geometric invariant theory and moduli spaces of mapsgeometric invariant theory and moduli spaces 2...

Geometric Invariant Theoryand Moduli Spaces of Maps

David J. SwinarskiBalliol College

Submitted in partial fulfillment of the requirements of theMaster of Science by Research

Trinity 2003

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Contents

1 Introduction 1

I Geometric Invariant Theory and Moduli Spaces 2

2 The Theory of Moduli 2

3 Geometric Invariant Theory 6

4 Cohomology of Quotients 8

5 Example: Hyperelliptic Curves 9

II Four Important Moduli Spaces 12

6 M(n, d) 126.1 Facts about vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 A G.I.T. construction of M(n, d) . . . . . . . . . . . . . . . . . . . . . . . . 136.3 The cohomology of M(n, d) . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Mg 167.1 Facts and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 U g(n, d) 19

9 Mg(X, β) 20

10 Motivation 21

III A G.I.T. Construction of the Moduli Space of Maps 24

11 Gieseker’s Construction of Mg Reviewed 2411.1 The Hilbert scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.2 The Hilbert point of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.3 Defining K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

12 Fitting Ideals and the Construction of Mg 27

13 Constructing Mg(Pr, d): General Information 29

14 The G.I.T. Set-up and the Numerical Criterion for Mg(Pr, d) 32

15 Maps from Smooth Curves Are G.I.T. Stable 36

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16 G.I.T. Semistable Maps Are Potentially Stable 4216.1 First properties of G.I.T. semistable maps . . . . . . . . . . . . . . . . . . . 4316.2 G.I.T. semistability implies that the only singularities are nodes . . . . . . . 5616.3 G.I.T. semistable curves are reduced . . . . . . . . . . . . . . . . . . . . . . 6916.4 Potential stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

17 The Construction Finished 75

18 Another Linearization 82

19 Closing: Toward Mg, U g(n, d), and Beyond 85

20 Appendix I: Equivariant Cohomology 87

21 Appendix II: Intersection Cohomology 89

22 Index of Terms and Notation 94

23 Acknowledgements 95

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1 Introduction

We begin with an introduction to the theory of moduli and geometric invariant theory

(G.I.T.), giving all the important definitions and results with references to the foundational

works. We also outline Kirwan’s techniques for studying the cohomology of G.I.T. quotients.

The second part of this paper is a quick tour of a menagerie of four important (families

of) moduli spaces. This may seem unusual considering that the primary result of this paper

is the construction of a single moduli space. However it is a philosophy we wish to emphasize

that lessons learned in studying the geometry and topology of the moduli spaces of vector

bundles over an algebraic curve as well as the moduli spaces of curves provide valuable

insight into the more recently studied moduli spaces of vector bundles over varying algebraic

curves and the moduli spaces of maps, and that there is a richness in considering all these

in relation to one another which does not emerge if each space is considered in and of itself.

A project proposed by Kirwan, our motivation for part three, illustrates this richness.

The third part is the heart of this paper. Working over the complex numbers, we construct

a G.I.T. quotient J//SL(W ) and prove that it is isomorphic to the Kontsevich-Manin moduli

space of maps Mg(Pr, d). We indicate how this construction could in turn be used to study

the moduli space of curves and perhaps even the moduli space of vector bundles over varying

curves.

An index of some of the terms and notation used for Part III is included.

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Part I

Geometric Invariant Theory andModuli Spaces

2 The Theory of Moduli

Often the set of geometric objects of a given type (or equivalence classes of geometric

objects of a given type) can be parametrized by another geometric object. For instance,

consider hyperelliptic curves, that is, compact Riemann surfaces C of genus g ≥ 2 admitting

a holomorphic map C → P1 of degree two. By the Riemann-Hurwitz Formula, C must

have 2g + 2 branch points in the Riemann sphere P1. Two hyperelliptic curves having

the same branch locus b1, . . . , b2g+2 are isomorphic since they are both isomorphic to the

normalization of the curve y2 =

2g+2∏i=1

(x − bi). In fact two hyperelliptic curves C1 and C2

are isomorphic if and only if their branch loci differ by an automorphism of P1. So an

unordered set of 2g + 2 distinct points in P1 specifies an isomorphism class of hyperelliptic

curves. We might therefore naively expect the set P1 (2g+2) − ∆ to classify hyperelliptic

curves, where ∆ is the union of all subdiagonals and the superscript (2g + 2) denotes the

symmetric product. Indeed this simple observation can be made precise (see Section 5 below),

yielding a quasiprojective complex algebraic variety each of whose points corresponds to an

isomorphism class of hyperelliptic curves of genus g.

Just how general is this phenomenon? When can the set of equivalence classes of some

type of geometric object itself be given the structure of a geometric object in a natural way?

The mathematical framework developed to study such questions is the theory of moduli;

classic references include [GIT] Ch. 5, [MS], and [New] Ch. 1. Moduli spaces arise naturally

in the setting of classification problems. In addition to their utility for classification, moduli

spaces are often interesting objects in their own right. Furthermore they give important

information regarding how geometric objects fit together in families.

So suppose A is the set of geometric objects we wish to classify under an equivalence

relation ∼ on A. Curiosity might lead us to ask whether A/∼ can be given for instance the

structure of a manifold or variety. More generally, we seek to define in a natural way on A/∼the structure of an object of Cat, where we denote by Cat one of the following categories:

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topological spaces, manifolds, varieties, schemes, or stacks. (In this paper all moduli spaces

will be locally noetherian schemes of finite type.)

To specify a moduli problem one must also provide a definition of a family X of objects in

A parametrized by an object S ∈ Ob(Cat). (We will denote families by uppercase calligraphic

letters.) It is often useful to define an equivalence relation ∼fam of families (though if one is

content with only a coarse moduli space as defined below then this is not strictly necessary).

The following formal properties must be satisfied:

i. A family parametrized by a single point S = s corresponds to a single

object Xs ∈ A.

ii. The equivalence relation ∼fam on families reduces to ∼ when S = s.

iii. For any morphism ϕ : S ′ → S and any family X parametrized by S,

there is an induced family ϕ∗X parametrized by S ′. This operation

is functorial in the sense that (ϕ1 ϕ2)∗ = ϕ∗2 ϕ∗1 and id∗SX = X .

Furthermore if X1 ∼fam X2 then ϕ∗X1 ∼fam ϕ∗X2.

We provide two examples to illustrate these definitions:

Example 2.1 A family X of projective curves parametrized by a scheme S is a flat mor-

phism X → S each of whose geometric fibers Xs is a projective curve of genus g. We take

the equivalence relation X1 ∼fam X2 to be algebraic isomorphism over S.

Example 2.2 We define a family over a scheme S of algebraic vector bundles over a fixed

nonsingular projective curve C to be a vector bundle X over C × S. A plausible candidate

for ∼fam would be to say that two families are equivalent if there is a bundle isomorphism

between them; however there is no fine moduli space for the moduli problem so defined (cf.

for instance [New] page 148)! The following choice of equivalence relation yields a moduli

problem which admits a fine moduli space if the rank and degree are coprime: We say that

X1 ∼fam X2 if there exists a line bundle Lπ→ S such that X1

∼= X2⊗π∗L.

Suppose the underlying point set of M ∈ Ob(Cat) is in bijective correspondence with

A/∼. For any base S ∈ Ob(Cat) and any family X of objects of A over S, there is a map

fX : S →M

s 7→ [Xs]

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where [Xs] is the equivalence class in A/∼ which Xs represents.

If our moduli space M is to be a truly “universal” object, then it ought to be compatible

with every family X over every S ∈ Ob(Cat). That is, the maps fX should be Cat-morphisms

for every X and S. What we are really dealing with is a contravariant functor F from Ob(Cat)

to the category of sets; here F(S) is the set of equivalence classes of families parametrized

by S, and the Cat-morphism F(ϕ) : F(S2) → F(S) for any ϕ : S2 → S is given by pullback

along ϕ.

Note that for any S the map fX defines a map

F (S) : F(S) → Hom(S,M)

F (S)([X ]) = fX .

This collection of maps defines a natural transformation

F : F → Hom(−,M).

Definition 2.3 A coarse moduli space for a given moduli problem is an object M ∈ Ob(Cat)

together with a natural transformation F : F → Hom(−,M) such that

i. F (s) is a bijection.

ii. For any object M ′ ∈ Ob(Cat) and natural transformation F ′ : F →Hom(−,M ′), there exists a unique natural transformation Ψ such that

F ′ = Ψ F .

If the natural transformation F is an isomorphism of functors, we say that M represents

F and call M a fine moduli space. In this case the moduli space has even better properties:

namely, there exists a universal family.

Definition 2.4 A universal family U is a family parametrized by M such that for every

family X parametrized by S there is a unique morphism ϕ : S →M such that X ∼fam ϕ∗U .

U represents the ∼fam equivalence class corresponding to the inverse image of the element

id ∈ Hom(M,M) under the (bijection) F (M) : F(M) → Hom(M,M).

Unfortunately many interesting moduli problems do not admit fine moduli spaces. There

are at least three ways around this problem. First, the functor may be representable in

a larger category. Second, a minor modification of the moduli problem may lead to the

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existence of a fine moduli space; for example while there is no fine moduli space of nonsingular

projective curves of genus g, there is a fine moduli space of curves marked with sufficiently

many points. Finally, a coarse moduli space often exists when a fine moduli space does not.

While the universal properties of a coarse moduli space are not as strong as those of a fine

moduli space, we shall see in the next section that good results for coarse moduli spaces can

still be obtained, especially if there exists a family X → S with the local universal property:

Definition 2.5 A family X → S is said to have the local universal property if for any family

X2 → S2 and any point s2 ∈ S2 there exists a neighbourhood U 3 s2 such that X|U ∼fam ϕ∗X

for some morphism ϕ : U → S. The morphism ϕ is not required to be unique.

Moduli spaces as quotients

Moduli spaces are often constructed as quotients; one fixes as many discrete invariants of

the moduli problem as possible and then parametrizes the set of objects A by S ∈ Ob(Cat)

so that the equivalence relation ∼ is given by a group action on S. More precisely:

Definition 2.6 Let G be an algebraic group acting on a scheme X. A categorical quotient

of X by G is a pair (Y, ϕ), where ϕ : X → Y is a morphism such that

i. ϕ is constant on the orbits of the G action

ii. for any scheme Y2 and morphism ϕ2 : X → Y2 which is constant on

orbits, there is a unique morphism ψ : Y → Y2 such that ψ ϕ = ϕ2.

A quotient (Y, ϕ) is an orbit space if in addition ϕ−1(y) consists of a single G-orbit for every

point y ∈ Y .

The relationship between orbit spaces and moduli spaces is described by the following

proposition:

Proposition 2.7 ([New] Proposition 2.13) Suppose for a given moduli problem there ex-

ists a family X → S which has the local universal property. Suppose a group G acts on S in

such a way that Xs ∼ Xt if and only if s and t belong to the same G-orbit. Then

i. any coarse moduli space is a quotient of S by G

ii. a quotient of S by G is a coarse moduli space if and only if it is an orbit

space.

In the next section we review an important technique for constructing orbit spaces.

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3 Geometric Invariant Theory

We will describe geometric invariant theory first in the category of complex projective

varieties. Let X be a projective variety embedded in CPn, and write I(X) for the ideal

generated by the set of homogeneous polynomials which vanish at every point of X. We call

the graded ring A(X) = C[x0, . . . , xn]/I(X) the homogeneous coordinate ring of X. Let G

be an algebraic group acting on X. We shall suppose further that the group action is linear,

that is, it is induced by a linear action of G on Cn+1. Then the group action gives rise to a

group action on A(X) as follows: g ∈ G takes f ∈ A(X) to f(gx) ∈ A(X). We denote by

A(X)G that portion of the ring which is fixed by the action of the group. The coordinate

ring of a variety is always finitely generated; if we add the hypothesis that the group G

is reductive, then by Nagata’s Theorem A(X)G is finitely generated also (see for instance

[New] Theorem 3.4), and there is a projective variety associated to this ring which we denote

X//G. We might hope that X → X//G would be a quotient in the sense of Definition 2.6

or that X//G would be the ordinary topological quotient X/G, but in general neither of

these statements is true. In fact there will be no morphism X → X//G at all if there are

any points of X at which every G-invariant polynomial vanishes; in this case such a map is

at best a rational transformation, that is, it is well-defined only when restricted to a dense

open subset. That subset can be described by an analysis of stability as follows. We call a

point x ∈ X G.I.T. semistable if there is a G-invariant homogeneous polynomial f ∈ A(X)G

such that f(x) 6= 0 . Write Xss for the set of semistable points. Then there is a surjective

morphism Xss → X//G. However, this map still may not constitute an orbit space, as one

orbit may be contained in the closure of another. We define a further subset Xs ⊆ Xss, the

stable locus. A point x is stable if the dimension of the orbit G.x is equal to the dimension

of G and if there is a G-invariant homogeneous polynomial f with deg f ≥ 1 such that the

action of G on the open set Xf := x ∈ X | f(x) 6= 0 is closed. Then Xs → Xs/G is an

orbit space. To summarize, we have:

Xs ⊆ Xss ⊆ X↓ ↓

Xs/G ⊆ X//G

Note that if all the semistable points are stable, X//G is the topological quotient Xs/G.

In the preceding paragraph we assumed that X was a variety embedded in Pn and the

group action was linear. More generally, then, let X be a scheme of finite type over k. Then

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to define the G.I.T. quotient we must specify a linearization of the group action.

Definition 3.1 A linearization of the action of G with respect to an ample line bundle L is

an action of G on Lp→ X such that

i. for all y ∈ L, g ∈ G, one has p(gy) = g.p(y).

ii. for all x ∈ X, g ∈ G, the map Lx → Lgx given by the rule y 7→ gy is

linear.

A linearization is thus a lifting of the group action to the line bundle L. We often abuse

language and call a linearization with respect to L “the linearization L.” Note that if G is

connected and admits no homomorphism to the multiplicative group k∗ and X is geomet-

rically reduced, then Mumford shows that each line bundle has at most one linearization

([GIT] Proposition 1.4), so this is not such a terrible abuse after all.

In this more general setting, stability and semistability are defined as follows: A point

x ∈ X is G.I.T. semistable with respect to the linearization L if for some integer n there

is a G-invariant section of Ln such that f(x) 6= 0 and Xf := x ∈ X|f(x) 6= 0 is affine.

A point x ∈ X is G.I.T. stable with respect to the linearization L if it is semistable and

furthermore the dimension of the orbit G.x is equal to the dimension of G and the action of

G on Xf is closed. The semistable locus Xss(L) and the stable locus Xs(L) are G-invariant

open sets. We write X//LG for the categorical quotient of Xss(L). Note that in general,

different linearizations carry different sets of semistable and stable points, and in particular

the G.I.T. quotients X//L1G and X//L2G need not be isomorphic; we shall have more to say

about this later (see page 36).

In general it is extremely difficult to verify G.I.T. stability or semistability directly, that

is, by studying G-invariant sections of L. Mumford established a numerical criterion ([GIT]

§2.1) which is highly useful for this purpose. The basic idea is to study the action of one-

parameter subgroups of G rather than the whole G-action. For simplicity let us assume

G = SL(W ) acts linearly on X. A one-parameter subgroup of SL(W ) is a homomorphism

λ : k∗ → SL(W ). Let

Wi = w ∈ W |the action of λ is given by λ(t)w = tiw

and let Sλ(W ) be the set of integers i such that Wi 6= 0. There is a decomposition (cf. [HM]

4.15)

W =⊕

i∈Sλ(W )

Wi.

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For any element x of W , write Sλ(x) = i ∈ Sλ(W )|xi 6= 0, where xi is the component

of x in Wi.

Definition 3.2 For any x ∈ W we define the integer µλ(x) = minSλ(x).

Theorem 3.3 (Hilbert-Mumford numerical criterion)

[x] ∈ P(W ) is SL(W )-semistable ⇐⇒ µλ(x) ≤ 0 for all λ

[x] ∈ P(W ) is SL(W )-stable ⇐⇒ µλ(x) < 0 for all λ.

Some advantages of geometric invariant theory are that its constructions work in positive

characteristic and the G.I.T. quotient X//G of a scheme X is again a scheme. Moreover, if

X is a projective scheme, X//G is also a projective scheme. However, note that X//G can

be badly singular even if X is nonsingular.

4 Cohomology of Quotients

See Appendix I for a discussion of equivariant cohomology.

A nonsingular complex projective variety may be thought of as a compact Kahler manifold

where the Kahler metric is given by the restriction of the Fubini-Study metric. The imaginary

part of this metric is a symplectic form, so in particular nonsingular complex projective

varieties are compact symplectic manifolds. Thus techniques developed by Frances Kirwan

to compute the cohomology of quotients of compact symplectic manifolds apply to G.I.T.

quotients of nonsingular projective varieties (see [Kir1]).

Let G be a connected complex reductive algebraic group acting linearly on a nonsingular

complex projective variety X ⊆ Pn. There is a stratification Sβ | β ∈ B of X such that

S0 = Xss. The cohomology of X can then be built up from the cohomology of the strata Sβ

using the equivariant Gysin sequence. One finds that the stratification Sβ is equivariantly

perfect, which by definition means that the equivariant Morse inequalities are equalities.

That is,

PGt (X) =

∑β∈B

t2dβPGt (Sβ)

where PGt (−) denotes the equivariant Poincare series of (−) and dβ is the complex codimen-

sion of Sβ in X.

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Observe that

H∗(X//G) ∼= H∗(Xss/G) if Xss = Xs

∼= H∗G(Xss) if stabG(x) is finite for every x ∈ Xss

(Note that if the equality Xss = Xs holds then stabG(x) is finite for every x ∈ Xss.)

In this situation we may compute

Pt(X//G) = PGt (Xss) = PG

t (X)−∑

β∈B, β 6=0

t2dβPGt (Sβ)

Throughout our discussion of geometric invariant theory and cohomology of quotients

we have focused our attention on complex projective varieties. It is possible to define the

stratification Sβ | β∈B in more general settings. However to use the preceding equations

it is essential that one be able to compute the equivariant cohomology H∗G(X). If X is a

compact symplectic manifold, Kirwan has shown [Kir1] that H∗G(X) ∼= H∗(X)⊗H∗(BG).

In the next section we return to our first example, the moduli space of hyperelliptic

curves, to illustrate in detail many of the ideas of the theory of moduli, geometric invariant

theory, and cohomology of quotients in algebraic geometry.

5 Example: Hyperelliptic Curves

Definition 5.1 A compact Riemann surface C of genus g ≥ 2 is hyperelliptic if there exists

a holomorphic map C → P1 of degree two.

By the Riemann-Hurwitz Formula, C must have 2g + 2 branch points in the Riemann

sphere P1. For convenience set n = 2g+2. Two hyperelliptic curves having the same branch

locus b1, . . . , bn are isomorphic, since they are both isomorphic to the normalization of the

curve y2 =n∏i=1

(x− bi). In fact,

Proposition 5.2 Two hyperelliptic curves C1 and C2 are isomorphic if and only if their

branch loci differ by an automorphism of P1, that is, by an element of PSL(2; C).

So an unordered set of n distinct points in P1 specifies an isomorphism class of hy-

perelliptic curves. Recall that PSL(2; C) = SL(2; C)/±I. It is technically more con-

venient to work with SL(2) than PSL(2) because there is an obvious linearization of the

SL(2) action on P1. We therefore lift the PSL(2) action to an SL(2) action. Observe

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next that there is a one-one correspondence between points of Pn and sets of n unordered

points in P1. This correspondence maps the point [a0, . . . , an] ∈ Pn to the set of zeroes

of the polynomial a0XnY 0 + · · · + anX

0Y n. Let S ⊂ Pn be the open set corresponding to

([x1 : y1], . . . , [xn : yn]) ∈ (P1)n | [xi : yi] 6= [xj : yj] if i 6= j/Σn where Σn is the symmet-

ric group on n elements. Then the points of S/SL(2) are in one-one correspondence with

isomorphism classes of hyperelliptic curves.

We exhibit a family over S/SL(2) which has the local universal property: Let P2 be the

blowup of P2 at the point [0 : 1 : 0]. Write

Z = ([x : y : z], (a0, . . . , an))∈P2 × S | y2z2g =∏i=0

(x− aiz),

and let Z be the proper transform of Z, i.e. Z is the closure of Z ∩ ((P2 − [0 : 1 : 0])× S)

in P2× S. Then Z is a family of hyperelliptic curves in which any singularities in the fibers

have been resolved. Define a map f : Z → P2 × P1 × S → P1 × S and a map π : Z → S

by p2 f . Then π is a family of hyperelliptic curves. The universal properties of a blow-up

allow us to show that π has the local universal property.

If we can also show that S/SL(2) is an orbit space then we can conclude from Propo-

sition 2.7 that S/SL(2) is a coarse moduli space. We show that S ⊂ (Pn)s; G.I.T. then

assures us that S/SL(2) is indeed an orbit space.

We study the Hilbert-Mumford numerical criterion in the context of this problem. Let λ

be a one-parameter subgroup (1-PS) of SL(2), that is, a non-trivial homomorphism

C \ 0 → SL(2). Any 1-PS in SL(2) is conjugate to one of the form

λr : t 7→(tr 00 t−r

)where r is a positive integer. An element t ∈ λ acts on Pn by the rule:

(t, x) 7→ diag(trn, tr(n−2), ...., tr(−n))

a0...an

where diag(trn, tr(n−2), ..., tr(−n)) denotes the (n+1)×(n+1) diagonal matrix whose diagonal

entries are trn, tr(n−2), ..., tr(−n). The numerical criterion in our situation may be stated as

follows:

Theorem 5.3 Define µ(x, λ) = maxr(n− 2i)|ai 6= 0. Then x is semistable if and only if

µ(gx, λ) ≥ 0 for every 1-PS λ of the form above and every g ∈ G. A point x is stable if and

only if µ(gx, λ) > 0 for every 1-PS λ of the form above and every g ∈ G.

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Let i0 be the smallest integer such that ai 6= 0. It follows that ai = 0 for all i < i0. Then x

is not stable⇔ µ ≤ 0 ⇔ n−2i0 ≤ 0 ⇔ i0 ≥ n/2. This implies that a point [a0 : . . . : an] ∈ Pn

is stable for the action of SL(2) if the polynomial∑aiX

n−iY i has no factors of multiplicity

greater than or equal to n/2. Similarly a point [a0 : . . . : an] ∈ Pn is semistable for the

action of SL(2) if the polynomial∑aiX

n−iY i has no factors of multiplicity greater than

n/2. Since points of S correspond to polynomials having distinct roots, we have S ⊂ (Pn)s,

and (Pn)//SL(2) is therefore a compactification of the moduli space of hyperelliptic curves.

Note that (Pn)ss = (Pn)s precisely when n is odd. But n = 2g + 2, so (Pn)ss 6= (Pn)s.

We are not in the good case; Pn//SL(2) 6= (Pn)ss/SL(2). We have three options:

i. The G.I.T. quotient Pn//SL(2) is a singular projective variety. It is

possible to calculate its intersection cohomology. See Appendix II for

a discussion of intersection cohomology.

ii. Whenever Xss 6= Xs and Xs 6= ∅ there exists a canonical partial

desingularization X//G of X//G due to Kirwan such that Xss = Xs.

The partial desingularization X//G is obtained by blowing up X//G

along a sequence of subvarieties. We can construct Pn//SL(2) and

compute H∗(Pn//SL(2)). See [Kir2] for details of the procedure; this

example is worked out in Section 9 of that work.

iii. If one is more interested in the topology of the uncompactified moduli

space S//SL(2), then the machinery we have set up offers a route to

H∗(S/SL(2)) = H∗G(S).

This situation is typical. The natural compactifications of several interesting moduli

spaces are singular varieties; it may be possible to obtain useful information about the

topology of these compactifications by calculating their intersection cohomology. One could

desingularize them but there is no a priori guarantee that the resulting space will have a

moduli interpretation. Of course there is always the non-compact moduli space to study.

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Part II

Four Important Moduli Spaces

In Part II we review four families of moduli spaces.

6 M(n, d)

Of the four moduli spaces we shall discuss in this paper, the moduli space of vector

bundles over a Riemann surface is perhaps the best understood. Although M(n, d) does

not play a direct role in the G.I.T. construction of the moduli space of maps, we review

its construction here both to show some of the previously described techniques in action

and because familiarity with this construction will be enlightening for later discussions of

U g(n, d), the moduli space of vector bundles over varying curves.

6.1 Facts about vector bundles

Let C be a non-singular projective curve of genus g. Vector bundles on C are classified

topologically by their rank n and degree d. If the moduli problem posed is to classify all

algebraic vector bundles up to isomorphism, it turns out there is not even a coarse moduli

space—there will be jump phenomena [HL]. The following definitions will permit us to pose

a moduli problem which admits a moduli space.

Definition 6.1 Let E be an algebraic vector bundle on C. We say E is slope semistable if

for every proper subbundle F , µ(F ) ≤ µ(E), where the slope µ(F ) :=deg(F)

rank(F). It is slope

stable if µ(F ) < µ(E) for every proper subbundle F .

Remark. If gcd(n, d) = 1, then a slope semistable bundle is slope stable. For suppose E is

a slope semistable bundle which is not stable. Then there exists a subbundle F ⊂ E such that

0 < rankF < rankE and degF ·rankE = degE·rankF , which implies gcd(rankE, degE) > 1.

Semistable bundles admit a moduli space. Restricting ourselves to semistable bundles is

not such a terribly unnatural thing to do: any algebraic vector bundle E on C has a unique

Harder-Narasimhan filtration [HL], that is, an increasing filtration

0 = HN0(E) ⊂ HN1(E) ⊂ · · · ⊂ HNj(E) = E

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such that each quotient HNi(E)/HNi−1(E), i = 1, . . . , j is a slope semistable bundle. So

slope semistable bundles may be viewed as the building blocks of all vector bundles.

Let E be a slope semistable bundle on C of rank n and degree d >> 0. (In the arguments

to follow we may assume d as large as we need, since tensoring a vector bundle over C with

any fixed line bundle of degree ` yields an isomorphism M(n, d) ∼= M(n, d + `n).) Then

H1(E) = 0 and E is generated by its global sections. By the Riemann-Roch theorem one

calculates dimH0(E) = d+ n(1− g) =: p.

For each slope semistable bundle E, there is a Jordan-Holder filtration

0 = E0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Eα = E

such that E0, E1/E0, . . . , Eα/Eα−1 are all slope stable and

µ(E0) = µ(E1/E0) = · · · = µ(Eα/Eα−1) = µ(E).

The Jordan-Holder filtration is not unique but the bundle

Gr (E) := E1/E0 ⊕ E2/E1 ⊕ · · · ⊕ Eα/Eα−1

is determined up to isomorphism by E. We say two semistable bundles E and E ′ are

equivalent if Gr (E) ∼= Gr (E ′) and write M(n, d) for the moduli space of equivalence classes

of slope semistable vector bundles over a fixed nonsingular projective curve C of genus g.

6.2 A G.I.T. construction of M(n, d)

It is possible to construct M(n, d) via geometric invariant theory. We outline one such

construction following [New] and highlighting the role of the Quot scheme as foundation

for later discussion of the space U g(n, d), the moduli space of vector bundles over varying

curves. First, fix a very ample line bundle H on C thus embedding C into PM for some M ,

and let h denote the degree of H. Let E be a semistable bundle, and assume the degree is

sufficiently large that H1(E) = 0. Then the Hilbert polynomial of E with respect to H is

P (m) = p+ nmh where p = d+ n(1− g) as in the previous subsection.

Next we pass from the category of algebraic vector bundles on C to the equivalent category

of locally free sheaves to pave the way for later generalization to singular base curves C. We

shall exploit the following fact proved by Grothendieck (see [HL] for a modern proof): For

any coherent sheaf E over C, the quotients of E which have a given Hilbert polynomial P

13

are parametrized by a Grothendieck Quot scheme Quot(E , P ) over which there is a coherent

sheaf U which has the universal property for flat families of such quotients of E . Since we

are interested in bundles E with dimH0(E) = p which are generated by their sections (see

previous subsection), we shall be interested in quotients of the sheaf

E =

p⊕OC

which have Hilbert polynomial

P (m) = p+ nmh.

Define a subset R ⊂ Quot(E , P ) to be those points for which U|q×C is locally free and

the map H0(⊕pOC) → H0(U|q×C) is an isomorphism. Let Rs and Rss denote the subsets

corresponding to stable and semistable vector bundles. If d is sufficiently large, then, properly

interpreted, U is a family over Rss with the local universal property for semistable bundles,

and over Rs has the local universal property for stable bundles. To produce an orbit space

using G.I.T., we must linearize the group action. This is most easily accomplished by

embedding R into projective space PN and using the standard linearization of SL(p) on

the hyperplane line bundle.

Proposition 6.2 For d sufficiently large, there is an immersion τ : R→ Z ⊂ PN such that

Rss and Rs coincide with the G.I.T. semistable and stable loci in Z for the action of SL(p).

Thus there exists a quotient M(n, d) := Z//SL(p) which has the structure of a projective

variety. Furthermore, associated to the open set Zs there is a quotient M(n, d)s which is an

open smooth subvariety ofM(n, d). In particular, if gcd(n, d) = 1, thenM(n, d) = M(n, d)s

is nonsingular.

6.3 The cohomology of M(n, d)

The cohomology of these spaces has been studied via several different approaches and

is fairly well-understood, at least when the rank and degree are coprime. Here we shall

use Atiyah and Bott’s construction of M(n, d) as the quotient of an infinite-dimensional

affine space by the infinite-dimensional gauge group G to describe generators for the coho-

mology ring [AB1]. In this section all cohomology groups are taken with respect to rational

coefficients.

14

The story begins with the ring H∗(BG; Q). There is a homotopy equivalence

BG'Map(M,BU(n)), where Map(M,BU(n)) is the space of continuous mapsM → BU(n).

Let V be the pullback of a universal classifying bundle EU(n) → BU(n) by the map

Map(C,BU(n))×Cev→ BU(n) so that V is a vector bundle of rank n over BG ×C. By the

Kunneth formula,

H2r(BG × C) ∼= H2r(BG)⊗H0(C) ⊕ H2r−1(BG)⊗H1(C) ⊕ H2r−2(BG)⊗H2(C).

Let α1, . . . , α2g be a basis of H1(C), and let ω be the standard generator of H2(C).

Then for each 1 ≤ r ≤ n, the rth Chern class has a decomposition of the following form:

cr(V ) = ar ⊗ 1 +∑

1≤j≤2g

bjr ⊗ αj + fr ⊗ ω.

Atiyah and Bott show that H∗(BG; Q) is generated by the elements

ar ∈ H2r(BG; Q), 1 ≤ r ≤ n

bjr ∈ H2r−1(BG; Q), 1 ≤ r ≤ n, 1 ≤ j ≤ 2g

fr ∈ H2r−2(BG; Q), 2 ≤ r ≤ n.

Let G be the quotient of the gauge group by its center U(1). Then the fibration

BU(1) → BG → BG

induces an isomorphism H∗(BG) ∼= H∗(BG)⊗H∗(BU(1)). The image of a1 can be expressed

by the images of the other generators, so we may omit it from the above list to obtain a set

of generators for H∗(BG).

There is a sequence of maps

H∗(BG)∼=→ H∗

G(C) → H∗G(C

ss)∼=→ H∗(M(n, d)). (1)

The first isomorphism exists because C is contractible. The second arrow is a surjection since

C has an equivariantly perfect stratification with Css as its open stratum. So H∗(M(n, d); Q)

is generated by the images of the elements

ar ∈ H2r(BG; Q), 2 ≤ r ≤ n

bjr ∈ H2r−1(BG; Q), 1 ≤ r ≤ n, 1 ≤ j ≤ 2g

15

fr ∈ H2r−2(BG; Q), 2 ≤ r ≤ n.

under the composite map of (1).

Atiyah and Bott calculate the Betti numbers of M(n, d) using infinite-dimensional ana-

logues of the formulas discussed in Section 4. Alternatively, Kirwan [K3] computes the Betti

numbers using the finite-dimensional G.I.T. construction described above.

Finally, a formula for the intersection pairing on H∗(M(n, d)) was conjectured by Witten

and rigorously proved by Jeffrey and Kirwan [JK] using Witten’s principle of nonabelian

localization.

One theme this section illustrates is that different constructions of moduli spaces may

be better suited to proving different properties of the space. Here for instance we have

seen that while G.I.T. establishes the existence of a moduli space as a projective variety,

the Atiyah-Bott construction far more readily yields generators for the cohomology and the

Betti numbers.

7 Mg

The moduli problem we wish to consider next is the classification of nonsingular projec-

tive curves of genus g up to isomorphism. Due to the existence of curves with nontrivial

automorphisms, there is no fine moduli space for this problem. However, a coarse moduli

space Mg exists. Mg is a quasi-projective variety which has finite quotient singularities at

points corresponding to curves with automorphisms.

7.1 Facts and definitions

We shall follow [Fant] for terminology: A curve C is a one-dimensional scheme, locally

of finite type over C. A curve is nonsingular if the stalk OC,p is a regular local ring for

every p ∈ C. In the language of algebraic geometry, a point p ∈ C is a node if the formal

completion OC,p of the local ring OC,p is isomorphic to C[[x, y]]/(y2 − x2). (A less technical

definition is that a node is an ordinary double point.) A connected reduced projective curve

is prestable if its only singularities are nodes. A prestable curve is Deligne-Mumford stable if

it has only finitely many nontrivial automorphisms. Equivalently, a prestable curve is stable

if and only if the following conditions are satisfied for every irreducible component E ⊂ C:

i. If E ∼= P1, then E must contain at least three nodes.

16

ii. If E has arithmetic genus 1, then E must contain at least one node.

The term Deligne-Mumford semistable curve describes a prestable curve which meets the

two conditions above with condition i. relaxed to two nodes rather than three.

The arithmetic genus of a curve is defined as h0(C,OC). If C is nonsingular, then by

Serre duality this is equal to the geometric genus h1(C, ωC).

The dualizing sheaf ωC of a curve is the analogue for singular curves of the canonical

divisor KC on a Riemann surface. A concise definition (assuming familiarity with the functor

f !) is that the dualizing sheaf is the unique nonzero cohomology group of the complex

f !(OSpec k), where f : C → Spec k is the structure morphism. This definition has the

advantage of extending readily to families X → S like so: we call the unique non-zero

cohomology group of the complex f !(OX/S) the relative dualizing sheaf ωX/S. A perhaps

more tangible definition of the dualizing sheaf is the following: Let ν : C → C be the

normalization of C, and write xi, yi, i = 1, . . . , n for the points on C which map to the nodes

of C and satisfy ν(xi) = ν(yi). Then ωC is the sheaf of 1-forms α on C which are regular

except for simple poles at the xi and yi and Resxiα+ Resyi

α = 0.

Let C be a Deligne-Mumford stable curve. Then ω⊗nC/k is very ample if n ≥ 3 (and this

property characterizes stable curves). Recall that a very ample line bundle L⊗j together

with a choice of basis of H0(C,L⊗j) determines an embedding C → PN−1 where

N = dim H0(C,L⊗j). Thus for any stable curve C a choice of basis of H0(C, ω⊗nC/k) deter-

mines an embedding C → Ps−1 called the n-canonical embedding. Here

s = dimH0(C, ω⊗nC/k) = (2n − 1)(g − 1) is the nth plurigenus of C. Write r = s − 1 and

d = degω⊗nC/k = 2n(g− 1). Then the Hilbert polynomial of an n-canonically embedded curve

C ⊆ Pr is P (m) = md − g + 1. These facts extend to families: any family of stable curves

X → S can be realized as a family of curves in Pr.

Mg is usually constructed using one of three approaches: Teichmuller theory, period

matrices, or geometric invariant theory. The Teichmuller theory and G.I.T. constructions

are particularly useful for studying the cohomology of Mg, and we shall discuss only these.

We briefly review the Teichmuller theory construction: Let S be a compact oriented

surface (that is, dimR(S) = 2) of genus g. Such an S can be given the structure of a

Riemann surface by pulling back the holomorphic structure of a Riemann surface C along

an orientation-preserving diffeomorphism f : C → S. Teichmuller space Tg is the set of

equivalence classes of Riemann surface structures C → S on S under the equivalence relation

17

f1 ∼ f2 if there is a biholomorphism Ψ : C1 → C2 such that Ψ f2 is isotopic to f1.

Bers showed more than fifty years ago that Tg is homeomorphic to a ball in C3g−3. Let

Γg be the mapping class group of S, the group of isotopy classes of orientation-preserving

diffeomorphisms of S. Then Γg acts on Tg and the quotient is the set of isomorphism classes

of complex structures on S. That is, Tg/Γg = Mg. Since Tg is contractible and the stabilizers

in Γg of points of Tg are finite, the rational cohomology of Mg is given by the cohomology

of Γg. Harer and others (see [HM] for an overview) have used this approach to calculate

H i(Mg) for g large and i small.

The Teichmuller theory construction thus gives a route to studying the quasiprojective

(noncompact) moduli spaceMg. As previously discussed a compact moduli space is desirable

for several reasons. It is possible to integrate over a compact space; studying how objects

degenerate begs for a closed space; and we wish to apply the many theorems of differential

or algebraic geometry which are valid only for compact manifolds or projective varieties.

Teichmuller theory is not particularly suited to producing a compactification which has a

modular interpretation, as any modular compactification ofMg must include singular curves.

The following example shows that nonsingular curves can degenerate to singular curves in

families:

Example 7.1 [MS] Let X → C be the family of curve in P2

y2z = x3 − t2axz2 − t3bz3

Then for t 6= 0 the curves Ct are smooth elliptic curves, all isomorphic to one other. On the

other hand, C0 is a rational cuspidal curve.

To obtain a compactification of Mg which has a moduli interpretation we must include

some singular curves. However, the space of all curves, singular and non-singular, does not

admit a moduli space due to jump phenomena, as the example above shows. So the trick

is to add just the right singular curves. If we add the Deligne-Mumford stable curves we

obtain a compactification Mg of Mg whose points have a moduli interpretation. Mg is an

irreducible projective variety ([DM, KM]) with orbifold singularities at points corresponding

to curves X which have nontrivial automorphism groups [Mum]. In section 11 we will outline

Gieseker’s construction of Mg.

18

8 U g(n, d)

We have introduced the moduli space of vector bundles over a fixed non-singular projec-

tive curve and the moduli space of curves. An obvious question is whether these spaces fit

together; that is, can we make sense of a moduli space of pairs (C,E) where C ranges over

all Deligne-Mumford stable curves and E is a slope semistable vector bundle over C?

Such a space has been constructed and compactified by Pandharipande ([Pand]). In

the literature it has been called the “universal moduli space” of vector bundles over curves;

however because it is only a coarse moduli space I prefer not to use that terminology. Concep-

tually Pandharipande’s construction is a straightforward combination of the constructions of

M(n, d) and Mg. Vector bundles in M(n, d) are taken over nonsingular curves C; for singu-

lar curves it will be more convenient to work in the category of torsion free sheaves. Thus we

consider the moduli problem of isomorphism classes of slope semistable torsion free sheaves

of uniform rank over Deligne-Mumford stable curves C. Let Quot be the Grothendieck Quot

scheme over the subscheme K ⊂ Hilb(PN) defined in Section 11. Then U g(n, d) is the G.I.T.

quotient Quot//(SL(r+1)×SL(n)). As we shall see in Section 11, the noncompactness of Kand hence of Quot → K is a major obstacle to applying Kirwan’s techniques to the moduli

spaces Mg and U g(n, d).

This gives a coarse moduli space U g(n, d) which is an irreducible singular projective

variety. Furthermore there is a commutative diagram

Ug(n, d) → U g(n, d)↓ ↓Mg → Mg

There has been a great deal of work on these relative moduli problems. For rank one

bundles, efforts to compactify the so-called relative Jacobian date back to the 1950s. Several

results for families containing reducible curves have appeared in the past decade. Caporaso

([Cap]) constructs the universal Picard variety Pd,g of line bundles of degree d on a (possibly

reducible) curve of genus g; in fact Pd,g ∼= U g(1, d). Jarvis ([Jar]) works in the category of

stacks to obtain a separated functor and thus a fine moduli space. Esteves’ compactification

([Est]) of the relative Jacobian is an algebraic space which becomes a scheme after an etale

base change. The advantage of his space is that it carries a Poincare sheaf. In particular,

if we find in future applications including Section 10 that U g(1, d) does not have all the

properties we desire, it may be possible to use one of these other constructions instead.

19

Caporaso’s work follows an approach first suggested by Gieseker. The moduli problem

of pairs is rigidified by choosing a basis for the space of sections H0(C,E). This determines

an embedding C → G(n, p) where p := h0(C,E) and G(n, p) is the Grassmannian of n-

dimensional linear subspaces of Pp. The embedded curves are then parametrized by the

Hilbert scheme of subschemes of G(n, p). For n = 2 this construction yields a space which

is not isomorphic to U g(2, d). Teixidor i Bigas is currently working on this approach to the

higher rank problem. See ([Teix]) for some preliminary results.

9 Mg(X, β)

Recall the following definitions from Section 7.1: A connected reduced projective curve is

prestable if its only singularities are nodes. Let X be a scheme locally of finite type over C.

A prestable map of genus g is a morphism f : C → X where C is a prestable curve of genus

g. A prestable map is stable if only finitely many automorphisms of the prestable curve C

commute with f . Note that a map f is stable if and only if for every irreducible component

E ⊂ C

i. If E ∼= P1 and f(E) is a point, then E must contain at least three nodes.

ii. If E has arithmetic genus 1 and f(E) is a point, then E must contain

at least one node.

In the 1990s Kontsevich introduced a compact moduli space of stable maps in his proof

of Witten’s conjecture for generating functions of Gromov-Witten invariants. Already in

their short lifetime these spaces have become the cornerstone of Gromov-Witten theory and

quantum cohomology.

Let X be a projective scheme over C. Let π : C → S be a family of genus g prestable

curves parametrized by S. A family of maps to X is given by a pair (π, µ) where µ : C → X

is a morphism. We say that two families of maps (π, µ) and (π′, µ′) are equivalent if there is

a scheme isomorphism τ : C → C′ such that π = π′ τ and µ = µ′ τ . A family of maps is

stable if for each fiber Cs =: C of π the map µ|Cs =: f is a stable map f : C → X.

There exists a projective schemeMg(X, β) which is a coarse moduli space of isomorphism

classes of stable maps f : C → X from prestable curves C of genus g to the space X such

that the pushforward f∗([C]) = β ∈ H2(X). A detailed construction can be found in [FP].

Very little is known in general about the spaces Mg(X, β). In the special case that X is

20

a nonsingular convex projective variety and g = 0 then M0(X, β) is a normal projective

variety of pure dimension dimX+∫βc1(TX)+n−3 (the “expected dimension”) and has the

structure of an orbifold. Also it is known that Mg(X, β) is connected if X is a homogeneous

space G/P where P is a parabolic subgroup of a connected complex semisimple algebraic

group G ([KP]). But even when X is projective space, the higher genus moduli spaces

Mg(Pk, d) are in general reducible, nonreduced, and singular ([GP] page 490). When g > 0

we are not aware of any published results concerning the number of irreducible components

of Mg(Pk, d) and their dimensions, how the boundary of Mg(P

k, d) meets Mg(Pk, d), or

the singularities of Mg(Pk, d). Though I am not able to answer any of these questions in

this paper, it is conceivable that G.I.T. could shed some light on these questions.

Fulton and Pandharipande hint how a G.I.T. construction of Mg(Pk, d) might proceed

but they do not construct Mg(X, β) as a G.I.T. quotient. Their process involves gluing

and showing that the result carries an ample line bundle and is hence projective. At least

two additional proofs of the projectivity of Mg(X, β) have been published ([Alex, Corn]).

However we are not aware of a published G.I.T. construction of Mg(X, β).

Two other spaces of maps deserve note. Tian ([Tian]) constructs a compact moduli space

of stable maps for the purpose of defining symplectic Gromov-Witten invariants. Bertram,

Daskalopolous, and Wentworth ([BDW]) have constructed a compactification B of the space

of holomorphic maps from a compact Riemann surface C to a Grassmannian. They ob-

tain a finite-dimensional Kahler manifold which is a gauge-theoretic quotient of an infinite-

dimensional space by an infinite-dimensional group.

10 Motivation

The following project proposed by Kirwan is one reason we sought to study the construc-

tion of Mg(Pr, d) as a quotient.

Let r = k − 1 where k = (2n − 1)(g − 1) is the nth plurigenus of a curve C. It is

convenient to take n = 10. Let H be the subset of Mg(Pr, d) consisting of isomorphism

classes of morphisms f : C → Pr which embed a nonsingular curve C 10-canonically into Pr.

Let [f ] be an element of H, let A ∈ SL(r + 1; C), and denote also by A the corresponding

element of Aut(Pr). We define a group action on H by the rule:

(A, [f ]) 7→ [A f ]

21

This map is well-defined and a morphism. Furthermore H/SL(r + 1; C) ∼= Mg.

Now let H be the closure of H in Mg(Pr, d). The group action on H extends to H. For

the purposes of geometric invariant theory we must also specify a linearization of the action.

This is almost always done by embedding the space to be quotiented in a projective space

and defining a group action on the hyperplane line bundle. However the construction of

Mg(Pr, d) does not explicitly embed it in projective space, though ample line bundles have

been constructed over Mg(Pr, d) (see [FP, Alex, Corn]).

After specifying a linearization we can make sense of H//SL(r+ 1; C). We would like to

relate H//SL(r + 1; C) to the Deligne-Mumford-Knudsen compactification Mg. If H ⊆ Hs

then H//SL(r + 1; C) is a compactification of Mg. A map H//SL(r + 1; C) → Mg is

obtained by mapping the map f : C → Pr to the isomorphism class of the curve f(C). It is

not immediately clear that this is an isomorphism. Recall that Kontsevich’s spaceMg(Pr, d)

contains maps whose domains are strictly prestable curves. By definition, the domain curves

of all maps in H are nonsingular hence stable. However the boundary ∂H could potentially

include maps whose domains are nodal curves which are not Deligne-Mumford stable curves.

The best case would occur if for every point [f : C → Pr] ∈ ∂Hss

, the domain curve C is

Deligne-Mumford stable and [f ] is 10-canonical. The situation will be more complicated if

there exist any points [f : C → Pr] ∈ ∂Hsssuch that

i. f is not 10-canonical and/or

ii. f is not an embedding and/or

iii. C is prestable but not stable.

So the first steps in this project are to study the closure H of H in Mg(Pr, d) and

linearizations of the SL(r+1) action onH. The following diagram may help us to understand

H:Mg(P

k−1, d)τ→U g(1, d)π

↑ σMg

Here τ([f ]) := [f ∗(L)] where L is the hyperplane line bundle on Pr, the map π is the

stabilization morphism (see [Manin] Ch.5) and σ([C]) := [ω⊗10C ].

On H, we have τ |H = σ π|H , so

H ⊆ [f ] ∈Mg(Pr, d) | τ([f ]) = σ(π([f ])).

22

The right hand side may equalH, or if it is reducible thenH may be its irreducible component

containing H.

Next, we would hope to find a linearization for which Hss//SL(W ) ∼= Mg. As noted

above, proofs of the existence of at least one ample line bundle on Mg(Pr, d) have been

published. More precisely,Mg(Pr, d) is constructed as a quotient J →Mg(P

r, d), and ample

line bundles are given on J . One of our main motivations for constructing Mg(Pr, d) as a

G.I.T. quotient is to study the line bundles (and hence possible different linearizations) on

Mg(Pr, d). We suspect that the G.I.T. construction to be given in Part III leads to the same

linearization on H as the natural linearization of Fulton and Pandharipande’s line bundle

DetQk ([FP] page 69; note that in their notation k is not the same k = r + 1 given above).

It is likely that Cornalba’s line bundle is not isomorphic to Fulton and Pandharipande’s and

hence gives rise to a different linearization; however we do not know whether it gives rise to

a different quotient.

We hope that studying H and Hss

(L) for one or more linearizations could yield new

results or new proofs of known results for the cohomology of Mg.

23

Part III

A G.I.T. Construction of the ModuliSpace of Maps

11 Gieseker’s Construction of Mg Reviewed

Our G.I.T. construction of Mg(Pr, d) closely parallels Gieseker’s construction of Mg.

Therefore we shall briefly review Gieseker’s construction here.

Grothendieck proved that there is a scheme HilbP,N parametrizing subschemes of PN

which have a given Hilbert polynomial P (m). Gieseker’s strategy is to “pick out” the sub-

schemes of PN which are n-canonically embedded Deligne-Mumford stable curves, then ac-

count for the choice of basis of H0(C, ω⊗nC/k)∼= kN+1 by quotienting by SL(N+1) (recall that

an embedding is determined by a very ample line bundle together with a choice of basis of its

space of sections). This “picking out” is accomplished by choosing a linearization for which

the G.I.T. stable points correspond exactly to subschemes of PN which are Deligne-Mumford

stable curves.

11.1 The Hilbert scheme

In fact, the Hilbert scheme is a fine moduli space, and so carries a universal family. To

make this precise, we follow [GIT] Ch. 0 Section 5. Let T be an arbitrary locally noetherian

scheme, let LocNoethSch/T be the category of locally noetherian schemes over T , and let X

and S be locally noetherian schemes over T .

Definition 11.1 An algebraic family of closed subschemes of X/T parametrized by S is a

closed subscheme Z ⊆ X ×T S. A family is flat if Z → S is flat. A geometric fiber of

the family is the pullback (1 × t)∗Z of Z to X ×S Spec(k(s)), where Spec(k(s)) → S is a

geometric point of Z.

Definition 11.2 We define a contravariant functor HilbX/T (−) : LocNoethSch/T → Sets by

the rule:

HilbX/T (S) = Z|Z is a flat algebraic family of closed subschemes of X/T

parametrized by S.

24

We also define a functor HilbP,X/T (−) : LocNoethSch/T → Sets as follows:

HilbP,X/T (S) = Z|Z is a flat algebraic family of closed subschemes of X/T

parametrized by S whose geometric fibers have Hilbert polynomial P.

HilbP,X/T (−) is an open and closed subfunctor of HilbX/T (−).

As previously mentioned, Grothendieck proved that HilbX/T (−) is represented by a

scheme HilbX/T which is projective over SpecZ. If P is a numerical polynomial we write

HilbP,X/T for the component of HilbX/T where the geometric fibers have Hilbert polynomial

P . Then HilbP,X/T represents HilbP,X/T (−). If X = PN we will compress the notation further

and write HilbP,N .

Since HilbP,N is a fine moduli space, we have HilbP,N(S) ∼= Hom(S,HilbP,N) for all locally

noetherian T -schemes S. In particular HilbP,N(HilbP,N) ∼= Hom(HilbP,N ,HilbP,N). Define the

universal family C ⊂ PN ×HilbP,N to be the scheme corresponding to the identity morphism

id ∈ Hom(HilbP,N ,HilbP,N). Note that the composition ϕ : C → PN × HilbP,N → HilbP,N is

a projective morphism.

11.2 The Hilbert point of a curve

Next we define the Hilbert point of a curve in projective space: Let C be a projective

curve in PN of genus g and degree e, let L = OPN (1)|C , and let

ϕm : H0(PN ,OPN (m)) → H0(C,Lm)

be the map induced by restriction. The Hilbert polynomial of C is P (m) = em− g + 1.

Form the P (m)th exterior power of the map ϕm. Then∧P (m) ϕm is a nonzero linear form

on∧P (m)H0(PN ,OPN (m)), hence an element of P(

∧P (m)H0(PN ,OPN (m))). We are using

the convention that for a vector space V , P(V ) denotes the space of equivalence classes of

nonzero linear forms on V . We call∧P (m) ϕm the mth Hilbert point of C, written Hm(C).

There is an induced SL(N + 1) action on P(∧P (m)H0(PN ,OPN (m))). If e0, . . . , eN is

a basis of H0(PN ,OPN (1)) then the induced action can be written in the following way: a

basis for H0(PN ,OPN (m)) is given by the degree m monomials Mi in the symbols e0, . . . , eN .

Then we define an SL(N + 1) action on H0(PN ,OPN (m)) by the rule

g.eγ00 · · · eγN

N = (ge0)γ0 · · · (geN)γN .

25

A basis of∧P (m)H0(PN ,OPN (m)) is given by elements of the form Mi1 ∧ · · · ∧MiP (m)

where 1 ≤ i1 < i2 < · · · < iP (m) ≤ α. We can define an SL(N + 1) action on the exterior

product∧P (m)H0(PN ,OPr(m)) by the rule

g.(Mi1 ∧ · · · ∧MiP (m)) = gMi1 ∧ · · · ∧ gMiP (m)

.

Then the dual action gives an action of SL(N + 1) on P(∧P (m)H0(PN ,OPr(m))).

The correspondence of a curve to its Hilbert point gives a pointwise embedding of the

Hilbert scheme. That is, for m sufficiently large, the map

ψm : HilbP,r → P(

P (m)∧H0(PN ,OPN (m)))

h 7→ Hm(Ch)

is a closed immersion, where for any h ∈ HilbP,N we write

Ch := C ×HilbP,NSpeck(h). (2)

So the SL(N + 1) action on P(∧P (m)H0(PN ,OPN (m))) restricts to ψm(HilbP,N) ∼= HilbP,N

to give an action on HilbP,N .

11.3 Defining K

Now take d = 2n(g − 1) and N = d − g. Define U ′ ⊂ HilbP,N to be the set of Hilbert

points which parametrize connected nodal curves of genus g and degree d = 2n(g−1) in PN .

Let C ϕ→ U ′ be the restriction to U ′ of the universal family C ϕ→ HilbP,N . Write OC(1) for the

pullback of OPN (1) by the morphism C → PN × U → PN . Let ω := ωC/U ′ be the relative

dualizing sheaf of ϕ (see section 7.1 above). Also Ch is a nodal curve in PN ; write ChιCh→ PN

for the inclusion morphism, and set OCh(1) := ι∗Ch

OPN (1). Let ωChbe the dualizing sheaf of

the curve Ch. Finally let U be the open subset of U ′ such that the multidegree of ωC/U ′|Chis

equal to the multidegree of OCh(1).

We define

K = h∈U | OCh(1) ∼= ω⊗nCh

. (3)

K is a locally closed subscheme of HilbP,N . However K is not a closed subscheme of

HilbP,N , and its closure is singular; these are major obstacles to using Kirwan’s techniques

to study the cohomology of Mg via Gieseker’s construction.

Let Kss = K ∩ ((P(∧P (m)H0(PN ,OPN (m))))ss). We have the following result:

26

Theorem 11.3 Every curve C ⊂ PN whose Hilbert point Hm(C) lies in Kss is Deligne-

Mumford stable, and every Deligne-Mumford stable curve of genus g has a model in Kss.

Then the G.I.T. quotient K//SL(N + 1) is Mg.

12 Fitting Ideals and the Construction of Mg

Using Fitting ideals it is possible to describe the locus K defined in line (3) above more

explicitly.

We begin with some preliminaries on Fitting ideals. Let I be a quasicoherent sheaf of

ideals on a scheme Y . Then I defines a closed subscheme V (I) whose underlying set is given

by V (I) = y∈Y | Iy 6= OY,y.

Definition 12.1 Let G be a quasicoherent sheaf of OY -modules on a scheme Y . Let

g be a nonnegative integer. The gth Fitting ideal FittgG of G is defined as follows: Let

E1Ψ→ E0 → G → 0 be a two-term free resolution for G, and write r := rankE0. Then we

define an ideal sheaf Ir−gϕ as follows:

Ir−gϕ =

the image of the map

r−g∧ E1⊗

r−g∧ E∗0 → OY induced by

r−g∧ Ψ if r − g > 0

OY if r − g ≤ 0.

Then FittgG := Ir−gϕ. By Fitting’s Lemma, FittgG is independent of the choice of two-term

resolution, so it is well-defined.

Proposition 12.2 Let G be a sheaf on Y . Then (FittgG)y ∼= Fittg(Gy).

In words, the stalk at a point y of the sheaf of ideals FittgG is isomorphic as an OY,y-module

to the gth Fitting ideal of the stalk Gy.Proof. Choose a two-term free resolution E1

Ψ→ E0 → G → 0. If r − g ≤ 0 then the

modules in question are both OY,y. If r − g > 0 we argue as follows:

FittgG := Ir−gϕ= Image(

r−g∧ E1⊗

r−g∧ E∗0 → OY )

(FittgG)y = (Image(r−g∧ E1⊗

r−g∧ E∗0 → OY ))y

= Image((r−g∧ E1⊗

r−g∧ E∗0 )y → OY,y) by [Hart] Exercise II.1.2

= Image(((r−g∧ E1)y ⊗ (

r−g∧ E∗0 )y) → OY,y) by [Liu] page 158

= Image((r−g∧ (E1)y⊗

r−g∧ (E∗0 )y) → OY,y)

= Fittg(Gy).

27

We apply these ideas to get a more explicit description of K. We work over C (though

it is likely that most of what follows is true over an arbitrary base). We take d = 2n(g − 1)

and N = d− g. Recall from page 26 that U is the open set where Ch is connected and nodal

and for every h ∈ U the multidegree of ωC/U |Chis equal to the multidegree of OCh

(1). Recall

also the definitions of ωC/U , and OC(1) given on page 26.

Let p1 be the projection map C ×U Speck(h)p1→ C. Then for an OC-module F we

shall often use the symbol F|Chto mean p∗1(F). Note in particular that with this notation

OCh(1) ∼= OC(1)|Ch

and ωCh∼= ω|Ch

.

We defined

K = h∈U | OCh(1) ∼= ω⊗nCh

.

We want to show

Proposition 12.3 K = V (Fittg−1R1ϕ∗(ω

⊗n ⊗OC(−1)) as closed subsets of U .

Remark. V (Fittg−1R1ϕ∗(ω

⊗n ⊗ OC(−1)) is a closed subscheme of U ; indeed this is how

Gieseker defines a scheme structure on K ([Gies] page 90), in modern language. Note that

we have written the index of the Fitting ideal as g − 1 rather than g as in [HM].

Proof. The relative dualizing sheaf of a family of nodal curves is invertible, so

ω⊗n ⊗OC(−1) is invertible. It is locally free of finite rank and therefore flat over C. The

map C ϕ→ U is flat. Therefore ω⊗n ⊗OC(−1) is flat on U .

The second cohomology H2(Ch, (ω⊗n⊗OC(−1))|Ch) = 0 for all h because Ch is a curve for

all h ∈ U . Then Φ2(h) : R2ϕ∗(ω⊗n ⊗ OC(−1)) ⊗ k(h) → H2(Ch, (ω⊗n ⊗ OC(−1))|Ch

) = 0 is

surjective. Here, to avoid confusion with the map C ϕ→ U , we have renamed from lowercase

ϕi(y) to uppercase Φi(h) the natural map appearing in the cohomology and base change

theorem (cf. [Hart] 12.11).

The vanishing second cohomology also implies by [Hart] Exercise 11.8 that

R2ϕ∗(ω⊗n ⊗OC(−1)) = 0 on a neighborhood of h. So R2ϕ∗(ω

⊗n⊗OC(−1)) is locally free of

rank 0 near h. This implies by [Hart] Theorem 12.11b that

Φ1(h) : R1ϕ∗(ω⊗n ⊗OC(−1))⊗ k(h) → H1(Ch, (ω⊗n ⊗OC(−1))|Ch

)

is surjective and hence an isomorphism at each h ∈ U .

We show that V (Fittg−1R1ϕ∗(ω

⊗n ⊗ OC(−1))) ⊆ K. More precisely, we show that if

h ∈ U \ K, then h 6∈ V (Fittg−1R1ϕ∗(ω

⊗n ⊗OC(−1))).

28

Let h 6∈ K. Then the sheaf (ω⊗n⊗OC(−1))|Chis nontrivial. Furthermore its multidegree

is zero since h ∈ U . Therefore H0(Ch, (ω⊗n ⊗OC(−1))|Ch) = 0, and we conclude that Φ0(h)

is surjective.

Since both maps Φ1(h) and Φ0(h) are surjective, we conclude by [Hart] 12.11b that

R1ϕ∗(ω⊗n⊗OC(−1)) is locally free on a neighborhood of h. Since this holds for all h ∈ U \K

we conclude that R1ϕ∗(ω⊗n ⊗ OC(−1)) is locally free on U \ K. Furthermore, the rank of

the stalk R1ϕ∗(ω⊗n⊗OC(−1))h is equal to rankH1(Ch, (ω⊗n⊗OC(−1))|Ch

). But this is g− 1

since (ω⊗n ⊗OC(−1))|Chis a degree zero nontrivial sheaf.

Proposition 12.4 ([St] 6.2) Let F be a coherent sheaf on a scheme S, and let r ∈ Z.

Then F is locally free of rank r if and only if Fittr(F) = OS and Fittr−1(F) = 0.

The proposition allows us to conclude that over U\K we have Fittg−1R1(ω⊗n⊗OC(−1)) ∼=

OU . Then the stalks (Fittg−1R1ϕ∗(ω

⊗n⊗OC(−1)))h ∼= OU,h are isomorphic for each h ∈ U\K.

We conclude that V (Fittg−1R1ϕ∗(ω

⊗n ⊗OC(−1))) ⊆ K.

We prove next that K ⊆ V (Fittg−1R1ϕ∗(ω

⊗n ⊗OC(−1))). Let h ∈ K so that

(ω⊗n ⊗OC(−1))|Chis trivial. Then appealing to the definition of the arithmetic genus we

have rankH1(Ch, (ω⊗n ⊗OC(−1))|Ch) = g.

Proposition 12.5 ([E], 20.8) A module M over a ring R is projective of constant rank g

if and only if Fittg(M) = R and Fittg−1(M) = 0.

This implies Fittg−1H1(Ch, (ω⊗n⊗OC(−1))|Ch

) = 0. Applying the isomorphism Φ1(h) we

have

Fittg−1((R1ϕ∗(ω

⊗n ⊗OC(−1)))h) = 0

⇒ (Fittg−1(R1ϕ∗(ω

⊗n ⊗OC(−1))))h = 0

⇒ (Fittg−1(R1ϕ∗(ω

⊗n ⊗OC(−1))))h 6= OU,h if h ∈ K.

That is, K ⊆ V (Fittg−1R1ϕ∗(ω

⊗n ⊗OC(−1))).

This shows the closed subsets K and V (Fittg−1R1ϕ∗(ω

⊗n ⊗OC(−1))) coincide.

13 Constructing Mg(Pr, d): General Information

We use the standard isomorphism H2(Pr) ∼= Z throughout what follows. Let Mg(P

r, d)

be the Kontsevich-Manin moduli space of isomorphism classes of stable maps of degree d

29

from genus g curves into Pr. We wish to apply the ideas of the previous section to this

moduli problem.

Notation. Given a stable map f : C → Pr, write L := ωC ⊗ f ∗(OPr(3)). Then L is

ample and ([BM] Lemma 3.12) there is a constant a depending on g,r,and d (but not on C

or f) such that La is very ample and h1(C,La) = 0. (Remark: Fulton and Pandharipande

denote this constant by the letter f ([FP] page 57).) It will be convenient to assume that

a is large. Write deg(La) = a(2g − 2 + 3d) =: e so h0(C,La) = e − g + 1. Set n = dg−1

so that e = (3n + 2)a(g − 1). Note that we do not require n to be an integer, and that

this clashes terribly with our choice of n in the previous two sections. Let W be a vector

space of dimension e− g + 1. Then an isomorphism W ∼= H0(C,La) induces an embedding

C → P(W ). (Recall that our convention, following Grothendieck and Gieseker, is that P(V )

is the set of equivalence classes of nonzero linear forms on V . This is dual to the convention

many authors use.) We will sometimes write N := e− g.

Let Hilb(P(W )×Pr) be the Hilbert scheme of genus g curves in P(W )×Pr of multidegree

(e, d). In analogy with the notation of the previous section we write C ϕ→ Hilb(P(W )×Pr) for

the universal family. Fulton and Pandharipande define an open subset U ⊂ Hilb(P(W )×Pr)

such that for each h ∈ U ,

i. Ch is a connected nodal curve.

ii. The projection map Ch → P(W ) is a non-degenerate embedding.

iii. The multidegree of (OP(W )(1) ⊗ OPr(1))|Chequals the multidegree of

(ω⊗aCh⊗OPr(3a+ 1))|Ch

.

By [FP] Proposition 1, there is a natural closed subscheme J ⊂ U where the sheaves in

line iii. above are isomorphic.

We give a Fitting ideal description of J in U :

Proposition 13.1 J = V (Fittg−1R1ϕ∗(ω

⊗aC ⊗ (OPr(3a))|C ⊗ (OP(W )(−1))|C)).

Proof. Same argument as for Proposition 12.3 above.

Fulton and Pandharipande observe that the quotient of J by PGL(W ) is the moduli space

Mg(Pr, d). Write J for the closure of J in Hilb(P(W )× Pr). We would like to establish

the following claim: there is a linearization of the group action on J such that the G.I.T.

quotient J//SL(W ) is isomorphic to Mg(Pr, d). The construction of J//SL(W ) and the

30

proof that J//SL(W ) ∼= Mg(Pr, d) will occupy Sections 14 to 17. However, to finish this

section we prove:

Proposition 13.2 Jj→Mg(P

r, d) is a categorical quotient for the SL(W ) action.

LetX f→ Pr

σ ↓S

be a family of stable maps to Pr. For any s ∈ S we write fs : Xs → Pr

for the stable map corresponding to s.

Lemma 13.3 R0σ∗(ω⊗aX/S ⊗ f ∗(OPr(3a))) is locally free.

Proof. Recall that we have chosen a so that H1(Xs;ω⊗aXs⊗f ∗s (OPr(3a))) = 0 for all s ∈ S.

Also since Xs is a curve for each s all higher cohomology groups are zero as well. Then by

[Hart] Exercise III.11.8, we have Riσ∗(ω⊗aX/S⊗f ∗(OPr(3a))) = 0 for all i > 0. The hypotheses

of [EGA] Corollary III.7.9.10 are satisfied, and we conclude that R0σ∗(ω⊗aX/S ⊗ f ∗(OPr(3a)))

is locally free.

Write C → J for the restriction of the universal family C ϕ→ Hilb(P(W )×Pr) to J .

Lemma 13.4 C ϕ→ J has the local universal property for the moduli problem Mg(Pr, d).

Proof. Suppose X σ→ S is a family of stable maps to Pr. For any s0 ∈ S we seek an open

neighborhood V 3 s0 and a morphism Vψ→ J such that ψ∗(C) ∼fam X|V .

Pick a basis for H0(Xs0 ;ω⊗aXs0⊗f ∗s0(OPr(3a))). We showed that R0σ∗(ω

⊗aX/S⊗f ∗(OPr(3a)))

is locally free. Choose a neighborhood V 3 s0 which is so small that σ∗(ω⊗aX/S⊗f ∗(OPr(3a)))

is free on V . Then by [Hart] III.12.11(b) and Lemma 13.3 there is an induced basis

of H0(Xs;ω⊗aXs⊗ f ∗s (OPr(3a))) for each s ∈ V . This defines a map Xs

ιs→ P(W ) for

each s ∈ V . These fit together as a morphism X|V → P(W ). For each s ∈ V form

(ιs, fs) : Xs → P(W )×Pr. These pairs fit together to give a family of stable maps

X|V(ι,f)→ P(W )×Pr → Pr.

↓V

But (ι, f) also gives X|V → V the structure of a family of curves in P(W )×Pr parametrized

by V as defined in Definition 11.1. By the universal properties of the Hilbert scheme, then,

31

there is a unique morphism ψ : V → Hilb(P(W )× Pr) such that X|V ∼fam ψ∗(C). Finally,

observe that ψ(V ) ⊂ J .

Proof of Proposition 13.2: Mg(Pr, d) is a coarse moduli space ([FP] Theorem 1). By

the previous lemma J carries a local universal family. The desired result follows by [New]

Proposition 2.13, which is quoted in this paper on page 5.

14 The G.I.T. Set-up and the Numerical Criterion for

Mg(Pr, d)

Now we construct a G.I.T. quotient J//SL(W ), following Gieseker’s construction of Mg

as a G.I.T quotient.

Recall Gieseker’s definition of the Hilbert point of a curve ([Gies] page 5 or page 25

above): If C ⊂ PN is a curve with Hilbert polynomial P (m) and

ϕm : H0(PN ,OPN (m)) → H0(C,OPN (m)|C)

is the map induced by restriction, then the mth Hilbert point of C is

Hm(C) :=∧P (m) ϕm ∈ P(

∧P (m)H0(PN ,OPN (m)), where P(V ) denotes the space of equiv-

alence classes of nonzero linear forms on the vector space V .

We adapt this as follows. Let h ∈ Hilb(P(W )×Pr). The Hilbert polynomial of

Ch ⊂ P(W )×Pr is em+ dm+ 1− g. We define Hm,m(h), the (m, m)−th Hilbert point of

h, as follows:

Definition 14.1 If m and m are sufficiently large, then H1(Ch,OP(W )(m)⊗OPr(m)|Ch) = 0

and the restriction map

ρm,m : H0(P(W )×Pr,OP(W )(m)⊗OPr(m)) → H0(Ch,OP(W )(m)⊗OPr(m)|Ch)

is surjective.

Then∧P (m)+dm ρm,m is a point of P(

∧P (m)+dmH0(P(W )×Pr,OP(W )(m)⊗OPr(m))).

We set Hm,m(h) :=∧P (m)+dm ρm,m for all m > m0.

For sufficiently large m, m, say m, m ≥ m′′′, the map

h 7→ Hm,m(h)

Hilb(P(W )×Pr) → P(∧P (m)+dmH0(P(W )×Pr,OP(W )(m)⊗OPr(m))

32

is a closed immersion (see 14.2 below).

There is an induced SL(W ) action on P(∧P (m)+dmH0(P(W )×Pr,OP(W )(m)⊗OPr(m)).

If w0, ..., wN is a basis of H0(P(W ),OP(W )(1)) then the SL(W ) action can be described as

follows: Let (aij) be a matrix representing g ∈ SL(W ). Then g acts by the rule

g.wp =∑N

j=0 apjwj. Let Bm = M1, ...,Mα be a monomial basis of H0(P(W ),OP(W )(m)).

We extend the previous action as follows: if Mi = wγ00 · · ·wγN

N then

g.Mi = (g.w0)γ0 · · · (g.wN)γN . Pick a basis f0, ..., fr of Cr+1 and let Bm be a monomial basis

of H0(Pr,OPr(m)). Tensor Bm with Bm to get a basis Bm,m of

H0(P(W ),OP(W )(m))⊗H0(Pr,OPr(m)) ∼= H0(P(W )×Pr,OP(W )(m)⊗OPr(m))

consisting of monomials having bidegree (m, m). Then if Mi ∈ Bm,m is given by

wγ00 · · ·wγN

N fΓ00 · · · fΓr

r , we have g.Mi = (g.w0)γ0 · · · (g.wN)γNfΓ0

0 · · · fΓrr .

A basis for∧P (m)+dmH0(P(W )×Pr,OP(W )(m)⊗OPr(m)) is given by

Mi1 ∧ · · · ∧ MiP (m)+dm|1 ≤ i1 < · · · < iP (m)+dm ≤ h0(P(W )×Pr,OP(W )(m)⊗OPr(m)).

The SL(W ) action on this space is given by

g.(Mi1 ∧ · · · ∧ MiP (m)+dm) = (g.Mi1) ∧ · · · ∧ (g.MiP (m)+dm

).

The dual action is the SL(W )-action on P(∧P (m)+dmH0(P(W )×Pr,OP(W )(m)⊗OPr(m))

we shall study.

Let λ′ be a 1-PS of SL(W ). We wish to state the Hilbert-Mumford numerical criterion

(cf. page 7) for our situation: Let w0, ..., wN be a basis of H0(P(W ),OP(W )(1)) diagonalizing

the action of λ′. There exist integers r0, ..., rN such that λ′(t)wi = triwi for all t ∈ C∗ and

0 ≤ i ≤ N . Then, referring to the notation of the previous paragraph, we define the

λ′- weight of a monomial Mi = wγ00 · · ·wγN

N fΓ00 · · · fΓr

r to be wλ′(Mi) =∑N

p=0 γprp. We define

the total λ′-weight of a collection of j monomials M1, ..., Mj to be∑j

i=1wλ′(Mi).

Recall that a basis for∧P (m)+dmH0(P(W )×Pr,OP(W )(m)⊗OPr(m)) is given by

Mi1 ∧ · · · ∧ MiP (m)+dm|1 ≤ i1 < · · · < iP (m)+dmh

0(P(W )×Pr,OP(W )(m)⊗OPr(m)). (4)

We follow Gieseker and write (Mi1 ∧ · · · ∧ MiP (m)+dm)∗ for elements of the basis of

(∧P (m)+dmH0(P(W )× Pr,OP(W )(m) ⊗ OPr(m)))∗ which is dual to the basis given in line

(4). Then the λ′ action on P(∧P (m)+dmH0(P(W )× Pr,OP(W )(m) ⊗ OPr(m)) is given by

λ′(t)((Mi1∧· · ·∧MiP (m)+dm)∗) = t−θ((Mi1∧· · ·∧MiP (m)+dm

)∗), where θ =∑P (m)+dm

j=1 wλ′(Mij).

33

Now write Hm,m(h) in this basis:

Hm,m(h) =∑

1≤i1<···<iP (m)+dm

ρm,m(Mi1 ∧ · · · ∧ MiP (m)+dm)(Mi1 ∧ · · · ∧ MiP (m)+dm

)∗.

The numerical criterion states that Hm,m(h) is λ′-semistable if and only if

µ(Hm,m(h), λ′) ≤ 0 where µ(Hm,m(h), λ′) = min∑P (m)+dm

j=1 wλ′(Mij) and the minimum is

taken over all sequences 1 ≤ i1 < · · · < iP (m)+dm such that ρm,m(Mi1 ∧ · · · ∧ MiP (m)+dm) 6= 0.

We will often obtain one-parameter subgroups λ′ of SL(W ) from one-parameter sub-

groups λ of GL(W ). The correspondence is as follows: Given a 1-PS λ of GL(W ), there is a

basis w0, ..., wN of H0(P(W ),OP(W )(1)) diagonalizing the action of λ so that the action of λ

is given by λ(t)wi = triwi where ri ∈ Z. Note that the sum∑N

p=0 rp is not necessarily zero.

Then we obtain a 1-PS λ′ of SL(W ) by the rule λ′(t)wi = tr′iwi where r′i = (N+1)ri−

∑Ni=0 ri.

Semistability for the λ′ action and the λ action are related in the following way. Let

Mi = wγ00 · · ·wγN

N fΓ00 · · · fΓr

r be a monomial in Bm,m. We define the total λ-weight of a

monomial and a collection of monomials in analogy with those defined above for a 1-PS of

SL(W ), namely wλ(Mi) =∑N

p=0 γprp and∑P (m)+dm

j=1 wλ(Mij). Then

P (m)+dm∑j=1

wλ′(Mij) ≤ 0 ⇐⇒

(N + 1)

P (m)+dm∑j=1

wλ(Mij)−m(P (m) + dm)N∑p=0

rp ≤ 0 ⇐⇒

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑p=0

rp

N + 1.

Thus if λ′ is the 1-PS of SL(W ) arising from a 1-PS λ of GL(W ), the numerical criterion may

be expressed as follows: the (m, m)th Hilbert point Hm,m(h) is λ′-semistable if and only if

there exist monomials Mi1 , ..., MiP (m)+dmin Bm,m such that ρm,m(Mi1), ..., ρm,m(MiP (m)+dm

)is a basis of H0(Ch,OP(W )(m)⊗OPr(m)|Ch

) and

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑p=0

rp

N + 1. (5)

We shall show that the quotient J//SL(W ) is projective and is isomorphic to J//SL(W )

and Mg(Pr, d). First, we show that if C ⊂ P(W )× Pr is smooth and the map C ⊂

34

P(W )× Pr → P(W ) is a nondegenerate embedding, then Hm,m(C) is SL(W )-stable. This

shows that J//SL(W ) is nonempty. Next we show that if Hm,m(C) is SL(W )-semistable,

then the map C → Pr must be very close to being a stable map. More precisely, C must

be reduced, nodal, the embedding in P(W )× Pr must satisfy a multidegree-multigenus

inequality, and any genus 0 components which collapse under f must meet the rest of the

curve in at least two points.

A few remarks on the notation we shall use: Let pW : P(W )×Pr → P(W ) be projection

onto the first factor, and pr : P(W )×Pr → Pr projection onto the second. Let

Cι→ P(W )×Pr be inclusion. We let LW denote ι∗p∗WOP(W )(1) and let Lr denote ι∗p∗rOPr(1).

We shall make free use of the following facts (some of which have already been asserted),

which are analogous to those stated by Gieseker:

Proposition 14.2 (cf. [Gies] page 25) There exist positive integers m′, m′′, m′′′, q1, q2,

q3, µ1, and µ2 such that

i. for all m, m > m′, H1(C,LmW ) = H1(C,Lmr ) = H1(C,LmW ⊗ Lmr ) = 0 and

the three restriction maps

H0(P(W ),OP(W )(m)) → H0(pW (C)OpW (C)(m))

H0(Pr,OPr(m)) → H0(pr(C),Opr(C)(m))

H0(P(W )×Pr,OP(W )(m)⊗OPr(m)) → H0(C,LmW ⊗ Lmr )

are 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 n ≥ 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

m > n ≥ m′′, H1(C, Im−nP ⊗ LmWC

) = 0, where IP is the ideal subsheaf

of OC defining P .

35

vii. For all integers m, m ≥ m′′′ the map

h 7→ Hm,m(h)

Hilb(P(W )×Pr) → P(

P (m)+dm∧H0(P(W )×Pr,OP(W )(m)⊗OPr(m))

is a closed immersion.

Remark. In the course of the construction we shall find it necessary to limit the range

of values mm

may take. Those familiar with Gieseker’s construction of Mg will note that his

construction works for any m ≥ m0. We are essentially linearizing the group action with

respect to the very ample invertible sheaf OP(W )(m) ⊗ OPr(m). The G.I.T. quotient does

not change if the linearization is replaced by a tensor power of itself, i.e. m and m may be

replaced by xm and xm for any x ∈ N and the quotient will be the same (cf. for instance [D]

page 51). However, if the ratio mm

is altered, then we have produced a different linearization.

In that case, the quotients need not be the same. Thaddeus [Th2] and Dolgachev and Hu

[DH] study this problem. The space of linearizations is divided into finitely many polyhedral

chambers. Two linearizations which lie in the same chamber give rise to the same quotient.

Two linearizations which lie in different chambers give rise to quotients which are related

by a flip. It would be interesting to study how different values of mm

might give rise to

nonisomorphic quotients J//L1SL(W ) and J//L2SL(W ).

15 Maps from Smooth Curves Are G.I.T. Stable

We shall use the following lemma of Gieseker:

Lemma 15.1 ([Gies] Lemma 0.2.4) Fix two integers g ≥ 2, e ≥ 20(g − 1) and write

N = e − g. Then there exists ε > 0 such that for all integers r0, ..., rN (not all zero) with∑ri = 0 and for all integers 0 = e0, ..., eN = e satisfying

i. If ej > 2g − 2 then ej ≥ j + g.

ii. If ej ≤ 2g − 2 then ej ≥ 2j.

there exists a sequence of integers 0 = i1, ..., ik = N verifying the following inequality:

k−1∑t=1

(rit+1 − rit)(eit+1 + eit) > 2rNeN + 2ε(rn − r0).

36

Theorem 15.2 Suppose that m >> 0 (this will be made more precise in the course of the

proof) and m ≥ 2g+1. If C ⊂ P(W )×Pr → Pr is a stable map, C is nonsingular, the map

H0(P(W ),OP(W )(1))ρ→ H0(pW (C),OpW (C)(1))

is an isomorphism, and LW is very ample (so that C ∼= pW (C)), then Hm,m(C) is SL(W )-

stable.

Proof. Let C ⊂ P(W )× Pr→Pr be such a map. Let λ′ be a 1-PS of SL(W ). There

exist a basis w0, ..., wN of H0(P(W ),OP(W )(1)) and integers r0 ≤ · · · ≤ rN such that∑ri = 0 and the action of λ′ is given by λ′(t)wi = triwi. By our hypotheses the map

pW∗ρ : H0(P(W ),OP(W )(1)) → H0(C,LW ) is injective. Write w′i := pW∗ρ(wi). Let Ej

be the invertible subsheaf of LW generated by w′0, ..., w

′j for 0 ≤ j ≤ N = e − g, and write

ej = degEj. Note that EN = LW since LW is very ample hence generated by global sections,

h0(C,LW ) = e−g+1, and w′0, ..., w

′N are linearly independent. The integers e0, ..., eN satisfy

the following two properties:

i. If ej > 2g − 2 then ej ≥ j + g.

ii. If ej ≤ 2g − 2 then ej ≥ 2j.

To see this, note that since by definition Ej is generated by j + 1 linearly independent

sections we have h0(C,Ej) ≥ j + 1. If ej = degEj > 2g − 2 then H1(C,Ej) = 0 so by

Riemann-Roch ej = h0− h1 + g− 1 ≥ j + g. If ej ≤ 2g− 2 then H0(C, ωC ⊗E−1j ) 6= 0 so by

Clifford’s theorem j + 1 ≤ h0 ≤ ej

2+ 1.

The hypotheses of Lemma 15.1 are satisfied with these ri and ej, so there exist integers

0 = i1, ..., ik = N such that

k−1∑t=1

(rit+1 − rit)(eit+1 + eit) > 2rNeN + 2ε(rN − r0).

Suppose p and n are large positive integers. (In this proof n is not dg−1

.) Recall that

H0(P(W ),OP(W )((p + 1)n)) has a basis consisting of monomials of degree (p + 1)n in

w0, ..., wN . For all 1 ≤ t ≤ k let Vit ⊂ H0(P(W ),OP(W )(1)) be the subspace spanned

by Sit := w0, ..., wit. For all triples (t1, t2, s) with 1 ≤ t1 < t2 ≤ k and 0 ≤ s ≤ p let

(V p−sit1

V sit2VN)n ⊂ H0(P(W ),OP(W )((p + 1)n)) be the subspace spanned by elements of the

form v1 · · · vn, where each vj is of the following form:

For s = 0, vj∈xj1 · · ·xjpzj |xjα∈Sit1 , zj∈H0(P(W ),OP(W )(1)).

For 0<s<p, vj∈xj1 · · ·xjp−syj1 · · · yjszj |xjα∈Sit1 , yjβ ∈Sit2 , zj∈H0(P(W ),OP(W )(1)).

For s = p, vj∈yj1 · · · yjpzj | yjα∈Sit2 , zj∈H0(P(W ),OP(W )(1)).

37

This gives a filtration of H0(P(W ),OP(W )((p+ 1)n)):

0 ⊂ (V pi1V 0i2VN)n ⊂ (V p−1

i1V 1i2VN)n ⊂ · · · ⊂ (V 1

i1V p−1i2

VN)n

⊂ (V pi2V 0i3VN)n ⊂ (V p−1

i2V 1i3VN)n ⊂ · · · ⊂ (V 1

i2V p−1i3

VN)n

......

......

......

......

⊂ (V pitV 0it+1

VN)n ⊂ (V p−1it

V 1it+1

VN)n ⊂ · · · ⊂ (V 1itV

p−1it+1

VN)n

......

......

......

......

⊂ (V pik−1

V 0ikVN)n ⊂ (V p−1

ik−1V 1ikVN)n ⊂ · · · ⊂ (V 1

ik−1V p−1ik

VN)n

⊂ (V 0ik−1

V pikVN)n = H0(P(W ),OP(W )((p+ 1)n)).

(6)

Note that while we have defined (V p−sit1

V sit2VN)n whenever t1 < t2 we only use consecutive

integers t1 and t1 + 1 in the filtration (6).

Tensor each (V p−sit1

V sit2V )n with H0(Pr,OPr(m)) to get a filtration of

H0(P(W )× Pr,OP(W )((p + 1)n) ⊗ OPr(m)). Assume that (p + 1)n and m are sufficiently

large that

ρ(p+1)n,m : H0(P(W )×Pr,OP(W )((p+ 1)n)⊗OPr(m)) → H0(C,L(p+1)nW ⊗ Lmr )

is surjective. Write

(V p−sit1

V sit2VN)n := ρ(p+1)n,m((V p−s

it1V sit2VN)n ⊗H0(Pr,OPr(m)). (7)

Then we have a filtration H0(C,L(p+1)nW ⊗Lmr ) given by subspaces of the form (V p−s

it1V sit2VN)n.

Claim 15.3 There exists an integer n′ which is independent of C, t1, and t2 such that if

n ≥ n′ then

ρ(p+1)n,m((V p−sit1

V sit2VN)n ⊗H0(Pr,OPr(m)) = H0(C, (Ep−s

it1⊗ Es

it2⊗ LW )n ⊗ Lmr ).

Proof of Claim. By hypothesis LW is very ample. Note that degLr = d > 0 since C is

nonsingular hence irreducible. Thus if m > 2g + 1 then Lmr is very ample.

It follows from the definitions of the sheaves Ej that the linear system

(V p−sit1

V sit2VN)n ⊗H0(Pr,OPr(m)) restricted to C generates

(Ep−sit1

⊗ Esit2⊗ LW )n ⊗ Lmr

and

ρ(p+1)n,m((V p−sit1

V sit2VN)n ⊗H0(Pr,OPr(m)) ⊆ H0(C, (Ep−s

it1⊗ Es

it2⊗ LW )n ⊗ Lmr ). (8)

38

Suppose that m = nm1 and m1 > 2g + 1 so that Lm1r is very ample hence gener-

ated by global sections. Eit1 and Eit2 are generated by global sections, so it follows that

ρp+1,m1(Vp−sit1

V sit2VN ⊗H0(Pr,OPr(m1))) is a very ample base point free linear system on C.

Much of the notation we are about to define will be abbreviated in displayed equa-

tions. Let ψ := ψp+1,m1 be the projective embedding corresponding to the linear system

ρp+1,m1(Vp−sit1

V sit2VN ⊗H0(Pr,OPr(m1))). Let IC/P be the ideal sheaf defining C as a closed

subscheme of P := P(ρp+1,m1(Vp−sit1

V sit2VN ⊗H0(Pr,OPr(m1)))). There is an exact sequence

of sheaves on P as follows:

0 → IC/P → OP → ψ∗OC → 0.

Tensoring by the very ample sheaf OP(n) we obtain

0 → IC/P(n) → OP(n) → (ψ∗OC)(n) → 0. (9)

Write

F := Ep−sit1

⊗ Esit2⊗ LW ⊗ Lm1

r

We have

(ψ∗OC)(n) := ψ∗OC ⊗OPOP(n)

∼= ψ∗OC ⊗OPOP(V)(1)n

∼= ψ∗(OC ⊗OCψ∗OP(V)(1)n) by the projection formula, cf. [Hart] page 124

∼= ψ∗(Fn) since ψ∗OP(V)(1) ∼= F .

Now the exact sequence (9) reads

0 → IC/P(n) → OP(n) → ψ∗(Fn) → 0. (10)

In the corresponding long exact sequence in cohomology we have

· · · → H0(P,OP(n)) → H0(P, ψ∗Fn) → H1(P, IC/P(n)) → · · · . (11)

The so-called “Uniform m Lemma” (cf. [HM] Lemma 1.11 or [St] Proposition 4.3) ensures

that there is an integer n′ > 0 depending on the Hilbert polynomial P but not on the curve

C such that H1(P, IC/P(n)) = 0 if n > n′. Then for such n the exact sequence (11) implies

that the map

H0(P,OP(n)) → H0(C,Fn)

39

is surjective. Recall that P := P(ρp+1,m1(Vp−sit1

V sit2VN ⊗H0(Pr,OPr(m1)))). Then

H0(P,OP(n)) ∼= Symn(ρp+1,m1(Vp−sit1

V sit2VN ⊗H0(Pr,OPr(m1)))).

Also there is a surjection

Symn(V p−sit1

V sit2VN ⊗H0(Pr,OPr(m1))) → Symn(ρp+1,m1(V

p−sit1

V sit2VN ⊗H0(Pr,OPr(m1))))

so putting this all together we have a surjection

Symn(V p−sit1

V sit2VN ⊗H0(Pr,OPr(m1))) → H0(C,Fn). (12)

There is a natural map

(V p−sit1

V sit2VN)n ⊗H0(Pr,OPr(m))

∼=→ Symn(V p−sit1

V sit2VN ⊗H0(Pr,OPr(m1)))

and in fact ρ(p+1)n,m factors as

(V p−sit1

V sit2VN)n ⊗H0(Pr,OPr(m)) −→ Symn(V p−s

it1V sit2VN ⊗H0(Pr,OPr(m1)))

ρ(p+1)n,m ↓ (12)

H0(C,Fn)

so

ρ(p+1)n,m : (V p−sit1

V sit2VN)n ⊗H0(Pr,OPr(m)) → H0(C, (Ep−s

it1⊗ Es

it2⊗ LW )n ⊗ Lmr ) (13)

is also surjective. It follows from lines (8) and (13) that

(V p−st1 V s

t2VN)n = H0(C, (Ep−sit1

⊗ Esit2⊗ LW )n ⊗ Lmr ).

40

Proof of Theorem 15.2 continued. Take n ≥ 2g + 1 so that

H1(C, (Ep−sit1

⊗ Esit2⊗ LW )n ⊗ Lmr ) = 0. We use Riemann-Roch to calculate

dim (V p−sit

V sit+1

V )n = h0(C, (Ep−sit

⊗ Esit+1

⊗ LW )n ⊗ Lmr )

= n((p− s)eit + seit+1 + eN) + dm− g + 1. (14)

We assume for the rest of the proof that p and n are sufficiently large that

p > maxe+ g,32e+ 1

ε,

n > maxp, (2g + 1)n′,

m := (p+ 1)n > m′′′.

Choose a basis B(p+1)n,m of H0(P(W )× Pr,OP(W )((p + 1)n) ⊗ OPr(m)) of monomials

Mi of bidegree ((p + 1)n, m) such that ρ(p+1)n,mM1, ..., ρ(p+1)n,mMP ((p+1)n)+dm is a basis of

H0(C,L(p+1)nW ⊗ Lmr ). Observe that a monomial M ∈ (V p−s

i1V si2V )n − (V p−s+1

i1V s−1i2

V )n has

λ-weight wλ(M) ≤ n((p− s)rit + srit+1 + rN).

We now estimate the total λ-weight of M1, ..., MP ((p+1)n)+dm:

P (m)+dm∑i=1

wλ(Mi) ≤ n(pri1 + rN) dim (V pi1V 0i2V )n

+∑

0 ≤ s ≤ p1 ≤ t ≤ k − 1

n((p−s)rit + srit+1 + rN)

[dim( (V p−s

itV sit+1

V )n − (V p−s+1it

V s−1it

V )n)

](15)

The first term on the right hand side of (15) is

n(pri1 +rN)(n(pei1 +eiN )+dm−g+1) = n(pri1 +rN)dm+n(pri1 +rN)(n(pei1 +eiN )−g+1).

The factor dim( (V p−sit

V sit+1


V s−1it

V )n) of the summand is(n((p− s)eit + seit+1 + eN) + dm− g + 1

)−

(n((p− s+ 1)eit + (s− 1)eit+1 + eN) + dm− g + 1

)= n(eit+1 − eit).

Note that nearly all of the terms having dm as a factor have “telescoped.” We have

RHS (15) = n(pri1 + rN) dim (V pi1V 0i2V )n

+∑

0 ≤ s ≤ p1 ≤ t ≤ k − 1

n((p−s)rit +srit+1 +rN)

[dim( (V p−s

itV sit+1


V s−1it

V )n)

]= n(pri1 + rN)dm+ n(pri1 + rN)(n(pei1 + eiN )− g + 1)

+∑

0 ≤ s ≤ p1 ≤ t ≤ k − 1

n((p−s)rit + srit+1 + rN)[n(eit+1 − eit)

].

41

The sum of the second two terms is exactly the expression Gieseker obtains at the bottom

of page 30. Gieseker calculates (page 34):

n(pri1 + rN)(n(pei1 + eiN )− g + 1) +∑

0≤s≤p

1≤t≤k−1

n((p− s)rit + srit+1 + rN)[n(eit+1 − eit)

]< n2p(rN − r0)

[−εp+ 3e

2+ e+g−1

p

]< 0.

where the last inequality follows because p > maxe+ g,32e+1

ε. Therefore we have

P (m)+dm∑i=1

wλ(Mi) ≤ 0 + n(pr0 + rN)dm < 0.

The last inequality follows because r0 < 0 and because by hypothesis p > | rNr0|.

(More precisely, the hypothesis states p > e+g. But we have∑N

i=0 ri = 0. If∑N−1

i=1 ri ≥ 0

then ∣∣∣∣rNr0∣∣∣∣ =

rN−r0

=rN

rN +∑N−1

i=1 ri≤ 1.

If∑N−1

i=1 ri < 0 then since ri ≥ r0 for all i > 0 we have∣∣∣∣rNr0∣∣∣∣ =

∣∣∣∣∣r0 +∑N−1

i=1 rir0

∣∣∣∣∣ ≤ 1 +

∣∣∣∣∣∑N−1

i=1 rir0

∣∣∣∣∣ ≤ 1 +N − 2 = e− g − 1 < e+ g.

So p > | rNr0|.)

By the numerical criterion, the point Hm,m(C) is λ′-stable. Nowhere in the proof have we

placed any conditions on the 1-PS λ′, so the result is true for every 1-PS of SL(W ). Then

Hm,m(C) is SL(W )-stable.

16 G.I.T. Semistable Maps Are Potentially Stable

In this section we study the locus of semistable points Hilb(P(W )× Pr)ss for certain

linearizations. The results 16.1-16.14 show that for certain linearizations, if Hm,m(C) is

SL(W )-semistable, then the abstract curve C ⊂ P(W )× Pr is reduced and nodal, and the

embedding C ⊂ P(W )× Pr must satisfy certain properties. Our investigation will uncover

exactly what these properties are, and this will guide us to the right definition of “potentially

stable map,” which is stated formally at the end of the section.

42

16.1 First properties of G.I.T. semistable maps

Proposition 16.1 (cf. [Gies] 1.0.2) Suppose that m, m > m′′′ and m > (q1+1)(e−g+1).

Then for all C such that Hm,m(C) is SL(W )-semistable, the composition

H0(P(W ),OP(W )(1)) → H0(pW (C)red,OpW (C)red(1)) → H0(Cred, LW red)

is injective.

Proof. Suppose the map

H0(P(W ),OP(W )(1)) → H0(Cred, LW red)

has nontrivial kernel W0. Write N0 = dimW0. Choose a basis w0, ..., wN of

H0(P(W ),OP(W )(1)) =: W1 relative to the filtration 0 ⊂ W0 ⊂ W1. Let λ be the 1-PS of

GL(W ) whose action is given by

λ(t)wi = wi, t ∈ C∗, 0 ≤ i ≤ N0 − 1λ(t)wi = twi, t ∈ C∗, N0 ≤ i ≤ N

and let λ′ be the associated 1-PS of SL(W ). Choose m and m sufficiently large as previ-

ously explained. Let Bm,m be a basis of H0(P(W )×Pr,OP(W )(m)⊗OPr(m)) consisting of

monomials of bidegree (m, m).

Suppose Hm,m(C) is λ′-semistable. Then there exist monomials Mi1 , ..., MiP (m)+dmin Bm,m

such that ρm,m(Mi1), ..., ρm,m(MiP (m)+dm) is a basis of H0(C,LmW ⊗ Lmr ) and

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1.

Let W q10 W

m−q11 denote the subspace of SymmW1

∼= H0(P(W ),OP(W )(m)) spanned by

elements of the form x1 · · ·xq1y1 · · · ym−q1 where xi ∈ W0 and yi ∈ W1. Let IC denote the

ideal sheaf of nilpotent elements of OC . Recall that the integer q1 satisfies Iq1C = 0. It follows

that the image of the vector space W q10 W

m−q11 under

ρm,m : H0(P(W )×Pr,OP(W )(m)⊗OPr(m)) → H0(C,LmW ⊗ Lmr )

is zero. Therefore, for all j, Mij 6∈ W q10 W

m−q11 since ρm,m(Mij) 6= 0. In particular if Mij is

written as wγ00 · · ·wγN

N , we must have γ0 < q1. It follows that wλ(Mij) ≥ m− q1 + 1, because

43

wλ(Mij) =∑γiri, where r0 = 0 and ri = 1 if i > 0, and

∑γi = m. Thus the total weight

of any P (m) + dm monomials Mi1 , ..., MiP (m)+dmwhose images under ρm,m form a basis of

H0(C,LmW ⊗ Lmr ) must be greater than or equal to (P (m) + dm)(m− q1 + 1). In symbols,

P (m)+dm∑j=0

wλ(Mij) ≥ (P (m) + dm)(m− q1 + 1).

Note too thatN∑i=0

wλ(wi) = dimW1 − dimW0 = e − g + 1 − dimW0 ≤ e − g because

dimW0 ≥ 1.

Combining these three inequalities, we have

(P (m) + dm)(m− q1 + 1)

m(P (m) + dm)≤

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1≤ e− g

e− g + 1

1− q1 − 1

m≤ 1− 1

e− g + 1m ≤ (q1 + 1)(e− g + 1)

But by hypothesis m > (q1 + 1)(e− g + 1). The contradiction implies that Hm,m(C) is not

λ′-semistable, and therefore Hm,m(C) is not SL(W )-semistable.

Let Ci be an irreducible component of C. If the morphism pW |Cidoes not collapse Ci to

a point then it is finite by [Liu] Lemma 7.3.10.

We introduce or recall the following notation: Let pW (C)i be the irreducible compo-

nents of pW (C), i = 1, ..., `. Recall that LW denotes ι∗p∗WOP(W )(1). We write C = C ′ ∪ Ywhere the multidegree of LW is zero on C ′ and nowhere zero on Y . That is, C ′ is the

union of all irreducible components of C which collapse under pW and Y = C − C ′. Let

C ′1,1, ..., C

′1,j′1, Y1,1, ..., Y1,j1 , C

′2,1, ..., C2,j′2

, Y2,1, ..., Y2,j2 , ..., C′`,1, ..., C

′`,j′`, Y`,1, ..., Y`,j` be the irre-

ducible components of C ordered so that

i. pW (C ′i,j′) ∈ pW (C)i, and if pW (C ′

−,−) lies on more than one component,

it is indexed by the smallest i

ii. pW (Yi,j) = pW (C)i

iii. deg pW |Yi,j red =: ni,j ≥ ni,j+1.

Let degpW (C)redOP(W )(1) =: eW and degpW (C)i red

OP(W )(1) =: eWi. Since pW (C) ⊂ P(W ) we

have eWi ≥ 1 for all i. Recall that LW denotes ι∗p∗WOP(W )(1) and Lr denotes ι∗p∗rOPr(1).

44

Let degYi,j redLW = ei,j, and let degYi red

Lr = di,j. Finally let ki,j = lengthOCi,j ,ξi,j and

ki = lengthOpW (C)i,ξi . Then we have:

e =∑

ki,jeij

ei,j = ni,jeWi

e =∑

ki,jni,jeWi

eW =∑

kieWi

Proposition 16.2 Suppose that m, m > m′′′ and

m > (g − 3

2+ e(q1 + 1) + q3 + µ1m

′′)(e− g + 1)

andm

m<

(3n+ 2)a− 5− 12(g−1)

4n.

Then for all curves C such that Hm,m(C) is SL(W )-semistable, in the notation above, the

morphism pW |Y red is generically 1-1, that is, ni,j = 1 and ji = 1 for all i = 1, ..., `. Further-

more Y is generically reduced.

Proof. Suppose not. We may assume that at least one of the following is true: n1,1 ≥ 2

or k1,j ≥ 2 for some 1 ≤ j ≤ j1 or j1 ≥ 2. The first condition implies that a component of Y

is a degree n1,1 cover of its image, the second condition implies that the subcurve Y is not

generically reduced, and the third condition implies two irreducible components of Y map

to the same irreducible component of pW (C). Let W0 be the kernel of the restriction map

H0(P(W ),OP(W )(1)) → H0(pW (C)1 red,OpW (C)1 red(1)).

We claim that W0 6= 0. To see this, suppose W0 = 0. Let D1 be a divisor on pW (C)1 red

corresponding to the invertible sheaf OpW (C)1 red(1) and having support in the smooth locus

of pW (C)1red. Consider the exact sequence 0 → OpW (C)1 red→ OpW (C)1 red

(1) → OD1 → 0.

Then the long exact sequence in cohomology implies that

h0(pw(C)1 red,OpW (C)1 red(1)) ≤ h0(pW (C)1 red,OD1) + h0(pW (C)1 red,OpW (C)1 red

).

Note that h0(pW (C)1 red,OpW (C)1 red) = 1, and

h0(pW (C)1 red,OD1) ≤ degD1 = degOpW (C)1 red(1) = eW1.

45

If W0 = 0, then

e− g + 1 = h0(P(W ),OP(W )(1))

≤ h0(pW (C)1 red,OpW (C)1 red(1))

≤ eW1 + 1 =e1,1n1,1

+ 1

⇒ k1,1n1,1(e− g) ≤ e−∑

(i,j) 6=(1,1)

ki,jni,jeWi

⇒ k1,1n1,1(e− g) < e−∑

(i,j) 6=(1,1)

ki,jni,jeWi < e

⇒ (k1,1n1,1 − 1)e ≤ k1,1n1,1g (16)

If n1,1 ≥ 2 or if k1,1 ≥ 2 then k1,1n1,1−1

k1,1n1,1≥ 1

2and we have e

2≤ g. But e > 2g so we have a

contradiction.

If n1,1 = k1,1 = 1 but j1 ≥ 2 we have as above (e − g) ≤ e1,1. But we also have

e1,1 = e −∑

6=1,1 ki,jni,jeWi ≤ e − k1,2n1,2eW1 < e − eW1 = e − e1,1. Adding the two

inequalities (e− g) ≤ e1,1 and (e− g) ≤ e− e1,1 we again obtain the contradiction e ≤ 2g.

This shows that W0 6= 0. Write N0 = dimW0. Choose a basis w0, ..., wN of

H0(P(W ),OP(W )(1)) relative to the filtration 0 ⊂ W0 ⊂ W1 = H0(P(W ),OP(W )(1)). Let λ

be the 1-PS of GL(W ) whose action is given by

λ(t)wi = wi, t ∈ C∗, 0 ≤ i ≤ N0 − 1λ(t)wi = twi, t ∈ C∗, N0 ≤ i ≤ N

and let λ′ be the associated 1-PS of SL(W ). Choose m and m sufficiently large. Let Bm,m be

a basis of H0(P(W )×Pr,OP(W )(m)⊗OPr(m)) consisting of monomials of bidegree (m, m).

Construct a filtration of H0(C,LmW ⊗Lmr ) as follows: For 0 ≤ p ≤ m let Wm−p0 W p

1 be the

subspace of H0(P(W ),OP(W )(m)) spanned by elements of the following type:

x1 · · ·xm|xi ∈ W0 if p = 0x1 · · ·xm−py1 · · · yp|xi ∈ W0, yi ∈ W1 if 0 < p < my1 · · · ym|yi ∈ W1 if p = m.

Set Wm−p0 W p

1 = Wm−p0 W p

1 ⊗H0(Pr,OPr(m)), and let

Wm−p0 W p

1 := ρm,m( Wm−p0 W p

1 ) ⊂ H0(C,LmW ⊗ Lmr ). (17)

Note that a monomial M ∈ Wm−p0 W p

1 −Wm−p+1

0 W p−11 has weight p. Therefore we have a

filtration of H0(C,LmW ⊗ Lmr ) in order of increasing weight:

0 ⊆ ¯Wm0 W

01 ⊆ Wm−1

0 W 11 ⊆ · · · ⊆ ¯W 0

0 Wm1 = H0(C,LmW ⊗ Lmr )

46

Write βp = dim Wm−p0 W p

1 .

Let C1 =⋃j′1j′=1C

′1,j′ ∪

⋃j1j=1 Y1,j and 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

R : H0(C,LmW ⊗ Lmr ) → H0(C, LmW C

⊗ Lmr C

) be the map induced by restriction.

Claim 16.3 C = C − C1 can be given the structure of a closed subscheme of C such that

for all 0 ≤ p ≤ m− q1,

Wm−p0 W p

1 ∩ kerR : H0(C,LmW ⊗ Lmr ) → H0(C, LmWC

⊗ LmrC

) = 0.

We shall prove the proposition assuming the claim and prove the claim afterward.

Let IP be the ideal subsheaf of OC defining the closed point P . We have an exact

sequence

0 → Im−pP ⊗ LmW C

⊗ Lmr C→ Lm

W C⊗ Lm

r C→ OC/I

m−pP ⊗ Lm

W C⊗ Lm

r C→ 0.

In cohomology we have

0 → H0(C, Im−pP ⊗ LmW C

⊗ Lmr C

) → H0(C, LmW C

⊗ Lmr C

)

→ H0(C,OC/Im−pP ⊗ Lm

W C⊗ Lm

r C) → H1(C, Im−pP ⊗ Lm

W C⊗ Lm

r C) → 0.

The following five facts are analogous to those stated by Gieseker in [Gies] (page 44):

I. h0(C, LmW C

⊗ Lmr C

) = χ(LmW C

⊗ Lmr C

) = degC LmW C

⊗ Lmr C

+ χ(OC)

≤ (e−∑k1,je1,j)m+ (d−

∑k1,jd1,j −

∑k1,j′d1,j′)m+ h0(C,OC)

≤ (e−∑k1,je1,j)m+ (d−

∑k1,jd1,j −

∑k1,j′d1,j′)m+ q3.

II. h0(C,OC/Im−pP ⊗ Lm

W C⊗ Lm

r C) ≥ m− p.

III. h0(C,OC/Im−pP ⊗ Lm

W C⊗ Lm

r C) ≤ µ1(m− p) + µ2, so

h1(C, Im−pP ⊗ LmW C

⊗ Lmr C

) ≤ µ1(m− p) + µ2.

IV. For p ≥ m′′, we have h1(C, Im−pP ⊗ LmW C

⊗ Lmr C

) = 0.

V. The image of Wm−p0 W p

1 under R is contained in the subspace

H0(C, Im−pP ⊗ LmW C

⊗ Lmr C

) of H0(C, LmW C

⊗ Lmr C

).

By the claim and fact V., we have βp = dim Wm−p0 W p

1 ≤ h0(C, Im−pP ⊗ LmW C

⊗ Lmr C

) if

0 ≤ p ≤ m− q1. From the exact sequence we have

h0(C, Im−pP ⊗ LmW C

⊗ Lmr C

) = h0(C, LmW C

⊗ Lmr C

)

− h0(C,OC/Im−pP ⊗ Lm

W C⊗ Lm

r C)

+ h1(C, Im−pP ⊗ LmW C

⊗ Lmr C

).

47

Thus, using the facts above, we have:

βp ≤

(e−

∑k1,je1,j)m+ (d−

∑k1,jd1,j−

∑k1,j′d1,j′)m+ q3 + p−m+ µ1(m−p) + µ2

if 0 ≤ p ≤ m′′ − 1(e−

∑k1,je1,j)m+ (d−

∑k1,jd1,j −

∑k1,j′d1,j′)m+ q3 + p−m

if m′′ ≤ p ≤ m− q1em+ dm− g + 1 if m− q1 + 1 ≤ p ≤ m.

Now suppose that Hm,m(C) is λ′-semistable. Then there exist monomials Mi1 , ..., Miα in

Bm,m such that ρm,m(Mi1), ..., ρm,m(Miα) is a basis of H0(C,LmW ⊗ Lmr ) and

α∑j=1

wλ(Mij)

m(em+ dm+ 1− g)≤

N∑i=0

wλ(wi)

e− g + 1.

Butα∑j=1

wλ(Mij) must be larger thanm∑p=1

p(βp − βp−1). We calculate:

m∑p=1

p(βp − βp−1) = mβm −m−1∑p=0

βp

≥ m(em+ dm+ 1− g)

−m−q1∑p=0

((e−

∑k1,je1,j)m+ (d−

∑k1,jd1,j −

∑k1,j′d1,j′)m+ q3 + p−m

)

−m′′−1∑p=0

(µ1(m− p) + µ2)−m−1∑

m−q1+1

(em+ dm+ 1− g)

=

(∑k1,je1,j +

1

2

)m2

+ m

[3

2− g +

(∑k1,jd1,j +

∑k1,j′d1,j′

)m− q3 −

∑k1,je1,j(q1 + 1)− µ1m

′′]

+ (q1 − 1)(g + q3 −(∑

k1,jd1,j −∑

k1,j′d1,j′

)m− q1

2− 1)− µ2m

′′ + µ1m′′(m′′ − 1)

2

≥(∑

k1,je1,j +1

2

)m2 − S2m+

(∑k1,jd1,j +

∑k1,j′d1,j′

)m(m− q1 + 1) + c2

≥(∑

k1,je1,j +1

2

)m2 − S2m (18)

where

S2 = g − 3

2+

∑k1,je1,j(q1 + 1) + q3 + µ1m

′′

c2 = (q1 − 1)(g + q3 −q12− 1)− µ2m

′′ + µ1m′′(m′′ − 1)

2.

48

The inequality (18) follows because the term (∑k1,jd1,j+

∑k1,j′d1,j′)m(m−q1+1) is positive

since the hypotheses imply m > q1 and because c2 ≥ 0 since q3 > q1 and µ1 > µ2 (see page

35).

Note also that

N∑i=0

wλ(wi) = dimW1 − dimW0 ≤ h0(pW (C1,1)red,OpW (C1,1)red(1))

≤ degOpW (C1,1) red(1) + 1 ≤ eW1 + 1.

The hypothesis on m implies m > S2(e − g + 1). Then we obtain a contradiction as

follows:

(∑k1,je1,j + 1

2)m2 − S2m

m(em+ dm+ 1− g)≤

P (m)+dm∑j=1

wλ(Mij)

m(em+ dm+ 1− g)≤

N∑i=0

wλ(wi)

e− g + 1≤ eW1 + 1

e− g + 1∑k1,je1,j + 1

2− S2

m

e+ dmm

≤ eW1 + 1

e− g + 1

(e− g + 1)(∑

k1,je1,j +1

2)− e(eW1 + 1) ≤ S2(e− g + 1)

m+

(eW1 + 1)dm

m

(e− g + 1)(∑

k1,je1,j +1

2)− e(eW1 + 1) ≤ 1 +

(eW1 + 1)dm

m(e− g + 1)(

∑k1,je1,j + 1

2)− e(eW1 + 1)− 1

(eW1 + 1)d≤ m

m

(e− g + 1)(eW1

∑k1,jn1,j + 1

2)− e(eW1 + 1)− 1

(eW1 + 1)d≤ m

m.

Note that since n1,1 ≥ 2 or j1 ≥ 2 we have∑k1,jn1,j ≥ 2. Thus

(e− g + 1)(eW1

∑k1,jn1,j +

1

2)− e(eW1 + 1)− 1 > 0.

Furthermore the quantity

(e− g + 1)(eW1

∑k1,jn1,j + 1

2)− e(eW1 + 1)− 1

(eW1 + 1)d

is minimized when eW1 takes its smallest value, that is, when eW1 = 1. Then

(e− g + 1)(eW1

∑k1,jn1,j + 1

2)− e(eW1 + 1)− 1

(eW1 + 1)d=

(3n+ 2)a− 5− 12(g−1)

4n.

But by hypothesis mm<

(3n+2)a−5− 12(g−1)

4n. The contradiction implies that Hm,m(C) is not

λ′-semistable, and therefore that Hm,m(C) is not SL(W )-semistable.

49

It remains to prove the claim:

Proof of Claim 16.3. Much of the notation in the proof that follows (particularly indices)

has no relation to what appears in the proof of the theorem above. Let P1, ..., Pt be the

associated points of C. Choose a finite open affine cover Ui of C such that each associated

point belongs to exactly one of the Ui and such that LW and Lr are trivialized on each Ui.

Suppose Ui ∼= SpecAi and Ui ∩ Uk ∼= SpecAik. In each Ai let (0) =⋂ni

j=1 qij be a

primary decomposition of the zero ideal, and suppose that each qij is pij-primary. We may

assume that the pij are ordered so that for all Ui such that Ui ∩ C1 6= 0, the components

C ′1,1red, ..., Y1,j1red correspond to pi1, ..., pi(j′1+j1). We define an ideal subsheaf I ofOC as follows:

If Ui ∩ C1 6= 0 then I(Ui) =⋂ni

j>j′1+j1qij, and if Ui ∩ C1 = 0 then I(Ui) =

⋂ni

j=1 qij = (0).

Let C be the subscheme of C defined by I. We show that C has the desired property: Let

s ∈ Wm−p0 W p

1 ∩kerR. Let si denote the restriction of s to Ui. For each Ui let ωi : LW |Ui→ Ai

and ρi : Lr|Ui→ Ai and γi : LmW ⊗ Lmr → Ai be the trivializing isomorphisms. Now if

Ui ∩ C1 = ∅ then si = 0 since si ∈ kerR. If Ui ∩ C1 6= ∅ then write ai := γi(si). Since

m−p ≥ q1, we have ai ∈ pq1ij for all j = 1, ..., j′1 + j1. Also pq1ij ⊂ qij for all j = 1, ..., j′1 + j1, so

ai ∈ qij for all j = 1, ..., j′1 + j1. Since si ∈ kerR we have ai ∈⋂j>j′1+j1

qij. Thus ai ∈⋂j qij,

so si = 0. Thus Wm−p0 W p

1 ∩ kerR = 0.

We shall need to introduce additional notation for the next proposition.

Notation. Suppose C is a curve which has at least two irreducible components, and

suppose it is generically reduced on any components which do not collapse under pW . Let

C ′ 6= C be a reduced, complete subcurve of C and let Y be the closure of C − C ′ in C with

the reduced structure. Let C ′ ιC′→ Cι→ P(W )×Pr and Y

ιY→ C be the inclusion morphisms.

LetLW C′ := ι∗C′ι

∗Cp

∗WOP(W )(1) LW Y := ι∗Y ι

∗Cp

∗WOP(W )(1)

Lr C′ := ι∗C′ι∗Cp

∗rOPr(1) Lr Y := ι∗Y ι

∗Cp

∗rOPr(1).

(19)

Let π : C → C be the normalization morphism. Let LW C′ := π∗LW C′ . and similarly for

the other three line bundles defined above. Define e′ := degC′ LW C′ = degC′ LW C′ and

d′ := degC′ Lr C′ = degC′ Lr C′ . Finally, write h0(pW (C ′),OpW (C′)(1)) =: h0.

Lemma 16.4 (cf. [Gies] 1.0.7) Suppose that m, m > m′′′ and

m > (g − 3

2+ e(q1 + 1) + q3 + µ1m

′′)(e− g + 1)

50

andm

m<

(3n+ 2)a− 5− 12(g−1)

4n.

Let C be a curve such that Hm,m(C) is SL(W )-semistable, and suppose C has at least two

irreducible components. Let C ′ and Y as above. Suppose there exist points P1, ..., Pk on Y

satisfying

i. π(Pi) ∈ Y ∩ C ′ for all 1 ≤ i ≤ k

ii. for each irreducible component Yj of Y ,

degYj(LW Y (−D)) ≥ 0,

where D = P1 + · · ·+ Pk.

Then(e′ + k

2)

e<h0 + (dh0−d′(e−g+1))m

em

e− g + 1+

S

em, (20)

where S = g + k(2g − 1) + q2 − gY + k2.

Remark. Gieseker derives the so-called Basic Inequality from an inequality very similar to

the one above. In his proof, his Basic Inequality implies that chains of rational components

must have length one. It follows that no points of Hilb(PN)ss parametrize curves which are

prestable but not semistable or Deligne-Mumford semistable curves with chains of rational

components of length greater than one. However, stable maps may have such curves as their

domains. We shall see that the inequality (20) places some restrictions on the embedding

in P(W ) (see Corollary 16.15 below) but every stable map has a model which satisfies (20)

(see Proposition 16.6 below) so we will obtain the whole moduli space of maps.

Remark. In all our applications the term Sem

will be made small by taking m, m >> 0.

Proof. Suppose first that C ′ is connected. Let

W0 := kerH0(P(W ),OP(W )(1)) → H0(pW (C ′),OpW (C′)(1)). (21)

Choose a basis w0, ..., wN of H0(P(W ),OP(W )(1)) relative to the filtration

0 ⊂ W0 ⊂ W1 = H0(P(W ),OP(W )(1)). Let λ be the 1-PS of GL(W ) whose action is given

byλ(t)wi = wi, t ∈ C∗, 0 ≤ i ≤ N0 − 1λ(t)wi = twi, t ∈ C∗, N0 ≤ i ≤ N

and let λ′ be the associated 1-PS of SL(W ).

51

For 0 ≤ p ≤ m let Wm−p0 W p

1 be the subspace of H0(P(W ),OP(W )(m)) spanned by

elements of the following type:


Set Wm−p0 W p

1 = Wm−p0 W p


Wm−p0 W p


1 ) ⊂ H0(C,LmW ⊗ Lmr ). (22)

Note that a monomial M ∈ Wm−p0 W p

1 −Wm−p+1

0 W p−11 has weight p. Therefore we have a

filtration of H0(C,LmW ⊗ Lmr ) in order of increasing weight:

0 ⊆ ¯Wm0 W

01 ⊆ Wm−1

0 W 11 ⊆ · · · ⊆ ¯W 0



1 .

The normalization morphism π : C → C induces a homomorphism

πm,m∗ : H0(C,LmW ⊗ Lmr ) → H0(C, LmW ⊗ Lmr ). (23)

There is a splitting H0(C, LmW ⊗ Lmr ) = H0(C ′, LmWC′

⊗ LmrC′

)⊕H0(Y , LmWY

⊗ LmrY

). By def-

inition the sections in πm,m∗(Wm−p0 W p

1 ) vanish on C ′ and to order ≥ m − p at the points

P1, ..., Pk. Thus πm,m∗(Wm−p0 W p

1 ) ⊆ H0(Y , LmWY

((p − m)D) ⊗ LmrY

) ⊆ H0(C, LmW ⊗ Lmr ).

Then

βp = dim Wm−p0 W p

1 ≤ h0(Y , LmWY

⊗ LmrY⊗OY ((p−m)D)) + dim kerπm,m∗

= (e− e′)m+ (d− d′)m+ k(p−m)− gY + 1+ h1(Y , Lm

WY⊗ Lm

rY⊗OY ((p−m)D)) + dim kerπm,m∗.

We have the following three estimates:

I. dim ker πm,m∗ < q2.

Proof. kerπm,m∗ ⊂ kerπYm,m∗ where πYm,m∗ : H0(Y, LmWY ⊗ LmrY ) → H0(Y , LmWY ⊗ LmrY )

is the homomorphism induced by the normalization of Y . Now let IY denote the ideal

sheaf of nilpotents in OY . We can choose an integer q2 such that h0(C, IC) < q2 hence

h0(Y, IY ) < q2 as well. Then dim ker πYm,m∗ < q2. To see this, recall that the normalization

morphism factors through Yred. We have π = ι π′, where ι : Yred → Y . Then we have

πm,m∗ = π′m,m∗ ιm,m∗. Now, π′m,m∗ is injective, so kerπm,m∗ = ker ιm,m∗. We have assumed

52

that Y is generically reduced, so IY has finite support. We have an exact sequence of sheaves

0 → IY ⊗ LmW ⊗ Lmr → LmW ⊗ Lmr → LmW red⊗ Lmrred → 0. This gives rise to an exact sequence

in cohomology. It follows that kerπm,m∗ = ker ιm,m∗ ∼= H0(Y, IY ⊗ LmW ⊗ Lmr ) and hence

dim ker πm,m∗ = h0(Y, IY ⊗ LmW ⊗ Lmr ) = h0(Y, IY ) < q2.

II. h1(Y , LmW Y

⊗ Lmr Y⊗OY ((p−m)D)) ≤ k(m− p) if 0 ≤ p ≤ 2g − 2.

Proof. There is a short exact sequence 0 → O(−(m − p)D) → OY → O(m−p)D → 0 of

sheaves on Y . Since LmW Y

⊗ Lmr Y

is an invertible sheaf, it is locally free and in particular flat.

Tensoring, we obtain a second short exact sequence

0 → LmW Y ⊗ Lmr Y ⊗OY (−(m− p)D) → LmW Y ⊗ Lmr Y → O(m−p)D → 0

of sheaves on Y . In the corresponding long exact sequence in cohomology we have

H1(Y , LmW Y

⊗ Lmr Y

) = 0 since m and m are large. It follows that the map

H0(Y ,O(m−p)D) → H1(Y , LmW Y ⊗ Lmr Y ⊗OY (−(m− p)D))

is a surjection. Therefore h1(Y , LmW Y

⊗ Lmr Y⊗ OY (−(m − p)D)) ≤ h0(Y ,O(m−p)D). But

O(m−p)D is just a skyscraper sheaf, and we calculate h0(Y ,O(m−p)D) = k(m− p).

III. h1(Y , LmW Y

⊗ Lmr Y⊗OY ((p−m)D)) = 0 if 2g − 1 ≤ p ≤ m− 1.

Proof. This follows because Y is an integral curve and deg LmW Y

((p−m)D)⊗Lmr Y≥ 2gY−1

whenever 2g − 1 ≤ p ≤ m− 1.

Combining this data with our previous formula (16.1) we have

βp ≤

(e− e′)m+ (d− d′)m+ k(p−m)− gY + 1 + q2 + k(m− p) if 0 ≤ p ≤ 2g − 2(e− e′)m+ (d− d′)m+ k(p−m)− gY + 1 + q2 if 2g − 1 ≤ p ≤ m− 1

We have supposed that Hm(C) is SL(W )-semistable, therefore λ′-semistable. Hence

there exist monomials Mi1 , ..., MiP (m)+dmin Bm,m such that ρm,m(Mi1), ..., ρm,m(MiP (m)+dm

)is a basis of H0(C,LmW ⊗ Lmr ) and

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1.

But

P (m)+dm∑j=1



53

m∑p=1

p(βp − βp−1) = mβm −m−1∑p=0

βp

≥ m(em+dm+ 1− g)−m−1∑p=0

(e−e′)m+(d− d′)m+ k(p−m)−gY +1+q2 −2g−2∑p=0

k(m−p)

≥ (e′ +k

2)m2 − Sm+ d′mm+

(2g − 1)(2g − 2)

2

≥ (e′ +k

2)m2 − Sm+ d′mm

where S = g + k(2g − 1) + q2 − gY + k2.

Note also thatN∑i=0

wλ(wi) = dimW1 − dimW0 ≤ h0(pW (C ′),OpW (C′)(1)) =: h0. We have

(e′ + k2)m2 − Sm+ d′mm

m(em+ dm+ 1− g≤ h0

e− g + 1

(e′ + k2)− S

m+ d′m

m

e+ dmm

≤ h0

e− g + 1

(e′ + k2)− S

m+ d′m

m

e≤

h0 + dh0mem

e− g + 1

(e′ + k2)− S

m

e≤

h0 + (dh0−d′(e−g+1))mem

e− g + 1.

This proves the lemma when C ′ is connected. If C ′ is not connected, write C ′ = ∪C ′i

where the C ′i are the connected components of C ′. Then we proved

(e′i +k′i2)

e<h0i +

(dh0i−d′i(e−g+1))m

em

e− g + 1+

S

em,

for each i, where h0i := h0(pW (C ′

i),OpW (C′i)(1)) =: h0. There are no points connecting any of

the C ′i to each other so k′ =

∑k′i. Also H0(pW (C ′),OpW (C′)(1)) = ⊕H0(pW (C ′

i),OpW (C′i)(1))

so h0 =∑h0i . Summing the inequalities over i we obtain the desired result.

Proposition 16.5 (cf. [Gies] 1.0.3) Suppose that m, m > m′′′ and

m > max(q1 + 1)(e− g + 1), (g − 3

2+

∑k1,je1,j(q1 + 1) + q3 + µ1m

′′)(e− g + 1)

and(1

2+ 3g+q2

m)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

(3n+ 2)a− 5− 12(g−1)

4n.

54

Then for all C such that Hm,m(C) is SL(W )-semistable, no irreducible components of C

collapse under pW .

Proof. Suppose that at least one component of C collapses under pW . Let C ′ be the

union of all irreducible components of C which collapse under pW and let Y be the union of

all irreducible components of C which are not collapsed under pW . Then C ′ ∩ Y 6= ∅ since

C is connected. Let P ∈ C ′∩Y. We have degYj(LW Y ) ≥ 1 so degYj

(LW Y (−P )) ≥ 0 for each

irreducible component of Y . The hypotheses of Lemma 16.4 are satisfied for k = 1. We have

h0(pW (C ′),OpW (C′)(1)) = 1 and d′ ≥ 1 since e′ = 0. Then the inequality reads

0 + 12− S

m

e≤

(e′ + k2)− S

m

e≤

h0 + (dh0−d′(e−g+1))mem

e− g + 1≤

1 + d−e+g−1e

mm

e− g + 1

(1

2− S

m)(e− g + 1)− (e− g + 1)− (g − 1) ≤ m

m(d− e+ g − 1)

−(1

2+S

m)(e− g + 1)− (g − 1) ≤ m

m(d− e+ g − 1)

(12

+ Sm

)(e− g + 1)− (g − 1)

e− g + 1− d≥ m

m

(12

+ Sm

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n≥ m

m

Recall that S = g + k(2g − 1) + q2 − gY + k2≤ 3g + q2 when k = 1. We have chosen

m

m>

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n≥

(12

+ Sm

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n

so the last line above is a contradiction.

Remark. We now know that for any curve C such that Hm,m(C) is SL(W )-semistable, the

map pW |C : C → pW (C) is surjective, finite, and generically 1-1. Therefore it is birational,

and the normalizations C and pW (C) are isomorphic.

We check that all stable maps have a model satisfying the inequality (20) of Lemma 16.4.

Given a stable map, Fulton and Pandharipande describe how to obtain a model such that

h1(C,LW ) = 0 and e′ = a(2g′− 2 + k) + 3ad′ for each irreducible component ([FP] page 58).

Proposition 16.6 If mm

= 3a2a−1

and m, m >> 0 then this model satisfies the inequality

e′ + k2

e≤h0 + (dh0−d′(e−g+1))m

em

e− g + 1. (24)

55

Proof. We have h0 = e′ − g′ + 1. We rewrite the desired result:

e′ + k2

e≤

h0 + (dh0−d′(e−g+1))mem

e− g + 1⇔

k

2(e−g+1) ≤ e′(d

m

m+ g − 1)− e(g′ − 1) +

m

m(−d′(e− g + 1) + d(1− g′)) ⇔

0 ≤ k(a(dm

m+ g − 1)− 1

2(e− g + 1)) + a(2g′ − 2)(d

m

m− g − 1) ⇔

+3ad′(dm

m+ g − 1)− e(g′ − 1) +

m

m(−d′(e− g + 1) + d(1− g′)) ⇔

0 ≤ n

((m

m− 3

2

)ka+

(m

m(2a−1)− 3a

)(g′−1)

)+k

2− d′

(m

m(2a−1)− 3a

)Note that if m

m= 3a

2a−1= 3

2+ 3

4a−2then the last line above reads 0 ≤ nk 3a

4a−2+ k

2which is

clearly true. In fact since 0 < nk 3a4a−2

+ k2

then by continuity the inequality must be verified

for some small range of mm

close to 3a2a−1

.

Remark. Note that the term Sem

appearing in the inequality (20) is not needed in the

statement of Proposition 16.6.

16.2 G.I.T. semistability implies that the only singularities arenodes

The next series of results shows that (for certain linearizations) if Hm,m(C) is SL(W )-

semistable and P ∈ C is a singular point, then P is a node.

Let g denote the minimum value of gC taken over all curves C of genus g.

Lemma 16.7 (cf. [Gies] 1.0.5) Suppose that m, m > m′′′ and

m > max

(g − 3

2+ e(q1 + 1) + q3 + µ1m

′′)(e− g + 1),(9g + 3q2 − 3g − 9

2)(e− g + 1)

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

For all C such that Hm,m(C) is SL(W )-semistable, the normalization morphism π : C → Cred

is unramified.

Remark.( 12+

3g+q2m

)((3n+2)a−1)−1

(3n+2)a−1−n ≈ 12

and12((3n+2)a−1)− 9

2− 1

g−1

4n≈ 3a

8when a is large.

56

Proof. Suppose π is ramified at P ∈ C. Then pW π : C → pW (C) is also ramified at P .

Recall that pW (C) ⊂ P(W ) is nondegenerate; then we can think of H0(P(W ),OP(W )(1)) as

a subspace of H0(pW (C),OpW (C)(1)). Define

W0 = s ∈ H0(P(W ),OP(W )(1))|π∗pW ∗s vanishes to order ≥ 3 at PW1 = s ∈ H0(P(W ),OP(W )(1))|π∗pW ∗s vanishes to order ≥ 2 at P

and write N0 := dimW0 and N1 := dimW1. Choose a basis

w0, ..., wN0 , wN0+1, ..., wN1 , wN1+1, ..., wN+1

of W3 := H0(P(W ),OP(W )(1)) relative to the filtration 0 ⊂ W0 ⊂ W1 ⊂ W3. Let λ be the

1-PS of GL(W ) whose action is given by

λ(t)wi = wi, t ∈ C∗, 0 ≤ i ≤ N0

λ(t)wi = twi, t ∈ C∗, N0 + 1 ≤ i ≤ N1

λ(t)wi = t3wi, t ∈ C∗, N1 + 1 ≤ i ≤ N + 1

and let λ′ be the associated 1-PS of SL(W ).

Choose m and m sufficiently large. Let Bm be a basis of H0(P(W ),OP(W )(m)) consisting

of monomials of degreem. For 0≤ i ≤ 3m write Ωmi for the subspace ofH0(P(W ),OP(W )(m))

spanned by M ∈ Bm|wλ(M) ≤ i. Let

Ωm,mi := Ωm

i ⊗H0(Pr,OPr(m)) (25)

ˆΩm,m

i := ρm,m(Ωm,mi ) ⊂ H0(C,LmW ⊗ Lmr ) (26)

where ρm,m : H0(P(W ),OP(W )(m)) ⊗ H0(P(r),OP(r)(m)) → H0(C,LmW ⊗ Lmr ) is the map

induced by restriction.

We show that πm,m∗(ˆΩm,m

i ) ⊆ H0(C, LmW ⊗ Lmr ⊗OC((−3m + i)P )) ⊂ H0(C, LmW ⊗ Lmr )

for 0 ≤ i ≤ 3m. When i = 0 this follows from the definitions. For 1 ≤ i ≤ 3m it is enough

to show that for any monomial M ∈ ˆΩm,m

i − ˆΩm,m

i−1 we have

πm,m∗(M) ∈ H0(C, LmW ⊗ Lmr OC((−3m+ i)P )). Suppose M has i0 factors from w0, ..., wN0,i1 factors from wN0+1, ..., wN1 and i3 factors from wN1+1, ..., wN+1. Then i0 + i1 + i3 = m

and i1 + 3i3 = i. By definition πm,m∗(M) ∈ H0(C, LmW ⊗ Lmr ⊗ OC((−3i0 − 2i1)P )). But

3i0 + 2i1 = 3(i0 + i1 + i3)− (i1 + 3i3) = 3m− i so we have

πm,m∗(M) ∈ H0(C, LmW ⊗ Lmr OC((−3m+ i)P )) as required.

By the claim and Riemann-Roch,

βi := dim ˆΩm,m

i ≤ h0(C, LmW ⊗ Lmr ⊗OC((−3m+ i)P )) + dim kerπm,m∗

= em+dm−3m+i−gC+1+h1(C, LmW ⊗ Lmr ⊗OC((−3m+i)P ))+dim kerπm,m∗

57

We have the following three estimates :


Proof. Let IC denote the ideal sheaf of nilpotents in OC . We can choose an integer q2

such that h0(C, IC) < q2. Then we also have dim kerπm,m∗ < q2. To see this, recall that the

normalization morphism factors through Cred. We have π = ι π′, where ι : Cred → C. Then

we have πm,m∗ = π′m,m∗ ιm,m∗. Now, π′m,m∗ is injective, so kerπm,m∗ = ker ιm,m∗. We have

shown that C is generically reduced, so IC has finite support. We have an exact sequence

of sheaves 0 → IC ⊗LmW ⊗Lmr → LmW ⊗Lmr → LmW red⊗Lmrred → 0. This gives rise to an exact

sequence in cohomology. It follows that ker πm,m∗ = ker ιm,m∗ ∼= H0(C, IC ⊗ LmW ⊗ Lmr ) and

hence dim ker πm,m∗ = h0(C, IC ⊗ LmW ⊗ Lmr ) = h0(C, IC) < q2.

II. h1(C, LmW ⊗ Lmr ⊗OC((−3m+ i)P )) ≤ 3m− i if 0 ≤ i ≤ 2g − 2.

Proof. There is a short exact sequence 0 → O(−3m + i)P ) → OC → O(−3m+i)P → 0 of

sheaves on C. Since LmW ⊗ Lmr is an invertible sheaf, it is locally free and in particular flat.

Tensoring, we obtain a second short exact sequence

0 → LmW ⊗ Lmr ⊗OC((−3m+ i)P ) → LmW ⊗ Lmr → O(−3m+i)P → 0

of sheaves on C. In the corresponding long exact sequence in cohomology we have

H1(C, LmW⊗Lmr )=0 sincem and m are large. AlsoH2(C, LmW⊗Lmr ⊗OC((−3m+i)P )) = 0. It

follows that the map H0(C,O(−3m+i)P ) → H1(C, LmW ⊗Lmr ⊗OC(−3m+i)P )) is a surjection.

Therefore h1(C, LmW ⊗ Lmr ⊗ OC(−3m + i)P )) ≤ h0(C,O(−3m+i)P ). But O(−3m+i)P is just a

skyscraper sheaf supported on P and we calculate h0(C,O(−3m+i)P ) = 3m− i.

III. h1(C, LmW ⊗ Lmr ⊗OC((−3m+ i)P )) = 0 if 2g − 1 ≤ i ≤ 3m− 1.

Proof. C is integral and if i ≥ 2g − 1 then deg LmW ⊗ Lmr ⊗ OC((−3m + i)P ) ≥ 2g − 1.

Therefore h1(C, LmW ⊗ Lmr ⊗OC((−3m+ i)P )) = 0.

Combining these inequalities we have

βi ≤q2 + em+ dm− 3m+ i− gC + 1 + 3m− i, 0 ≤ p ≤ 2g − 2q2 + em+ dm− 3m+ i− gC + 1, 2g − 1 ≤ p ≤ m

Now suppose that Hm,m(C) is λ′-semistable. Then there exist monomials

Mi1 , ..., MiP (m)+dmin Bm,m such that ρm,m(Mi1), ..., ρm,m(MiP (m)+dm

) is a basis of

58

H0(C,LmW ⊗ Lmr ) andP (m)+dm∑

j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1.

But

P (m)+dm∑j=1

wλ(Mij) must be larger than3m∑i=1

i(βi − βi−1). We calculate:

3m∑i=1

i(βi − βi−1) = 3mβ3m −3m−1∑i=0

βi

≥ 3m(em+ dm+ 1− g)−3m−1∑i=0

q2 + em+ dm− 3m+ i− gC + 1−2g−2∑i=0

3m− i

=9

2m2 − S7m+ (g − 1)(2g − 1)

≥ 9

2m2 − S7m

where S7 = 9g + 3q2 − 3gC − 92.

Next we show thatN∑i=0

wλ(wi) ≤ 4. Note that the image of W0 under π∗ is contained

in H0(C, LW (−3P )), and the image of W1 under π∗ is contained in H0(C, LW (−2P )). We

have two exact sequences

0 → LW (−P ) → LW → k(P ) → 00 → LW (−3P ) → LW (−2P ) → k(P ) → 0.

which give rise to long exact sequences in cohomology

0 → H0(C, LW (−P )) → H0(C, LW ) → H0(C, k(P )) → · · ·0 → H0(C, LW (−3P )) → H0(C, LW (−2P )) → H0(C, k(P ) → · · ·

The second long exact sequence implies that dimW1/W0 ≤ 1. Recall that LW := π∗(LW )

and π is ramified at P . The ramification index must be at least two, so we have

H0(C, LW (−P )) = H0(C, LW (−2P )). Then the first long exact sequence implies that

dimW3/W1 ≤ 1. ThusN∑i=0

wλ(wi) ≤ 1 + 3 = 4.

By hypothesism > S7(e−g+1) and mm<

12((3n+2)a−1)− 9

2− 1

g−1

4n. Combining these inequalities

leads to a contradiction as follows:

92m2 − S7m

m(em+ dm+ 1− g)≤

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1≤ 4

e− g + 1

59

92− S7

m

e+ dmm

≤ 4

e− g + 1

9

2(e− g + 1)− 4e ≤ S7(e− g + 1)

m+

4dm

m12((3n+ 2)a− 1)− 9

2− 1

g−1

4n≤ m

m.

The contradiction implies that Hm,m(C) is not λ′-semistable, and therefore that Hm,m(C)

is not SL(W )-semistable.

In particular, Lemma 16.7 implies that


m > max

(g − 3

2+ e(q1 + 1) + q3 + µ1m

′′)(e− g + 1),(9g + 3q2 − 3g − 9

2)(e− g + 1)

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

Then if Hm,m(C) is SL(W )-semistable, Cred does not have a cusp singularity.

Proof. Suppose Q ∈ Cred is a cusp. Write π−1(Q) = P . Then π is ramified at P .


m > max

(g − 3

2+ e(q1 + 1) + q3 + µ1m

′′)(e− g + 1),(9g + 3q2 − 3g − 9

2)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1)

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

For all curves C such that Hm,m(C) is SL(W )-semistable, all singular points of Cred are

double points.

Proof. Suppose there exists a point P ∈ C with multiplicity ≥ 3 on Cred. Let

ev : H0(P(W ),OP(W )(1)) → k(P ) be the evaluation map. Let W0 = ker ev. We have

60

N0 := dimW0 = e−g. Choose a basis of W0 and extend it to a basis of H0(P(W ),OP(W )(1)).

Let λ be the 1-PS of GL(W ) whose action is given by

λ(t)wi = wi, t ∈ C∗, 0 ≤ i ≤ e− g − 1λ(t)we−g = twe−g, t ∈ C∗



As in the previous proof, construct a filtration of H0(C,LmW ⊗ Lmr ) as follows: For

0 ≤ p ≤ m let Wm−p0 W p

1 be the subspace of H0(P(W ),OP(W )(m)) spanned by elements of

the following type:


Set Wm−p0 W p

1 = Wm−p0 W p


Wm−p0 W p


1 ) ⊂ H0(C,LmW ⊗ Lmr ) (27)

where ρm,m is the homomorphism induced by restriction (see page 32). Note that a monomial

M ∈ Wm−p0 W p

1 −Wm−p+1

0 W p−11 has weight p. Therefore we have a filtration of

H0(C,LmW ⊗ Lmr ) in order of increasing weight:

0 ⊆ ¯Wm0 W

01 ⊆ Wm−1

0 W 11 ⊆ · · · ⊆ ¯W 0



1 .

Since P is a point of multiplicity ≥ 3 on Cred, we have the following three cases:

1. There is exactly one component of Cred passing through P .

2. There are exactly two components of Cred passing through P , say C1

and C2.

3. There are at least three components of Cred passing through P , say

C1, C2, C3.

Define a divisor D = P1 + P2 + P3 on C as follows: Let π : C → C be the normalization

morphism. The multiplicity of a point P is the sum of the ramification indices at the

preimages of P . The hypotheses of Lemma 16.7 are satisfied. This implies π is unramified,

so in all three cases there are at least three distinct points in π−1(P ). In case 3, choose

61

Pi ∈ π−1(P ) ∩ Ci for i = 1, 2, 3. In case 2, at least one of the components, say C1, satisfies

deg pW (C1) ≥ 3, and P is a singular point of C1. Choose P1, P2 ∈ π−1(P ) ∩ C1 and P3 ∈π−1(P ) ∩ C2. In case 1, choose three distinct points P1, P2, P3 from the fiber π−1(P ).

The normalization morphism induces a homomorphism

πm,m∗ : H0(C,LmW ⊗ Lmr ) → H0(C, LmW ⊗ Lmr ).

Note that πm,m∗(Wm−p0 W p

1 ) ⊆ H0(C, LmW ⊗ Lmr ⊗OC((p−m)D)). We have

βp := dim Wm−p0 W p

1 ≤ h0(C, LmW ⊗ Lmr ⊗OC((p−m)D)) + dim kerπm,m∗

= em+ dm+ 3(p−m)− gC + 1 + h1(C, LmW ⊗ Lmr ⊗OC((p−m)D)) + dim kerπm,m∗

The following estimates may be established by arguments entirely analogous to the proofs

given on page 58:

I. h0(C, IC) < q2.

II. h1(C, LmW ((p−m)D)⊗ Lmr ) ≤ 3(m− p).

III. h1(C, LmW ((p−m)D)⊗ Lmr ) = 0 if p ≥ 2g − 1.

Entering these quantities into the previous formulae, we have

βp ≤q2 + em+ dm+ 3(p−m)− gC + 1 + 3(m− p), 0 ≤ p ≤ 2g − 2q2 + em+ dm+ 3(p−m)− gC + 1, 2g − 1 ≤ p ≤ m



) is a basis of


j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1.

But

P (m)+dm∑j=1



m∑p=1

p(βp − βp−1) = mβm −m−1∑p=0

βp

≥ m(em+dm+1−g)−m−1∑p=0

em+dm+3(p−m)−gC+1+q2−2g−2∑p=0

3(m−p)

=3

2m2 − S9m+ (g − 1)(2g − 1) ≥ 3

2m2 − S9m

62

where S9 = 7g − gC + q2 − 152.


wλ(wi) = 1 since wλ(w0) = 1 and wλ(wi) = 0, i > 0.

By hypothesis m > S9(e− g + 1). Combining these inequalities leads to a contradiction

as follows:

32m2 − S9m

m(em+ dm+ 1− g)≤

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1≤ 1

e− g + 132− S9

m

e+ dmm

≤ 1

e− g + 1

3

2(e− g + 1)− e ≤ S9(e− g + 1)

m+dm

m12e− 3

2(g − 1)− 1

d<

m

m12(3n+ 2)a− 3

2− 1

g−1

n<

m

m(28)

But the last line contradicts the hypothesis that mm<

12((3n+2)a−1)− 9

2− 1

g−1

4n. The contradic-

tion implies that Hm,m(C) is not λ′-semistable, and therefore that Hm,m(C) is not SL(W )-

semistable.

Remark. To contradict (28) it suffices that mm≤

12(3n+2)a− 3

2− 1

g−1

n. The stronger hypothesis

that mm<

12((3n+2)a−1)− 9

2− 1

g−1

4nwas imposed so that Lemma 16.7 could be used on page 61.


m > max

(g − 3

2+ e(q1 + 1) + q3 + µ1m

′′)(e− g + 1),(9g + 3q2 − 3g − 9

2)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

For all curves C such that Hm,m(C) is SL(W )-semistable, Cred does not have a tacnode.

Proof. Suppose Cred has a tacnode at P . Decompose C into its irreducible components.

Note that C must have at least two irreducible components. There exist two components,

63

say C1 and C2, of the normalization C and points Q1 ∈ C1 red and Q2 ∈ C2 red such that

π(Q1) = π(Q2) = P and C1 red and C2 red have a common tangent at P .

Once again we think of H0(P(W ),OP(W )(1)) as a subspace of H0(pW (C),OpW (C)(1)) and

define subspaces

W0 = s ∈ H0(P(W ),OP(W )(1))|π∗pW ∗s vanishes to order ≥ 2 at Q1 and Q2W1 = s ∈ H0(P(W ),OP(W )(1))|π∗pW ∗s vanishes to order ≥ 1 at Q1 and Q2

and write N0 := dimW0 and N1 := dimW1. Choose a basis

w0, ..., wN0 , wN0+1, ..., wN1 , wN1+1, ..., wN+1

of W2 := H0(P(W ),OP(W )(1)) relative to the filtration 0 ⊂ W0 ⊂ W1 ⊂ W2. Let λ be the

1-PS of GL(W ) whose action is given by

λ(t)wi = wi, t ∈ C∗, 0 ≤ i ≤ N0

λ(t)wi = twi, t ∈ C∗, N0 + 1 ≤ i ≤ N1

λ(t)wi = t2wi, t ∈ C∗, N1 + 1 ≤ i ≤ N + 1



As in the previous proof, construct a filtration of H0(C,LmW ⊗ Lmr ) as follows: For all

0 ≤ p ≤ m let Wm−p0 W p

1 be the subspace of H0(P(W ),OP(W )(m)) spanned by elements of

the following type:


Similarly for 0 ≤ p ≤ m let Wm−p1 W p

2 be the subspace of H0(P(W ),OP(W )(m)) spanned by

elements of the following type:


Set Wm−p0 W p

1 = Wm−p0 W p

1 ⊗H0(Pr,OPr(m)) and Wm−p1 W p

2 = Wm−p1 W p

2 ⊗H0(Pr,OPr(m)),

and let

Wm−p0 W p


1 ) ⊂ H0(C,LmW ⊗ Lmr )

Wm−p1 W p


2 ) ⊂ H0(C,LmW ⊗ Lmr ).

64

where ρm,m is the homomorphism induced by restriction (see page 32). Note that a monomial

M ∈ Wm−p0 W p

1 −Wm−p+1

0 W p−11 has weight p and a monomial M ∈ Wm−p

1 W p2 −

Wm−p+11 W p−1

2

has weight m + p. Therefore we have a filtration of H0(C,LmW ⊗ Lmr ) in order of increasing

weight:

0 ⊆ ¯Wm0 W

01 ⊆ Wm−1

0 W 11 ⊆ · · · ⊆ ¯W 0

0 Wm1

= ¯Wm1 W

02 ⊆ Wm−1

1 W 12 ⊆ · · · ⊆ ¯W 0


(29)

Write γ := dim Wm−p0 W p

1 and βp = dim Wm−p1 W p

2 .

We shall divide the proof into the following cases:

1. degC1 redLW ≥ 2 and degC2 red

LW ≥ 2.

2. degC1 redLW = 1 and degC2 red

LW ≥ 2.

3. degC1 redLW ≥ 2 and degC2 red

LW = 1.

Note that these are the only possibilities, since P is a tacnode. Also, it is enough to prove

1. and 2., for then 3. will follow by interchanging the roles of C1 and C2 in 2.

Proof of Case 1. The normalization morphism π : C → C induces a homomorphism

πm,m∗ : H0(C,LmW ⊗ Lmr ) → H0(C, LmW ⊗ Lmr ). By definition,

πm,m∗(Wm−p0 W p

1 ) ⊂ H0(C, LmW ⊗ Lmr ⊗OC((p− 2m)(Q1 +Q2)))


2 ) ⊂ H0(C, LmW ⊗ Lmr ⊗OC((p−m)(Q1 +Q2))).

Then we use Riemann-Roch to calculate

γp := dim Wm−p0 W p

1 ≤ h0(C, LmW ⊗ Lmr ⊗OC((p− 2m)(Q1 +Q2))) + dim kerπm,m∗

= em+ dm+ 2p− 4m− gC + 1

+h1(C, LmW ⊗ Lmr ⊗OC((p− 2m)(Q1 +Q2))) + dim kerπm,m∗.


2 ≤ h0(C, LmW ⊗ Lmr ⊗OC((p−m)(Q1 +Q2))) + dim kerπm,m∗

= em+ dm+ 2p− 2m− gC + 1

+h1(C, LmW ⊗ Lmr ⊗OC((p−m)(Q1 +Q2))) + dim kerπm,m∗.


given on page 58:


65

II. h1(C, LmW ⊗ Lmr ⊗OC((p− 2m)(Q1 +Q2))) ≤ 4m− 2p if 0 ≤ p ≤ 2g− 2.

III. h1(C, LmW ⊗ Lmr ⊗OC((p− 2m)(Q1 +Q2))) = 0 if 2g − 1 ≤ p ≤ m− 1.

IV. h1(C, LmW ⊗ Lmr ⊗OC((p−m)(Q1 +Q2))) = 0 for all 0 ≤ p ≤ m− 1.

We have

γp ≤q2 + em+ dm+ 2p− 4m− gC + 1 + 4m− 2p, 0 ≤ p ≤ 2g − 2q2 + em+ dm+ 2p− 4m− gC + 1, 2g − 1 ≤ p ≤ m− 1

βp ≤ q2 + em+ dm+ 2p− 2m− gC + 1, 0 ≤ p ≤ m− 1.



) is a basis of


j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1.

But

P (m)+dm∑j=1


(m + p)(βp − βp−1) +m∑p=1

p(γp − γp−1). We

calculate:

m∑p=1

(m+ p)(βp − βp−1) +m∑p=1

p(γp − γp−1)

= 2mβm −m−1∑p=0

βp −m−1∑p=0

γp

≥ 2m(em+ dm+ 1− g)−m−1∑p=0

q2 + em+ dm+ 2p− 2m− gC + 1

−m−1∑p=0

q2 + em+ dm+ 2p− 4m− gC + 1−2g−2∑p=0

4m− 2p

= 4m2 − S10m+ (2g − 2)(2g − 1) ≥ 4m2 − S10m

where S10 = 10g − 2gC + 2q2 − 6.


wλ(wi) ≤ 3. To see this note that the image ofW0 under π∗ is contained

in H0(C, LW (−Q1)), and the image of W1 under π∗ is contained in H0(C, LW (−Q1)). We

have two exact sequences

0 → LW (−Q1) → LW → k(Q1) → 00 → LW (−2Q1) → LW (−Q1) → k(Q1) → 0.

which give rise to long exact sequences in cohomology

66

0 → H0(C, LW (−Q1)) → H0(C, LW ) → H0(C, k(Q1)) → · · ·0 → H0(C, LW (−2Q1)) → H0(C, LW (−Q1)) → H0(C, k(Q1) → · · ·

The first long exact sequence implies that dimW2/W1 ≤ 1, and the second long exact

sequence implies that dimW1/W0 ≤ 1. ThusN∑i=0

wλ(wi) ≤ 1 + 2 = 3.

By hypothesis m > S10(e− g+ 1). Combining these inequalities leads to a contradiction

as follows:

4m2 − S10m

m(em+ dm+ 1− g)≤

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1≤ 3

e− g + 1

4− S10

m

e+ dmm

≤ 3

e− g + 1

4(e− g + 1)− 3e ≤ S10(e− g + 1)

m+

3dm

me− 4(g − 1)− 1

3d≤ m

m(3n+ 2)a− 4− 1

g−1

n≤ m

m

But the last line contradicts the hypothesis that

m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

The contradiction implies that Hm,m(C) is not λ′-semistable, and therefore that Hm,m(C) is

not SL(W )-semistable.

Proof of Case 2. Recall that Case 2. means that degC1 redLW = 1 and degC2 red

LW ≥ 2.

We continue to think of H0(P(W ),OP(W )(1)) as a subspace of H0(pW (C),OpW (C)(1)). Let

Y be the closure of C − C1 in C. Now, if s ∈ W0, then π∗pW∗s vanishes on C1red. Therefore


1 ) ⊂ H0(Yred, LmWYred

⊗ LmrYred

⊗OY ((p− 2m)Q2)). Therefore

γp := dim Wm−p0 W p

1 ≤ h0(Yred, LmWYred

⊗ LmrYred⊗OY ((p− 2m)Q2)) + dim kerπm,m∗

= (e− 1)m+ (d− d1)m+ p− 2m− gY + 1

+ h1(Yred, LmWYred

⊗ LmrYred⊗OY ((p− 2m)Q2)) + dim kerπm,m∗.


2 ≤ h0(C, LmW ⊗ Lmr ⊗OY ((p−m)(Q1 +Q2)) + dim kerπm,m∗

67

= em+ dm+ 2p− 2m− gC + 1

+ h1(C, LmW ⊗ Lmr ⊗OY ((p−m)(Q1 +Q2))) + dim kerπm,m∗.


given on page 58:


II. h1(Yred, LmWYred

⊗ LmrYred

⊗OY ((p− 2m)Q2)) ≤ 2m− p if 0 ≤ p ≤ 2g− 2.

III. h1(Yred, LmWYred

⊗ LmrYred

⊗OY ((p− 2m)Q2)) = 0 if 2g − 1 ≤ p ≤ m− 1.

IV. h1(C, LmW ⊗ Lmr ⊗OY ((p−m)(Q1 +Q2))) ≤ 2m− 2p if 0 ≤ p ≤ 2g− 2.

V. h1(C, LmW ⊗ Lmr OY ((p−m)(Q1 +Q2))) = 0 for all 2g− 1 ≤ p ≤ m− 1.

Therefore

γp ≤q2 + (e− 1)m+ (d− d1)m+ p− 2m− gY + 1 + 2m− p, 0 ≤ p ≤ 2g − 2q2 + (e− 1)m+ (d− d1)m+ p− 2m− gY + 1, 2g − 1 ≤ p ≤ m− 1

βp ≤q2 + em+ dm+ 2p− 2m− gC + 1 + 2m− 2p, 0 ≤ p ≤ 2g − 2q2 + em+ dm+ 2p− 2m− gC + 1, 2g − 1 ≤ p ≤ m− 1



) is a basis of


j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1.

But

P (m)+dm∑j=1


(m + p)(βp − βp−1) +m∑p=1

p(γp − γp−1). We

calculate:

m∑p=1

(m+ p)(βp − βp−1) +m∑p=1

p(γp − γp−1)

= 2mβm −m−1∑p=0

βp −m−1∑p=0

γp

≥ 2m(em+ dm+ 1− g)−m−1∑p=0

q2 + em+ dm+ 2p− 2m− gC + 1−2g−2∑p=0

2m− 2p

−m−1∑p=0

q2 + (e− 1)m+ (d− d1)m+ p− 2m− gY + 1−2g−2∑p=0

2m− p

= 72m2 + d1mm− S10′m+ (3g − 3)(2g − 1) ≥ 7

2m2 − S10′m

68

where S10′ = 10g + 2q2 − 112− gC − gY .


wλ(wi) ≤ 3 (cf. proof of case 1, page 66).

By hypothesis m > S10′(e− g+1). Combining these inequalities leads to a contradiction

as follows:

72m2 − S10′m

m(em+ dm+ 1− g)≤

P (m)+dm∑j=1

wλ(Mij)

m(P (m) + dm)≤

N∑i=0

wλ(wi)

e− g + 1≤ 3

e− g + 172− S10′

m

e+ dmm

≤ 3

e− g + 1

7

2(e− g + 1)− 3e ≤ S10′(e− g + 1)

m+

3dm

m12e− 7

2(g − 1)− 1

3d≤ m

m12(3n+ 2)a− 7

2− 1

g−1

3n≤ m

m

But the last line contradicts the hypothesis that mm<

12((3n+2)a−1)− 9

2− 1

g−1

4n. The contradic-

tion implies that Hm,m(C) is not λ′-semistable, and therefore that Hm,m(C) is not SL(W )-

semistable.

16.3 G.I.T. semistable curves are reduced

The next three results show that SL(W )-semistability implies that the curve C is reduced.

We begin with a generalized Clifford’s theorem.

Lemma 16.11 (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) 6= 0 then there is a subcurve C ′ ⊂ C such that

h0(C ′, L) ≤ degC′(L)

2+ 1. (30)

Furthermore C ′ ∼= P1 only if L is trivial on C ′.

Proof. Gieseker proves nearly all of this. It remains only to show that C ′ 6∼= P1 if L is

not trivial on C ′. So suppose that C ′ ∼= P1. Now, every line bundle on P1 is isomorphic to

69

Hm for some m ∈ Z, where H is the hyperplane line bundle. By hypothesis L is generated

by global sections; this implies that m ≥ 0. But h0(C ′, L) = m+ 1 when m ≥ 0. Combining

this with the inequality (30) we have m+ 1 ≤ m2

+ 1 which implies m = 0. Then L is trivial

on C ′.


m > max

(g − 32

+ e(q1 + 1) + q3 + µ1m′′)(e− g + 1),

(9g + 3q2 − 3g − 92)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

If Hm,m(C) is SL(W )-semistable, then H1(Cred, LW red) = 0.

Proof. Since Cred is nodal, it has a dualizing sheaf ω. If H1(Cred, LW red) 6= 0 then by

duality H0(Cred, ω ⊗ L−1W red

) ∼= H1(Cred, LW red) 6= 0. By Proposition 16.5 LW red is not trivial

on any component of Cred. Then by Lemma 16.11 there is a subcurve C ′ 6∼= P1 of Cred for

which e′ > 1 and h0(C,LWC′)− 1 ≤ e′

2. We apply Lemma 16.4 with k = 1 to obtain:

e′ + k2− S

m

e≤

h0 + (dh0−d′(e−g+1))mem

e− g + 1≤

e′

2+ 1 +

(d( e′2

+1+)−d′(e−g+1))m

em

e− g + 1

(e− g + 1)

(e′ +

k

2− S

m

)≤ e

(e′

2+ 1

)+

(d

(e′

2+ 1

)− d′(e− g + 1)

)m

m

e′(e− g + 1− e

2− dm

2m

)≤

(S

m− k

2

)(e− g + 1) + e+ (d− d′(e− g + 1))

m

m

Since mm<

12((3n+2)a−1)− 9

2− 1

g−1

4nthe quantity e− g + 1− e

2− dm

2m> 0 and we proceed:

e′ ≤2e+ 2d m

m+ (e− g + 1)(2S

m− k − 2d′ m

m)

e− 2(g − 1)− d mm

=2(3n+ 2)a+ 2n m

m+ ((3n+ 2)a− 1)(2S

m− k − 2d′ m

m)

(3n+ 2)a− 2− n mm

=((3n+ 2)a− 2− n m

m)(2S

m− k − 2d′ m

m+ 2) + (1 + n m

m)(2S

m− k − 2d′ m

m) + 4n m

m+ 4

(3n+ 2)a− 2− n mm

70

=2S

m− k − 2d′

m

m+ 2 +

(n mm

+ 1)(4 + 2Sm− k − 2d′ m

m)

(3n+ 2)a− 2− n mm

≤ 2− k +2S

m+

(n mm

+ 1)(4− k + 2Sm

)

(3n+ 2)a− 2− n mm

. (31)

Note that k = 1 and e′ ≥ 2. Furthermore the hypothesis

m

m≤

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n=

(3n+ 2)a− 10− 2g−1

8n

implies that

1

(3n+ 2)a− 2− n mm

≤ 1

(3n+ 2)a− 2− 18((3n+ 2)a− 10− 2

g−1)

(3 + 2Sm

)(n mm

+ 1)

(3n+ 2)a− 2− n mm

≤(3 + 2S

m)(1

8((3n+ 2)a− 10− 2

g−1) + 1)

(3n+ 2)a− 2− 18((3n+ 2)a− 10− 2

g−1)

=(3 + 2S

m)((3n+ 2)a− 6

7+ 2

7(g−1)− 64

7− 16

7(g−1)+ 8)

7((3n+ 2)a− 67

+ 2g−1

)

≤3 + 2S

m

7.

Combining these facts with the inequality (31) we obtain

2 ≤ 1 +2S

m+

3

7+

2S

7m.

Since by hypothesis m > 4S this is a contradiction.


m > max

(g − 32

+ e(q1 + 1) + q3 + µ1m′′)(e− g + 1),

(9g + 3q2 − 3g − 92)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

If Hm,m(C) is SL(W )-semistable, then C is reduced.

71

Proof. Let ι : Cred → C be the canonical inclusion. The exact sequence of sheaves on C

0 → IC ⊗ LW → LW → ι∗LW red → 0

gives rise to a long exact sequence in cohomology

· · · → H1(C, IC ⊗ LW ) → H1(C,LW ) → H1(C, ι∗LW red) → 0.

Since C is generically reduced, IC has finite support, hence H1(C, IC ⊗ LW ) = 0. By

the previous lemma, H1(C, ι∗LW red) = H1(Cred, LW red) = 0. The exact sequence implies

H1(C,LW ) = 0 as well. Next, the map

H0(P(W ),OP(W )(1)) → H0(pW (C)red,OpW (C)red(1)) → H0(Cred, LW red)

is injective by Proposition 16.1. Then e− g + 1 = h0(P(W ),OP(W )(1)) ≤ h0(Cred, LW red) =

h0(C,LW )− h0(C, IC ⊗ LW ) = e− g + 1− h0(C, IC ⊗ LW ). Therefore h0(C, IC ⊗ LW ) = 0.

Since IC ⊗ LW has finite support, IC = 0, so C is reduced.

Summary. We have shown that if C is a connected curve such that Hm,m(C) is SL(W )-

semistable, then C is reduced, has at worst nodes as singularities, and H1(C,LW ) = 0.

Next, we improve Lemma 16.4. If C is nodal and Hm,m(C) is SL(W )-semistable, Lemma

16.4 is true even if condition ii. there does not hold.

Amplification 16.14 (cf. [Gies] 1.0.7 and Lemma 16.4 above) Suppose that

m, m > m′′′ and

m > max

(g − 32

+ e(q1 + 1) + q3 + µ1m′′)(e− g + 1),

(9g + 3q2 − 3g − 92)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

Let C be a curve such that Hm,m(C) is SL(W )-semistable; then C has at worst nodes as

singularities. Suppose C has at least two irreducible components. Let C ′ 6= C be a reduced,

complete subcurve of C and let Y be the closure of C − C ′ in C with the reduced structure.

72

Suppose there exist points P1, ..., Pk on Y satisfying π(Pi) ∈ Y ∩C ′ for all 1 ≤ i ≤ k. Write

h0(pW (C ′),OpW (C′)(1)) =: h0. Then

(e′ + k2)

e<h0 + (dh0−d′(e−g+1))m

em

e− g + 1+

S

em,

where S = g + k(2g − 1) + q2 − gY + k2.

Proof. The argument given in [Gies] pages 83-5 works here too.

16.4 Potential stability

We begin with the following corollary to Amplification 16.14:

Corollary 16.15 Suppose that m, m > m′′′ and

m > max

(g − 32

+ e(q1 + 1) + q3 + µ1m′′)(e− g + 1),

(9g + 3q2 − 3g − 92)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

If Hm,m(C) is SL(W )-semistable, and C ′ is a subcurve of C such that g′ = 0 and d′ = 0,

then #C ′ ∩ Y = k ≥ 2. Furthermore if g′ = d′ = 0 and k = 2 then e′ = 1.

Proof. We apply Amplification 16.14 to C ′. Since C is nodal, we have gY = g − k + 1.

Also dY = d− d′ = d− 0 = d.

(eY +k

2)(e− g + 1) ≤ e(eY − gY + 1) + (d(eY − gY + 1)− dY (e− g + 1))

m

m+

S

em

0 ≤ (eY − e+k

2)(g − 1) + d

m

m(eY − e+ k − 1) +

S

em(32)

Since Hm,m(C) is SL(W )-semistable, by Proposition 16.5 e′ ≥ 1. Thus eY − e ≤ −1, so

if k = 1, then the terms (eY − e+ k2)(g− 1) and d m

m(eY − e+ k− 1) on the right hand side of

equation (32) are negative. In fact (eY − e+ k2)(g − 1) + d m

m(eY − e+ k − 1) ≤ −1

2whereas

the hypothesis on m imply that Sem

< 12

so we obtain a contradiction.

73

If k = 2, then eY − e+ 1 ≤ 0, so again we obtain a contradiction unless eY = e− 1 and

e′ = 1.

The corollary implies in particular that if C is SL(W )-semistable and C ′ is a subcurve

which is a chain of rational components and satisfies d′ = 0 and k = 2, then C ′ ∼= P1. For

if C ′ were irreducible, then we must have ei > 0 for each irreducible component of C ′ by

Proposition 16.5. But we have 1 = e′ =∑Ci⊂C′

ei so C ′ must consist of exactly one irreducible

component.

We recall terminology and results of Gieseker, Harris, Morrison, and Caporaso and review

Corollary 16.15 in light of them.

Definition 16.16 ([HM] p.224) A connected curve C of genus g and degree e in P(W )

where dimW = e−g+1 is potentially stable if the embedded curve C is nondegenerate, the

abstract curve C is Deligne-Mumford semistable, the linear series embedding C is complete

and nonspecial (i.e. h0(C,OC(1)) = e− g and h1(C,OC(1)) = 0) and any complete subcurve

Y ⊂ C satisfies the inequality

eY + k2

e≤ h0(C,OC(1))

e− g + 1. (33)

Note that while Deligne-Mumford (semi)stability is a property of an abstract curve,

potential stability is a property of an embedded curve.

It follows from the inequality (33) that if Y is a subcurve and gY = 0 then k ≥ 2.

Furthermore, if gY = 0 and k = 2 then eY = 1; this means that any such destabilizing

components are embedded as lines and occur as chains of length at most one. Caporaso

[Cap] studies their behavior under the G.I.T. quotient map of Gieseker’s construction.

In the title of this section I have coined the term “potentially stable map.” The results

of this section suggest what the definition (given precisely below) ought to be: a potentially

stable map should be one whose domain is a Deligne-Mumford semistable curve whose desta-

bilizing components are embedded as lines in P(W ) and occur as chains of length at most

one. This terminology is not standard.

We summarize the results of this section with the following definition and theorem:

Definition 16.17 Let h ∈ Hilb(P(W )×Pr). The map Chfh→ Pr is potentially stable if the

following conditions are satisfied:

74

i. Ch is a reduced, connected, nodal curve

ii. the map Ch → P(W ) induces an injective map

H0(P(W ),OP(W )(1)) → H0(Cred, LW red)

iii. h1(C,LW ) = 0

iv. any complete subcurve C ′ ⊂ C with C ′ 6= C satisfies the inequality

(e′ + k2)

e<h0 + (dh0−d′(e−g+1))m

em

e− g + 1+

S

em

of Amplification 16.14.

Theorem 16.18 Suppose that m, m > m′′′ and

m > max

(g − 32

+ e(q1 + 1) + q3 + µ1m′′)(e− g + 1),

(9g + 3q2 − 3g − 92)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

For all h ∈ Hilb(P(W )×Pr)ss the map Chfh→ Pr is potentially stable.

17 The Construction Finished

In the previous section we studied Hilb(P(W )× Pr)ss. In this section we focus on Jss.

Recall the definitions of U and J from Section 13: U ⊂ Hilb(P(W )× Pr) is the open set

such that for each h ∈ U ,

i. Ch is a connected nodal reduced curve.

ii. The projection map Ch → P(W ) is a non-degenerate embedding.

iii. The multidegree of (OP(W )(1) ⊗ OPr(1))|Chequals the multidegree of

(ω⊗aCh⊗OPr(3a+ 1))|Ch

.

and J is the closed subscheme of U where the sheaves in line iii. above are isomorphic.

75


m > max

(g − 32

+ e(q1 + 1) + q3 + µ1m′′)(e− g + 1),

(9g + 3q2 − 3g − 92)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

and

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<m

m<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

For all h ∈ Jss the map Chfh→ Pr is stable.

Proof. By Theorem 16.18 in the previous section, under these hypotheses, for all h ∈Hilb(P(W )× Pr)ss, the map Ch

fh→ Pr is potentially stable. Suppose that h ∈ Jss and that

fh is a potentially stable map but not a stable map. Then there is at least one irreducible

component C ′ of Ch for which d′ = g′ = 0 and k ≤ 2. Then

(OP(W )(1)⊗OPr(1))|Ch∼= (ω⊗aCh

⊗OPr(3a+ 1))|Ch

since h ∈ J . But

degC′(OP(W )(1)⊗OPr(1))|Ch= 1 + 0 = 1

while

degC′ ω⊗aCh⊗OPr(3a+ 1))|Ch

= 2 · 0− 2 + k + 0 ≤ 0.

The contradiction implies that fh is a stable map.

Now we shall construct Mg(Pr, d). We will assume m

m= 3a

2a−1because the proof is most

transparent for this choice. Note that

(12

+ 3g+q2m

)((3n+ 2)a− 1)− 1

(3n+ 2)a− 1− n<

3a

2a− 1<

12((3n+ 2)a− 1)− 9

2− 1

g−1

4n.

so Theorem 16.18 and Proposition 17.1 apply. After completing the construction with this

value of mm

we shall make some remarks on extending the range of mm

.

76


m > max

(g − 32

+ e(q1 + 1) + q3 + µ1m′′)(e− g + 1),

(9g + 3q2 − 3g − 92)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

and m

m= 3a

2a−1. Then Jss is closed in Hilb(P(W )×Pr)ss.

Proof. We argue very similarly to [HM] Proposition 4.55. As noted above J is a closed

subscheme of an open set U ⊂ Hilb(P(W )×Pr) so Jss is a locally closed subset of Hilb(P(W )×Pr)ss. We will use the valuative criterion of properness to show that Jss is proper hence

closed in Hilb(P(W )×Pr)ss. Let R be a discrete valuation ring, let η ∈ SpecR be the generic

point, and let α : SpecR → Hilb(P(W )× Pr)ss be a morphism such that α(η) ⊂ Jss. Then

we will show that α(0) ∈ Jss.Define a family D of curves in P(W )×Pr by the following pullback diagram:

D → C↓ ↓

SpecR → Hilb(P(W )×Pr)ss

By the definition of J we have

(OP(W )(1)⊗OPr(1))|Dη∼= (ω⊗a

D/SpecR ⊗OPr(3a+ 1))|Dη .

We will write D0 =: C. Decompose D0 = C =⋃Ci into its irreducible components.

Then we can write

(OPW (1)⊗OPr(1))|D ∼= ω⊗aD/SpecR ⊗OPr(3a+ 1))⊗OD(−

∑aiCi) (34)

where the ai are integers. OD(−C) ∼= OD so we can normalize the integers ai so that they

are all nonnegative and at least one of them is zero. Divide C into two subcurves Y =⋃ai=0

Ci

and C ′ =⋃ai>0

Ci. Since at least one of the ai is zero, we have Y 6= ∅ and C ′ 6= C. Suppose for

purposes of contradiction that C ′ 6= ∅ (hence Y 6= C); then the hypotheses of Amplification

16.14 are satisfied for C ′ and k.

Any local equation for the divisor OD(−∑aiCi) must vanish identically on every com-

ponent of C ′ and on no component of Y . Such an equation is zero therefore at each of the

77

k nodes in Y ∩ C ′. Thus we obtain the inequality

k ≤ degY (OD(−∑

aiCi)

= degY (OP(W )(1)⊗ ω⊗−a ⊗OPr(−3a))

= degY (OP(W )(1)|D0 ⊗ (ω⊗−a ⊗OPr(−3a))|D0

= degY (OP(W )(1)|D0 − a degY (ω ⊗OPr(−3))|D0

= eY − a(2gY − 2 + k)− a(3dY ) (35)

Substituting e′ = e− eY , d′ = d− dY , g′ = g− gY − k+ 1 and e = a(2g− 2 + 3d), inequality

(35) is equivalent to

e′ − 3ad′ − 2a(g′ − 1) ≤ ak − k. (36)

We show below that Amplification 16.14 applied to C ′ yields the inequality

eY − a(2gY − 2 + k)− a(3dY ) < k.

Together with line (35) this implies k < k. The contradiction implies that C ′ = ∅ and

Y = C. This implies that all the coefficients ai are zero and α(0) ∈ Jss, and the proof is

then complete.

To reiterate, we want to show that

eY − a(2gY − 2 + k)− a(3dY ) < k

⇐⇒ eY − a(2gY − 2)− a(3dY ) < k(a+ 1)

Substituting e′ = e− eY , d′ = d− dY , g′ = g − gY − k + 1 and e = a(2g − 2 + 3d), we get

⇐⇒ (e− e′)− a(2(g − g′ − k + 1)− 2)− a(3(d− d′)) < k(a+ 1)

⇐⇒ e− 2a(g − 1)− 3ad− e′ + 2a(g′ − 1) + 3ad′ + 2ak < k(a+ 1)

⇐⇒ ak − k < e′ − 2a(g′ − 1)− 3ad′

⇐⇒ k

2<

e′

2a− (g′ − 1)− 3

2d′ +

k

2a(37)

We apply Amplification 16.14 to C ′ to establish inequality (37).

e′ + k2

e<

e′ − g′ + 1 + (d(e′−g′+1)−d′(e−g+1))mem

e− g + 1+

S

emk

2(e− g + 1) ≤ e′(

dm

m+ g − 1) + (e+

dm

m)(1− g′)− d′(e− g + 1)

m

m+S

m

78

k

2≤

e′(dmm

+ g − 1)

e− g + 1+ 1− g′ +

(d′mm

+ g − 1)(1− g′)

e− g + 1− d′m

m+

S

(e− g + 1)m

k

2≤

e′(1 + nmm

)

(3n+ 2)a− 1+ 1− g′ +

(nmm

+ 1)(1− g′)

(3n+ 2)a− 1− d′m

m+

S

(e− g + 1)m

k

2≤

(e′ − g′ + 1)(1 + nmm

)

(3n+ 2)a− 1+ 1− g′ − d′m

m+

S

(e− g + 1)m

k

2≤ e′ − g′ + 1

3a

m

m(

1 + mnm

1 + 23n− 1

3na

)− d′m

m+ 1− g′ +

S

(e− g + 1)m(38)

We showed in line (36) that e′ − 3ad′ − 2a(g′ − 1) ≤ ak − k. This implies that

e′ − 3ad′ − 2a(g′ − 1) +S

(e− g + 1)m≤ ak − k +

S

(e− g + 1)m.

The hypotheses on m imply that S(e−g+1)m

< 1 so ak−k+ S(e−g+1)m

< ak−k+1 < (2a−1)k.

Thus

e′ − 3ad′ − 2a(g′ − 1) +S

(e− g + 1)m< (2a− 1)k

e′ − g′ + 1

3a

3a

2a− 1

(1 + 2a−1

3an

1 + 23n− 1

3an

)− d′

3a

2a− 1+

S

(e− g + 1)m<

e′

2a− 3d′

2+

k

2a. (39)

Then if mm

= 3a2a−1

the inequalities (38) and (39) imply that

k

2≤ e′ − g′ + 1

3a

m

m

(1 + m

nm

1 + 23n− 1

3na

)− d′m

m+ 1− g′ +

S

(e− g + 1)m

=e′ − g′ + 1

3a

3a

2a− 1

(1 + 2a−1

3an

1 + 23n− 1

3na

)− 3ad′

2a− 1+ 1− g′ +

S

(e− g + 1)m

<e′

2a− 3d′

2+

k

2a+ 1− g′.

This is what we needed to show, for as remarked earlier substituting e′ = e− eY ,

d′ = d− dY , g′ = g − gY − k + 1 and e = a(2g − 2 + 3d) this implies

eY − 3adY − a(2gY − 2 + k) < k

which together with line (35) yields the contradiction k < k. The contradiction implies that

we cannot decompose C into two strictly smaller subcurves C ′ and Y as described. Thus all

the coefficients ai must be zero, and we have an isomorphism

(OP(W )(1)⊗OPr(1))|D ∼= (ω⊗aD/SpecR ⊗OPr(3a+ 1))|D.

79

In particular, D0 ∈ Jss, so Jss is closed in Hilb(P(W )×Pr)ss.

Corollary 17.3 Suppose that m, m > m′′′ and

m > max

(g − 32

+ e(q1 + 1) + q3 + µ1m′′)(e− g + 1),

(9g + 3q2 − 3g − 92)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

and m

m= 3a

2a−1. Then J//SL(W ) = J//SL(W ). In particular the quotient J//SL(W ) is

projective.

Proof. By Proposition 17.2, Jss = J ∩Hilb(P(W )×Pr)ss is closed in Hilb(P(W )×Pr)ss,

so J ∩ Hilb(P(W )× Pr)ss = J ∩ Hilb(P(W )× Pr)ss. Then J//SL(W ) and J//SL(W ) are

canonically isomorphic since they both denote the categorical quotient of the same set.

Finally, we know that Hilb(P(W )×Pr) is projective. Then J is projective since it is a closed

subscheme of the projective scheme Hilb(P(W )× Pr). The G.I.T. quotient of a projective

scheme is projective, so J//SL(W ) ∼= J//SL(W ) is projective.

We have constructed a projective quotient J//SL(W ). We now want to relate this quo-

tient to the Kontsevich-Manin moduli space of maps.

Theorem 17.4 Suppose that m, m > m′′′ and

m > max

(p+ 1)n(g − 3

2+ e(q1 + 1) + q3 + µ1m

′′)(e− g + 1),(9g + 3q2 − 3g − 9

2)(e− g + 1),

(7g − g + q2 − 152)(e− g + 1),

10g − 2g + 2q2 − 6,10g + 2q2 − 11

2− g

12g + 4q2 − 4g

where this p and n are those defined on page 41. If m

m= 3a

2a−1, then J//SL(W ) ∼= Mg(P

r, d).

Proof. The hypotheses on m, m, a, and mm

ensure that all previous results in Part III of

this paper hold. Fulton and Pandharipande have shown that the quotient of J by SL(W )

is Mg(Pr, d). Graber and Pandharipande note that the action of PGL(W ) on J is proper

80

([GP] Appendix A). Then the SL(W ) action is proper as well. Therefore j : J →Mg(Pr, d)

is a closed morphism. Jss is an open subset of J , so J \Jss is closed in J . Therefore j(J \Jss)is closed in Mg(P

r, d). But Jss is also SL(W )-invariant, so j(Jss)∩j(J \Jss) = ∅. Therefore

j(J) \ j(Jss) = j(J) \ j(Jss) is closed in Mg(Pr, d). But J

j→ Mg(Pr, d) is surjective, so

j(J)\j(Jss) = Mg(Pr, d)\j(Jss) is closed, and we conclude that j(Jss) is open inMg(P

r, d).

The property of being a categorical quotient is local, i.e. j|Jss : Jss → j(Jss) is a

categorical quotient of Jss for the SL(W ) action. The categorical quotient is unique up to

isomorphism, so J//SL(W ) ∼= j(Jss). We have that J//SL(W ) is projective so j(Jss) is also

projective, hence closed. Then j(Jss) is open and closed, and by Theorem 15.2 it is nonempty,

so it must be a connected component ofMg(Pr, d). But Pandharipande and Kim have shown

(main theorem, [KP]) that Mg(Pr, d) is connected, so we have J//SL(W ) ∼= Mg(P

r, d).

Applying the previous corollary we have J//SL(W ) ∼= Mg(Pr, d)

Remark. In Gieseker’s construction of Mg, the first three steps are analogous to ours:

he shows that nonsingular curves have G.I.T. stable Hilbert points, that G.I.T. semistable

Hilbert points correspond to potentially stable curves, and that Kss is closed in Hilb(P(W )).

The end of his argument however is different from the proof of Theorem 17.4 above. Gieseker

uses a deformation argument, the projectivity of the quotient, and semistable reduction for

curves to show that Deligne-Mumford stable curves also have SL(W )-semistable Hilbert

points.

That approach is more complicated for stable maps (not all stable maps can be smoothed),

so we gave an alternative argument. One undesirable feature of our proof of Theorem 17.4

is that our argument depends on the cited results of Alexeev, Fulton, Graber, Kim, and

Pandharipande that a projective scheme Mg(Pr, d) exists, is a coarse moduli space for this

moduli problem, is connected, and is the quotient of a proper action J →Mg(Pr, d); in the

literature these facts are only presented over C. It seems that Alexeev proves the existence of

Mg(Pr, d) as an algebraic space in characteristic 0; he notes ([Alex] 5.7) all the places where

he uses this hypothesis, and is optimistic that it could be removed. The papers [FP], [GP],

and [KP] present their results only over C, and it is not always obvious how this hypothesis

is used.

We note that the G.I.T. construction of J//SL(W ) presented above should work over

any base. However our argument comparing the quotient J//SL(W ) to Mg(Pr, d) relies on

81

results which have only been published over C. So while Gieseker constructs Mg over Z,

we can only claim to have constructed the moduli space of maps over C. This may not be

sharp.

Extending the Range of mm

To prove Theorem 17.2, Corollary 17.3, and Theorem 17.4 we fixed the value mm

= 3a2a−1

.

Recall that this was used in the proof to establish the inequality

e′ − g′ + 1

3a

m

m

(1 + m

nm

1 + 23n− 1

3na

)− d′m

m<

e′

2a− 3d′

2+

k

2a(40)

of page 79. So we now ask: can we find a wider range of mm

for which (40) is verified so that

the proof of Theorem 17.2 goes through as written?

The inequality (40) is equivalent to

m

m

((e′ − g′ + 1)n

3an+ 2a− 1− d′

)≤ e′

2a− 3d′

2+

k

2a− 3a

3an+ 2a− 1. (41)

It is tempting to take

m

m≤

e′

2a− 3d′

2+ k

2a− 3a

3an+2a−1

(e′−g′+1)n3an+2a−1

− d′(42)

but this is not justified, because it is not clear whether each side of the inequality (41) is

positive or negative. Furthermore, even assuming that both sides of (41) are positive and

that (42) holds, it is not clear what the minimum over all appropriate e′, g′, d′ of the quantity

e′

2a− 3d′

2+ k

2a− 3a

3an+2a−1

(e′−g′+1)n3an+2a−1

− d′

will be. Hence for the time being we are left with mm

= 3a2a−1

as the only linearization for

which we claim the quotient J//SL(W ) ∼= Mg(Pr, d).

18 Another Linearization

We mentioned (page 36) that G.I.T quotients depend on a choice of linearization. In

our construction of Mg(Pr, d) we continually refined our choice of m

m; recall that this value

determines the linearization used. The following argument suggests that for a different range

of values of mm

, namely whenm >> m, then stable maps whose domains are Deligne-Mumford

stable curves ought to have SL(W )-semistable Hilbert points.

82

The following argument relates SL(W )-semistability in Gieseker’s construction and

SL(W )-semistability for our construction. A useful hint in keeping track of the two notations

is that in our construction we often use the same letters Gieseker uses but add a hat.

Recall Gieseker’s definition of the Hilbert point Hm(C) (see page 25 or [Gies] pages 6-9)

and recall our definition of the Hilbert point Hm,m(C) from page 32: Let h ∈ Hilb(P(W )×Pr).

The Hilbert polynomial of Ch ⊂ P(W )×Pr is em+ dm+ 1− g. We define Hm,m(Ch), the

(m, m)−th Hilbert point of Ch, as follows:

Definition 18.1 If m and m are sufficiently large, then H1(Ch,OP(W )(m)⊗OPr(m)|Ch) = 0

and the restriction map

ρm,m : H0(P(W )×Pr,OP(W )(m)⊗OPr(m)) → H0(Ch,OP(W )(m)⊗OPr(m)|Ch)

is surjective.

Then∧P (m)+dm ρm,m is a point of P(

∧P (m)+dmH0(P(W )× Pr,OP(W )(m) ⊗ OPr(m))).

We set Hm,m(Ch) :=∧P (m)+dm ρm,m for all m > m0.

Recall that there is an induced SL(W ) action on

P(∧P (m)+dmH0(P(W )×Pr,OP(W )(m)⊗OPr(m)). We will make this more explicit than in

Section 14. If w0, ..., wN is a basis of H0(P(W ),OP(W )(1)) then the SL(W ) action can be

described as follows: Let (aij) be a matrix representing g ∈ SL(W ). Then g acts by the rule

g.wp =∑N

j=0 apjwj.

Write wγ0,γ1,...,γN

0,1,...,N := wγ00 wγ11 · · ·wγN

N where γ0 + · · · + γN = m. Let Bm be the resulting

monomial basis of H0(P(W ),OP(W )(m)). The SL(W ) action is as follows:

g.wγ0,γ1,...,γN

0,1,...,N . = (g.w0)γ0 · · · (g.wN)γN .

Pick a basis f0, ..., fr of Cr+1. Set fΓ0,Γ1,...,Γr

0,1,...,r := fΓ00 fΓ1

1 · · · fΓrr where Γ0+· · ·+Γr = m. Let

Bm be the resulting monomial basis of H0(Pr,OPr(m)). Tensor Bm with Bm to get a basis

Bm,m of H0(P(W ),OP(W )(m))⊗H0(Pr,OPr(m)) ∼= H0(P(W )×Pr,OP(W )(m)⊗OPr(m))

consisting of monomials having bidegree (m, m). The SL(W ) action may be written

g.wγ0,γ1,...,γN

0,1,...,N fΓ0,Γ1,...,Γr

0,1,...,r = (g.w0)γ0 · · · (g.wN)γNfΓ0

0 · · · fΓrr .

A basis for∧P (m)+dmH0(P(W )× Pr,OP(W )(m) ⊗ OPr(m)) is given by elements of the

form

wγ0,1,...,γN,1

0,...,N fΓ0,1,...,Γr,1

0,...,r ∧ · · · ∧ wγ0,P (m)+dm,...,γN,P (m)+dm

0,...,N fΓ0,P (m)+dm,...,Γr,P (m)+dm

0,...,r (43)

83

such that for all j, γ0,j + · · · γN,j = m and Γ0,j + · · ·+ Γr,j = m and such that

(γ0,j, . . . , γN,j,Γ0,j, . . . ,Γr,j) < (γ0,j+1, . . . , γN,j+1,Γ0,j+1, . . . ,Γr,j+1) with respect to lexico-

graphical order. The basis may in turn be ordered lexicographically.

Fix a maximal torus of SL(W ). Let α0, . . . , αN be the standard weights for the action

on W with respect to the basis w0, . . . , wN . Then the weight of the element given in line

(43) above is P (m)+dm∑j=1

γ0,j

α0 + · · ·+

P (m)+dm∑j=1

γN,j

αN (44)

Let w0, . . . , wN be the images of w0, . . . , wN under the map

H0(P(W ),OP(W )(1)) → H0(pW (C),OpW (C)(1)) → H0(C,LW ).

Choose a section s ∈ H0(C,Lr) which does not vanish identically on any component of

C. Then the map

H0(C,LmW ) → H0(C,LmW ⊗ Lmr )

σ 7→ σ ⊗ sm

is injective. Let S = H0(C,LmW )⊗ sm ⊂ H0(C,LmW ⊗ Lmr ).

Complete S to a basis of H0(C,LmW ⊗ Lmr ) by choosing a set of vectors

S = w ˜γ0,j ,...,γN,j

0,...,N f˜Γ0,j ,...,Γr,j

0,...,r |1 ≤ j ≤ dm

so that S ∪ S is a basis of H0(C,LmW ⊗ Lmr ). Now if Hm(C) := ∧P (m)ϕm(C) (see page 25

and [Gies] pages 6-9) has a nonzero coefficient corresponding to weight (

P (m)∑j=1

γ0,j)α0 + · · ·+

(

P (m)∑j=1

γN,j)αN there is a basis of H0(P(W ),OP(W )(1)) which induces a basis of H0(C,LmW )

of the form w ˜γ0,j ,...,γN,j

0,...,N |1 ≤ j ≤ P (m). Then, writing S with this basis, we obtain a basis

S ∪ S of H0(C,LmW ⊗ Lmr ) which may be written

w ˜γ0,j ,...,γN,j

0,...,N |1 ≤ j ≤ P (m) ∪ w ˜γ0,j ,...,γN,j

0,...,N f˜Γ0,j ,...,Γr,j

0,...,r |1 ≤ j ≤ dm.

That this is a basis implies that Hm,m(C) has a nonzero coefficient corresponding to

weight (

P (m)∑j=1

γ0,j)α0 + · · ·+ (

P (m)∑j=1

γN,j)αN + (dm∑j=1

γ0,j)α0 + · · ·+ (dm∑j=1

γN,j)αN .

84

Let CH denote the convex hull of the set of weights such that the corresponding coefficient

of Hm(C) is nonzero, and let CH denote the convex hull of the set of weights such that the

corresponding coefficient of Hm,m(C) is nonzero. It is well-known in geometric invariant

theory that if Hm(C) is SL(W )-stable then in every coordinate system on W , 0 lies in the

interior of the convex hull CH.

By the argument above, the set of weights such that the coefficient of Hm,m(C) is nonzero

contains the set

(

P (m)∑j=1

γ0,j)α0 + · · ·+ (

P (m)∑j=1

γN,j)αN + (dm∑j=1

γ0,j)α0 + · · ·+ (dm∑j=1

γN,j)αN (45)

Since the γi,j and the γi,j are nonnegative integers and∑N

i=1 γi,j = m for all 1 ≤ j ≤ P (m)

while∑N

i=1 γi,j = m for all 1 ≤ j ≤ dm we find that if m >> m then 1mP (m)

CH is a

small perturbation of 1mP (m)

CH. If 0 lies in the interior of the convex hull CH, and if the

perturbation is sufficiently small, then 0 will lie in the interior of the convex hull CH as well.

Deligne-Mumford stable curves have SL(W )-stable Hilbert points, so this suggests that if

m >> m and C is a Deligne-Mumford stable curve then Hm,m(C) is SL(W )-stable. Then

for such linearizations J//SL(W ) 6= ∅.However, if m >> m, the inequality

(e′ + k2)

e<h0 + (dh0−d′(e−g+1))m

em

e− g + 1,

fails for all maps whose domains are not potentially stable curves in the sense of Definition

16.16. This suggests that for such linearizations the quotient J//SL(W ) 6∼= Mg(Pr, d).

Thaddeus [Th1] exploits the existence of different linearizations with nonisomorphic quo-

tients to study H∗(M(2, d)), the cohomology ring of the moduli space of rank 2 vector

bundles over a (fixed) curve, yielding a proof of the Verlinde formula. If there are indeed

different quotients J//LSL(W ) then perhaps similar ideas could be applied to them, yielding

results on the cohomology of Mg and Mg(Pr, d).

19 Closing: Toward Mg, U g(n, d), and Beyond

One of the primary motivations for constructing Mg(Pr, d) as a G.I.T. quotient was to

help us study other moduli spaces. We turn our thoughts in this direction before closing.

Let Mg(Pr, d) be the moduli space of isomorphism classes of morphisms C

f→ Pr

from nonsingular curves C of genus g to Pr such that f∗([C]) = d. There is an inclusion

85

Mg(Pr, d) →Mg(P

r, d) into the Kontsevich-Manin moduli space of isomorphism classes of

stable maps. Define H ⊂Mg(Pr, d) as the locus of nondegenerate 10-canonical embedding

maps. Identify H with its image under inclusion into Mg(Pr, d). Let H be the closure in

Mg(Pr, d) of H. As stated in Section 10 we would like to study H and H.

Let Jj→Mg(P

r, d) be the quotient map as before. It is natural to wonder whether we

can compare the closed subscheme V (Fittg−1R1ϕ∗(ω

⊗10C ⊗ ((OPr(1)|C)−1) of J to j−1(H) or

j−1(H) or else its image j(V (Fittg−1R1ϕ∗(ω

⊗10C ⊗ ((OPr(1)|C)−1)) to H or H.

The proof of Proposition 12.3 shows that

V (Fittg−1R1ϕ∗(ω

⊗10C ⊗ ((OPr(1)|C)−1) ⊃ h ∈ J |ω10

Ch

∼= OPr(1)|Ch,

which allows us to conclude that every 10-canonical stable map C → Pr with C nonsingular

has a model in V (Fittg−1R1ϕ∗(ω

⊗10C ⊗ ((OPr(1)|C)−1), and by Theorem 15.2 such points

are SL(W )-stable so their behavior under the quotient map is straightforward. Thus we can

conclude that j(V (Fittg−1R1ϕ∗(ω

⊗10C ⊗((OPr(1)|C)−1)) is a closed set ofMg(P

r, d) containing

H.

A word of caution: to use [FP] Proposition 1 to show

V (Fittg−1R1ϕ∗(ω

⊗10C ⊗ ((OPr(1)|C)−1) = h ∈ J |ω10

Ch

∼= OPr(1)|Ch (46)

we must have that the multidegrees of ω10Ch

and OPr(1)|Chare equal for all h ∈ J . But this

multidegree condition is only satisfied on an open set of J . So the stronger statement (46)

may not be true. So we have produced a closed set of Mg(Pr, d) containing H though we

do not have as natural a description of it as we might desire.

In closing we would like to remark that it may be possible to find a locus inMg(G(n, p), β)

whose G.I.T. quotient is U g(n, d). It may also be possible to construct a moduli space

“U g(X, β)(n, d)” of triples (C,E, f) where C is a prestable curve of genus g, E is a slope

semistable torsion free sheaf on C, and f : C → X is a stable map to a projective scheme

X. Indeed Pandharipande constructs U g(n, d) as a G.I.T. quotient of a Quot scheme over

K; perhaps a similar construction beginning with J would yield U g(X, β)(n, d).

86

20 Appendix I: Equivariant Cohomology

Definition 20.1 Let X be a topological G-space, and fix a universal classifying bundle

EG → BG. The G-equivariant cohomology of X, denoted H∗G(X), is the ordinary coho-

mology H∗(EG×GX). Here EG×GX is by convention the quotient by the diagonal action

of G acting on EG on the right and on X on the left.

The G-bundle EG → BG is unique up to homotopy, so equivariant cohomology is well-

defined. Note also that H∗G(X) is a module over H∗(BG). Furthermore, if G is compact

and the action of G on X is free (the stabilizer at every point is trivial) or locally free (the

stabilizer of the action at every point is discrete) then H∗G(X) ∼= H∗(X/G).

EG×G X is a “better” homotopy-theoretic quotient than X/G. One result we shall use

is that if E → B is an equivariant vector bundle of rank q + 1 and E denotes the sphere

bundle inside E, there is an equivariant Thom-Gysin sequence

· · · → HrG(B) → Hr

G(E) → Hr−qG (B)

e(E)→ Hr+1G (B) → · · ·

There is a de Rham version of equivariant cohomology which is very useful for compu-

tations. (In fact there are two models for this theory, the Cartan and Weil models, which

give rise to isomorphic equivariant cohomology groups but which are defined via distinct

complexes. We describe the Cartan model.) Let G be a Lie group acting smoothly on a

smooth manifold M . Let g be the Lie algebra of G, and g∗ its dual. Let Ω(M) denote the

de Rham complex on M . We form the complex of G-equivariant polynomial maps on the

Lie algebra g taking values in the de Rham complex:

A := (S(g∗)⊗ Ω(M))G

For each X ∈ g, let XM denote the vector field given by the infinitesimal action of X on M .

Let d be exterior differentiation, and let i(v) denote contraction by a vector field v. Then

we can define a differential operator D : A→ A by the rule:

(Dω)(X) = d(ω(X))− i(XM)(ω(X)).

Another useful tool is the abelian localization theorem for the equivariant cohomology of

torus actions:

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Theorem 20.2 Let T be a torus acting on a manifold M , let F index the components F of

the fixed point set MT , and let eF be the equivariant euler class of the normal bundle to F

in M . Let η ∈ H∗T (M). Then ∫

M

η(X) =∑F∈F

∫F

η(X)

eF (X).

See ([AB2, BGV]) for more details.

88

21 Appendix II: Intersection Cohomology

Several powerful theorems on the cohomology of nonsingular varieties fail for singular

varieties. These include Hodge decomposition, Poincare duality, the Lefschetz hyperplane

theorem, the Hard Lefschetz theorem, and the Hodge signature theorem.

A cohomology theory called intersection cohomology has been developed by Goresky and

MacPherson which is isomorphic to ordinary cohomology for nonsingular varieties but which

satisfies (or is conjectured to satisfy) the five theorems listed above for singular varieties.

See [Kir6] for an introduction.

Roughly speaking intersection homology can be constructed from the simplicial chain

complex by leaving out those chains which intersect the singular set in sets of too great

dimension.

Specifically, suppose the space X is given a Whitney stratification X = Xn ⊇ Xn−1 ⊇· · · ⊇ X0 and a triangulation compatible with the Whitney stratification. Let ξ be an i-chain

in the simplicial chain complex C∗(X) and let |ξ| denote the support of ξ on X with respect

to the triangulation. We call ξ admissible with respect to the middle perversity if it satisfies

the following two conditions:

dimR(|ξ| ∩Xn−k) ≤ i− k − 1

dimR(|∂ξ| ∩Xn−k) ≤ i− k − 2.

In each degree i let ICi(X) ⊂ Ci(X) denote the subgroup of admissible chains . The

usual boundary map ∂ : Ci(X) → Ci−1(X) induces a boundary map ICi(X) → ICi−1(X),

making IC∗(X) a sub-chain complex of C∗(X). The ith intersection homology group of X

with respect to the middle perversity is defined to be

IHi(X) =ker ∂ : ICi(X) → ICi−1(X)

im ∂ : ICi+1(X) → ICi(X).

It is sufficient for the purposes of this paper to define intersection cohomology ICi(X; Q)

as the dual vector space of the intersection homology group ICi(X).

Many results which hold for the cohomology of a nonsingular variety also hold for the

intersection (co)homology of a singular variety. However, there are three major differences

(see [Kir6]):

Warning 21.1 A continuous map f : X → Y does not, in general, induce homomorphisms

f∗ : IH∗(X) → IH∗(Y ) or f ∗ : IH∗(Y ) → IH∗(X).

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However, the situation is better for birational maps of varieties:

Theorem 21.2 [BBD] If f : A → B be a proper projective birational map of complex

varieties, then IH i(B; Q) is a direct summand of IH i(A; Q).

In light of the first warning, the second is not terribly surprising:

Warning 21.3 Intersection (co)homology is not homotopy invariant.

Finally,

Warning 21.4 There is no natural ring structure on IH∗(X).

90

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22 Index of Terms and Notation

categorical quotient, 5coarse moduli space, 4Deligne-Mumford (semi)stable curve, 16dualizing sheaf, 17fine moduli space, 4Fitting ideal, 27Hilbert functor, 24Hilbert point of a curve, 25,Hilbert scheme, 24linearization, 7local universal property, 5orbit space, 5potentially stable curve, 74potentially stable map, 74prestable curve, 16prestable map, 20semistability, G.I.T. 7stable map, 20universal family, 4Xss(L), the locus of G.I.T. semistable points with respect to a linearization L, 7X//LG, the categorical quotient of Xss(L), 7Mg, the moduli space of nonsingular curves of genus g, 16Mg, the moduli space of Deligne-Mumford stable curves of genus g, 18Mg(X, β), the moduli space of stable maps, 20HilbP,X/T , the Hilbert scheme of subschemes of X with Hilbert polynomial P , 24HilbP,N , the Hilbert scheme of subschemes of PN with Hilbert polynomial P , 24Hm(C), the Hilbert point of C in Gieseker’s construction, 25

Hm,m(C), the Hilbert point of a map C → Pr, 32

K, the locus of n-canonically embedded curves in the Hilbert scheme, 26Fittg(G), the gth Fitting ideal of an ideal sheaf G, 27V (I), the closed subscheme corresponding to the quasicoherent sheaf of ideals I, 27Φi(), the cohomology and base change theorem map, 28Riϕ∗, higher direct image functor, 28

C ϕ→ Hilb, the universal family over the Hilbert scheme, 25, 31Hilb(P(W )×Pr), the Hilbert scheme of subschemes of P(W )×Pr having bidegree (e, d)e, 29d, the degree of the stable map to Pr, 29n, denotes d

2(g−1)on pages 21-29, and d

g−1on pages 29-86

N , frequently denotes e− ga, constant relating e and d, 29J , 29wλ′(Mi), the λ′-weight of a monomial Mi, 33∑j

i=1wλ′(Mi), the total λ′-weight of a collection of j monomials, 33LW , Lr, 35m′,m′′,m′′′, q1, q2, q3, µ1, µ2, constants, 35C1, 47g, the minimum value of gC taken over all curves C of genus g, 56

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23 Acknowledgements

I would like to thank my supervisor, Frances Kirwan, for her extraordinary patience as

she introduced me to the exciting world of moduli spaces and the techniques mathematicians

use to study their topology and geometry.

I would also like to thank the faculty and staff of the Mathematical Institute and my

fellow geometry graduate students for their assistance. Sami Assaf, Brent Doran, and Davesh

Maulik earned my special thanks for explaining technical points in algebraic geometry to me.

I am grateful to Aaron Bertram, Brian Conrad, Barbara Fantechi, Jun Li, Joseph Lipman,

Ian Morrison, Rahul Pandharipande, Michael Thaddeus, and Angelo Vistoli with whom I

consulted in the course of this project. I would like to thank my undergraduate advisors

Frank Connolly and Jeff Diller for their continued advice, and Seth Bodnar, Tom McCaleb,

and Tim Strabbing for their support.

Finally I would like to thank the Marshall Aid Commemoration Commission and the

citizens of the U.K., who funded my studies at Oxford with a British Marshall Scholarship.

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