theorem 2.8.1). Recalling lines (2.58) and (2.60) from Section 2.7,

it is sufficient to take

ε ≤ 1

2N(N+1) (N − 2g + 3− 2b) if n ≥ 11

2N(N+1) (N − 2g + 3) if n = 0.

Thus B and ε are suitable to be used in Theorem 2.8.1.


Step 3. Now that we have g, n, A, ν, d, N, J, B, and ε, next choose m sufficiently large

to satisfy the requirements of the GIT-Stability Theorem (2.8.1) and Theorems A.5.3 and

B.0.5 from the Appendix. (Note: all the other constants appearing in the statements in the

Appendix are determined by the Hilbert polynomial, which was fixed in Step 1.) Also, choose

m sufficiently divisible that bim2 is an integer. Finally, set m′i = bim2.

Condition 1. in choosing ν together with these choices of m and B is used in the Potential

Stability Theorem (A.5.3) to rule out singularities other than nodes. Conditions 3. and 4.

ensure that the Potential Stability Theorem conditions on marked points hitting nodes and

marked points colliding with each other precisely match the description of a DM-stable

weighted pointed curve.

With these choices, the GIT-stability Theorem (2.8.1), and the Potential Stability Theorem

(A.5.3) and Theorem B.0.5 from the Appendix, hold. We have: smooth curves are GIT-stable;

anything GIT-stable is potentially stable (see Definition A.5.4); and Jss is closed in Iss .

We now show that the quotient J//SL(N + 1) Mg,A. The argument is very similar to

[Gies] Theorem 2.0.2 and [Mum3] p. 93–94.

Let (C, P1, . . . , Pn,A) be an arbitrary DM-stable weighted pointed curve of genus g.

Then by the Smoothability Lemma (Lemma 3.2.1), we may find a family π : C → S :=

Spec k[[t]] whose generic fiber is smooth and whose central fiber is C. Choose a frame for

π∗ωνC /S(

∑νaiσi(S)). This determines an embedding C ⊂ PN and hence a map β : S → J.

By the GIT-Stability Theorem (2.8.1), we know that CK is GIT-stable. We also know by

Theorem B.0.5 that Jss = Jss. Thus, since Jss

is projective, we may apply the GIT semistable

replacement (Lemma 3.2.3) to conclude that there is a ramified cover S′ → S such that S′

maps to Jss . Pulling back the universal family to S′, we get a family C ′ whose general fiber

is isomorphic to CK and whose special fiber has the same stable reduction as C. But C ′0 is in

J, so it must be a DM-stable weighted pointed curve. Thus, C C ′0 , and hence there is a

GIT-stable point representing C.

Chapter 4

Polarizations on Mg

As discussed in Section 1.2, GIT requires two ingredients: a parameter space with group

action as well as a linearization of the group action. If the parameter space is a projective

scheme and the line bundle which is linearized is ample, then the quotient is really a pair:

the quotient scheme together with a polarization in its ample cone corresponding to the

O(1) on Proj(⊕H0(X, L⊗n)G).

In [Mum3] Mumford computes the polarization on Mg coming from his GIT construction,

which uses a locus in the Chow variety as the parameter space and the natural linearization

on the Chow variety. In Section 4.1 I compute the polarization on Mg coming from Gieseker’s

or my GIT construction, which use a locus in a Hilbert scheme as the parameter space. Using

the universal properties of the Chow variety and Hilbert scheme, one can show that the loci

in the Chow variety and in the Hilbert scheme are isomorphic, so I will denote both by J;

what differs in the two constructions is the linearization. I will refer to the two different

constructions as “via the Chow linearization” or “via Hilbert linearizations.” (Note: the

terms “Chow quotient” and “Hilbert quotient” already exist in the literature with a different

meaning, due to Kapranov, and that is why I have avoided these terms.) The calculation of

the polarization associated to Hilbert linearizations is practically a byproduct of Mumford’s

calculation, but when we examine the resulting formula with a view toward studying the

ample cone of Mg via GIT, we will see that using Hilbert linearizations potentially offers a

much broader view than using Chow linearizations alone. We begin by recalling the Picard

group of Mg and Mumford’s picture.

The Picard group of Mg

It is well-known that Pic(Mg) ⊗Q is generated by the classes λ and δirr, δ1, . . . , δbg/2c,

and that these classes are independent (if g ≥ 3). In this chapter we will really only use

λ and the total boundary divisor δ = δirr + δ1 + · · · + δbg/2c. In light of the rest of this

79

80 CHAPTER 4. POLARIZATIONS ON Mg

dissertation perhaps the most efficient way for us to define them is as follows:

Let π : C → J be the universal family of the Hilbert scheme restricted to J. Then

λ := c1(π∗ωC /J), and δ is the divisor on J consisting of all points j where the fiber of π

above j has at least one node. These are SL(N+1)-equivariant and hence descend to classes

on Mg .

Mumford’s early results on the ample cone

Looking ahead, we will see in (4.3) that polarizations arising from GIT quotients of J

via Chow or Hilbert linearizations must lie in the (δ, λ)-plane inside the ample cone of

Mg. Following [Mum3], we will take the axes to be −δ and λ. So, for instance, the class

κ = 12λ− δ corresponds to the ray in the first quadrant with slope 12.

Mumford’s construction of Mg via Chow linearizations with ν ≥ 5 established that

classes of slope 11.2 to slope 12 were ample. He also knew Arakelov’s result that λ is nef.

Hence, he knew that the slice of the ample cone in the (δ, λ)-plane ranged at least from

slope 11.2 to the λ axis. Mumford also found that the class 11λ− δ is not ample on Mg ; it

is trivial on a family which has a fixed component of genus g − 1 and an elliptic tail whose j

invariant is allowed to vary. Thus, the GIT quotient for the Chow linearization with ν = 4 is

not Mg , and slope 11 is a lower bound for the ample cone in the (δ, λ)-plane. (Later Cornalba

and Harris [CH] proved that slope 11 is indeed the sharp lower bound, but they did not

exhibit GIT quotients whose polarizations approach slope 11.) Thus, GIT quotients using

Chow linearizations only produce a narrow sector of the ample cone in the (δ, λ)-plane.

4.1 The Polarization Formula for Mg

In this section we calculate the polarization on Mg coming from Gieseker’s or my GIT

construction via the Hilbert scheme, and compare it to the polarization on Mg coming from

Mumford’s GIT construction via the Chow variety.

For the reader’s convenience, I have included a dictionary summarizing how to translate

the notation of [Mum3] into mine.

In the last entry of the table, κ is defined as π∗(c1ωC /J)2, and I am applying the

well-known relation on the stack κ = 12λ − δ rather than introducing another letter for

4.1. THE POLARIZATION FORMULA FOR Mg 81

Figure 4.1: Mumford’s picture of the ample cone

1

1

12

Ample

δ

λ

From Chow linearizations

Table 4.1: Dictionary

Quantity Mumford’s notation My notation

genus g g

power of canonical bundle e νused to embed curve

(projective) dimension of ν − 1 Npluricanonical linear series

degree of pluricanonical series d d

parameter specifying linearization n mon Hilbert scheme

locus in Hilbert scheme of H Jpluricanonically embedded curves

top exterior power of∧max det

a vector bundle

the group law in Pic is multiplicative additive

the class on the moduli stack δ δof the divisor of nodal curves

the class 12λ− δ µ κ


12λ− δ.

The following two lines are translations into my notation of [Mum3] page 106, line (5.17)

and the line following it:

det π∗OC (m) =(

νm2

)κ + λ+mP(m)Q (4.1)

(N + 1)Q = −(

ν2

)κ − λ (4.2)

Thus we get: for any m such that the quotient J//SL(N + 1) Mg ,

the polarization on Mg

=(

νm2

)κ + λ+mP(m)Q

=(

νm2

)κ + λ+ mP(m)

N + 1(−(

ν2

)κ − λ)

=((

νm2

)−(

ν2

)mP(m)N + 1

)κ +

(1− mP(m)

N + 1

)λ

=((

νm2

)−(

ν2

)mP(m)N + 1

)(12λ− δ)+

(1− mP(m)

N + 1

)λ

=[

12((

νm2

)−(

ν2

)mP(m)N + 1

)+ 1− mP(m)

N + 1

]λ−

[(νm2

)−(

ν2

)mP(m)N + 1

]δ

= g − 1N + 1

[(6ν2 − 2ν)m2 + (−6ν2 + 1)m+ (2ν − 1)

]λ

+ g − 1N + 1

[−1

2ν2m2 + 1

2ν2m

]δ

= 12ν − 1

[(6ν2 − 2ν)m2 + (−6ν2 + 1)m+ (2ν − 1)

]λ (4.3)

+ 12ν − 1

[−1

2ν2m2 + 1

2ν2m

]δ.

When m is asymptotically large, the m-Hilbert linearizations approach the Chow lin-

earization. More precisely, recall the following result of Mumford, Fogarty, and Knudsen:

Proposition 4.1.1 (cf. [Mum3] Proposition 5.16 and references there). Let J be a locally

closed subscheme of a Hilbert scheme parametrizing pluricanonically embedded varieties of

dimension r , and let C be the universal family over J. Then for m >> 0, there exist invertible

sheaves µi such that

det π∗OC (m) =r+1Oi=0

µ(mi )

i , (4.4)

and the Chow linearization is given by µr+1.

4.2. INTERPRETING THE POLARIZATION FORMULA 83

Following Mumford, we use Proposition 4.1.1 to find the polarization on Mg coming

from the Chow GIT construction: Using the identities

m2 = 2(

m2

)+(

m1

)(4.5)(

νm2

)= ν2

(m2

)+ ν(ν − 1)

2

(m1

)(4.6)

a little algebraic manipulation on (4.1) yields:

µ2 = ν2κ + 2dQ (4.7)

µ1 = 12

ν(ν − 1)κ − (d + g − 1)Q (4.8)

µ0 = λ. (4.9)

Further simplification yields

µ2 =ν(g − 1)

N + 1(νκ − 4λ) .

and dividing by ν(g−1)N+1 yields the familiar formula for the Chow polarization:

(12ν − 4)λ− νδ. (4.10)

We compare this to the formula (4.3). If we multiply the terms of order m2 in (4.3) by

2(N + 1)/(g − 1)(ν), we obtain

(12ν − 4)λ− νδ,

which indeed matches the polarization Mumford computes coming from the Chow lineariza-

tion.

4.2 Interpreting the Polarization Formula

Isopolarizations

Michael Thaddeus suggested an approach to (4.3) that I have found very helpful: We plot

the curves of isopolarization, that is, the level curves of

−(6ν2 − 2ν)m2 + (−6ν2 + 1)m+ (2ν − 1)−1

2 ν2m2 + 12 ν2m

= c (4.11)

for different contours c, where the numerator and denominator are the coefficients of λ and

δ in (4.3), and the minus sign is introduced to make our discussion more compatible with

[Mum3] and [HM].


Figure 4.2: Isopolarization curves

5 10 15 200

10

20

30

40

50

This contour plot was produced in Mathematica. From left to right, the contours for

c = 11, 11.1, . . . , 11.7 are shown.

The formula (4.11) and the picture suggest the following proposition:

Proposition 4.2.1. 1. Arbitrarily large value of c are obtained as m → 0 and ν →∞.

2. The first quadrant branch of each level curve has a vertical asymptote and a horizontal

asymptote.

Proof. The first statement is an easy limit computation. For the second statement, we

compute an implicit formula for the derivative:

dmdν

= νc(m2 −m)− (12ν − 2)m2 + 12νm− 2(12ν2 − 4ν)m− (6ν2 + 1)− c

2 ν2(2m− 1)(4.12)

From the degrees in ν and m in the numerator and denominator of this formula, it is clear

that dmdν → 0 as ν →∞, and dm

dν →∞ as m →∞.

The formula (4.3) and the picture above must be interpreted carefully. We do not

know that the quotients for these values of m and ν are Mg with the polarization given by

this contour. Rather, the formula only tells us that if the quotient is Mg, then it has the

4.2. INTERPRETING THE POLARIZATION FORMULA 85

polarization indicated. Note too that not every point on a contour corresponds to a GIT

quotient. To me, the parameter space J used for our GIT quotients only makes sense for

positive integer values of ν . On the other hand, m can take rational values, since fractional

linearizations make sense in GIT.

The quotients when ν = 4

Mumford and Gieseker’s work shows that when ν ≥ 5 then the GIT quotient

J//SL(N + 1) with either the Chow or Hilbert linearization is Mg. Schubert showed that

when ν = 3, the quotient of the Chow parameter space is not Mg. He named this quotient

Mpsg , the moduli space of “pseudostable curves”; it is the contraction of Mg where the

divisor δ1 of curves with an elliptic tail maps to the class of a curve with a cusp at the

singular point ([Schubert]).

I was surprised to discover recently that the case ν = 4 is not explicitly covered by the

existing literature. As noted previously, 11λ− δ is not ample on Mg . From the Polarization

Formula, when ν = 4, the polarization increases to 11 as m → ∞. Thus, the quotients for

these linearizations cannot be Mg. It seems likely that these quotients are isomorphic to

Mpsg ; perhaps the arguments of [Schubert] can all be adapted to the case ν = 4, but I have

not checked this.

Small m

Proposition 4.2.1 part 1. suggests that as m approaches 0 and ν approaches ∞, if the

GIT quotients are Mg , then their polarizations would fill in the part of the ample cone lying

between slope 12 and the λ axis. Note the “if”—I do not know a reason why we should

expect that these quotients are isomorphic to Mg , nor a reason why we should expect that

they are not.

Studying small m linearizations when ν is also small would be interesting for applications

to the Log Minimal Model Program for Mg begun by Hassett and Hyeon . They find that the

first birational map of the sequence is the contraction to Mpsg ; the divisor of curves having

an elliptic tails is blown down to a curve with a cusp ([HH]). The second birational map is a

flip (a blowup followed by a blowdown); curves with an elliptic bridge flip to curves with a


tacnode.

It is intriguing that all the spaces they encounter can be described as GIT quotients;

I know of no reason to predict a priori that this should occur. For the first two maps,

the parameter space is different for each of the quotients in question. However, the next

map, conjecturally, flips genus 2 tails to ramphoid cusps (singularities locally given by the

equation y2 = x5), and that the spaces in question are GIT quotients of Hilbert schemes of

bicanonically embedded curves, with linearizations given by m decreasing from 6+ ε to 6

to 6− ε.

To my knowledge, there is still a large gap between what is predicted and what is proven.

For instance, for g ≥ 4, it is not even known whether these GIT quotients are nonempty for

such small m!

Studying GIT quotients when m is small seems a challenging question, and existing

techniques (those employed by Mumford, Gieseker, Schubert, Swinarski, and Baldwin) are

not well-suited for it. In trying to adapt the Stability Proof, we lose the key Lemma 2.3.1

right off the bat. We also lose the Potential Stability Theorem; we assumed m >> 0 in every

step of the proof, beginning with Proposition A.2.1, and I do not see how to remove this.

Summary

GIT quotients of J with Chow linearizations produce only a narrow sector of the ample

cone in the λ− δ plane. But GIT quotients of J with Hilbert linearizations might produce all

of the slice of the ample cone in the λ− δ plane. This suggests the following problems:

Problems 4.2.2. 1. Study GIT quotients of J with ν = 4 as m → ∞ to fill this gap in the

literature.

2. Study GIT quotients of J for small m to fill out the ample cone in the (δ, λ)-plane.

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Appendix A

The potential stability theorem

As mentioned in the introduction, in previous GIT constructions of moduli spaces of

curves or maps, it is not proved directly that nodal Deligne-Mumford stable curves are

GIT-stable. Instead, the proofs proceed indirectly. First, one shows that smooth objects are

GIT-stable. Second, one shows that anything which is GIT-semistable is DM-stable or very

nearly so. Finally, a deformation argument is used to establish GIT-stability of the nodal

objects of the moduli problem.

The second step has come to be called the Potential Stability Theorem. The most general

version of the Potential Stability Theorem currently available is in the preprint [BS]. Although

weights are not considered there, with very minor adaptations, the proof there would cover

the case at hand. However, differences in notation between that paper and this dissertation

make them very difficult to read alongside one another. Therefore, I have included the full

proof here in notation which matches the rest of this dissertation to show that the details

have been checked. On the other hand, the differences between this proof and [BS] are

superficial, and the relevant section of [BS] was written jointly by Elizabeth Baldwin and me.

Therefore I have relegated this proof to an appendix, as I do not make for this proof the

claims of originality and authorship which apply to the rest of my dissertation. In March

2008 the paper [BS] was accepted for publication in International Mathematics Research

Papers; Baldwin and the journal have given me permission to include this material here.

There is one more point to be discussed at the outset. The theorem we prove here is a

result about what happens for specific linearizations; referring back to Section 2.2, we will

ultimately set m′i = bim2. However, we shall go as long as possible without imposing this

condition, getting partial results about more general linearizations than just those used to

construct the moduli spaces Mg,A.

89

90 APPENDIX A. THE POTENTIAL STABILITY THEOREM

A.1 Strategy and setup for the Potential Stability Theorem

Many of the results in this section are proved by a common strategy, so we outline it

here in detail and refer back to this subsection as needed.

Many of the proofs will be by contradiction. We will suppose that (C, P1, . . . , Pn) ⊂ PN

is connected and SL(N + 1)-semistable with respect to a given linearization, and also that

C has some “undesirable” geometric property. We will then exhibit a 1-PS λ′ of SL(N + 1)

which we claim is destabilizing. We will not describe λ′ explicitly; instead we obtain it

implicitly from a 1-PS λ of GL(N + 1). The 1-PS will all have a special form: they will give

rise to a two- or three-stage weighted filtration 0 ⊂ V2 ⊆ V1 ⊆ V0 := H0(C,O(1)). We may

choose a basis v0, . . . , vN diagonalizing the λ action and adapted to this filtration. Let rj be

the weight of the basis elements corresponding to the stage Vj . That is, λ acts as follows:

λ(t)vi = tr0 vi , t ∈ C∗, 0 ≤ i ≤ codim V1 − 1λ(t)vi = tr1 vi , t ∈ C∗, codim V1 ≤ i ≤ codim V2 − 1λ(t)vi = tr2 vi , t ∈ C∗, codim V2 ≤ i ≤ N

For our purposes, to specify the λ action, it is enough to specify V2 and V1, and the weights

r0, r1, r2 (which appear with multiplicities in λ, in general). Then there is a 1-PS λ′ of

SL(N + 1) associated to λ obtained by normalizing the weights. Specifically, now listing the

weights as many times as they appear (that is, without multiplicities), the λ′ weights r ′i are

obtained by the rule

r ′i = (N + 1)ri −N∑

j=0rj .

This leads to the following restatement of the numerical criterion for the λ′-action in terms

of the λ action:

Lemma A.1.1 (cf. [Gies] page 10). Let λ be a 1-PS of GL(N +1), with λ′ the associated 1-PS of

SL(N + 1). There exist monomials M1, . . . , MP(m) in Bm such that ρCm(M1), . . . , ρC

m(MP(m))

is a basis of H0(C,O(m)), and there exist basis elements v1, . . . , vn such that vi(Pi) ≠ 0, with

µL((C, P1, . . . , Pn), λ′) =P(m)∑j=1

wtλ′(Mj )+n∑

i=1r ′i m′

i

=P(m)∑

j=1wtλ(Mj )+

n∑i=1

wtλ(vi)m′i

(d − g + 1)−mP(m)+

n∑i=1

m′i

N∑j=0

wtλ(vj )

.

(A.1)

A.1. STRATEGY AND SETUP FOR THE POTENTIAL STABILITY THEOREM 91

We shall find a lower bound for µL((C, P1, . . . , Pn), λ′) by filtering the vector space

H0(C,O(m)) according to the weight with which λ acts. The filtration is constructed in the

same way every time; we describe it now. Assume rN ≥ 0 (this will be the case in all our

applications). Let R be a positive integer such that∑N

j=0 rj ≤ R. For 0 ≤ p ≤ m let Ωmp be the

subspace of H0(PN ,O(m)) spanned by monomials of weight less than or equal to p. Let

Ωmp := ρC

m(Ωmp ) ⊂ H0(C,O(m)).

We have a filtration of H0(C,O(m)) in order of increasing weight:

0 ⊆ Ωm0 ⊆ Ωm

1 ⊆ · · · ⊆ Ωmm = H0(C,O(m)). (A.2)

Note that the weight increases with these indices, whereas it decreased for the Vj ’s. Write

δp = dim Ωmp .

We will get bounds on δp which depend on the problem at hand. However, these bounds

will always have the same format (described below). The next lemma shows how to estimate

the minimal weight µL((C, P1, . . . , Pn), λ′) given these bounds on δp:

Lemma A.1.2. In the setup described above, suppose that

δp ≤ (d −α)m+ βp + εp

where α, β, εp are constants. Set

ε := 1m

r0m−1∑p=0

εp.

Write v1, . . . , vn for the elements of minimal weight satisfying vi(Pi) ≠ 0. Then

µL((C, P1, . . . , Pn), λ′) ≥((

r0α− r 20

β2

)(d − g + 1)− Rd

)m2

+n∑

i=1

((d − g + 1) wtλ(vi) − R

)m′

i

−(

(r0(g − 1)− r0β2+ ε)(d − g + 1)+ R

)m. (A.3)

Proof. Suppose M1, . . . , MP(m) are monomials such that the set ρCm(M1), . . . , ρC

m(MP(m)) is

a basis of H0(C,O(m)). As our filtration is in order of increasing weight, a lower bound for


∑P(m)j=1 wtλ(Mj ) is

∑mr0p=1 p(δp − δp−1). We calculate:

mr0∑p=1

p(δp − δp−1) = r0mδmr0 −mr0−1∑

p=0δp

≥ r0m(dm− g + 1)−r0m−1∑

p=0((d −α)m+ βp + εp)

=(

r0α− r 20 β2

)m2 −

(r0(g − 1)− r0β

2+ ε

)m,

where ε := 1m∑mr0−1

p=0 εp. Let λ′ be the associated 1-PS of SL(N + 1). Thus, using Lemma

A.1.1, we calculate:

µL((C, P1, . . . , Pn), λ′)

≥(r0α− r 2

0β2

)m2 −

(r0(g − 1)− r0β

2+ ε

)m+

n∑i=1

wtλ(vi)m′i

(d − g + 1)

−m(dm− g + 1)+

n∑i=1

m′i

N∑j=0

rj

=

((r0α− r 2

0β2

)(d − g + 1)− Rd

)m2 +

n∑i=1

(wtλ(vi(d − g + 1) − R

)m′

i

−((

r0(g − 1)− r0β2+ ε

)(d − g + 1)+ R

)m,

where we have used the bounds 0 ≤∑N

i=0 ri ≤ R to estimate appropriately, according to the

sign of each term.

The following lemma is also included in this section for future reference.

If C is a general curve, we have an inclusion i : Cred C. The reduced curve Cred

has normalization π ′ : Cred → Cred. Following Gieseker in [Gies] page 22, we define the

normalization π : C → C by letting C := Cred and π := i π ′. Then, whatever the properties

of C, the curve C is smooth and integral (though possibly disconnected). With these

conventions, we may show:

Claim A.1.3 (cf. [Gies] p.52).

I. Let C be a generically reduced curve over k; we do not assume it has genus g. Let

π : C → C be the normalization morphism, and let IC be the sheaf of nilpotents. Suppose

that C ⊂ PN . Let L := π∗LC = π∗O(1). Let

πm∗ : H0(C,O(m)) → H0(C, Lm) (A.4)

A.1. STRATEGY AND SETUP FOR THE POTENTIAL STABILITY THEOREM 93

be the induced morphism. Then

dim ker πm = h0(C, IC).

II. Suppose that C is a reduced curve. Let D be an effective divisor on C, and let M be an

invertible sheaf on C such that H1(C, M) = 0. Then h1(C, M(−D)) ≤ deg D.

III. Suppose that C is an integral and smooth curve with genus gC . Let M be an invertible

sheaf on C with deg M ≥ 2gC − 1. Then H1(C, M) = 0.

Finally, we need the following facts which are analogous to those given by Gieseker:

Proposition A.1.4 (cf. [Gies] page 25). Let C ⊂ PN have genus g and degree d. There exist

positive integers m1, m2, m3, q1, q2, q3, µ1, and µ2 satisfying the following properties.

(i) For all m > m1, H1(C, Lm) = 0. Also H1(C, LmW ) = 0 and the restriction map

H0(PN ,OPN (m)) → H0(C,O(m))

is surjective.

(ii) Iq1C = 0 where IC is the sheaf of nilpotents in OC .

(iii) h0(C, IC) ≤ q2.

(iv) For every complete subcurve C of C, h0(C,OC) ≤ q3 and q3 ≥ q1.

(v) µ1 > µ2 and for every point P ∈ C and for all integers x ≥ 0,

dimOC,P

mxC,P

≤ µ1x+ µ2,

where OC,P is the local ring of C at P and mC,P is the maximal ideal of OC,P .

(vi) For every subcurve C of C, for every point P ∈ C, and for all integers i such that

m2 ≤ i < m, the cohomology H1(C, Im−iP ⊗ Lm) = 0, where IP is the ideal subsheaf of

OC defining P .


(vii) For all integers m ≥ m3 the map

h , Hm(h)

Hilb(PN) → P( P(m)∧

H0(PN ,OPN (m)))

is a closed immersion.

In addition, we define a constant not used by Gieseker:

g := min0, gY | Y is the normalization of a complete subcurve Y contained

in a connected fiber Ch for some h ∈ Hilb(PN).

Y need not be a proper subcurve. The maximum number of irreducible components of Y is

d, as each must have positive degree. Hence a lower bound for g is given by −d + 1. One

would expect g to be negative for most (g, n, d), but we have stipulated in particular that

g ≤ 0 as this will be convenient in our calculations.

A.2 First properties of GIT-semistable curves

First, we want to prove that degenerately embedded curves are GIT-unstable when m

and m′i are large.

Proposition A.2.1 (cf. [Gies] 1.0.2, [BS] Proposition 5.1). Suppose m, m′i satisfy m > m3,

m′i ≥ 1 for all i, and m > (q1 − 1)(d − g + 1). Suppose that C is connected and SL(N + 1)-

semistable. Then C is a nondegenerate curve in PN , i.e. C is not contained in any hyperplane.

Proof. It is enough to prove that the composition

H0(PN ,OPN (1)) → H0(Cred,OCred(1)) → H0(Cred, Lred)

is injective. So suppose that it has nontrivial kernel V1. Referring to the setup described in

A.1, let λ be the 1-PS of GL(N + 1) which acts with weight 0 on V1 = V2 and weight 1 on V0,

and let λ′ be the associated 1-PS of SL(N + 1).

We wish to show that (C, P1, . . . , Pn) is GIT-unstable. For this proof we do not follow all

parts of the strategy outlined in Section A.1, as a simpler proof is available.

A.2. FIRST PROPERTIES OF GIT-SEMISTABLE CURVES 95

Let Bm be a basis of H0(PN ,O(m)) consisting of monomials of degree m. Then the first

part of the conclusion of Lemma A.1.1 is that there exist monomials M1, . . . , MP(m) in Bm

such that ρCm(M1), . . . , ρC

m(MP(m)) is a basis of H0(C,O(m)). For each of the Mj ’s, write

Mj in the form ve0,j0 · · ·veN,j

N .

Recall that, if IC denotes the ideal sheaf of nilpotent elements of OC , then the integer q1

was chosen such that it satisfies Iq1C = 0. Now suppose that

∑N1−1j=0 ej ≥ q1. It follows that

ρCm(Mj ) = 0, and so cannot be in a basis for H0(C, Lm). Thus

∑N1−1j=0 ej ≤ q1 − 1. The 1-PS λ

acts with weight 1 on the factors vN1 , . . . , vN , and so

wtλ(Mj ) ≥ m− q1 + 1.

Our basis consists of P(m) such monomials, so we can estimate their total weight:

P(m)∑j=1

wtλ(Mj ) ≥ P(m)(m− q1 + 1).

We assumed that the n marked points lie on the curve. Hence if vi(Pi) ≠ 0 then wtλ(vi)

must be equal to 1, so∑n

i=1 wtλ(vi)m′i =

∑ni=1 m′

i . Finally,

N∑j=0

wtλ(vj ) = dim V0 − dim V1 = d − g + 1− dim V1 ≤ d − g,

because dim V1 ≥ 1.

Combining these estimates with Lemma A.1.1, we obtain:

µ((C, P1, . . . , Pn), λ′)

≥P(m)(m− q1 + 1)+

n∑i=1

m′i

(d − g + 1)−mP(m)+

n∑i=1

m′i

(d − g)

≥ P(m)(m− (q1 − 1)(d − g + 1)).

However, P(m) is positive and by hypothesis m > (q1 − 1)(d − g + 1); thus

µ((C, P1, . . . , Pn), λ′) > 0. Thus the curve (C, P1, . . . , Pn) is GIT-unstable.

Next, we would like to show that things which are not generically reduced are GIT-

unstable. The analogous result in [Gies] is 1.0.3; it is almost but not quite strong enough for

us—Gieseker uses an integer argument at the end which cannot be used for the linearizations

studied here. On the other hand, the analogous result in [BS], Proposition 5.2, is quite a bit

more involved than what we need.


Let C1, . . . , Cp be the irreducible components of C. Define:

d` := degC` redOPN (1).

Since C ⊂ PN we have d` ≥ 1 for ` = 1, . . . , p.

Recall that L denotes ι∗OPN (1). Finally, let ξ` be the generic point of C`. Write

k` := length OC`,ξ` .

Then

d =∑

k`d`.

Proposition A.2.2 ([Gies] 1.0.3). Suppose that m > m3 and

m > (g − 12+ d(q1 + 1)+ q3 + µ1m2)(d − g + 1)

with ∑m′

im2 <

14

d − 54

g + 34

.

Suppose that (C, Pi) is connected and SL(N + 1) semistable for the linearization under

consideration. Then, C is generically reduced, i.e. k` = 1 for all ` = 1, . . . , p.

Proof. Suppose not. Then, reordering the components if necessary, we may assume that

k1 ≥ 2.

Let V1 be the kernel of the restriction map

H0(PN ,OPN (1)) → H0(C1 red,OC1 red(1)).

Step 1. We claim that V1 ≠ 0. To see this, suppose V1 = 0. Let D1 be a divisor on C1 red

corresponding to the invertible sheaf OC1 red(1) and having support in the smooth locus of

C1red. We have an exact sequence:

0 → OC1 red → OC1 red(1) → OD1 → 0.

Then the long exact sequence in cohomology implies that

h0(C1 red,OC1 red(1)) ≤ h0(C1 red,OD1 )+ h0(C1 red,OC1 red).


Note that h0(C1 red,OC1 red) = 1, and

h0(C1 red,OD1 ) ≤ deg D1 = degOC1 red(1) = d1.

If V1 = 0, then

d − g + 1 = h0(PN ,OPN (1)) ≤ h0(C1 red,OC1 red(1)) ≤ d1 + 1

⇒ d − g ≤ d1. (A.5)

We show that this statement leads to a contradiction. Recall that k1,1 ≥ 2. Now we rearrange

(A.5) to find:

k1(d − g) ≤ k1d1 = d −∑`>1

k`d` ≤ d

⇒ k1(d − g) < d −∑`>1

k`d`

⇒ (k1 − 1)d < k1g −∑`>1

k`d` ≤ k1g

⇒ d2≤ g.

The contradiction implies that V1 ≠ 0.

Step 2. By Step 1 we have that V1 ≠ 0, and in particular that d1 < d − g, as it is line (A.5)

which leads to the contradiction. Referring to the setup described in Section A.1, let λ be

the 1-PS of GL(N + 1) which acts with weight 0 on V2 = V1 and weight 1 on V0 and let λ′ be

the associated 1-PS of SL(N + 1).

Suppose that (C, P1, . . . , Pn) is GIT-stable or GIT-semistable, so that

µ((C, P1, . . . , Pn), λ′) ≤ 0. We shall show that this leads to a contradiction.

We have a filtration of H0(C, Lm) in order of decreasing weight as discussed in Section

Section A.1:

Ωmm = H0(C, Lm) ⊃ Ωm

m−1 ⊃ · · · ⊃ 0,

and we write δp = dim Ωmp .

Let C be the closure of C − C1 in C. Since C is connected, there is at least one closed

point in C1 ∩ C. Choose one such point P . Let

ρC,C : H0(C, Lm) → H0(C, LmC

)


be the map induced by restriction.

The following claim is analogous to one of Gieseker and may be proved using a similar

argument:

Claim A.2.3 (cf. [Gies] page 43). C = C − C1 can be given the structure of a closed subscheme

of C such that for all 0 ≤ p ≤ m− q1,

Ωmp ∩ kerρC,C : H0(C, Lm) → H0(C, Lm

tildeC) = 0.

Let IP be the ideal subsheaf of OC defining the closed point P . We have an exact sequence

0 → Im−pP ⊗ Lm

C→ Lm

C→ OC/Im−p

P ⊗ LmC→ 0.

In cohomology this induces

0 → H0(C, Im−pP ⊗ Lm

C) → H0(C, Lm

C)

→ H0(C,OC/Im−pP ⊗ Lm

C) → H1(C, Im−p

P ⊗ LmC⊗) → 0. (A.6)

The following facts are analogous to those stated by Gieseker in [Gies] page 44:

I. h0(C, LmC

) = χ(LmC

) = degC LmC+ χ(OC)

≤ (d − 2e1)m+ q3.

II. Since OC/Im−pP ⊗ Lm

Chas support only at the point P ∈ C, we make the estimate

h0(C,OC/Im−pP ⊗ Lm

C) ≥ m− p.

III. For 0 ≤ p ≤ m2−1, Proposition A.1.4 says that h0(C,OC/Im−pP ⊗Lm

C) ≤ µ1(m−p)+µ2,

so from the long exact sequence in cohomology, h1(C, Im−pP ⊗ Lm

C) ≤ µ1(m− p)+ µ2.

IV. For m2 ≤ p < m, we have h1(C, Im−pP ⊗ Lm

C) = 0 (cf. Proposition A.1.4).

V. ρC,Cm (Ωm

p ) ⊂ H0(C, Im−pP ⊗ Lm

C) ⊂ H0(C, Lm

C).

If p > m−q1 we may make no useful estimate, but if 0 ≤ p ≤ m−q1 then by Claim A.2.3

and fact V, we have

δp = dim Ωmp ≤ h0(C, Im−p

P ⊗ LmC

).


Now the exact sequence (A.6) tells us

h0(C, Im−pP ⊗ Lm

C) = h0(C, Lm

C)− h0(C,OC/Im−p

P ⊗ LmC

)

+ h1(C, Im−pP ⊗ Lm

C).

Thus, using the facts above, we have:

δp ≤

(d −

∑2d1) m+ q3 + p −m+ µ1(m− p)+ µ2 if 0 ≤ p ≤ m2 − 1

(d − 2d1) m+ q3 + p −m if m2 ≤ p ≤ m− q1

dm− g + 1 if m− q1 + 1 ≤ p ≤ m.

This is not exactly of the form required to use Lemma A.1.2, but we can easily calculate

what we need.

Step 3. We wish to estimateP(m)∑j=1

wtλ(Mj ). This is larger thanm∑

p=1p(δp − δp−1), and we

proceed:

m∑p=1

p(δp − δp−1) (A.7)

= mδm −m−1∑p=0

δp

≥ m(dm+ 1− g)−m−q1∑p=0

((d − 2d1)m+ q3 + p −m

)

−m2−1∑p=0

(µ1(m− p)+ µ2)−m−1∑

m−q1+1(dm+ 1− g)

=(

2d1 +12

)m2 +

(32− g − q3 − 2d1(q1 + 1)− µ1m2

)m

+ (q1 − 1)(

g + q3 −q1

2− 1

)− µ2m2 + µ1

m2(m2 − 1)2

≥(

2d1 +12

)m2 − S1m+ c2, (A.8)

where

S1 = g − 32+ 2d1(q1 + 1)+ q3 + µ1m2

c2 = (q1 − 1)(g + q3 −q1

2− 1)− µ2m2 + µ1

m2(m2 − 1)2

.

We may estimate c2 ≥ 0 since q3 > q1 and µ1 > µ2 (see Proposition A.1.4), to obtain

P(m)∑j=1

wtλ(Mj ) ≥(

2d1 +12

)m2 − S1m.


Step 4. We estimate the weight coming from the marked points. We know nothing about

which components each marked point lies on, so we can simply state that∑n

i=1 wtλ(vi)m′i ≥ 0.

Finally, we estimate the sum of the weights:

N∑j=0

wtλ(vj ) = dim V0 − dim V1 ≤ h0(C1 red,OC1 red(1))

≤ degOC1 red(1)+ 1 ≤ d1 + 1.

Step 5. We combine the inequalities in Lemma A.1.1 to obtain a contradiction as follows:

((2d1 +

12

)m2 − S1m)

(d − g + 1)− (mP(m)−∑

m′i )(d1 + 1)

≤ µL((C, P1, . . . , Pn), λ′) ≤ 0

=⇒ (2d1 +12

)(d − g + 1)m2 − (d + g − 1m

)(d1 + 1)m2 − S1(d − g + 1)m

≤ (d1 + 1)(∑

m′i

m2 )m2.

We showed that d1 < d−g, and by hypothesis m > (g− 12+d(q1+1)+q3+µ1m2)(d−g+1) ≥

(S1 + 1)(d − g + 1) > (S1 − g + 1)(d − g + 1), so we may estimate:

(2d1 + 12 )(d − g + 1)− d(d1 + 1)− 1

(d1 + 1)≤

∑m′

im2 . (A.9)

Note that the numerator is positive:

(2d1 +12

)(d − g + 1)− d(d1 + 1)− 1 > 0.

Furthermore the quantity

(2d1 + 12 )(d − g + 1)− d(d1 + 1)− 1

(d1 + 1)

is minimized when d1 takes its smallest value, that is, when d1 = 1. So

(2d1 + 12 )(d − g + 1)− d(d1 + 1)− 1

(d1 + 1)≥

52 (d − g + 1)− 2d − 1

2

= 14

d − 54

g + 34

.

But by hypothesis∑

m′i

m2 < 14 d − 5

4 g + 34 ; combining this result with (A.9) gives a contra-

diction. This implies that µL((C, P1, . . . , Pn), λ′) > 0. It follows that (C, P1, . . . , Pn) is not

λ-semistable. This completes the proof of Proposition A.2.2.


Next we derive the “Weighted Pointed Basic Inequality” (A.10), which is similar to

Gieseker’s so-called Basic Inequality. This turns out to be an extremely useful tool. Later we

will show that this inequality holds in more generality than is stated here (see Proposition

A.4.4).

Notation. Suppose C is a curve which has at least two irreducible components, and

suppose it is generically reduced on all components. Let Y be a complete subcurve of C,

suppose Y ≠ C, and let Z be the closure of C − Y in C. Let Z ιZ→ C ιC→ PN and Y ιY→ C be the

inclusion morphisms. Let

LY := ι∗Y ι∗COPN (1) LZ := ι∗Z ι∗COPN (1)

Let π : C → C be the normalization morphism. Let LY := π∗LY and similarly define

LZ := π∗LZ . Normalization induces a homomorphism

πm∗ : H0(C, Lm) → H0(C, Lm).

Define

dZ := degZ LZ = degZ LZ .

Write h0 := h0(Z,O(Z)(1)). Recall that we defined g to be

g := min0, gY | Y is the normalization of a complete subcurve Y contained

in a connected fiber Ch for some h ∈ Hilb(PN).

Proposition A.2.4 ([Gies] 1.0.7). Suppose m and the m′i satisfy m > m3 and

m > (g − 12+ d(q1 + 1)+ q3 + µ1m2)(d − g + 1)

with ∑m′

im2 <

14

d − 54

g + 34

.

Let (C, P1, . . . , Pn) be a connected weighted pointed curve which is SL(N + 1)-semistable.

Suppose C has at least two irreducible components. Let Z and Y be as above. Suppose there

exist distinct points Q1, . . . , Qκ on Y satisfying


(i) π(Qi) ∈ Y ∩ Z for all 1 ≤ i ≤ κ

(ii) for each irreducible component Y j of Y ,

degY j (LY (−D)) ≥ 0,

where D = Q1 + · · · +Qκ .

Then

dZ +κ2

<h0d +

((h0)

(∑C m′

i)−(∑

Z m′i)

(d − g + 1)) 1

m2

d − g + 1+ S

m, (A.10)

where S = g + κ(2g − 32 )+ q2 − g + 1.

Remark. The marked points Pi are allowed to coincide with the points Qj .

Proof. We start by defining the key 1-PS for this case. Let

V1 := kerH0(PN ,OPN (1)) → H0(Z,OZ (1)).

Following the setup described in Section A.1 let λ be the 1-PS of GL(N + 1) with weight

0 on V2 = V1 and weight 1 on V0, and let λ′Z be the associated 1-PS of SL(N + 1). (λ is

therefore a 1-PS which acts with positive weight on Y and negative weight on Z.) Suppose

that µ((C, P1, . . . , Pn), λ′Z ) ≤ 0. We shall derive the fundamental inequality from this, using

Lemma A.1.1.

We estimate the weights for λZ coming from the marked points as∑n

i=1 wtλZ (vi)m′i =∑

Pi∈Z m′i . Also, we estimate the sum of the weights:

∑Nj=0 wtλZ (vj ) ≤ h0.

Now we look at the weight coming from the curve. See Section A.1 for details: We wish

to estimate the sum∑P(m)

j=1 wtλZ (Mj ).

We construct a filtration of H0(C, Lm) in order of decreasing weight:

H0(C, Lm) = Ωmm ⊇ · · · ⊃ Ωm

1 ⊇ Ωm0 ⊇ 0

and write δp = dim Ωmp .

For p = m, it is clear that δm = h0(C, Lm) = dm − g + 1. We estimate δp in the case

p ≠ m. Restriction to Y induces a homomorphism

ρY ,Cm : H0(C, Lm) → H0(Y , Lm

Y ).


We restrict this to Ωmp , where 0 ≤ p < m. Note that if M is a monomial in Ωm

p and p < m

then M has at least one factor from V1 and hence by definition M vanishes on Z . If such M

also vanishes on Y then M is zero on C. Hence the restriction ρY ,Cm |Ωm

phas zero kernel, so is

an isomorphism of vector spaces, and thus:

dim ρY ,Cm (Ωm

p ) = dim Ωmp = δp.

We denote ρY ,Cm (Ωm

p ) by Ωmp |Y .

The normalization morphism πY : Y → Y induces a homomorphism

πY m : H0(Y , Lm) → H0(Y , Lm).

By definition the sections in πY m∗(Ωmp |Y ) vanish to order at least m − p at the points

Q1, . . . , Qk. Thus

πY m∗(Ωmp |Y ) ⊆ H0(Y , Lm

Y (−(m− p)D)).

Then

δp = dim(Ωmp )Y ≤ h0(Y , Lm

Y (−(m− p)D))+ dim ker πY m∗

= (d − dZ )m− κ(m− p)− g + 1+ h1(Y , LmY (−(m− p)D))+ dim ker πY m∗.(A.11)

We apply the estimates of Claim A.1.3 to our current situation.

1. I. dim ker πY m∗ < q2.

By Proposition A.2.2 the curve Y is generically reduced. Claim A.1.3.I may be applied to

Y ⊂ PN . Let IY denote the ideal sheaf of nilpotents in OY . Then dim ker πY m∗ < h0(Y , IY ).

In Proposition A.1.4, the constant q2 was defined to have the property h0(C, IC) < q2; hence

h0(Y , IY ) < q2 as well.

1. II. h1(Y , LmY (−(m− p)D)) ≤ κ(m− p) ≤ κm if 0 ≤ p ≤ 2g − 2.

The sheaf LY m is locally free on Y , and we have chosen m so that H1(Y , LmY ) = 0. The

hypotheses of Claim A.1.3.II hold, and we calculate deg(m− p)D = κ(m− p). We make a

coarser estimate than we could as this will be sufficient for our purposes.


1. III. h1(Y , LmY (−(m− p)D)) = 0 if 2g − 1 ≤ p ≤ m− 1.

Y is reduced, and is a union of disjoint irreducible (and hence integral) components Y j of

genus gYj ≤ g. We apply Claim A.1.3.III separately to each component. Our assumption (ii)

was that degY j (LY (−D)) ≥ 0, so

degY j (LY ) ≥ degY j D.

If degY j D ≥ 1 then

degY j (LmY (−(m− p)D)) ≥ m(degY j D)− (m− p)(degY j D)

≥ p ≥ 2g − 1 ≥ 2gYj − 1,

as required. On the other hand, suppose that degY j D = 0. Since Y ⊂ PN , the degree

degY j (LY ) ≥ 1. Thus again:

degY j (LmY (−(m− p)D)) ≥ m ≥ 2gYj − 1.

Combining this data with our previous formula (A.11) we have shown:

δp ≤

(d − dZ − κ)m+ κp − g + 1+ q2 + κm 0 ≤ p ≤ 2g − 2(d − dZ − κ)m++κp − g + 1+ q2, 2g − 1 ≤ p ≤ m− 1.

Thus, we may use Lemma A.1.2, setting α = dZ + κ, β = k, ε = − g + 1 + q2 + (2g − 1)κ,

rN = 1 and R = h0. Following Lemma A.1.2, we see

µ((C, P1, . . . , Pn), λ′) ≥(

(dZ +κ2

)(d − g + 1)− h0d)

m2

+(d − g + 1)(

∑Pi∈Z

m′i − (h0)(

∑m′

i )

−(

g − κ2− g + q2 + (2g − 1)κ

)(d − g + 1)m− h0m.

Thus, since we assume that µ((C, P1, . . . , Pn), λ) ≤ 0, we may conclude that

dZ +κ2

<h0d +

((h0)

(∑C m′

i)−(∑

Z m′i)

(d − g + 1)) 1

m2

d − g + 1+ S

m,

where S = g + κ(2g − 32 )+ q2 − g + 1.

A.3. THE ONLY SINGULARITIES ALLOWED ARE NODES 105

A.3 The only singularities allowed are nodes

The next series of results shows that if the connected curve (C, P1, . . . , Pn) is GIT-

semistable, then any singularities of C are nodes. First we show that C has no cusps by

showing that the normalization morphism π : C → C is unramified. Singular points are

shown to be double points by showing that the inverse image under π of any P ∈ C contains

at most two points. Next we rule out tacnodes.

Proposition A.3.1 (cf. [Gies] 1.0.5). Let d be sufficiently large that d−9g+7 > 0, and suppose

m > m3 and

m > max

(g − 1

2 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),(9g + 3q2 − 3g)(d − g + 1)

and ∑m′

im2 <

18

d − 98

g + 78

.

If (C, P1, . . . , Pn) is connected and SL(N + 1)-semistable, then the normalization morphism

π : C → Cred is unramified. In particular, C possesses no cusps.

Proof. Suppose π is ramified at P ∈ C. Recall that by Proposition A.2.1, the curve C ⊂ PN is

nondegenerate; we can think of H0(PN ,OPN (1)) as a subspace of H0(C, L). Define

V2 = s ∈ H0(PN ,OPN (1))|π∗s vanishes to order ≥ 3 at PV1 = s ∈ H0(PN ,OPN (1))|π∗s vanishes to order ≥ 2 at P.

Let λ be the 1-PS of GL(N + 1) which acts with weight 0 on V2, weight 1 on V1 and weight 3

on V0, and let λ′ be the associated 1-PS of SL(N + 1).

We make an estimate for µ((C, P1, . . . , Pn), λ′) using the filtration of (A.2) and Lemma

A.1.2. As usual, we need to find an estimate for δi := dim Ωmi .

As in Proposition A.2.4 we use the homomorphism

πm : H0(C, Lm) → H0(C, Lm)

induced by the normalization morphism. We show that

πm(Ωmi ) ⊆ H0(C, Lm ⊗OC((−3m+ i)P)), (A.12)


for 0 ≤ i ≤ 3m. When i = 0 this follows from the definitions. For 1 ≤ i ≤ 3m it is enough to

show that a monomial M ∈ (Ωmi −Ωm

i−1) vanishes at P to order at least 3m− i. Suppose that

such M has i2 factors from V2, i1 factors from V1 and i0 factors from V2. Then i0+ i1+ i2 = m

and i1 + 3i0 = i. By definition, M vanishes at P to order at least 3i0 + 2i1. But

3i0 + 2i1 = 3(i0 + i1 + i2)− (i1 + 3i0) = 3m− i,

so the monomial vanishes as required and hence (A.12) is satisfied.

By (A.12) and Riemann-Roch,

δi := dim Ωmi ≤ h0(C, Lm ⊗OC(−(3m− i)P))+ dim ker πm

≤ dm− 3m+ i − g + 1

+ h1(C, Lm ⊗OC(−(3m− i)P))+ dim ker πm.

We may use Claim A.1.3 in a straightforward way to show that dim ker πm < q2 and that

h1(

C, Lm ⊗OC(−(3m− i)P))≤ 3m− i ≤ 3m if 0 ≤ i ≤ 2g − 2. More care is needed to show

that the h1 term vanishes for higher values of i.

Let C i be an irreducible component of C. Suppose C i does not contain P ∈ C. Then

degC iL = degCi L ≥ 1. Thus

degC i

(Lm ⊗OC(−(3m− i)P)

)= degC i

(Lm) ≥ m ≥ 2gCi − 1.

On the other hand, suppose that C i is the component of C on which P lies. The morphism

C i → Ci red is ramified at P , so Ci red is singular and integral in PN . Were Ci red an integral

curve of degree 1 or 2 in PN , it would be either a line or a conic and hence nonsingular. We

conclude that degC iL = degCi red

L ≥ 3. Then

degC i

(Lm ⊗OC(−(3m− i)P)

)≥ 3m− 3m+ i = i.

Thus, Claim A.1.3.III shows that h1(

C, Lm ⊗OC(−(3m− i)P))= 0 if 2g − 1 ≤ i ≤ 3m− 1.

Combining these inequalities we have

δi ≤

(d − 3)m+ i − g + q2 + 1+ 3m, 0 ≤ i ≤ 2g − 2(d − 3)m+ i − g + q2 + 1, 2g − 1 ≤ i ≤ 3m− 1.

Thus, in the language of Lemma A.1.2, we shall set α = 3, β = 1, and ε = − 3g + 3q2 + 6g.

We may estimate the minimum weight of the action of λ on the marked points Pi as zero.

We know that r0 = 3. It remains to find an upper bound for∑

wtλ(vi).


Recall that we are regarding V as a subspace of H0(C, L). Note that the image of V2

under π∗ is contained in H0(C, L(−3P)), and the image of V1 under π∗ is contained in

H0(C, L(−2P)). We have two exact sequences

0 → L(−P) → L → k(P) → 00 → L(−3P) → L(−2P) → k(P) → 0,

which give rise to long exact sequences in cohomology

0 → H0(C, L(−P)) → H0(C, L) → H0(C, k(P)) → ·· ·0 → H0(C, L(−3P)) → H0(C, L(−2P)) → H0(C, k(P)) → ·· · .

The second long exact sequence implies that dim V1/V2 ≤ 1. Now recall that L := π∗(L)

and π is ramified at P . The ramification index must be at least two, so we have

H0(C, L(−P)) = H0(C, L(−2P)). Then the first long exact sequence implies that

dim V0/V1 ≤ 1. We conclude that∑N+1

i=1 wtλ(vi) ≤ 1+ 3 = 4 =: R.

We may now estimate the λ′-weight for (C, P1, . . . , Pn,A), using Lemma A.1.2:

µ((C, P1, . . . , Pn), λ′) ≥(

92

(d − g + 1)− 4d)

m2 − 4∑

m′i

−(

(9g − 3g + 3q2 −92

)(d − g + 1)+ 4)

m

≥(

12

d − 92

(g − 1)− 4∑

m′i

m2

)m2 −

(9g − 3g + 3q2

)(d − g + 1)m.

We assumed that(9g − 3g + 3q2

)(d − g + 1) < m, so we have shown that

µ((C, P1, . . . , Pn), λ′) ≥(

12

d − 92

g + 72− 4

∑m′

im2

)m2.

This is clearly positive, as we assumed that

18

d − 98

g + 78

>∑

m′i

m2 ,

so m2 has a positive coefficient.

Thus µ((C, P1, . . . , Pn), λ′) > 0. and thus (C, P1, . . . , Pn) is not λ-semistable. But this

contradicts the hypothesis that (C, P1, . . . , Pn) is SL(N + 1)-semistable

Proposition A.3.2 (cf. [Gies] 1.0.4). Suppose that d − 9g + 7 > 0, and m > m3 and

m > max

(g − 1

2 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),(9g + 3q2 − 3g)(d − g + 1)


with ∑m′

im2 <

18

d − 98

g + 78

.

If (C, P1, . . . , Pn) is connected and GIT-semistable, then all singular points of Cred are double

points.

Proof. Suppose there exists a point P ∈ C with multiplicity ≥ 3 on Cred. Let

ev : H0(PN ,OPN (1)) → k(P) be the evaluation map. Let V1 = V2 = ker ev. Let λ be the 1-PS

of GL(N + 1) with weight 0 on V1 and weight 1 on V0, and let λ′ be the associated 1-PS of

SL(N + 1).

We estimate µL((C, P1, . . . , Pn), λ′) using Lemma A.1.2. Construct a filtration of

H0(C, Lm) of increasing weight as in (A.2). We need to find an upper bound for δp := dim Ωmp .

Define a divisor D on C as follows: Let π : C → C be the normalization morphism. The

hypotheses of Proposition A.3.1 are satisfied, so π is unramified; as P has multiplicity at

least 3 there must be at least three distinct points in the preimage π−1(P). Let Q1, Q2, Q3

be three such points, and set D := Q1 +Q2 +Q3. Note that if any two of these points lie

on the same component C1 ⊂ C, then the corresponding component C1 ⊂ C must have

degC1 L ≥ 3, by the same argument as in the proof of Proposition A.3.1.

The normalization morphism induces a homomorphism

πm∗ : H0(C, Lm) → H0(C, Lm).

Note that πm∗(Ωmp ) ⊆ H0(C, LmOC(−(m− p)D)). We have

δp := dim Ωmp ≤ h0(C, Lm ⊗⊗OC(−(m− p)D))+ dim ker πm∗

≤ dm− 3(m− p)− g + 1

+ h1(C, Lm ⊗OC(−(m− p)D))+ dim ker πm∗.

We may use Claim A.1.3 parts I and II to estimate that h0(C, IC) < q2 and that

h1(C, Lm(−(m− p)D)⊗) ≤ 3(m− p) ≤ 3m if 0 ≤ p ≤ 2g − 2. To show that

h1(C, Lm(−(m − p)D)) = 0 if p ≥ 2g − 1, we may verify that the degree of L on any

component C1 meeting P implies that the hypothesis of Claim A.1.3.III is satisfied.

Thus:

δp ≤

(d − 3)m+ 3p − g + q2 + 1+ 3m, 0 ≤ p ≤ 2g − 2(d − 3)m+ 3p − g + q2 + 1, 2g − 1 ≤ p ≤ m.


We may apply Lemma A.1.2, setting α = 3, β = 3 and ε = − g + q2 + 6g − 2. We know r0 = 1

and may estimate the weight of the action of λ on the marked points P1, . . . , Pn as greater

than or equal to zero. Finally, note that∑N

j=0 wtλ(vj ) = 1 =: R. Now, substituting in these

values:

µL((C, P1, . . . , Pn), λ′) ≥(

32

(d − g + 1)− d)

m2 −∑

m′i

−(

(7g − g + q2 −92

)(d − g + 1)+ 1)

m

≥(

12

d − 32

(g + 1)−∑

m′i

m2

)m2 −

(7g − g + q2

)(d − g + 1)m.

Our assumptions imply that (7g − g + q2)(d − g + 1) < m. We have shown that

µL((C, P1, . . . , Pn), λ′) ≥(

12

d − 32

g + 12−∑

m′i

m2

)m2.

However, we assumed that ∑m′

im2 <

18

d − 98

g + 78

,

and 18 d − 9

8 g + 78 < 1

2 d − 32 g + 1

2 since d − g ≥ 4, so the coefficient of m2 is positive.

Thus µL((C, P1, . . . , Pn), λ′) > 0, so (C, P1, . . . , Pn) is not GIT-semistable.

The remaining singularity we must rule out is that C possesses a tacnode. The analogous

proposition in [Gies] is 1.0.6, but the proof there contains at least two errors (one should

use Ωmi accounting rather than the Tata notes’ W m−r

i W rj when the filtration has more than

two stages; and tacnodes need not be separating). These may be avoided if we simply follow

[HM] 4.53 instead.

Proposition A.3.3 (cf. [HM] 4.53). Suppose that d − 9g + 7 > 0, and

m > max

m3

(g − 12 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),

(10g + 3q2 − 3g)(d − g + 1)

with ∑

m′i

m2 <18

d − 98

g + 78

.

If C is connected and (C, P1, . . . , Pn) is GIT-semistable, then Cred does not have a tacnode.


Proof. Suppose that Cred has a tacnode at P . Let π : C → C be the normalization. There exist

two distinct points, Q1, Q2 ∈ C, such that π(Q1) = π(Q2) = P . Moreover the two tangent

lines to C at P are coincident. Define the divisor D := Q1 +Q2 on C.

We consider H0(PN ,OPN (1)) as a subspace of H0(C, L). Thus we may define subspaces:

V2 := s ∈ H0(PN ,OPN (1))|π∗pW∗s vanishes to order ≥ 2 at Q1 and Q2

V1 := s ∈ H0(PN ,OPN (1))|π∗pW∗s vanishes to order ≥ 1 at Q1 and Q2.

Let λ be the 1-PS which acts with weight 0 on V2, weight 1 on V1 and weight 2 on V0, and let

λ′ be the associated 1-PS of SL(W ).

Construct a filtration of H0(C, Lm) by subspaces of the form Ωmi , following (A.2) as

usual. We wish to estimate δi = dim Ωmi in order to apply Lemma A.1.2.

As in Proposition A.2.4 we use the homomorphism πm∗ induced by the normalization

morphism. Similarly to as in Proposition A.3.1 we show that

πm∗(Ωmi ) ⊆ H0(C, Lm ⊗OC(−(2m− i)D)), (A.13)

for 0 ≤ i ≤ 2m. When i = 0 this follows from the definitions. For 1 ≤ i ≤ 2m it is enough to

show that for any monomial M ∈ Ωmi −Ωm

i−1 we have

πm∗(M) ∈ H0(C, Lm ⊗OC(−(2m− i)D)).

Suppose that such M has jk factors from Vk, for k = 0, 1, 2. Then j0 + j1 + j2 = m and

j1 + 2j0 = i. By definition, πm∗(M) vanishes at both Q1 and Q2 to order at least 2j2 + j1.

But

2j2 + j1 = 2(j0 + j1 + j2)− (j1 + 2j0) = 2m− i,

giving the required vanishing of πm∗(M).

We shall also prove at this stage that∑N+1

j=1 wtλ(vj ) ≤ 3. We have two exact sequences

0 → L(−Q1) → L → k(Q1) → 00 → L(−2Q1) → L(−Q1) → k(Q1) → 0,

which give rise to long exact sequences in cohomology

0 → H0(C, L(−Q1)) → H0(C, L) → H0(C, k(P))0 → H0(C, L(−2Q1)) → H0(C, L(−Q1)) → H0(C, k(P)).


If we let V 0 be the image of V0 = H0(PN ,OPN (1)) in H0(C, L), we may intersect each of

the spaces in the sequences with V 0. Note that the image of V1 in H0(C, L) is precisely

V 2 ∩ H0(C, L(−Q1)); if a section of L which vanishes at Q1 is the pullback a section of

C, then it automatically vanishes at Q2 as well. Similarly the image of V0 in H0(C, L) is

V 0 ∩H0(C, L(−2Q1)). We have obtained:

0 → V 1 → V 2 → V 2 ∩H0(C, k(P))0 → V 0 → V 1 → V 2 ∩H0(C, k(P)).

It follows that dim V0/V1 ≤ 1 and dim V1/V2 ≤ 1. Thus∑N+1

j=1 wtλ(vj ) ≤ 1+ 2 = 3 =: R.

The weight coming from the marked points will always be estimated as zero. In order

to estimate δp, there are unfortunately various cases to consider, which shall give rise to

different inequalities:

1. there is one irreducible component C1 of C passing through P ;

2. there are two irreducible components C1 and C2 of C passing through P ,

and degCi redL ≥ 2 for i = 1, 2;

3. there are two irreducible components C1 and C2 of C passing through P ,

and degC1 redL = 1, while degC2 red

L ≥ 2.

There cannot be two degree 1 curves meeting at a tacnode so these are the only cases. Note

that in Case 1, since C1 is an irreducible curve with a tacnode, degC1 redL ≥ 4.

Cases 1 and 2.

We estimate δp. By (A.13) and Riemann-Roch,

δi := dim Ωmi ≤ h0(C, Lm ⊗OC(−(2m− i)D))+ dim ker πm∗

≤ dm− 2(2m− i)− g + 1+ h1(C, Lm ⊗OC(−(2m− i)D))+ dim ker πm∗.

Claim A.1.3 allows us to estimate, as usual, the upper bound dim ker πm∗ < q2 and a coarse

upper bound h1(

C, Lm ⊗OC(−(2m− i)D))≤ 4m− 2i ≤ 4m if 0 ≤ i ≤ 2g − 2. To show that

the h1 term vanishes for larger values of i, in Case 1 we have degC1Lm ⊗OC(−(2m− i)D) ≥

4m − 2 · 2m + 2i ≥ 2g − 2 if 2g − 2 ≤ i ≤ 3m − 1. (Note the bounds on i here are coarse.)

The other components of the curve do not meet D and so one sees from Claim A.1.3.III that

h1 is zero there. Case 2 follows similarly.


Combining these inequalities we have

δi ≤

(d − 4)m+ 2i − g + 1+ q2 + 4m, 0 ≤ i ≤ 2g − 2(d − 4)m+ 2i − g + 1+ q2, 2g − 1 ≤ i ≤ 3m− 1.

In the language of Lemma A.1.2, we set α = 4, β = 2, ε = 8g − 3g + 3q2 − 1, rN = 2 and

R = 3. Now:

µL((C, P1, . . . , Pn), λ′) ≥ (4(d − g + 1)− 3d)m2 − 3∑

m′i

−((10g − 3g + 3q2 − 5)(d − g + 1)+ 3

)m

≥(

d − 4(g − 1)− 3∑

m′i

m2

)m2 −

(10g − 3g + 3q2

)(d − g + 1)m.

This is clearly positive for large m. In particular, as we set m > (10g − 3g + 3q2)(d − g + 1),

we must infer that

µL((C, P1, . . . , Pn), λ′) ≥(

d − 4g + 3− 3∑

m′i

m2

)m2.

Then, since ∑m′

im2 <

18

d − 98

g + 78

<13

d − 43

g + 1

(where the latter inequality may be seen to hold, since d − g ≥ 4), we conclude that

µL((C, P1, . . . , Pn), λ′) > 0.

Case 3.

Now that degC1L = 1, we need a new way to estimate δi := dim Ωm

p . However we do

know that the genus of C1 is zero. Let Y := C − C1. Noting in line (A.13) that C1 and Y are

disjoint, we may write

πm∗(Ωmp ) ⊂ H0(C, Lm ⊗OC(−(2m− i)D))

= H0(C1, Lm(−(2m− i)Q1))⊕H0(Y , Lm(−(2m− i)Q2)).

Thus,

δi := dim Ωmi ≤ h0

i + (d − 1)m− (2m− i)− gC + 1

+ h1(Y , Lm ⊗OY (−(2m− i)Q2))+ dim ker πm∗,

where we write h0i for h0(C1, Lm ⊗OC1

(−(2m− i)Q1)).


As usual, Claim A.1.3 allows us to estimate the upper bounds dim ker πm∗ < q2 and

h1(

Y , Lm ⊗OY (−(2m− i)Q2))≤ 2m− i ≤ 2m if 0 ≤ i ≤ 2g − 2. The assumptions for Case

3 tell us directly that we may apply Claim A.1.3.III and obtain

h1(

Y , Lm ⊗OY (−(2m− i)Q2))= 0 if 2g − 1 ≤ i ≤ 2m− 1.

Combining these inequalities we have:

δi ≤

(d − 3)m+ i − g + 1+ q2 + 2m+ h0i , 0 ≤ i ≤ 2g − 2

(d − 3)m+ i − g + 1+ q2 + h0i , 2g − 1 ≤ i ≤ 2m− 1.

(A.14)

To calculate the ε term in the language of Lemma A.1.2, we must calculate∑2m−1

i=0 h0i . We

recall that C1 P1 and that degC1L = 1. Thus LC1

= OP1 (1), so

h0i = h0(P1,OP1 (m)⊗OP1 (−(2m− i)Q1)) =

0 i ≤ m− 1−m+ i + 1 i ≥ m.

Thus in particular2m−1∑

i=0h0

i =m∑

j=1j = 1

2m2 + 1

2m.

Hence, we calculate ε to be = 4g − 2g + 2q2 + 12 +

12 m. For the rest of the dictionary for

Lemma A.1.2, we set α = 4, β = 1, r0 = 2 and R = 3. Then

µL((C, P1, . . . , Pn), λ′) ≥(4(d − g + 1)− 3d

)m2 − 3

∑m′

i

−(

(6g − 2g + 2q2 −52+ 1

2m)(d − g + 1)+ 3

)m

≥(

12

d − 72

(g − 1)− 3∑

m′i

m2

)m2 −

(6g − 2g + 2q2

)(d − g + 1)m.

Our estimates for m imply that m > (d − g + 1)(6g − 2g + 2q2), so we have shown that

µL((C, P1, . . . , Pn), λ′) ≥(

12

d − 72

g + 52− 3

∑m′

im2

)m2.

This is clearly positive, as

∑m′

im2 <

18

d − 98

g + 78

<16

d − 76

g + 56

,

where the second inequality holds since d − g ≥ 4.

Thus µL((C, P1, . . . , Pn), λ) > 0, and hence we see that (C, P1, . . . , Pn) is GIT-unstable.


A.4 GIT-semistable curves are reduced

The next three results show that, if (C, P1, . . . , Pn) is GIT-semistable, then the curve C is

reduced. We begin with a generalized Clifford’s theorem.

Lemma A.4.1 (cf. [Gies] page 18). Let C be a reduced curve with only nodes, and let L be a

line bundle generated by global sections which is not trivial on any irreducible component of

C. If H1(C, L) ≠ 0 then there is a connected subcurve Z ⊂ C such that

h0(Z, L) ≤ degZ (L)2

+ 1. (A.15)

Furthermore C′ 6 P1.

Proof. Gieseker proves nearly all of this. It remains only to show that that Z may be taken

to be connected and Z 6 P1. Firstly, if (A.15) is satisfied by Z ⊂ C then to is clear that

(A.15) must be satisfied by some connected component of Z . So assume Z is connected and

suppose that Z P1. Now, every line bundle on P1 is isomorphic to OP1 (`) for some ` ∈ Z.

By hypothesis L is generated by global sections and nontrivial on Z ; this implies that ` > 0.

However, combining this with (A.15) implies that ` + 1 = h0(Z, L) ≤ `2 + 1 which implies

` ≤ 0, a contradiction.

Lemma A.4.2 ([Gies] page 79). Suppose that d − 9g + 7 > 0, and

m > max

m3

(g − 12 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),

(10g + 3q2 − 3g)(d − g + 1)

with

m′i

m2 <18

d − 98

g + 78

.

If (C, P1, . . . , Pn) is connected and GIT-semistable, then H1(Cred, Lred) = 0.

Proof. Since Cred is nodal, it has a dualizing sheaf ω. Suppose H1(Cred, Lred) ≠ 0. Then by

duality,

H0(Cred, ω⊗ L−1red) H1(Cred, Lred) ≠ 0.

The line bundle Lred is not trivial on any component of Cred , so by Lemma A.4.1 there is a

connected subcurve Z 6 P1 of Cred for which dZ > 1 and h0(Z, LZ ) ≤ dZ2 + 1.

A.4. GIT-SEMISTABLE CURVES ARE REDUCED 115

Let Y := C − Zred and pick a point P on the normalization Y so that π(P) ∈ Z ∩ Y . We

know that degY j L(−P) ≥ 0 for every component Yj of Y , so we may apply Proposition

A.2.4, setting κ = 1 and bZ = 1. Estimate∑

Z m′i ≥ 0. Recall that in (WPBI) we have

S = 3g + q2 − g′ + 12 , and so the hypotheses on m certainly imply that S

m (d − g + 1) ≤ 12 . We

obtain:

dZ +12= dZ +

κ2

≤

(dZ2 + 1

)d +

(∑C m′

i) (dZ

2 + 1)

1m2

d − g + 1+ 1

2(d − g + 1)

⇒ 0 < −(

dZ +12

)(d − g + 1)+

(12

dZ + 1)(

d +∑

C m′i

m2

)+ 1

2.

Use the bound∑

m′i

m2 < 18 d − 9

8 g + 78 to obtain

0 < −(

dZ +12

)(d − g + 1)+

(12

dZ + 1)(

98

d − 98

g + 78

)+ 1

2

= −(

716

dZ −58

)(d − g)− 9

16dZ +

78

.

Since dZ > 1 we may substitute in dZ ≥ 2:

0 < − 14

(d − g)− 14

,

a contradiction. Thus H1(Cred, LW red) = 0.

We now show that GIT-semistable curves are reduced.

Proposition A.4.3 (cf. [Gies] 1.0.8). Suppose that d − 9g + 7 > 0, and

m > max

m3

(g − 12 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),

(10g + 3q2 − 3g)(d − g + 1),

with ∑

m′i

m2 <18

d − 98

g + 78

.

If (C, P1, . . . , Pn) is connected and GIT-semistable, then C is reduced.

Proof. Let ι : Cred → C be the canonical inclusion. The exact sequence of sheaves on C

0 → IC ⊗ L → L → ι∗Lred → 0

gives rise to a long exact sequence in cohomology

· · · → H1(C, IC ⊗ L) → H1(C, L) → H1(C, ι∗Lred) → 0.


Since C is generically reduced, IC has finite support, hence H1(C, IC ⊗ L) = 0. Lemma

A.4.2 tells us us that H1(C, ι∗Lred) = H1(Cred, Lred) = 0. The exact sequence implies

H1(C, L) = 0 as well. Next, the map

H0(PN ,OPN (1)) → H0(Cred,OCred(1)) → H0(Cred, Lred)

is injective by Proposition A.2.1. Then

d − g + 1 = h0(PN ,OPN (1)) ≤ h0(Cred, Lred)

= h0(C, L)− h0(C, IC ⊗ L) = d − g + 1− h0(C, IC ⊗ L).

Therefore h0(C, IC ⊗ L) = 0. Since IC ⊗ L has finite support, IC = 0, so C is reduced.

Next, we improve on Proposition A.2.4. If C is connected and (C, P1, . . . , Pn) is GIT-

(semi)stable, and if Z is a connected subcurve of C, then we show that the fundamental

inequality (A.10) is satisfied even without condition (ii) of the Proposition.

We repeat the definition of λ′Z : Let

V1 := kerH0(PN ,OPN (1)) → H0(Z,OZ (1)).

Let λ be the 1-PS of GL(N + 1) which acts with weight 1 on V0 and weight 0 on V1 = V2, and

let λ′Z be the associated 1-PS of SL(N + 1).

Lemma A.4.4 (cf. [Gies] page 83 and Proposition A.2.4 above). Suppose that d − 9g + 7 > 0,

and that

m > max

m3

(g − 12 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),

(10g + 3q2 − 3g)(d − g + 1),

with

m′i

m2 <18

d − 98

g + 78

.

Let (C, P1, . . . , Pn) ⊂ PN be a connected curve whose only singularities are nodes. Suppose C

has at least two irreducible components. Let Z ≠ C be a reduced complete subcurve of C and

let z denote the number of connected components of Z. Let Y be the closure of C − Z in C

with the reduced structure. Suppose there exist points Q1, . . . , Qκ on Y , the normalization of

Y , satisfying π(Qi) ∈ Y ∩ Z for all 1 ≤ i ≤ κ. Write h0(Z,OZ (1)) =: h0.

A.4. GIT-SEMISTABLE CURVES ARE REDUCED 117

Finally, suppose that

µL((C, P1, . . . , Pn), λ′Z ) ≤ 0.

Then

dZ +κ2

<h0d +

((h0)

(∑C

m′i

m2

)−(∑

Zm′

im2

)(d − g + 1)

)d − g + 1

+ zSm

, (A.16)

where S = g + κ(2g − 32 )+ q2 − g + 1.

Proof. First, assume Z is connected. Arguing similarly to [Gies] pages 83–85, we prove the

result by contradiction. We first assume that κ = #(Y ∩ Z) and then show that this implies

the general case for Z connected.

Let Z be a connected subcurve of C, and let Q1, . . . , Qκ be all the points on Y satisfying

π(Qi) ∈ Y ∩ Z . We assume that (A.16) is not satisfied for Z , and further that Z is maximal

with this property. Namely, if Z′ is complete and connected, and Z ⊊ Z′ ⊂ C, then (A.16)

does hold for Z′. Since (A.16) does not hold for Z ,

(dZ +

κ2

)(d − g + 1) (A.17)

≥ (dZ − gZ + 1)d +∑

C

m′i

m2

(dZ − gZ + 1)−∑

Z

m′i

m2

(d−g+1)

+ Sm

(d−g+1).

As all other hypotheses of Proposition A.2.4 have been met, we must conclude that condition

(ii) there fails. Thus there is some irreducible component Y j of Y , the normalization of Y ,

such that

degY j (LY (−(Q1 + · · · +Qκ))) < 0.

Let Yj be the corresponding irreducible component of Y . We have degY j (LY ) = degYj (L) > 0.

Putting this together,

0 < degY j (LY ) < #(Y j ∩ Q1, . . . , Qκ) = #(Yj ∩ Z) =: i(Yj , Z).

Thus, i(Yj , Z) ≥ 2. We define D to be the connected subcurve Yj ∪ Z. By the maximality

assumption on Z, it follows that (A.16) does hold for D. We define constants dD, κD, h0D,∑

D m′i and SD in analogy with the constants for Z . Similarly we define constants pertaining

to Yj . Then:

dD +κD

2<

h0Dd +

((∑C

m′i

m2

)h0

D −(∑

Dm′

im2

)(d − g + 1)

)d − g + 1

+ SD

m. (A.18)


Observe that dD = dZ + dYj and∑

Dm′

im2 =

∑Z

m′i

m2 +∑

Yjm′

im2 −

∑Z∩Yj

m′i

m2 . The curve C is nodal,

so we conclude:

(i) κD = #((Yj ∪ Z) ∩ (Y − Yj )) = #(Z ∩ Y ) + #(Yj ∩ Y − Yj ) − #(Z ∩ Yj ); if we set

i(Yj , Y ) := #(Yj ∩ Y ) then κD = κ + i(Yj , Y )− i(Yj , Z);

(ii) gD = gZ + gYj + i(Yj , Z)− 1;

(iii) h0D = dD − gD + 1 = dZ + dYj − gZ − gYj − i(Yj , Z)+ 2.

Note in particular that, since i(Yj , Z) ≥ 2 and since Z and Yj are connected, (ii) implies

that gD ≥ 1. But gD ≤ g, so if g = 0 then we already have the required contradiction. We

henceforth assume that g ≥ 1. Equation (A.18) may be rearranged to form

(dZ + dYj +

κ + i(Yj , Y )− i(Yj , Z)2

)(d − g + 1)

< (dZ + dYj − gZ − gYj − i(Yj , Z)+ 2)d

+∑

C

m′i

m2

(dZ + dYj − gZ − gYj − i(Yj , Z)+ 2)

−∑

Z

m′i

m2 +∑Yj

m′i

m2 −∑

Z∩Yj

m′i

m2

(d − g + 1)+ SD

m(d − g + 1).

We subtract our assumption, line (A.17):

(dYj +

i(Yj , Y )− i(Yj , C′)2

)(d − g + 1)

< (dYj − gYj − i(Yj , C′)+ 1)d +∑

C

m′i

m2

(dYj − gYj − i(Yj , C′)+ 1) (A.19)

−∑

Yj

m′i

m2 −∑

Z∩Yj

m′i

m2

(d − g + 1)+(i(Yj , Y )− i(Yj , C′))(2g − 1

2 )m

(d − g + 1).

We rearrange, and use the inequality dYj ≤ i(Yj , C′)− 1. i(Yj , Y )+ i(Yj , C′)2

+ gYj − 1+∑Yj

m′i

m2 +(i(Yj , C′)− i(Yj , Y ))(2g − 1

2 )m

d

<

i(Yj , Y )+ i(Yj , C′)2

− 1+∑Yj

m′i

m2 +(i(Yj , C′)− i(Yj , Y ))(2g − 1

2 )m

(g−1)

−gYj

∑C

m′i

m2

,

A.5. THE BEHAVIOR OF WEIGHTED MARKED POINTS ON A GIT-STABLE CURVE 119

so that finally we may estimate i(Yj , Y )+ i(Yj , C′)2

+ gYj − 1+∑Yj

m′i

m2 +(i(Yj , C′)− i(Yj , Y ))(2g − 1

2 )m

(d − g + 1)

< −gYj

∑C

m′i

m2

≤ 0. (A.20)

Recall that d − g + 1 > 0, that i(Yj , Z) ≥ 2 and that g ≥ 1. Thus the left hand side of (A.20)

is strictly positive. This contradiction implies that no such Z exists, i.e. all subcurves Z of C

satisfy inequality (A.16), provided that κ = #(Z ∩ Y ).

Finally, suppose that we choose any κ points Q1 . . . , Qκ on Y such that π(Yi) ∈ (Z ∩ Y ).

Then κ ≤ #(Z ∩ Y ) := κ′. We proved that (A.16) is true for κ′, and though we must take

a little care with the dependence of S on κ, it follows that (A.16) is true for κ. We have

therefore proved the result when z = 1, i.e. Z is connected.

Now, let Z1, . . . , Zz be the connected components of Z . We may prove a version of (A.16)

for each Zi , for i = 1, . . . , z. When we sum these inequalities over i, it follows that

Z + κ2

<h0d ++

((∑C

m′i

m2

)(h0)−

(∑Z

m′i

m2

)(d − g + 1)

)d − g + 1

+ zSm

.

A.5 The behavior of weighted marked points on a GIT-stablecurve

We now turn our attention to the weighted marked points. Ideally GIT-(semi)stability

would require these to behave exactly according to the moduli description given in the

introduction. This is addressed in the following two propositions.

Proposition A.5.1. Suppose d − 9g + 7 > 0, and

m > max

m3

(g − 12 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),

(10g + 3q2 − 3g)(d − g + 1)

with ∑

m′i

m2 <18

d − 98

g + 78

.


Suppose that C is connected. Let I ⊆ 1, . . . , n with #I ≥ 2, and suppose that the marked

points Pi : i ∈ I all coincide at a smooth point P ∈ C. If∑i∈I m′

im2 >

∑i 6∈I m′

im2 + 1

2 d + 12 g + 1

2d − g

(A.21)

then (C, P1, . . . , Pn) is GIT-unstable.

Proof. Let ev : H0(PN ,OPN (1)) → k(P) be the evaluation map. Let V1 = V2 = ker ev. Let λ

be the 1-PS of GL(N + 1) whose weight on V1 is and whose weight on V0 is 1, and let λ′ be


Note that∑N

j=0 wtλ(vi) = 1 =: R. Construct a filtration of H0(C, Lm) of increasing weight

as in (A.2). We need to find an upper bound for δp := dim Ωmp .

This time, since by hypothesis P is a smooth point of C, we do not need to use the

normalization to estimate δp. It is clear that Ωmp ⊆ H0(C, Lm ⊗OC(−(m− p)P)). We have

δp := dim Ωmp ≤ h0(C, Lm ⊗OC(−(m− p)P))

= dm− (m− p)− g + 1+ h1(C, Lm ⊗OC(−(m− p)P)).

We use Claim A.1.3 to estimate that h1(C, Lm(−(m − p)P)) ≤ (m − p) ≤ m and that

h1(C, Lm(−(m− p)P)) = 0 if p ≥ 2g − 1.

These give us upper bounds for δp:

δp ≤

(d − 1)m+ p − g + 1+m, 0 ≤ p ≤ 2g − 2(d − 1)m+ p − g + 1, 2g − 1 ≤ p ≤ m− 1.

We may apply Lemma A.1.2, setting α = 1, β = 1, ε = g − 1, r0 = 1 and R = 1. Thus we

estimate:

µL((C, P1, . . . , Pn), λ′) ≥(

12

(d − g + 1)− d)

m2

+∑i∈I

((d − g + 1)− 1

)m′

i +∑i 6∈I

(−1)m′i

−(

(2g − 52

)(d − g + 1)+ 1)

m

≥(−1

2d − 1

2g − 1

2

)m2

+∑i∈I

(d − g)m′i −

∑i 6∈I

m′i

where we have used the fact that our assumptions imply m > (2g − 52 )(d − g + 1)+ 1.


However, this is positive, as we assumed that

∑i∈I

(d − g)m′

im2 >

∑i 6∈I m′

im2 + 1

2 d + 12 g + 1

2d − g

Thus µL((C, P1, . . . , Pn), λ′) > 0, and therefore (C, P1, . . . , Pn) is not GIT-semistable.

Proposition A.5.2. Suppose that d − 9g + 7 > 0, and that

m > max

m3

(g − 12 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),

(10g + 3q2 − 3g)(d − g + 1)

with ∑

m′i

m2 <18

d − 98

g + 78

.

Suppose that C is connected. Let I ⊆ 1, . . . , n with #I ≥ 2, and suppose that the marked

points Pi : i ∈ I all coincide at a singular point P ∈ C (which by the previous propositions

must be a node of C). If ∑i∈I

m′i

m2 >g∑

i 6∈Im′

im2

d − g. (A.22)

then (C, P1, . . . , Pn) is GIT-unstable.

Proof. Let ev : H0(PN ,OPN (1)) → k(P1) be the evaluation map. Let V2 = V1 = ker ev. Let λ

be the 1-PS of GL(N + 1) which acts with weight 0 on V1 and weight 1 on V0, and let λ′ be


Now, if v ∈ V1 then v(Pi) = 0 for all i ∈ I, so vi must be vd−g, whose λ-weight is 1, for

these values of i. Hence∑n

i=1 wtλ(vi)m′i =

∑i∈I m′

i . Note also thatN∑

j=0wtλ(vj ) = 1 =: R. As

usual, construct a filtration of H0(C, Lm) of increasing weight as in (A.2). We need to find

an upper bound for δp = dim Ωmp .

Let π : C → C be the normalization morphism, which is unramified as the hypotheses

of Proposition A.3.1 are satisfied. There are two distinct points in π−1(P1), by Proposition

A.3.2. Let the divisor D := Q1 +Q2 on C consist of these points. If Q1 and Q2 lie on the

same component C1 of C, then degC1L ≥ 3.

As is familiar by now, the normalization morphism induces a homomorphism πm∗ with

πm∗(Ωmp ) ⊆ H0(C, Lm ⊗OC(−(m− p)D)). We have

δp := dim Ωmp ≤ h0(C, Lm ⊗OC(−(m− p)D))+ dim ker πm∗

= dm− 2(m− p)− gC + 1+ h1(C, Lm ⊗OC(−(m− p)D))+ dim ker πm∗.


We may use Claim A.1.3 as usual to establish that dim ker πm∗ < q2, that

h1(C, Lm ⊗ (−(m− p)D)) ≤ 2(m− p) ≤ 2m if 0 ≤ p ≤ 2g − 2, and that

h1(C, Lm ⊗ (−(m− p)D)) = 0 if p ≥ 2g − 1.

We can now estimate δp:

δp ≤

(d − 2)m+ 2p − g + q2 + 1+ 2m, 0 ≤ p ≤ 2g − 2(d − 2)m+ 2p − g + q2 + 1, 2g − 1 ≤ p ≤ m.

We may apply Lemma A.1.2, setting α = 2, β = 2, ε = −g + q2 + 4g − 1, r0 = 1 and R = 1;

thus we estimate:

µL((C, P1, . . . , Pn), λ′) ≥((d − g + 1)− d

)m2

+∑i∈I

((d − g + 1)− 1)m′i +

∑i 6∈I

(−1)m′i

−((5g − g + q2 − 3)(d − g + 1)+ 1

)m

≥ (−g)m2 +∑i∈I

(d − g)m′i −

∑i 6∈I

m′i

where we have used the fact that our assumptions imply that m > (5g − g + q2)(d − g + 1).

The right hand side above is positive, as we assumed that

∑i∈I

m′i

m2 >g +

∑i 6∈I

m′i

md − g

.

Thus µL((C, P1, . . . , Pn), λ′) > 0, and it follows that (C, P1, . . . , Pn) is not GIT-semistable.

Summary: Potential Stability

For convenient reference, we summarize the results of Sections A.2 to A.5 in the theorem

and definition below.

Theorem A.5.3 (Potential Stability Theorem). Suppose that d − 9g + 7 > 0, and

m > max

m3

(g − 12 + d(q1 + 1)+ q3 + µ1m2)(d − g + 1),

(10g + 3q2 − 3g)(d − g + 1),

with ∑

m′i

m2 <18

d − 98

g + 78

.

Let (C, P1, . . . , Pn) be a connected curve which is GIT-semistable. Then:


(i) C is a reduced curve whose only nodes are singularities

(ii) C is nondegenerately embedded in PN

(iii) h1(C, L) = 0;

(iv) any complete subcurve Z ⊂ C with Z ≠ C satisfies the inequality

dZ +κ2

<h0d +

((h0)

(∑C m′

i)−(∑

Z m′i)

(d − g + 1)) 1

m2

d − g + 1+ S

m,

of Lemma A.4.4, where Z consists of Zb connected components and S = g+κ(2g− 32 )+

q2 − g + 1.

Furthermore, a subset of the marked points Pi : i ∈ I cannot collide if

∑i∈I m′

im2 >

∑i 6∈I m′

im2 + 1

2 d + 12 g + 1

2d − g

,

and cannot collide at a node if ∑i∈I

m′i

m2 >g +

∑i 6∈I

m′i

m2

d − g.

Definition A.5.4. If (C, P1, . . . , Pn) satisfies conditions (i)-(vi) of Theorem A.5.3, then we call it

a potentially stable pointed curve.

Appendix B

Linearizations for which Jss is closedin Jss

The result of this appendix is derived from [BS]. See the introduction to Appendix A for

more discussion.

Proposition B.0.5. Suppose that all the hypotheses of Theorem 2.8.1 and Theorem A.5.3 are

satisfied. Write

ηi := m′i

m2 − bi (B.1)

and suppose that ∑C|ηi| ≤

14ν − 2

−3g + q2 − g − 1

2m

(B.2)

Then Jss is closed in Jss.

Proof. We use the valuative criterion of properness to show that the inclusion J∩Jss Jss

is

proper. It then follows that J ∩ Jssis closed in Jss

, as desire. So, let R be a DVR with generic

point K and closed point 0. Let α : Spec R → Jssbe a morphism such that α(K) ⊂ J ∩ Jss

.

Then we will show that α(0) ∈ J ∩ Jssalso.

Let C be the universal family over I, and C |Jss its restriction to Jss. Define a family D of

weighted pointed curves in PN by the following pullback diagram:

D //

C |Jss

Spec R α // Jss

Write σ1, . . . , σn : Spec R → D for the pullbacks of the sections giving the marked points of

C . The images of the σi in D are divisors, denoted σi(Spec R). By definition of J we have

(OPN (1)) |DK (ωDK (a1σ1(K)+ · · · + anσn(K)))ν|DK .

We will write (D0, σ1(0), . . . , σn(0),A) =: (C, P1, . . . , Pn,A), and show that its represen-

tative in the universal family over I in fact lies over Jss .

124

125

The curve C is connected, as a limit of connected curves. We have α(0) ∈ Jss = J ∩ Iss ,

so by A.5.3 C is reduced, and any singularities it has are nodes. In particular, ωC is defined

and is an invertible sheaf. Also by A.5.3, the curve C ⊂ PN is nondegenerately embedded.

The rest of the proof will be devoted to showing that C is “pluricanonically embedded,” i.e.

OPN (1)|C (ωC(a1P1 + · · · + anPn))ν (B.3)

Then we will know that (C, P1, . . . , Pn,A) is in J.

Decompose C =⋃

Ci into its irreducible components. Then we can write

OPN (1)|D (ωD/ Spec R(a1σ1(Spec R)+ · · · + aNσn(Spec R))

)⊗ν |D ⊗OD(∑

eiCi)

where the ei are integers. As OD -modules, OD(C) OD so we can normalize the integers

ei so that they are all nonnegative and at least one of them is zero. Separate C into two

subcurves Y :=⋃

ei=0Ci and Z :=

⋃ei>0

Ci . Since at least one of the ei is zero, we have Y ≠ and

Z ≠ C. Suppose for purposes of contradiction that Z ≠ (hence Y ≠ C). Let κ = #(Y ∩ Z)

and let tZ be the number of connected components of Z. Since C is connected, we must

have κ ≥ tZ . We will obtain our contradiction by showing that κtZ

< 1.

Any local equation for the divisor OD(∑

eiCi) must vanish identically on every compo-

nent of Z and on no component of Y . Such an equation is zero therefore at each of the κ

nodes in Y ∩ Z . Thus we obtain the inequality

κ ≤ degY

(OD(−

∑eiCi)

)= degY

(OPN (1)⊗ω⊗−ν

D0(−νaiσ1(0)− · · · − νanσn(0))

)= dY − ν(2gY − 2+ aY + κ).

Substituting dZ = d − dY , gZ = g − gY − κ + 1 and d = ν(2g − 2+ a), this is equivalent to

dZ − ν(2gZ − 2+ aZ ) ≤ (ν − 1)κ. (B.4)

The hypotheses of Lemma A.4.4 are satisfied for Z and κ = #(Y ∩Z). Write γ = ν2ν−1 and

m′i

m2 = bi + ηi .

dZ +κ2

<(dZ − gZ + 1)d

d−g+1+ ((dZ − gZ + 1)(

∑C ai)− (d−g+1)(

∑Z ai))

d−g+1+ tZ S

m

126 APPENDIX B. LINEARIZATIONS FOR WHICH Jss IS CLOSED IN Jss

aκ2

(d−g+1)

< dZ (g − 1)− d(gZ − 1)+ ν2ν−1

(dZ−gZ+1)

∑C

ai

− (d−g+1)

∑Z

ai

+(dZ − gZ + 1)

∑C

ηi

− (d−g+1)

∑Z

ηi

+ tZ Sm

(d−g+1). (B.5)

The final terms of the right hand side are already in the form we want them, so for brevity

we shall only work on the first line.

dZ (g − 1)− d(gZ − 1)+ ν2ν−1

(dZ − gZ + 1)

∑C

ai

− (d−g−1)

∑Z

ai

= dZ

2ν−1

(2ν−1)(g−1)+ ν

∑C

ai

− (gZ − 1)

d−g+1+ 12ν−1

(2ν−1)(g − 1)+ ν

∑C

ai

− ν

2ν−1(d−g−1)

∑Z

ai

= dZ

2ν−1(d−g−1)− (gZ − 1)(d−g−1)

(1+ 1

2ν−1

)− ν

2ν−1(d−g−1)

∑Z

ai

,

where for the last equality we have recalled that (2ν−1)(g − 1)+ ν(∑

C ai) = d−g−1. We

substitute this into line (B.5) and then multiply everything by 2ν−1d−g−1 to obtain:

(2ν − 1)κ2

< dZ − ν

2gZ − 2+∑Z

ai

+ (2ν − 1)

∑C

ηi

(dZ − gZ + 1d−g−1

−∑

Z ηi∑C ηi

)+ (2ν−1)

tZ Sm

.

Now use (B.4) to see

κ2

< (2ν−1)

∑C

ηi

(dZ − gZ + 1d−g−1

−∑

Z ηi∑C ηi

)+ (2ν − 1)

tZ Sm

⇒ κ2tZ

< (2ν−1)

∑C

η

(dZ − gZ + 1d−g−1

−∑

Z ηi∑C ηi

)+ (2ν−1)

Sm

(B.6)

We must take care as S varies with κ; explicitly, S = g + κ(2g − 32 )+ q2 − g + 1. Thus the

inequality we wish to contradict becomes

κtZ

14ν − 2

−(2g − 3

2 )m

<

∑C

ηi

(dZ − gZ + 1d−g−1

−∑

Z ηi∑C ηi

)+ g + q2 − g + 1

m. (B.7)

127

It is time to use our bounds for ηi . Note that

−1 ≤ dZ − gZ + 1d−g−1

−∑

Z ηi∑C ηi

≤ 1

We assumed that∑

C |ηi| ≤ 14ν−2 −

3g+q2−g− 12

m . It follows that∑C

ηi

(dZ − gZ + 1d−g−1

−∑

Z ηi∑C ηi

)≤ 1

4ν − 2−

3g + q2 − g − 12

m.

Hence line (B.7) says

κtZ

14ν − 2

−(2g − 3

2 )m

<1

4ν − 2−

2g − 32

m.

By hypothesis m > (6g + 2q2 − 2g + 1)(2ν − 1) > (2g − 32 )(4ν − 2) and so we know that

14ν−2 −

2g− 32

m > 0. Thus we have proved that κtZ

< 1, a contradiction.

The contradiction implies that we cannot decompose C into two strictly smaller sub-

curves Z and Y as described. Thus all the coefficients ei must be zero, and we have an

isomorphism

OPN (1)|D ωνD/ Spec R(νa1σ1(Spec R)+ · · · + νanσn(Spec R))|D .

In particular, (D0, σ1(0), . . . , σn(0)) is in J. Thus α(0) ∈ J ∩ Jss. Hence Jss = J ∩ Jss

is

closed in Jss, which completes the proof.

Remark. The allowable range printed for∑

C |ηi| is not sharp. Note that

dZ−gZ+1d−g−1 −

∑Z ηi∑C ηi

> −1, enabling us to drop our lower bound to below 14ν−1 . It is not clear

whether the upper bound can be improved.

Index

A, 15

Avir, 34, 36

Aviri , 48

Ak,i , 41, 46

Avirk,i , 39, 48

Avirk , 39

Bi , 20

Dj , 20

I, 12, 67

J, 7, 67

N, 12, 67

P(t), 12, 67

Qi , 20

T , 15

T vir, 36, 48, 49

U , 67

V•, 9, 20

W , 31, 42

Zj , 50

∆, 34, 39, 46

∆k,i , 39, 47

Ki , 29

Λ, 43

Mg,A, 5

Ωmp , 91

α

in Appendix, 91

Ωmp , 91

β

in Appendix, 91

δ

as a class in Pic(Mg), 79

δp

in Appendix, 91

ε, 53, 56

in Appendix, 91

εp

in Appendix, 91

η, 44

1〈uξ〉=0, 44

1ζ≠0, 44

κ


in Appendix, 101

λ


λ′, 90

A, 4

B, 13

IC , 93

Ij , 49

µL(x, λ), 3, 8, 14

µ1, 93

µ2, 93

128

129

ν , 67, 77

C, 92

Mpsg , 85

N, 29

g, 94

σj,i , 61

τj,i , 61

G•, 9, 19

N, 24, 29

V•, 9, 23

X•, 9, 30, 37

ck,i , 23

f (k), 35

fi(k), 35

x(k, w), 31

x(k, w, i), 42

ξ, 42

ζ, 42

a, 67

ai , 4

b, 13

bi , 13, 77

cj,i , 20

d, 67

dj , 20

e, 50

g, 4

j(i, `), 29

jCliff , 21, 50

jRR, 21, 50

k(i,•), 48

k(i, `), 29

m, 13

m1, 93

m2, 93

m3, 93

m′i , 13

in Appendix, 89

n, 4

q, 20

q1, 93

q2, 93

q3, 93

rj , 15

s(k, i), 28

t(k, i), 28

u, 21

u0, 58

v , 21

v0(u), 58

w , 21

zj , 20

x(k, w, i), 30

X• , 30

Alexeev, V., 4, 73

ample cone of Mg , 80

angle brackets 〈〉, 43

Baldwin, E., 2, 8, 24, 66, 89

130 INDEX

Basic Inequality, 102, 117

Caporaso, L., 71

Cases 0–3, 51

Cases A–E, 53

Cases I–IV, 28, 39

Combinatorial problem, 60

DM-stable, def., 4

Dolgachev, 3, 68

Example 1, 24, 36, 37

profile, 31

GIT semistable replacement, 71

Guy, G. M., 4, 73

Hassett, B., 4, 69, 73, 85

Hu, Y., 3, 68

Hyeon, D., 85

isopolarizations, 83

linearization, 3, 13, 77

in Appendix, 89

Mayer, A., 1

Morrison, I., 19, 26, 64

Multiplication Lemma, 22

Murphy’s Law, 76

numerical criterion, 15

parameter space, 3, 12

polarization formula, 82

potential stability, 123

profile, 9, 17, 37

for X•, 31

progress report, 38

pseudostable curves, 85

Schubert, D., 85

Seshadri, C., 71

Span Lemma, 26

strategy

in Appendix, 90

Thaddeus, M., 3, 68, 83

Vakil, R., 76

VGIT, 3

virtual profile, 11, 19, 34, 36, 37

Wall Street, 50

weighted pointed curve, definition, 4

worst 1-PS, putative, 64

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Geometric invariant theory and moduli spaces of pointed curves · 2014-11-06 · Geometric invariant theory and moduli spaces of pointed curves David Swinarski Ph.D dissertation Columbia