Geometric Invariant Theory and Moduli Spaces of Maps

David J. Swinarski Balliol College

Submitted in partial fulﬁllment of the requirements of the Master of Science by Research Trinity 2003

1 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) 12 6.1 Facts about vector bundles ...... 12 6.2 A G.I.T. construction of M(n, d) ...... 13 6.3 The cohomology of M(n, d) ...... 14

7 Mg 16 7.1 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 24 11.1 The Hilbert scheme ...... 24 11.2 The Hilbert point of a curve ...... 25 11.3 Deﬁning K˜ ...... 26

12 Fitting Ideals and the Construction of Mg 27

r 13 Constructing Mg(P , d): General Information 29

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

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

i 16 G.I.T. Semistable Maps Are Potentially Stable 42 16.1 First properties of G.I.T. semistable maps ...... 43 16.2 G.I.T. semistability implies that the only singularities are nodes ...... 56 16.3 G.I.T. semistable curves are reduced ...... 69 16.4 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

ii 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 r space of maps Mg(P , 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.

1 Part I Geometric Invariant Theory and Moduli 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 2g+2 2 Y normalization of the curve y = (x − bi). In fact two hyperelliptic curves C1 and C2 i=1 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:

2 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 : S0 → S and any family X parametrized by S, there is an induced family ϕ∗X parametrized by S0. This operation ∗ ∗ ∗ ∗ is functorial in the sense that (ϕ1 ◦ ϕ2) = ϕ2 ◦ ϕ1 and idSX = 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 morphism 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 π ∼ ∗ X1 ∼fam X2 if there exists a line bundle L → 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]

3 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 0 ∈ Ob(Cat) and natural transformation F 0 : F → Hom(−,M 0), there exists a unique natural transformation Ψ such that F 0 = 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

4 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 exists 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.

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

6 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 p an action of G on L → 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 n is a G-invariant section of L 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 ss s G on Xf is closed. The semistable locus X (L) and the stable locus X (L) are G-invariant ss open sets. We write X//LG for the categorical quotient of X (L). Note that in general, different linearizations carry different sets of semistable and stable points, and in particular the G.I.T. quotients X//L1 G and X//L2 G 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

i Wi = {w ∈ W |the action of λ is given by λ(t)w = t w} and let Sλ(W ) be the set of integers i such that Wi 6= 0. There is a decomposition (cf. [HM] 4.15) M W = Wi.

i∈Sλ(W )

7 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) = min Sλ(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 Kähler manifold where the Kählermetric 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 n complex projective variety X ⊆ P . There is a stratification {Sβ | β ∈ B} of X such that ss S0 = X . 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,

G X 2dβ G Pt (X) = t Pt (Sβ) β∈B

G where Pt (−) denotes the equivariant Poincar´eseries of (−) and dβ is the complex codimen- sion of Sβ in X.

8 Observe that

H∗(X//G) ∼= H∗(Xss/G) if Xss = Xs ∼ ∗ ss ss = HG(X ) if stabG(x) is ﬁnite for every x ∈ X

ss s ss (Note that if the equality X = X holds then stabG(x) is ﬁnite for every x ∈ X .) In this situation we may compute

G ss G X 2dβ G Pt(X//G) = Pt (X ) = Pt (X) − t Pt (Sβ) β∈B, β6=0

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 HG(X). If X is a ∗ ∼ ∗ ∗ compact symplectic manifold, Kirwan has shown [Kir1] that HG(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 n 2 Y curve y = (x − bi). In fact, i=1

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 hyperelliptic curves. Recall that PSL(2; C) = SL(2; C)/{±I}. It is technically more convenient 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

9 next that there is a one-one correspondence between points of Pn and sets of n unordered 1 n points in P . This correspondence maps the point [a0, . . . , an] ∈ P to the set of zeroes n 0 0 n n of the polynomial a0X Y + ··· + anX Y . Let S ⊂ P be the open set corresponding to 1 n {([x1 : y1],..., [xn : yn]) ∈ (P ) | [xi : yi] 6= [xj : yj] if i 6= j}/Σn where Σn is the symmetric 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 P˜ 2 be the blowup of P2 at the point [0 : 1 : 0]. Write

2 2 2g Y Z = {([x : y : z], (a0, . . . , an))∈P × S | y z = (x − aiz)}, i=0 and let Z˜ be the proper transform of Z, i.e. Z˜ is the closure of Z ∩ ((P2 − [0 : 1 : 0]) × S) in P˜ 2 × 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 tr 0 λ : t 7→ r 0 t−r where r is a positive integer. An element t ∈ λ acts on Pn by the rule:   a0 rn r(n−2) r(−n)  .  (t, x) 7→ diag(t , t , ...., t )  .  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) = max{r(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.

10 Let i0 be the smallest integer such that ai 6= 0. It follows that ai = 0 for all i < i0. Then x n is not stable ⇔ µ ≤ 0 ⇔ n−2i0 ≤ 0 ⇔ i0 ≥ n/2. This implies that a point [a0 : ... : an] ∈ P P n−i i is stable for the action of SL(2) if the polynomial aiX Y has no factors of multiplicity n greater than or equal to n/2. Similarly a point [a0 : ... : an] ∈ P is semistable for the P n−i i action of SL(2) if the polynomial aiX Y 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 X˜ ss = X˜ s. The partial desingularization X//G˜ is obtained by blowing up X//G along a sequence of subvarieties. We can construct Pñ//SL(2) and compute H∗(Pñ//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)) = HG(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.

11 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 deg(F) for every proper subbundle F , µ(F ) ≤ µ(E), where the slope µ(F ) := . It is slope rank(F) 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 deg F ·rankE = deg E·rankF , which implies gcd(rankE, deg E) > 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

12 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 dim H0(E) = d + n(1 − g) =: p. For each slope semistable bundle E, there is a Jordan-Hölder 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-H¨olderﬁltration 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 E0 are equivalent if Gr (E) ∼= Gr (E0) 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 dim H0(E) = p which are generated by their sections (see previous subsection), we shall be interested in quotients of the sheaf

p M E = 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 0 Lp 0 s ss the map H ( OC ) → H (U|{q}×C ) is an isomorphism. Let R and R 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 of M(n, d). In particular, if gcd(n, d) = 1, then M(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 cohomology 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 maps M → BU(n). Let V be the pullback of a universal classifying bundle EU(n) → BU(n) by the map Map(C,BU(n)) × C →ev BU(n) so that V is a vector bundle of rank n over BG × C. By the K¨unnethformula,

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

1 2 Let {α1, . . . , α2g} be a basis of H (C), and let ω be the standard generator of H (C). Then for each 1 ≤ r ≤ n, the rth Chern class has a decomposition of the following form:

X j cr(V ) = ar ⊗ 1 + br ⊗ αj + fr ⊗ ω. 1≤j≤2g

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

2r ar ∈ H (BG; Q), 1 ≤ r ≤ n

j 2r−1 br ∈ H (BG; Q), 1 ≤ r ≤ n, 1 ≤ j ≤ 2g

2r−2 fr ∈ H (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

∗ =∼ ∗ ∗ ss =∼ ∗ H (BG) → HG(C) → HG(C ) → 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

2r ar ∈ H (BG; Q), 2 ≤ r ≤ n

j 2r−1 br ∈ H (BG; Q), 1 ≤ r ≤ n, 1 ≤ j ≤ 2g

15 2r−2 fr ∈ H (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 projective 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 ˆ 2 2 completion OC,p of the local ring OC,p is isomorphic to C[[x, y]]/(y − x ). (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. 0 The arithmetic genus of a curve is deﬁned as h (C, OC ). If C is nonsingular, then by 1 Serre duality this is equal to the geometric genus h (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 Resxi α + Resyi α = 0. ⊗n Let C be a Deligne-Mumford stable curve. Then ωC/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 0 ⊗j 0 ⊗n N = dim H (C, L ). Thus for any stable curve C a choice of basis of H (C, ωC/k) determines an embedding C → Ps−1 called the n-canonical embedding. Here 0 ⊗n s = dim H (C, ωC/k) = (2n − 1)(g − 1) is the nth plurigenus of C. Write r = s − 1 and ⊗n d = deg ωC/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: Teichm¨uller theory, period matrices, or geometric invariant theory. The Teichm¨uller 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 Teichm¨ullertheory 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. Teichm¨ullerspace 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. 3g−3 Bers showed more than ﬁfty years ago that Tg is homeomorphic to a ball in C . 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 i H (Mg) for g large and i small. The Teichm¨ullertheory construction thus gives a route to studying the quasiprojective

(noncompact) moduli space Mg. 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. Teichmüllertheory is not particularly suited to producing a compactification which has a

modular interpretation, as any modular compactiﬁcation of Mg 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 projective 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 singular 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 ˜ N scheme over the subscheme K ⊂ Hilb(P ) 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 K˜ and 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 étale base change. The advantage of his space is that it carries a Poincarésheaf. 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 (π0, µ0) are equivalent if there is a scheme isomorphism τ : C → C0 such that π = π0 ◦ τ and µ = µ0 ◦ 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 scheme Mg(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 R variety of pure dimension dim X + β c1(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 k Mg(P , 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 k k k of Mg(P , d) and their dimensions, how the boundary of Mg(P , d) meets Mg(P , d), or k the singularities of Mg(P , 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. k Fulton and Pandharipande hint how a G.I.T. construction of Mg(P , 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 obtain a finite-dimensional Kähler 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- r tion of Mg(P , d) as a quotient. Let r = k − 1 where k = (2n − 1)(g − 1) is the nth plurigenus of a curve C. It is r convenient to take n = 10. Let H be the subset of Mg(P , 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. r Now let H be the closure of H in Mg(P , 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 r Mg(P , d) does not explicitly embed it in projective space, though ample line bundles have r been constructed over Mg(P , d) (see [FP, Alex, Corn]). After specifying a linearization we can make sense of H//SL(r + 1; C). We would like to s relate H//SL(r + 1; C) to the Deligne-Mumford-Knudsen compactification Mg. If H ⊆ H

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 r not immediately clear that this is an isomorphism. Recall that Kontsevich’s space Mg(P , 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. ss The best case would occur if for every point [f : C → Pr] ∈ ∂H , the domain curve C is Deligne-Mumford stable and [f] is 10-canonical. The situation will be more complicated if ss there exist any points [f : C → Pr] ∈ ∂H such that

i. f is not 10-canonical and/or ii. f is not an embedding and/or iii. C is prestable but not stable.

r So the ﬁrst steps in this project are to study the closure H of H in Mg(P , d) and linearizations of the SL(r+1) action on H. The following diagram may help us to understand H: k−1 τ Mg(P , d) → U g(1, d) π & ↑ σ Mg Here τ([f]) := [f ∗(L)] where L is the hyperplane line bundle on Pr, the map π is the ⊗10 stabilization morphism (see [Manin] Ch.5) and σ([C]) := [ωC ].

On H, we have τ|H = σ ◦ π|H , so

r H ⊆ {[f] ∈ Mg(P , d) | τ([f]) = σ(π([f]))}.

22 The right hand side may equal H, or if it is reducible then H may be its irreducible component containing H. ss ∼ Next, we would hope to find a linearization for which H //SL(W ) = Mg. As noted r above, proofs of the existence of at least one ample line bundle on Mg(P , d) have been r r published. More precisely, Mg(P , d) is constructed as a quotient J → Mg(P , d), and ample r line bundles are given on J. One of our main motivations for constructing Mg(P , d) as a G.I.T. quotient is to study the line bundles (and hence possible different linearizations) on r Mg(P , 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. ss We hope that studying H and H (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 Moduli Space of Maps

11 Gieseker’s Construction of Mg Reviewed

r Our G.I.T. construction of Mg(P , d) closely parallels Gieseker’s construction of Mg. Therefore we shall brieﬂy review Gieseker’s construction here. N Grothendieck proved that there is a scheme HilbP,N parametrizing subschemes of P which have a given Hilbert polynomial P (m). Gieseker’s strategy is to “pick out” the subschemes of PN which are n-canonically embedded Deligne-Mumford stable curves, then ac- 0 ⊗n ∼ N+1 count for the choice of basis of H (C, ωC/k) = k 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 N P . Then HilbP,X/T represents HilbP,X/T (−). If X = P 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 N universal family C ⊂ P × HilbP,N to be the scheme corresponding to the identity morphism N id ∈ Hom(HilbP,N , HilbP,N ). Note that the composition ϕ : C ,→ P × 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 N curve in P of genus g and degree e, let L = OPN (1)|C , and let

0 N 0 m ϕm : H (P , OPN (m)) → H (C,L ) be the map induced by restriction. The Hilbert polynomial of C is P (m) = em − g + 1. VP (m) Form the P (m)th exterior power of the map ϕm. Then ϕm is a nonzero linear form VP (m) 0 N VP (m) 0 N on H (P , OPN (m)), hence an element of P( H (P , OPN (m))). We are using the convention that for a vector space V , P(V ) denotes the space of equivalence classes of VP (m) nonzero linear forms on V . We call ϕm the mth Hilbert point of C, written Hm(C). VP (m) 0 N There is an induced SL(N + 1) action on P( H (P , OPN (m))). If e0, . . . , eN is 0 N a basis of H (P , OPN (1)) then the induced action can be written in the following way: a 0 N basis for H (P , OPN (m)) is given by the degree m monomials Mi in the symbols e0, . . . , eN . 0 N Then we deﬁne an SL(N + 1) action on H (P , OPN (m)) by the rule

γ0 γN γ0 γN g.e0 ··· eN = (ge0) ··· (geN ) .

25 P (m) 0 N V N A basis of H (P , OP (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 VP (m) 0 N product H (P , OPr (m)) by the rule

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

VP (m) 0 N Then the dual action gives an action of SL(N + 1) on P( H (P , 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

P (m) ^ 0 N ψm : HilbP,r → P( H (P , OPN (m)))

h 7→ Hm(Ch)

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

Ch := C × Speck(h). (2) HilbP,N

VP (m) 0 N ∼ So the SL(N + 1) action on P( H (P , OPN (m))) restricts to ψm(HilbP,N ) = HilbP,N

to give an action on HilbP,N .

11.3 Deﬁning K˜

0 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 . ϕ 0 0 ϕ Let C → U be the restriction to U of the universal family C → HilbP,N . Write OC(1) for the N N pullback of OPN (1) by the morphism C → P × U → P . Let ω := ωC/U 0 be the relative ι N Ch N dualizing sheaf of ϕ (see section 7.1 above). Also Ch is a nodal curve in P ; write Ch → P ∗ for the inclusion morphism, and set O (1) := ι O N (1). Let ω be the dualizing sheaf of Ch Ch P Ch 0 0 the curve Ch. Finally let U be the open subset of U such that the multidegree of ωC/U |Ch is

equal to the multidegree of OCh (1). We deﬁne K˜ = {h∈U | O (1) ∼= ω⊗n}. (3) Ch Ch ˜ ˜ 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. ˜ ss ˜ VP (m) 0 N ss Let K = K ∩ ((P( H (P , OPN (m)))) ). We have the following result:

26 N ˜ ss Theorem 11.3 Every curve C ⊂ P whose Hilbert point Hm(C) lies in K is Deligne- Mumford stable, and every Deligne-Mumford stable curve of genus g has a model in K˜ ss.

˜ 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 Ψ E1 → 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:

( r−g r−g r−g ∗ the image of the map ∧ E1⊗ ∧ E → OY induced by ∧ Ψ if r − g > 0 Ir−gϕ = 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. Ψ Proof. Choose a two-term free resolution E1 → 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 r−g r−g ∗ = Image( ∧ E1⊗ ∧ E0 → OY ) r−g r−g ∗ (FittgG)y = (Image( ∧ E1⊗ ∧ E0 → OY ))y r−g r−g ∗ = Image(( ∧ E1⊗ ∧ E0 )y → OY,y) by [Hart] Exercise II.1.2 r−g r−g ∗ = Image((( ∧ E1)y ⊗ ( ∧ E0 )y) → OY,y) by [Liu] page 158 r−g r−g ∗ = Image(( ∧ (E1)y⊗ ∧ (E0 )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 |Ch is equal to the multidegree of OCh (1). Recall

also the deﬁnitions of ωC/U , and OC(1) given on page 26. p1 Let p1 be the projection map C ×U Speck(h) → C. Then for an OC-module F we ∗ shall often use the symbol F|Ch to mean p1(F). Note in particular that with this notation ∼ ∼ OCh (1) = OC(1)|Ch and ωCh = ω|Ch . We deﬁned K˜ = {h∈U | O (1) ∼ ω⊗n}. Ch = Ch We want to show

˜ 1 ⊗n Proposition 12.3 K = V (Fittg−1R ϕ∗(ω ⊗ OC(−1)) as closed subsets of U.

1 ⊗n Remark. V (Fittg−1R ϕ∗(ω ⊗ 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 ⊗n ω ⊗ OC(−1) is invertible. It is locally free of finite rank and therefore flat over C. The ϕ ⊗n map C → U is flat. Therefore ω ⊗ OC(−1) is flat on U. 2 ⊗n The second cohomology H (Ch, (ω ⊗OC(−1))|Ch ) = 0 for all h because Ch is a curve for 2 2 ⊗n 2 ⊗n all h ∈ U. Then Φ (h): R ϕ∗(ω ⊗ OC(−1)) ⊗ k(h) → H (Ch, (ω ⊗ OC(−1))|Ch ) = 0 is ϕ surjective. Here, to avoid confusion with the map C → 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 2 ⊗n 2 ⊗n R ϕ∗(ω ⊗ OC(−1)) = 0 on a neighborhood of h. So R ϕ∗(ω ⊗ OC(−1)) is locally free of rank 0 near h. This implies by [Hart] Theorem 12.11b that

1 1 ⊗n 1 ⊗n Φ (h): R ϕ∗(ω ⊗ OC(−1)) ⊗ k(h) → H (Ch, (ω ⊗ OC(−1))|Ch )

is surjective and hence an isomorphism at each h ∈ U. 1 ⊗n ˜ We show that V (Fittg−1R ϕ∗(ω ⊗ OC(−1))) ⊆ K. More precisely, we show that if ˜ 1 ⊗n h ∈ U \ K, then h 6∈ V (Fittg−1R ϕ∗(ω ⊗ OC(−1))).

28 ˜ ⊗n Let h 6∈ K. Then the sheaf (ω ⊗ OC(−1))|Ch is nontrivial. Furthermore its multidegree 0 ⊗n 0 is zero since h ∈ U. Therefore H (Ch, (ω ⊗ OC(−1))|Ch ) = 0, and we conclude that Φ (h) is surjective. Since both maps Φ1(h) and Φ0(h) are surjective, we conclude by [Hart] 12.11b that 1 ⊗n ˜ R ϕ∗(ω ⊗OC(−1)) is locally free on a neighborhood of h. Since this holds for all h ∈ U \K 1 ⊗n ˜ we conclude that R ϕ∗(ω ⊗ OC(−1)) is locally free on U \ K. Furthermore, the rank of 1 ⊗n 1 ⊗n the stalk R ϕ∗(ω ⊗ OC(−1))h is equal to rankH (Ch, (ω ⊗ OC(−1))|Ch ). But this is g − 1 ⊗n since (ω ⊗ OC(−1))|Ch is 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.

˜ 1 ⊗n ∼ The proposition allows us to conclude that over U \K we have Fittg−1R (ω ⊗OC(−1)) = 1 ⊗n ∼ ˜ OU . Then the stalks (Fittg−1R ϕ∗(ω ⊗OC(−1)))h = OU,h are isomorphic for each h ∈ U\K. 1 ⊗n ˜ We conclude that V (Fittg−1R ϕ∗(ω ⊗ OC(−1))) ⊆ K. ˜ 1 ⊗n ˜ We prove next that K ⊆ V (Fittg−1R ϕ∗(ω ⊗ OC(−1))). Let h ∈ K so that ⊗n (ω ⊗ OC(−1))|Ch is trivial. Then appealing to the deﬁnition of the arithmetic genus we 1 ⊗n have rankH (Ch, (ω ⊗ 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.

1 ⊗n 1 This implies Fittg−1H (Ch, (ω ⊗ OC(−1))|Ch ) = 0. Applying the isomorphism Φ (h) we have

1 ⊗n Fittg−1((R ϕ∗(ω ⊗ OC(−1)))h) = 0

1 ⊗n ⇒ (Fittg−1(R ϕ∗(ω ⊗ OC(−1))))h = 0 1 ⊗n ˜ ⇒ (Fittg−1(R ϕ∗(ω ⊗ OC(−1))))h 6= OU,h if h ∈ K.

˜ 1 ⊗n That is, K ⊆ V (Fittg−1R ϕ∗(ω ⊗ OC(−1))). ˜ 1 ⊗n This shows the closed subsets K and V (Fittg−1R ϕ∗(ω ⊗ OC(−1))) coincide.

r 13 Constructing Mg(P , d): General Information

r ∼ r We use the standard isomorphism H2(P ) = Z throughout what follows. Let Mg(P , 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. r ∗ Notation. Given a stable map f : C → P , 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 0 a d a is large. Write deg(L ) = a(2g − 2 + 3d) =: e so h (C, L ) = e − g + 1. Set n = g−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 ϕ (e, d). In analogy with the notation of the previous section we write C → 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.

r iii. The multidegree of (OP(W )(1) ⊗ OP (1))|Ch equals the multidegree of ⊗a (ω ⊗ O r (3a + 1))| . Ch P 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:

1 ⊗a Proposition 13.1 J = V (Fittg−1R ϕ∗(ωC ⊗ (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 P GL(W ) is the moduli space r ¯ r Mg(P , d). Write J for the closure of J in Hilb(P(W )× P ). We would like to establish the following claim: there is a linearization of the group action on J¯ such that the G.I.T. ¯ r ¯ quotient J//SL(W ) is isomorphic to Mg(P , d). The construction of J//SL(W ) and the

30 ¯ ∼ r proof that J//SL(W ) = Mg(P , d) will occupy Sections 14 to 17. However, to ﬁnish this section we prove:

j r Proposition 13.2 J → Mg(P , d) is a categorical quotient for the SL(W ) action.

f X → Pr r r Let σ ↓ be a family of stable maps to P . For any s ∈ S we write fs : Xs → P S for the stable map corresponding to s.

0 ⊗a ∗ Lemma 13.3 R σ∗(ωX /S ⊗ f (OPr (3a))) is locally free.

1 ⊗a ∗ Proof. Recall that we have chosen a so that H (X ; ω ⊗ f (O r (3a))) = 0 for all s ∈ S. s Xs s P

Also since Xs is a curve for each s all higher cohomology groups are zero as well. Then by i ⊗a ∗ [Hart] Exercise III.11.8, we have R σ∗(ωX /S ⊗f (OPr (3a))) = 0 for all i > 0. The hypotheses 0 ⊗a ∗ of [EGA] Corollary III.7.9.10 are satisﬁed, and we conclude that R σ∗(ωX /S ⊗ f (OPr (3a))) is locally free.

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

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

σ r Proof. Suppose X → S is a family of stable maps to P . For any s0 ∈ S we seek an open ψ ∗ neighborhood V 3 s0 and a morphism V → J such that ψ (C) ∼fam X |V . 0 ⊗a ∗ 0 ⊗a ∗ r r Pick a basis for H (Xs0 ; ω ⊗f (OP (3a))). We showed that R σ∗(ω ⊗f (OP (3a))) Xs0 s0 X /S ⊗a ∗ is locally free. Choose a neighborhood V 3 s0 which is so small that σ∗(ωX /S ⊗f (OPr (3a))) is free on V . Then by [Hart] III.12.11(b) and Lemma 13.3 there is an induced basis 0 ⊗a ∗ ιs of H (X ; ω ⊗ f (O r (3a))) for each s ∈ V . This deﬁnes a map X → P(W ) for s Xs s P s

each s ∈ V . These ﬁt together as a morphism X |V → P(W ). For each s ∈ V form r (ιs, fs): Xs → P(W ) × P . These pairs ﬁt together to give a family of stable maps

(ι,f) r r X |V → P(W )× P → P . ↓ V

r But (ι, f) also gives X |V → V the structure of a family of curves in P(W )×P parametrized by V as deﬁned in Deﬁnition 11.1. By the universal properties of the Hilbert scheme, then,

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

r Proof of Proposition 13.2: Mg(P , 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 r Mg(P , 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

0 N 0 ϕm : H (P , OPN (m)) → H (C, OPN (m)|C )

is the map induced by restriction, then the mth Hilbert point of C is VP (m) VP (m) 0 N Hm(C) := ϕm ∈ P( H (P , OPN (m)), where P(V ) denotes the space of equivalence 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 r ˆ Ch ⊂ P(W )× P is em + dmˆ + 1 − g. We deﬁne Hm,mˆ (h), the (m, mˆ )−th Hilbert point of h, as follows:

1 r Deﬁnition 14.1 If m and mˆ are suﬃciently large, then H (Ch, OP(W )(m)⊗OP (m ˆ )|Ch ) = 0 and the restriction map

0 r 0 r r ρˆm,mˆ : H (P(W )× P , OP(W )(m) ⊗ OP (m ˆ )) → H (Ch, OP(W )(m) ⊗ OP (m ˆ )|Ch )

is surjective. VP (m)+dmˆ VP (m)+dmˆ 0 r Then ρˆm,mˆ is a point of P( H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ ))). ˆ VP (m)+dmˆ We set Hm,mˆ (h) := ρˆm,mˆ for all m > m0.

For suﬃciently large m, mˆ , say m, mˆ ≥ m000, the map ˆ h 7→ Hm,mˆ (h) r VP (m)+dmˆ 0 r Hilb(P(W )× P ) → P( H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ ))

32 is a closed immersion (see 14.2 below). VP (m)+dmˆ 0 r There is an induced SL(W ) action on P( H (P(W )×P , OP(W )(m)⊗OPr (m ˆ )). 0 If w0, ..., wN is a basis of H (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 PN 0 g.wp = j=0 apjwj. Let Bm = {M1, ..., Mα} be a monomial basis of H (P(W ), OP(W )(m)). γ0 γN We extend the previous action as follows: if Mi = w0 ··· wN then

γ0 γN r+1 g.Mi = (g.w0) ··· (g.wN ) . Pick a basis f0, ..., fr of C and let Bmˆ be a monomial basis 0 r ˆ of H (P , OPr (m ˆ )). Tensor Bm with Bmˆ to get a basis Bm,mˆ of

0 0 r ∼ 0 r H (P(W ), OP(W )(m)) ⊗ H (P , OPr (m ˆ )) = H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ ))

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

γ0 γN Γ0 Γr ˆ γ0 γN Γ0 Γr w0 ··· wN f0 ··· fr , we have g.Mi = (g.w0) ··· (g.wN ) f0 ··· fr . VP (m)+dmˆ 0 r A basis for H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) is given by

0 r ˆ ˆ r {Mi1 ∧ · · · ∧ MiP (m)+dmˆ |1 ≤ i1 < ··· < iP (m)+dmˆ ≤ h (P(W )× P , OP(W )(m) ⊗ OP (m ˆ ))}.

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

ˆ ˆ ˆ ˆ g.(Mi1 ∧ · · · ∧ MiP (m)+dmˆ ) = (g.Mi1 ) ∧ · · · ∧ (g.MiP (m)+dmˆ ).

VP (m)+dmˆ 0 r The dual action is the SL(W )-action on P( H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) we shall study. Let λ0 be a 1-PS of SL(W ). We wish to state the Hilbert-Mumford numerical criterion 0 (cf. page 7) for our situation: Let w0, ..., wN be a basis of H (P(W ), OP(W )(1)) diagonalizing

0 0 ri ∗ the action of λ . There exist integers r0, ..., rN such that λ (t)wi = t wi for all t ∈ C and 0 ≤ i ≤ N. Then, referring to the notation of the previous paragraph, we deﬁne the

0 ˆ γ0 γN Γ0 Γr ˆ PN λ - weight of a monomial Mi = w0 ··· wN f0 ··· fr to be wλ0 (Mi) = p=0 γprp. We deﬁne 0 ˆ ˆ Pj ˆ the total λ -weight of a collection of j monomials M1, ..., Mj to be i=1 wλ0 (Mi). VP (m)+dmˆ 0 r Recall that a basis for H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) is given by

0 r ˆ ˆ r {Mi1 ∧ · · · ∧ MiP (m)+dmˆ |1 ≤ i1 < ··· < iP (m)+dmˆ h (P(W )× P , OP(W )(m) ⊗ OP (m ˆ ))}. (4)

ˆ ˆ ∗ We follow Gieseker and write (Mi1 ∧ · · · ∧ MiP (m)+dmˆ ) for elements of the basis of VP (m)+dmˆ 0 r ∗ ( H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ ))) which is dual to the basis given in line 0 VP (m)+dmˆ 0 r (4). Then the λ action on P( H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) is given by 0 ∗ −θ ∗ P (m)+dmˆ ˆ ˆ ˆ ˆ P 0 ˆ λ (t)((Mi1 ∧· · ·∧MiP (m)+dmˆ ) ) = t ((Mi1 ∧· · ·∧MiP (m)+dmˆ ) ), where θ = j=1 wλ (Mij ).

33 ˆ Now write Hm,mˆ (h) in this basis:

ˆ X ˆ ˆ ˆ ˆ ∗ Hm,mˆ (h) = ρˆm,mˆ (Mi1 ∧ · · · ∧ MiP (m)+dmˆ )(Mi1 ∧ · · · ∧ MiP (m)+dmˆ ) .

1≤i1<···

ˆ 0 The numerical criterion states that Hm,mˆ (h) is λ -semistable if and only if 0 0 P (m)+dmˆ ˆ ˆ P 0 ˆ µ(Hm,mˆ (h), λ ) ≤ 0 where µ(Hm,mˆ (h), λ ) = min{ j=1 wλ (Mij )} and the minimum is ˆ ˆ taken over all sequences 1 ≤ i1 < ··· < iP (m)+dmˆ such thatρ ˆm,mˆ (Mi1 ∧ · · · ∧ MiP (m)+dmˆ ) 6= 0. We will often obtain one-parameter subgroups λ0 of SL(W ) from one-parameter subgroups λ of GL(W ). The correspondence is as follows: Given a 1-PS λ of GL(W ), there is a 0 basis w0, ..., wN of H (P(W ), OP(W )(1)) diagonalizing the action of λ so that the action of λ

ri PN is given by λ(t)wi = t wi where ri ∈ Z. Note that the sum p=0 rp is not necessarily zero. 0 N 0 0 ri 0 P Then we obtain a 1-PS λ of SL(W ) by the rule λ (t)wi = t wi where ri = (N +1)ri− i=0 ri. Semistability for the λ0 action and the λ action are related in the following way. Let

ˆ γ0 γN Γ0 Γr ˆ Mi = w0 ··· wN f0 ··· fr be a monomial in Bm,mˆ . 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 ˆ PN PP (m)+dmˆ ˆ SL(W ), namely wλ(Mi) = p=0 γprp and j=1 wλ(Mij ). Then

P (m)+dmˆ X 0 ˆ wλ (Mij ) ≤ 0 ⇐⇒ j=1

P (m)+dmˆ N X ˆ X (N + 1) wλ(Mij ) − m(P (m) + dmˆ ) rp ≤ 0 ⇐⇒ j=1 p=0

P (m)+dmˆ N X ˆ X wλ(Mij ) rp j=1 p=0 ≤ . m(P (m) + dmˆ ) N + 1 Thus if λ0 is the 1-PS of SL(W ) arising from a 1-PS λ of GL(W ), the numerical criterion may ˆ 0 be expressed as follows: the (m, mˆ )th Hilbert point Hm,mˆ (h) is λ -semistable if and only if ˆ ˆ ˆ ˆ ˆ there exist monomials Mi1 , ..., MiP (m)+dmˆ in Bm,mˆ such that {ρˆm,mˆ (Mi1 ), ..., ρˆm,mˆ (MiP (m)+dmˆ )} 0 r is a basis of H (Ch, OP(W )(m) ⊗ OP (m ˆ )|Ch ) and

P (m)+dmˆ N X ˆ X wλ(Mij ) rp j=1 p=0 ≤ . (5) m(P (m) + dmˆ ) N + 1 We shall show that the quotient J//SL(W ) is projective and is isomorphic to J//SL¯ (W ) r r and Mg(P , d). First, we show that if C ⊂ P(W ) × P is smooth and the map C ⊂

34 r ˆ P(W )× P → P(W ) is a nondegenerate embedding, then Hm,mˆ (C) is SL(W )-stable. This ˆ shows that J//SL(W ) is nonempty. Next we show that if Hm,mˆ (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. r A few remarks on the notation we shall use: Let pW : P(W )× P → P(W ) be projection r r onto the ﬁrst factor, and pr : P(W )× P → P projection onto the second. Let ι r ∗ ∗ ∗ ∗ C → P(W )×P be inclusion. We let LW denote ι pW OP(W )(1) and let Lr denote ι prOPr (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:

0 00 000 Proposition 14.2 (cf. [Gies] page 25) There exist positive integers m , m , m , q1, q2, q3, µ1, and µ2 such that

0 1 m 1 mˆ 1 m mˆ i. for all m, mˆ > m , H (C,LW ) = H (C,Lr ) = H (C,LW ⊗ Lr ) = 0 and the three restriction maps

0 0 H (P(W ), OP(W )(m)) → H (pW (C)OpW (C)(m)) 0 r 0 r H (P , OP (m ˆ )) → H (pr(C), Opr(C)(m ˆ )) 0 r 0 m mˆ H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) → H (C,LW ⊗ Lr )

are surjective.

q1 ii. IC = 0 where IC is the sheaf of nilpotents in OC . 0 iii. h (C, IC ) ≤ q2. ˜ 0 ˜ iv. For every complete subcurve C of C, h (C, OC˜) ≤ q3 and q3 ≥ q1.

v. µ1 > µ2 and for every point P ∈ C and for all integers n ≥ 0,

OC,P dim x ≤ µ1x + µ2, mC,P

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 ≥ m00, H1(C,˜ Im−n ⊗ Lm ) = 0, where I is the ideal subsheaf P W C˜ P

of OC˜ deﬁning P .

35 vii. For all integers m, mˆ ≥ m000 the map

ˆ h 7→ Hm,mˆ (h) P (m)+dmˆ r ^ 0 r Hilb(P(W )× P ) → P( H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ ))

is a closed immersion.

Remark. In the course of the construction we shall find it necessary to limit the range mˆ of values m 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 ˆ ). The G.I.T. quotient does not change if the linearization is replaced by a tensor power of itself, i.e. m andm ˆ may be replaced by xm and xmˆ for any x ∈ N and the quotient will be the same (cf. for instance [D] mˆ page 51). However, if the ratio m 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 mˆ by a flip. It would be interesting to study how different values of m might give rise to ¯ ¯ nonisomorphic quotients J//L1 SL(W ) and J//L2 SL(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 > 0 such that for all integers r0, ..., rN (not all zero) with P 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 X (rit+1 − rit )(eit+1 + eit ) > 2rN eN + 2(rn − r0). t=1

36 Theorem 15.2 Suppose that m >> 0 (this will be made more precise in the course of the proof) and mˆ ≥ 2g + 1. If C ⊂ P(W )× Pr → Pr is a stable map, C is nonsingular, the map

0 ρ 0 H (P(W ), OP(W )(1)) → H (pW (C), OpW (C)(1)) ∼ ˆ is an isomorphism, and LW is very ample (so that C = pW (C)), then Hm,mˆ (C) is SL(W )- stable.

Proof. Let C ⊂ P(W )× Pr→Pr be such a map. Let λ0 be a 1-PS of SL(W ). There 0 exist a basis {w0, ..., wN } of H (P(W ), OP(W )(1)) and integers r0 ≤ · · · ≤ rN such that

P 0 0 ri ri = 0 and the action of λ is given by λ (t)wi = t wi. By our hypotheses the map 0 0 0 pW ∗ρ : H (P(W ), OP(W )(1)) → H (C,LW ) is injective. Write wi := pW ∗ρ(wi). Let Ej 0 0 be the invertible subsheaf of LW generated by w0, ..., wj for 0 ≤ j ≤ N = e − g, and write

ej = deg Ej. Note that EN = LW since LW is very ample hence generated by global sections, 0 0 0 h (C,LW ) = e−g +1, and w0, ..., wN 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 0 1 sections we have h (C,Ej) ≥ j + 1. If ej = deg Ej > 2g − 2 then H (C,Ej) = 0 so by 0 1 0 −1 Riemann-Roch ej = h − h + g − 1 ≥ j + g. If ej ≤ 2g − 2 then H (C, ωC ⊗ Ej ) 6= 0 so by 0 ej Cliﬀord’s theorem j + 1 ≤ h ≤ 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 X (rit+1 − rit )(eit+1 + eit ) > 2rN eN + 2(rN − r0). t=1 d Suppose p and n are large positive integers. (In this proof n is not g−1 .) Recall that 0 H (P(W ), OP(W )((p + 1)n)) has a basis consisting of monomials of degree (p + 1)n in 0 w0, ..., wN . For all 1 ≤ t ≤ k let Vit ⊂ H (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 p−s s n 0 (V V VN ) ⊂ H (P(W ), OP(W )((p + 1)n)) be the subspace spanned by elements of the it1 it2 form v1 ··· vn, where each vj is of the following form: For s = 0, v ∈{x ··· x z | x ∈S , z ∈H0(P(W ), O (1))}. j j1 jp j jα it1 j P(W ) For 0

37 0 This gives a ﬁltration of H (P(W ), OP(W )((p + 1)n)):

0 ⊂ (V pV 0V )n ⊂ (V p−1V 1V )n ⊂ · · · ⊂ (V 1V p−1V )n i1 i2 N i1 i2 N i1 i2 N p 0 n p−1 1 n 1 p−1 n ⊂ (Vi Vi VN ) ⊂ (Vi Vi VN ) ⊂ · · · ⊂ (Vi Vi VN ) . . 2 3 . . 2 3 . . . . 2 3 ...... ⊂ (V pV 0 V )n ⊂ (V p−1V 1 V )n ⊂ · · · ⊂ (V 1V p−1V )n it it+1 N it it+1 N it it+1 N (6) ...... ⊂ (V p V 0 V )n ⊂ (V p−1V 1 V )n ⊂ · · · ⊂ (V 1 V p−1V )n ik−1 ik N ik−1 ik N ik−1 ik N ⊂ (V 0 V p V )n = H0(P(W ), O ((p + 1)n)). ik−1 ik N P(W )

p−s s n Note that while we have defined (V V VN ) whenever t1 < t2 we only use consecutive it1 it2 integers t1 and t1 + 1 in the filtration (6). p−s s n 0 r Tensor each (V V V ) with H (P , OPr (m ˆ )) to get a filtration of it1 it2 0 r H (P(W )× P , OP(W )((p + 1)n) ⊗ OPr (m ˆ )). Assume that (p + 1)n andm ˆ are sufficiently large that

0 r 0 (p+1)n mˆ ρˆ(p+1)n,mˆ : H (P(W )× P , OP(W )((p + 1)n) ⊗ OPr (m ˆ )) → H (C,LW ⊗ Lr ) is surjective. Write

¯ p−s ¯ s ¯ n p−s s n 0 r (V \V VN ) :=ρ ˆ(p+1)n,mˆ ((V V VN ) ⊗ H (P , OPr (m ˆ )). (7) it1 it2 it1 it2

0 (p+1)n mˆ ¯ p−s ¯ s ¯ n Then we have a ﬁltration H (C,L ⊗L ) given by subspaces of the form (V \V VN ) . W r it1 it2

0 Claim 15.3 There exists an integer n which is independent of C, t1, and t2 such that if n ≥ n0 then

p−s s n 0 r 0 p−s s n mˆ ρˆ(p+1)n,mˆ ((V V VN ) ⊗ H (P , OPr (m ˆ )) = H (C, (E ⊗ E ⊗ LW ) ⊗ L ). it1 it2 it1 it2 r

Proof of Claim. By hypothesis LW is very ample. Note that deg Lr = d > 0 since C is mˆ nonsingular hence irreducible. Thus ifm ˆ > 2g + 1 then Lr is very ample.

It follows from the deﬁnitions of the sheaves Ej that the linear system p−s s n 0 r (V V VN ) ⊗ H (P , OPr (m ˆ )) restricted to C generates it1 it2

p−s s n mˆ (E ⊗ E ⊗ LW ) ⊗ L it1 it2 r and

p−s s n 0 r 0 p−s s n mˆ ρˆ(p+1)n,mˆ ((V V VN ) ⊗ H (P , OPr (m ˆ )) ⊆ H (C, (E ⊗ E ⊗ LW ) ⊗ L ). (8) it1 it2 it1 it2 r

38 mˆ 1 Suppose thatm ˆ = nmˆ 1 andm ˆ 1 > 2g + 1 so that Lr is very ample hence generated by global sections. E and E are generated by global sections, so it follows that it1 it2 p−s s 0 r ρˆp+1,mˆ (V V VN ⊗ H (P , OPr (m ˆ 1))) is a very ample base point free linear system on C. 1 it1 it2 Much of the notation we are about to deﬁne will be abbreviated in displayed equations. Let ψ := ψp+1,mˆ 1 be the projective embedding corresponding to the linear system p−s s 0 r ρˆp+1,mˆ (V V VN ⊗ H (P , OPr (m ˆ 1))). Let IC/P be the ideal sheaf deﬁning C as a closed 1 it1 it2 p−s s 0 r subscheme of P := P(ˆρp+1,mˆ (V V VN ⊗ H (P , OPr (m ˆ 1)))). There is an exact sequence 1 it1 it2 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

p−s s mˆ 1 F := E ⊗ E ⊗ LW ⊗ L it1 it2 r We have

(ψ∗OC )(n) := ψ∗OC ⊗OP OP(n) ∼ n = ψ∗OC ⊗OP OP(V)(1) ∼ ∗ n = ψ∗(OC ⊗OC ψ OP(V)(1) ) by the projection formula, cf. [Hart] page 124 ∼ n ∗ ∼ = ψ∗(F ) since ψ OP(V)(1) = F.

Now the exact sequence (9) reads

n 0 → IC/P(n) → OP(n) → ψ∗(F ) → 0. (10)

In the corresponding long exact sequence in cohomology we have

0 0 n 1 · · · → H (P, OP(n)) → H (P, ψ∗F ) → H (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 n0 > 0 depending on the Hilbert polynomial P but not on the curve 1 0 C such that H (P, IC/P(n)) = 0 if n > n . Then for such n the exact sequence (11) implies that the map 0 0 n H (P, OP(n)) → H (C, F )

39 p−s s 0 r is surjective. Recall that P := P(ˆρp+1,mˆ (V V VN ⊗ H (P , OPr (m ˆ 1)))). Then 1 it1 it2

0 ∼ n p−s s 0 r H (P, OP(n)) = Sym (ˆρp+1,mˆ (V V VN ⊗ H (P , OPr (m ˆ 1)))). 1 it1 it2

Also there is a surjection

n p−s s 0 r n p−s s 0 r Sym (V V VN ⊗ H (P , OPr (m ˆ 1))) → Sym (ˆρp+1,mˆ (V V VN ⊗ H (P , OPr (m ˆ 1)))) it1 it2 1 it1 it2 so putting this all together we have a surjection

n p−s s 0 r 0 n Sym (V V VN ⊗ H (P , OPr (m ˆ 1))) → H (C, F ). (12) it1 it2

There is a natural map

p−s s n 0 r =∼ n p−s s 0 r (V V VN ) ⊗ H (P , OPr (m ˆ )) → Sym (V V VN ⊗ H (P , OPr (m ˆ 1))) it1 it2 it1 it2 and in factρ ˆ(p+1)n,mˆ factors as

p−s s n 0 r n p−s s 0 r (V V VN ) ⊗ H (P , OPr (m ˆ )) −→ Sym (V V VN ⊗ H (P , OPr (m ˆ 1))) it1 it2 it1 it2

ρˆ(p+1)n,mˆ & ↓ (12) H0(C, F n) so

p−s s n 0 r 0 p−s s n mˆ ρˆ(p+1)n,mˆ :(V V VN ) ⊗ H (P , OPr (m ˆ )) → H (C, (E ⊗ E ⊗ LW ) ⊗ L ) (13) it1 it2 it1 it2 r is also surjective. It follows from lines (8) and (13) that

¯ p−s ¯ s ¯ n 0 p−s s n mˆ (V \V VN ) = H (C, (E ⊗ E ⊗ LW ) ⊗ L ). t1 t2 it1 it2 r

40 Proof of Theorem 15.2 continued. Take n ≥ 2g + 1 so that 1 p−s s n mˆ H (C, (E ⊗ E ⊗ LW ) ⊗ L ) = 0. We use Riemann-Roch to calculate it1 it2 r dim (V¯ p−\sV¯ s V¯ )n = h0(C, (Ep−s ⊗ Es ⊗ L )n ⊗ Lmˆ ) it it+1 it it+1 W r

= n((p − s)eit + seit+1 + eN ) + dmˆ − g + 1. (14)

We assume for the rest of the proof that p and n are suﬃciently large that 3 e + 1 p > max{e + g, 2 }, n > max{p, (2g + 1)n0}, m := (p + 1)n > m000.

ˆ 0 r Choose a basis B(p+1)n,mˆ of H (P(W )× P , OP(W )((p + 1)n) ⊗ OPr (m ˆ )) of monomials ˆ ˆ ˆ Mi of bidegree ((p + 1)n, mˆ ) such thatρ ˆ(p+1)n,mˆ M1, ..., ρˆ(p+1)n,mˆ MP ((p+1)n)+dmˆ is a basis of H0(C,L(p+1)n ⊗ Lmˆ ). Observe that a monomial Mˆ ∈ (V¯ p\−sV¯ sV¯ )n − (V¯ p−s\+1V¯ s−1V¯ )n has W r i1 i2 i1 i2 ˆ λ-weight wλ(M) ≤ n((p − s)rit + srit+1 + rN ). ˆ ˆ We now estimate the total λ-weight of M1, ..., MP ((p+1)n)+dmˆ :

P (m)+dmˆ X w (Mˆ ) ≤ n(pr + r ) dim (V¯\pV¯ 0V¯ )n λ i i1 N i1 i2 i=1 X + n((p−s)r + sr + r ) dim((V¯ p−\sV¯ s V¯ )n − (V¯ p−s\+1V¯ s−1V¯ )n) (15) it it+1 N it it+1 it it 0 ≤ s ≤ p 1 ≤ t ≤ k − 1 The ﬁrst term on the right hand side of (15) is

n(pri1 +rN )(n(pei1 +eiN )+dmˆ −g +1) = n(pri1 +rN )dmˆ +n(pri1 +rN )(n(pei1 +eiN )−g +1).

The factor dim((V¯ p−\sV¯ s V¯ )n − (V¯ p−s\+1V¯ s−1V¯ )n) of the summand is it it+1 it it n((p − s)ei + sei + eN ) + dmˆ − g + 1 t t+1 − n((p − s + 1)eit + (s − 1)eit+1 + eN ) + dmˆ − g + 1 = n(eit+1 − eit ). Note that nearly all of the terms having dmˆ as a factor have “telescoped.” We have

RHS (15) = n(pr + r ) dim (V¯\pV¯ 0V¯ )n i1 N i1 i2 X + n((p−s)r +sr +r ) dim((V¯ p−\sV¯ s V¯ )n − (V¯ p−s\+1V¯ s−1V¯ )n) it it+1 N it it+1 it it 0 ≤ s ≤ p 1 ≤ t ≤ k − 1

= n(pri1 + rN )dmˆ + n(pri1 + rN )(n(pei1 + eiN ) − g + 1) X + n((p−s)rit + srit+1 + rN ) n(eit+1 − eit ) . 0 ≤ s ≤ p 1 ≤ t ≤ k − 1

41 The sum of the second two terms is exactly the expression Gieseker obtains at the bottom of page 30. Gieseker calculates (page 34): X n(pri1 + rN )(n(pei1 + eiN ) − g + 1) + n((p − s)rit + srit+1 + rN ) n(eit+1 − eit ) 0≤s≤p 1≤t≤k−1 2 h 3e e+g−1 i < n p(rN − r0) −p + 2 + p < 0.

3 2 e+1 where the last inequality follows because p > max{e + g, }. Therefore we have

P (m)+dmˆ X ˆ wλ(Mi) ≤ 0 + n(pr0 + rN )dmˆ < 0. i=1

rN The last inequality follows because r0 < 0 and because by hypothesis p > | |. r0 PN PN−1 (More precisely, the hypothesis states p > e+g. But we have i=0 ri = 0. If i=1 ri ≥ 0 then

rN rN rN = = ≤ 1. r −r PN−1 0 0 rN + i=1 ri PN−1 If i=1 ri < 0 then since ri ≥ r0 for all i > 0 we have

r r + PN−1 r PN−1 r N 0 i=1 i i=1 i = ≤ 1 + ≤ 1 + N − 2 = e − g − 1 < e + g. r0 r0 r0

So p > | rN |.) r0 ˆ 0 By the numerical criterion, the point Hm,mˆ (C) is λ -stable. Nowhere in the proof have we placed any conditions on the 1-PS λ0, so the result is true for every 1-PS of SL(W ). Then ˆ Hm,mˆ (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 ˆ linearizations. The results 16.1-16.14 show that for certain linearizations, if Hm,mˆ (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

000 Proposition 16.1 (cf. [Gies] 1.0.2) Suppose that m, mˆ > m and m > (q1 +1)(e−g+1). ˆ Then for all C such that Hm,mˆ (C) is SL(W )-semistable, the composition

0 0 0 H (P(W ), OP(W )(1)) → H (pW (C)red, OpW (C)red (1)) → H (Cred,LW red)

is injective.

Proof. Suppose the map

0 0 H (P(W ), OP(W )(1)) → H (Cred,LW red)

has nontrivial kernel W0. Write N0 = dim W0. Choose a basis w0, ..., wN of 0 H (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 − 1 ∗ λ(t)wi = twi, t ∈ C ,N0 ≤ i ≤ N and let λ0 be the associated 1-PS of SL(W ). Choose m andm ˆ suﬃciently large as previ- ˆ 0 r ously explained. Let Bm,mˆ be a basis of H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) consisting of monomials of bidegree (m, mˆ ). ˆ 0 ˆ ˆ ˆ Suppose Hm,mˆ (C) is λ -semistable. Then there exist monomials Mi1 , ..., MiP (m)+dmˆ in Bm,mˆ ˆ ˆ 0 m mˆ such that {ρˆm,mˆ (Mi1 ), ..., ρˆm,mˆ (MiP (m)+dmˆ )} is a basis of H (C,LW ⊗ Lr ) and

P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) j=1 ≤ i=0 . m(P (m) + dmˆ ) e − g + 1

q1 m−q1 m ∼ 0 Let W0 W1 denote the subspace of Sym W1 = H (P(W ), OP(W )(m)) spanned by elements of the form x1 ··· xq1 y1 ··· ym−q1 where xi ∈ W0 and yi ∈ W1. Let IC denote the

q1 ideal sheaf of nilpotent elements of OC . Recall that the integer q1 satisﬁes IC = 0. It follows q1 m−q1 that the image of the vector space W0 W1 under

0 r 0 m mˆ ρˆm,mˆ : H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) → H (C,LW ⊗ Lr )

ˆ q1 m−q1 ˆ ˆ is zero. Therefore, for all j, Mij 6∈ W0 W1 sinceρ ˆm,mˆ (Mij ) 6= 0. In particular if Mij is γ0 γN ˆ written as w0 ··· wN , we must have γ0 < q1. It follows that wλ(Mij ) ≥ m − q1 + 1, because

43 P P wλ(Mij ) = γiri, where r0 = 0 and ri = 1 if i > 0, and γi = m. Thus the total weight ˆ ˆ of any P (m) + dmˆ monomials Mi1 , ..., MiP (m)+dmˆ whose images underρ ˆm,mˆ form a basis of 0 m mˆ H (C,LW ⊗ Lr ) must be greater than or equal to (P (m) + dmˆ )(m − q1 + 1). In symbols,

P (m)+dmˆ X ˆ wλ(Mij ) ≥ (P (m) + dmˆ )(m − q1 + 1). j=0

N X Note too that wλ(wi) = dim W1 − dim W0 = e − g + 1 − dim W0 ≤ e − g because i=0 dim W0 ≥ 1. Combining these three inequalities, we have

P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) (P (m) + dmˆ )(m − q + 1) j=1 e − g 1 ≤ ≤ i=0 ≤ m(P (m) + dmˆ ) m(P (m) + dmˆ ) e − g + 1 e − g + 1 q − 1 1 1 − 1 ≤ 1 − m e − g + 1 m ≤ (q1 + 1)(e − g + 1) ˆ But by hypothesis m > (q1 + 1)(e − g + 1). The contradiction implies that Hm,mˆ (C) is not 0 ˆ λ -semistable, and therefore Hm,mˆ (C) is not SL(W )-semistable.

Let Ci be an irreducible component of C. If the morphism pW |Ci does 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- ∗ ∗ 0 nents of pW (C), i = 1, ..., `. Recall that LW denotes ι pW OP(W )(1). We write C = C ∪ Y 0 0 where the multidegree of LW is zero on C and nowhere zero on Y . That is, C is the 0 union of all irreducible components of C which collapse under pW and Y = C − C . Let 0 0 0 0 0 0 C1,1, ..., C 0 ,Y1,1, ..., Y1,j1 ,C2,1, ..., C2,j ,Y2,1, ..., Y2,j2 , ..., C , ..., C 0 ,Y`,1, ..., Y`,j` be the irre- 1,j1 2 `,1 `,j` ducible components of C ordered so that

0 0 i. pW (Ci,j0 ) ∈ 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 deg O (1) =: e and deg O (1) =: e . Since p (C) ⊂ P(W ) we pW (C)red P(W ) W pW (C)i red P(W ) W i W ∗ ∗ ∗ ∗ have eW i ≥ 1 for all i. Recall that LW denotes ι pW OP(W )(1) and Lr denotes ι prOPr (1).

44 Let deg L = e , and let deg L = d . Finally let k = lengthO and Yi,j red W i,j Yi red r i,j i,j Ci,j ,ξi,j

ki = lengthOpW (C)i,ξi . Then we have: X e = ki,jeij

ei,j = ni,jeW i X e = ki,jni,jeW i X eW = kieW i

Proposition 16.2 Suppose that m, mˆ > m000 and 3 m > (g − + e(q + 1) + q + µ m00)(e − g + 1) 2 1 3 1 and 1 mˆ (3n + 2)a − 5 − < 2(g−1) . m 4n ˆ Then for all curves C such that Hm,mˆ (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

0 0 H (P(W ), OP(W )(1)) → H (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

0 0 0 h (pw(C)1 red, OpW (C)1 red (1)) ≤ h (pW (C)1 red, OD1 ) + h (pW (C)1 red, OpW (C)1 red ).

0 Note that h (pW (C)1 red, OpW (C)1 red ) = 1, and

0 h (pW (C)1 red, OD1 ) ≤ deg D1 = deg OpW (C)1 red (1) = eW 1.

45 If W0 = 0, then

0 e − g + 1 = h (P(W ), OP(W )(1))

0 ≤ h (pW (C)1 red, OpW (C)1 red (1)) e1,1 ≤ eW 1 + 1 = + 1 n1,1 X ⇒ k1,1n1,1(e − g) ≤ e − ki,jni,jeW i (i,j)6=(1,1) X ⇒ k1,1n1,1(e − g) < e − ki,jni,jeW i < e (i,j)6=(1,1)

⇒ (k1,1n1,1 − 1)e ≤ k1,1n1,1g (16)

k1,1n1,1−1 1 e If n1,1 ≥ 2 or if k1,1 ≥ 2 then ≥ and we have ≤ g. But e > 2g so we have a k1,1n1,1 2 2 contradiction.

If n1,1 = k1,1 = 1 but j1 ≥ 2 we have as above (e − g) ≤ e1,1. But we also have P e1,1 = e − 6=1,1 ki,jni,jeW i ≤ e − k1,2n1,2eW 1 < e − eW 1 = 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 = dim W0. Choose a basis w0, ..., wN of 0 0 H (P(W ), OP(W )(1)) relative to the ﬁltration 0 ⊂ W0 ⊂ W1 = H (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 − 1 ∗ λ(t)wi = twi, t ∈ C ,N0 ≤ i ≤ N 0 ˆ and let λ be the associated 1-PS of SL(W ). Choose m andm ˆ suﬃciently large. Let Bm,mˆ be 0 r a basis of H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) consisting of monomials of bidegree (m, mˆ ). 0 m mˆ m−p p Construct a ﬁltration of H (C,LW ⊗ Lr ) as follows: For 0 ≤ p ≤ m let W0 W1 be the 0 subspace of H (P(W ), OP(W )(m)) spanned by elements of the following type:

{x1 ··· xm|xi ∈ W0} if p = 0 {x1 ··· xm−py1 ··· yp|xi ∈ W0, yi ∈ W1} if 0 < p < m {y1 ··· ym|yi ∈ W1} if p = m.

\m−p p m−p p 0 r Set W0 W1 = W0 W1 ⊗ H (P , OPr (m ˆ )), and let

¯\m−p ¯ p \m−p p 0 m mˆ W0 W1 :=ρ ˆm,mˆ (W0 W1 ) ⊂ H (C,LW ⊗ Lr ). (17) ˆ ¯\m−p ¯ p ¯ m−\p+1 ¯ p−1 Note that a monomial M ∈ W0 W1 − W0 W1 has weight p. Therefore we have a 0 m mˆ ﬁltration of H (C,LW ⊗ Lr ) in order of increasing weight:

¯ m ¯ 0 ¯\m−1 ¯ 1 ¯ 0 ¯ m 0 m mˆ 0 ⊆ W\0 W1 ⊆ W0 W1 ⊆ · · · ⊆ W\0 W1 = H (C,LW ⊗ Lr )

46 ˆ ¯\m−p ¯ p Write βp = dim W0 W1 . j0 S 1 0 Sj1 ˜ Let C1 = j0=1 C1,j0 ∪ j=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,Lm ⊗ Lmˆ ) → H0(C,L˜ m ⊗ Lmˆ ) be the map induced by restriction. W r W C˜ r C˜

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,

¯\m−p ¯ p 0 m mˆ 0 ˜ m mˆ W0 W1 ∩ ker{R : H (C,LW ⊗ Lr ) → H (C,LW C˜ ⊗ LrC˜)} = 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

m−p m mˆ m mˆ m−p m mˆ 0 → IP ⊗ LW C˜ ⊗ Lr C˜ → LW C˜ ⊗ Lr C˜ → OC˜/IP ⊗ LW C˜ ⊗ Lr C˜ → 0.

In cohomology we have

0 → H0(C,˜ Im−p ⊗ Lm ⊗ Lmˆ ) → H0(C,L˜ m ⊗ Lmˆ ) P W C˜ r C˜ W C˜ r C˜ → H0(C,˜ O /Im−p ⊗ Lm ⊗ Lmˆ ) → H1(C,˜ Im−p ⊗ Lm ⊗ Lmˆ ) → 0. C˜ P W C˜ r C˜ P W C˜ r C˜ The following ﬁve facts are analogous to those stated by Gieseker in [Gies] (page 44):

I. h0(C,L˜ m ⊗ Lmˆ ) = χ(Lm ⊗ Lmˆ ) = deg Lm ⊗ Lmˆ + χ(O ) W C˜ r C˜ W C˜ r C˜ C˜ W C˜ r C˜ C˜ P P P 0 ˜ ≤ (e − k1,je1,j)m + (d − k1,jd1,j − k1,j0 d1,j0 )m ˆ + h (C, O ˜) P P P C ≤ (e − k1,je1,j)m + (d − k1,jd1,j − k1,j0 d1,j0 )m ˆ + q3. II. h0(C,˜ O /Im−p ⊗ Lm ⊗ Lmˆ ) ≥ m − p. C˜ P W C˜ r C˜ III. h0(C,˜ O /Im−p ⊗ Lm ⊗ Lmˆ ) ≤ µ (m − p) + µ , so C˜ P W C˜ r C˜ 1 2 h1(C,˜ Im−p ⊗ Lm ⊗ Lmˆ ) ≤ µ (m − p) + µ . P W C˜ r C˜ 1 2 IV. For p ≥ m00, we have h1(C,˜ Im−p ⊗ Lm ⊗ Lmˆ ) = 0. P W C˜ r C˜ ¯\m−p ¯ p V. The image of W0 W1 under R is contained in the subspace H0(C,˜ Im−p ⊗ Lm ⊗ Lmˆ ) of H0(C,L˜ m ⊗ Lmˆ ). P W C˜ r C˜ W C˜ r C˜ By the claim and fact V., we have βˆ = dim W¯\m−pW¯ p ≤ h0(C,˜ Im−p ⊗ Lm ⊗ Lmˆ ) if p 0 1 P W C˜ r C˜

0 ≤ p ≤ m − q1. From the exact sequence we have

0 ˜ m−p m mˆ 0 ˜ m mˆ h (C, IP ⊗ LW C˜ ⊗ Lr C˜) = h (C,LW C˜ ⊗ Lr C˜) 0 ˜ m−p m mˆ − h (C, OC˜/IP ⊗ LW C˜ ⊗ Lr C˜) 1 ˜ m−p m mˆ + h (C, IP ⊗ LW C˜ ⊗ Lr C˜).

47 Thus, using the facts above, we have:  P P P (e− k1,je1,j)m + (d− k1,jd1,j − k1,j0 d1,j0 )m ˆ + q3 + p − m + µ1(m−p) + µ2  00  if 0 ≤ p ≤ m − 1 ˆ  P P P βp ≤ (e − k1,je1,j)m + (d − k1,jd1,j − k1,j0 d1,j0 )m ˆ + q3 + p − m 00  if m ≤ p ≤ m − q1   em + dmˆ − g + 1 if m − q1 + 1 ≤ p ≤ m.

ˆ 0 ˆ ˆ Now suppose that Hm,mˆ (C) is λ -semistable. Then there exist monomials Mi1 , ..., Miα in ˆ ˆ ˆ 0 m mˆ Bm,mˆ such that {ρˆm,mˆ (Mi1 ), ..., ρˆm,mˆ (Miα )} is a basis of H (C,LW ⊗ Lr ) and

α N X ˆ X wλ(Mij ) wλ(wi) j=1 ≤ i=0 . m(em + dmˆ + 1 − g) e − g + 1

α m X ˆ X ˆ ˆ But wλ(Mij ) must be larger than p(βp − βp−1). We calculate: j=1 p=1 m m−1 X ˆ ˆ ˆ X ˆ p(βp − βp−1) = mβm − βp p=1 p=0

≥ m(em + dmˆ + 1 − g)

m−q1 X X X X − (e − k1,je1,j)m + (d − k1,jd1,j − k1,j0 d1,j0 )m ˆ + q3 + p − m p=0 m00−1 m−1 X X − (µ1(m − p) + µ2) − (em + dmˆ + 1 − g)

p=0 m−q1+1 X 1 = k e + m2 1,j 1,j 2 3 X X X 00 + m − g + k d + k 0 d 0 mˆ − q − k e (q + 1) − µ m 2 1,j 1,j 1,j 1,j 3 1,j 1,j 1 1 00 00 X X q1 00 m (m − 1) + (q − 1)(g + q − k d − k 0 d 0 mˆ − − 1) − µ m + µ 1 3 1,j 1,j 1,j 1,j 2 2 1 2 X 1 2 X X ≥ k e + m − S m + k d + k 0 d 0 mˆ (m − q + 1) + c 1,j 1,j 2 2 1,j 1,j 1,j 1,j 1 2 X 1 ≥ k e + m2 − S m (18) 1,j 1,j 2 2 where

3 X S = g − + k e (q + 1) + q + µ m00 2 2 1,j 1,j 1 3 1 q m00(m00 − 1) c = (q − 1)(g + q − 1 − 1) − µ m00 + µ . 2 1 3 2 2 1 2

48 P P The inequality (18) follows because the term ( k1,jd1,j + k1,j0 d1,j0 )m ˆ (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 X 0 wλ(wi) = dim W1 − dim W0 ≤ h (pW (C1,1)red, OpW (C1,1)red (1)) i=0

≤ deg OpW (C1,1) red(1) + 1 ≤ eW 1 + 1.

The hypothesis on m implies m > S2(e − g + 1). Then we obtain a contradiction as follows:

P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) P 1 2 ( k e + )m − S m j=1 e + 1 1,j 1,j 2 2 ≤ ≤ i=0 ≤ W 1 m(em + dmˆ + 1 − g) m(em + dmˆ + 1 − g) e − g + 1 e − g + 1 P k e + 1 − S2 e + 1 1,j 1,j 2 m ≤ W 1 dmˆ e − g + 1 e + m X 1 S2(e − g + 1) (eW 1 + 1)dmˆ (e − g + 1)( k e + ) − e(e + 1) ≤ + 1,j 1,j 2 W 1 m m X 1 (eW 1 + 1)dmˆ (e − g + 1)( k e + ) − e(e + 1) ≤ 1 + 1,j 1,j 2 W 1 m (e − g + 1)(P k e + 1 ) − e(e + 1) − 1 mˆ 1,j 1,j 2 W 1 ≤ (eW 1 + 1)d m (e − g + 1)(e P k n + 1 ) − e(e + 1) − 1 mˆ W 1 1,j 1,j 2 W 1 ≤ . (eW 1 + 1)d m P Note that since n1,1 ≥ 2 or j1 ≥ 2 we have k1,jn1,j ≥ 2. Thus X 1 (e − g + 1)(e k n + ) − e(e + 1) − 1 > 0. W 1 1,j 1,j 2 W 1 Furthermore the quantity

P 1 (e − g + 1)(eW 1 k1,jn1,j + 2 ) − e(eW 1 + 1) − 1 (eW 1 + 1)d

is minimized when eW 1 takes its smallest value, that is, when eW 1 = 1. Then

(e − g + 1)(e P k n + 1 ) − e(e + 1) − 1 (3n + 2)a − 5 − 1 W 1 1,j 1,j 2 W 1 = 2(g−1) . (eW 1 + 1)d 4n

(3n+2)a−5− 1 mˆ 2(g−1) ˆ But by hypothesis m < 4n . The contradiction implies that Hm,mˆ (C) is not 0 ˆ λ -semistable, and therefore that Hm,mˆ (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. ∼ ∼ Tni Suppose Ui = SpecAi and Ui ∩ Uk = SpecAik. In each Ai let (0) = 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 0 C , ..., Y correspond to p , ..., p 0 . We deﬁne an ideal subsheaf I of O as follows: 1,1red 1,j1red i1 i(j1+j1) C Tni Tni If Ui ∩ C1 6= 0 then I(Ui) = 0 qij, and if Ui ∩ C1 = 0 then I(Ui) = qij = (0). j>j1+j1 j=1 Let C˜ be the subscheme of C deﬁned by I. We show that C˜ has the desired property: Let ¯\m−p ¯ p ˜ s ∈ W0 W1 ∩ker R. Let si denote the restriction of s to Ui. For each Ui let ωi : LW |Ui → Ai ˜ m mˆ ˜ and ρi : Lr|Ui → Ai and γi : LW ⊗ Lr → Ai be the trivializing isomorphisms. Now if

Ui ∩ C1 = ∅ then si = 0 since si ∈ ker R. If Ui ∩ C1 6= ∅ then write ai := γi(si). Since

q1 0 q1 0 m − p ≥ q1, we have ai ∈ pij for all j = 1, ..., j1 + j1. Also pij ⊂ qij for all j = 1, ..., j1 + j1, so 0 T T ai ∈ qij for all j = 1, ..., j + j1. Since si ∈ ker R we have ai ∈ 0 qij. Thus ai ∈ qij, 1 j>j1+j1 j ¯\m−p ¯ p so si = 0. Thus W0 W1 ∩ ker R = 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 C0 6= C be a reduced, complete subcurve of C and let Y be the closure of C − C0 in C with ι the reduced structure. Let C0 →C0 C →ι P(W )× Pr and Y →ιY C be the inclusion morphisms. Let ∗ ∗ ∗ ∗ ∗ ∗ LWC0 := ιC0 ιC pW OP(W )(1) LWY := ιY ιC pW OP(W )(1) ∗ ∗ ∗ ∗ ∗ ∗ (19) Lr C0 := ιC0 ιC prOPr (1) Lr Y := ιY ιC prOPr (1). ¯ ¯ ∗ Let π : C → C be the normalization morphism. Let LW C¯0 := π LW C¯0 . and similarly for 0 ¯ the other three line bundles deﬁned above. Deﬁne e := degC¯0 LW C¯0 = degC0 LWC0 and 0 0 0 0 ¯ 0 0 d := degC¯0 Lr C¯0 = degC0 Lr C . Finally, write h (pW (C ), OpW (C )(1)) =: h .

Lemma 16.4 (cf. [Gies] 1.0.7) Suppose that m, mˆ > m000 and 3 m > (g − + e(q + 1) + q + µ m00)(e − g + 1) 2 1 3 1

50 and 1 mˆ (3n + 2)a − 5 − < 2(g−1) . m 4n ˆ Let C be a curve such that Hm,mˆ (C) is SL(W )-semistable, and suppose C has at least two 0 ¯ irreducible components. Let C and Y as above. Suppose there exist points P1, ..., Pk on Y satisfying

0 i. π(Pi) ∈ Y ∩ C for all 1 ≤ i ≤ k ¯ ¯ ii. for each irreducible component Yj of Y ,

¯ ¯ degY¯j (LW Y (−D)) ≥ 0,

where D = P1 + ··· + Pk.

Then 0 0 (e0 + k ) h0 + (dh −d (e−g+1))m ˆ S 2 < em + , (20) e e − g + 1 em k where S = g + k(2g − 1) + q2 − gY¯ + 2 .

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. S Remark. In all our applications the term em will be made small by taking m, mˆ >> 0. Proof. Suppose ﬁrst that C0 is connected. Let

0 0 0 0 W0 := ker{H (P(W ), OP(W )(1)) → H (pW (C ), OpW (C )(1))}. (21)

0 Choose a basis w0, ..., wN of H (P(W ), OP(W )(1)) relative to the ﬁltration 0 0 ⊂ W0 ⊂ W1 = H (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 − 1 ∗ λ(t)wi = twi, t ∈ C ,N0 ≤ i ≤ N and let λ0 be the associated 1-PS of SL(W ).

51 m−p p 0 For 0 ≤ p ≤ m let W0 W1 be the subspace of H (P(W ), OP(W )(m)) spanned by elements of the following type:

{x1 ··· xm|xi ∈ W0} if p = 0 {x1 ··· xm−py1 ··· yp|xi ∈ W0, yi ∈ W1} if 0 < p < m {y1 ··· ym|yi ∈ W1} if p = m.

\m−p p m−p p 0 r Set W0 W1 = W0 W1 ⊗ H (P , OPr (m ˆ )), and let

¯\m−p ¯ p \m−p p 0 m mˆ W0 W1 :=ρ ˆm,mˆ (W0 W1 ) ⊂ H (C,LW ⊗ Lr ). (22)

¯\m−p ¯ p ¯ m−\p+1 ¯ p−1 Note that a monomial M ∈ W0 W1 − W0 W1 has weight p. Therefore we have a 0 m mˆ ﬁltration of H (C,LW ⊗ Lr ) in order of increasing weight:

¯ m ¯ 0 ¯\m−1 ¯ 1 ¯ 0 ¯ m 0 m mˆ 0 ⊆ W\0 W1 ⊆ W0 W1 ⊆ · · · ⊆ W\0 W1 = H (C,LW ⊗ Lr )

ˆ ¯\m−p ¯ p Write βp = dim W0 W1 . The normalization morphism π : C¯ → C induces a homomorphism

0 m mˆ 0 ¯ ¯m ¯mˆ πm,mˆ ∗ : H (C,LW ⊗ Lr ) → H (C, LW ⊗ Lr ). (23)

0 ¯ ¯m ¯mˆ 0 ¯0 ¯m ¯mˆ 0 ¯ ¯m ¯mˆ There is a splitting H (C, LW ⊗ Lr ) = H (C , LW C¯0 ⊗ LrC¯0 ) ⊕ H (Y, LW Y¯ ⊗ LrY¯ ). By def- \m−p p ¯0 inition the sections in πm,mˆ ∗(W0 W1 ) vanish on C and to order ≥ m − p at the points \m−p p 0 ¯ ¯m ¯mˆ 0 ¯ ¯m ¯mˆ P1, ..., Pk. Thus πm,mˆ ∗(W0 W1 ) ⊆ H (Y, LW Y¯ ((p − m)D) ⊗ LrY¯ ) ⊆ H (C, LW ⊗ Lr ). Then

ˆ ¯\m−p ¯ p 0 ¯ ¯m ¯mˆ βp = dim W0 W1 ≤ h (Y, LW Y¯ ⊗ LrY¯ ⊗ OY¯ ((p − m)D)) + dim ker πm,mˆ ∗ 0 0 = (e − e )m + (d − d )m ˆ + k(p − m) − gY¯ + 1 1 ¯ ¯m ¯mˆ + h (Y, LW Y¯ ⊗ LrY¯ ⊗ OY¯ ((p − m)D)) + dim ker πm,mˆ ∗. We have the following three estimates:

I. dim ker πm,mˆ ∗ < q2.

Y Y 0 m mˆ 0 ¯ ¯m ¯mˆ Proof. ker πm,mˆ ∗ ⊂ ker πm,mˆ ∗ where πm,mˆ ∗ : H (Y,LWY ⊗ LrY ) → H (Y, LWY ⊗ LrY ) is the homomorphism induced by the normalization of Y . Now let IY denote the ideal 0 sheaf of nilpotents in OY . We can choose an integer q2 such that h (C, IC ) < q2 hence 0 Y h (Y, IY ) < q2 as well. Then dim ker πm,mˆ ∗ < q2. To see this, recall that the normalization 0 morphism factors through Yred. We have π = ι ◦ π , where ι : Yred ,→ Y . Then we have 0 0 πm,mˆ ∗ = πm,mˆ ∗ ◦ ιm,mˆ ∗. Now, πm,mˆ ∗ is injective, so ker πm,mˆ ∗ = ker ιm,mˆ ∗. We have assumed

52 that Y is generically reduced, so IY has ﬁnite support. We have an exact sequence of sheaves m mˆ m mˆ m mˆ 0 → IY ⊗ LW ⊗ Lr → LW ⊗ Lr → LW red ⊗ Lrred → 0. This gives rise to an exact sequence ∼ 0 m mˆ in cohomology. It follows that ker πm,mˆ ∗ = ker ιm,mˆ ∗ = H (Y, IY ⊗ LW ⊗ Lr ) and hence 0 m mˆ 0 dim ker πm,mˆ ∗ = h (Y, IY ⊗ LW ⊗ Lr ) = h (Y, IY ) < q2.

1 ¯ ¯m ¯mˆ II. h (Y, LW Y¯ ⊗ Lr 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 ¯ ¯m ¯mˆ sheaves on Y . Since LW Y¯ ⊗ Lr Y¯ is an invertible sheaf, it is locally free and in particular ﬂat. Tensoring, we obtain a second short exact sequence

¯m ¯mˆ ¯m ¯mˆ 0 → LW Y¯ ⊗ Lr Y¯ ⊗ OY¯ (−(m − p)D) → LW Y¯ ⊗ Lr Y¯ → O(m−p)D → 0

of sheaves on Y¯ . In the corresponding long exact sequence in cohomology we have 1 ¯ ¯m ¯mˆ H (Y, LW Y¯ ⊗ Lr Y¯ ) = 0 since m andm ˆ are large. It follows that the map

0 ¯ 1 ¯ ¯m ¯mˆ H (Y, O(m−p)D) → H (Y, LW Y¯ ⊗ Lr Y¯ ⊗ OY¯ (−(m − p)D))

1 ¯ ¯m ¯mˆ 0 ¯ is a surjection. Therefore h (Y, LW Y¯ ⊗ Lr Y¯ ⊗ OY¯ (−(m − p)D)) ≤ h (Y, O(m−p)D). But 0 ¯ O(m−p)D is just a skyscraper sheaf, and we calculate h (Y, O(m−p)D) = k(m − p).

1 ¯ ¯m ¯mˆ III. h (Y, LW Y¯ ⊗ Lr Y¯ ⊗ OY¯ ((p − m)D)) = 0 if 2g − 1 ≤ p ≤ m − 1.

¯ ¯m ¯mˆ Proof. This follows because Y is an integral curve and deg LW Y¯ ((p−m)D)⊗Lr Y¯ ≥ 2gY¯ −1 whenever 2g − 1 ≤ p ≤ m − 1. Combining this data with our previous formula (16.1) we have

0 0 ˆ (e − e )m + (d − d )m ˆ + k(p − m) − gY¯ + 1 + q2 + k(m − p) if 0 ≤ p ≤ 2g − 2 βp ≤ 0 0 (e − e )m + (d − d )m ˆ + k(p − m) − gY¯ + 1 + q2 if 2g − 1 ≤ p ≤ m − 1

ˆ 0 We have supposed that Hm(C) is SL(W )-semistable, therefore λ -semistable. Hence ˆ ˆ ˆ ˆ ˆ there exist monomials Mi1 , ..., MiP (m)+dmˆ in Bm,mˆ such that {ρˆm,mˆ (Mi1 ), ..., ρˆm,mˆ (MiP (m)+dmˆ )} 0 m mˆ is a basis of H (C,LW ⊗ Lr ) and

P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) j=1 ≤ i=0 . m(P (m) + dmˆ ) e − g + 1

P (m)+dmˆ m X ˆ X ˆ ˆ But wλ(Mij ) must be larger than p(βp − βp−1). We calculate: j=1 p=1

53 m m−1 X ˆ ˆ ˆ X ˆ p(βp − βp−1) = mβm − βp p=1 p=0

m−1 2g−2 X 0 0 X ≥ m(em+dmˆ + 1− g) − (e−e )m+(d− d)m ˆ + k(p−m)−gY¯ +1+q2 − k(m−p) p=0 p=0 k (2g − 1)(2g − 2) ≥ (e0 + )m2 − Sm + d0mmˆ + 2 2 k ≥ (e0 + )m2 − Sm + d0mmˆ 2

k where S = g + k(2g − 1) + q2 − gY¯ + 2 . N X 0 0 0 0 Note also that wλ(wi) = dim W1 − dim W0 ≤ h (pW (C ), OpW (C )(1)) =: h . We have i=0

(e0 + k )m2 − Sm + d0mmˆ h0 2 ≤ m(em + dmˆ + 1 − g e − g + 1 0 (e0 + k ) − S + d mˆ h0 2 m m ≤ dmˆ e − g + 1 e + m (e0 + k ) − S + d0mˆ h0 + dh0mˆ 2 m m ≤ em e e − g + 1 0 0 (e0 + k ) − S h0 + (dh −d (e−g+1))m ˆ 2 m ≤ em . e e − g + 1

0 0 0 0 This proves the lemma when C is connected. If C is not connected, write C = ∪Ci 0 0 where the Ci are the connected components of C . Then we proved

k0 (dh0−d0 (e−g+1))m ˆ (e0 + i ) h0 + i i S i 2 < i em + , e e − g + 1 em

0 0 0 0 for each i, where h := h (p (C ), O 0 (1)) =: h . There are no points connecting any of i W i pW (Ci) 0 0 P 0 0 0 0 0 the C to each other so k = k . Also H (p (C ), O 0 (1)) = ⊕H (p (C ), O 0 (1)) i i W pW (C ) W i pW (Ci) 0 P 0 so h = hi . Summing the inequalities over i we obtain the desired result.

Proposition 16.5 (cf. [Gies] 1.0.3) Suppose that m, mˆ > m000 and 3 X m > max{(q + 1)(e − g + 1), (g − + k e (q + 1) + q + µ m00)(e − g + 1)} 1 2 1,j 1,j 1 3 1 and 1 ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 mˆ (3n + 2)a − 5 − 2 m < < 2(g−1) . (3n + 2)a − 1 − n m 4n

54 ˆ Then for all C such that Hm,mˆ (C) is SL(W )-semistable, no irreducible components of C collapse under pW .

0 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 0 all irreducible components of C which are not collapsed under pW . Then C ∩ Y 6= ∅ since 0 ¯ ¯ ¯ ¯ C is connected. Let P ∈ C ∩ Y. We have degY¯j (LW Y ) ≥ 1 so degY¯j (LW Y (−P )) ≥ 0 for each irreducible component of Y . The hypotheses of Lemma 16.4 are satisﬁed for k = 1. We have 0 0 0 0 0 h (pW (C ), OpW (C )(1)) = 1 and d ≥ 1 since e = 0. Then the inequality reads

0 0 0 + 1 − S (e0 + k ) − S h0 + (dh −d (e−g+1))m ˆ 1 + d−e+g−1 mˆ 2 m ≤ 2 m ≤ em ≤ e m e e e − g + 1 e − g + 1 1 S mˆ ( − )(e − g + 1) − (e − g + 1) − (g − 1) ≤ (d − e + g − 1) 2 m m 1 S mˆ −( + )(e − g + 1) − (g − 1) ≤ (d − e + g − 1) 2 m m ( 1 + S )(e − g + 1) − (g − 1) mˆ 2 m ≥ e − g + 1 − d m ( 1 + S )((3n + 2)a − 1) − 1 mˆ 2 m ≥ (3n + 2)a − 1 − n m

k Recall that S = g + k(2g − 1) + q2 − gY¯ + 2 ≤ 3g + q2 when k = 1. We have chosen mˆ ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 ( 1 + S )((3n + 2)a − 1) − 1 > 2 m ≥ 2 m m (3n + 2)a − 1 − n (3n + 2)a − 1 − n so the last line above is a contradiction.

ˆ Remark. We now know that for any curve C such that Hm,mˆ (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 1 0 0 0 h (C,LW ) = 0 and e = a(2g − 2 + k) + 3ad for each irreducible component ([FP] page 58).

mˆ 3a Proposition 16.6 If m = 2a−1 and m, mˆ >> 0 then this model satisﬁes the inequality

0 0 e0 + k h0 + (dh −d (e−g+1))m ˆ 2 ≤ em . (24) e e − g + 1

55 Proof. We have h0 = e0 − g0 + 1. We rewrite the desired result:

0 0 e0 + k h0 + (dh −d (e−g+1))m ˆ 2 ≤ em ⇔ e e − g + 1 k mˆ mˆ (e−g+1) ≤ e0(d + g − 1) − e(g0 − 1) + (−d0(e − g + 1) + d(1 − g0)) ⇔ 2 m m mˆ 1 mˆ 0 ≤ k(a(d + g − 1) − (e − g + 1)) + a(2g0 − 2)(d − g − 1) ⇔ m 2 m mˆ mˆ +3ad0(d + g − 1) − e(g0 − 1) + (−d0(e − g + 1) + d(1 − g0)) ⇔ m m mˆ 3 mˆ k mˆ 0 ≤ n − ka + (2a−1) − 3a (g0 −1) + − d0 (2a−1) − 3a m 2 m 2 m

mˆ 3a 3 3 3a k Note that if m = 2a−1 = 2 + 4a−2 then the last line above reads 0 ≤ nk 4a−2 + 2 which is 3a k clearly true. In fact since 0 < nk 4a−2 + 2 then by continuity the inequality must be veriﬁed mˆ 3a for some small range of m close to 2a−1 . S Remark. Note that the term em 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 are nodes

ˆ The next series of results shows that (for certain linearizations) if Hm,mˆ (C) is SL(W )- semistable and P ∈ C is a singular point, then P is a node.

Letg ¯ 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, mˆ > m000 and

3 00 (g − 2 + e(q1 + 1) + q3 + µ1m )(e − g + 1), m > max 9 (9g + 3q2 − 3¯g − 2 )(e − g + 1) and ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 mˆ 1 ((3n + 2)a − 1) − 9 − 1 2 m < < 2 2 g−1 . (3n + 2)a − 1 − n m 4n ˆ ¯ For all C such that Hm,mˆ (C) is SL(W )-semistable, the normalization morphism π : C → Cred is unramiﬁed.

1 3g+q2 1 ((3n+2)a−1)− 9 − 1 ( 2 + m )((3n+2)a−1)−1 1 2 2 g−1 3a Remark. (3n+2)a−1−n ≈ 2 and 4n ≈ 8 when a is large.

56 ¯ ¯ Proof. Suppose π is ramified at P ∈ C. Then pW ◦ π : C → pW (C) is also ramified at P . 0 Recall that pW (C) ⊂ P(W ) is nondegenerate; then we can think of H (P(W ), OP(W )(1)) as 0 a subspace of H (pW (C), OpW (C)(1)). Define 0 W0 = {s ∈ H (P(W ), OP(W )(1))|π∗pW ∗s vanishes to order ≥ 3 at P } 0 W1 = {s ∈ H (P(W ), OP(W )(1))|π∗pW ∗s vanishes to order ≥ 2 at P } and write N0 := dim W0 and N1 := dim W1. Choose a basis

w0, ..., wN0 , wN0+1, ..., wN1 , wN1+1, ..., wN+1

0 of W3 := H (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 3 ∗ λ(t)wi = t wi, t ∈ C ,N1 + 1 ≤ i ≤ N + 1 and let λ0 be the associated 1-PS of SL(W ). 0 Choose m andm ˆ suﬃciently large. Let Bm be a basis of H (P(W ), OP(W )(m)) consisting m 0 of monomials of degree m. For 0≤i ≤ 3m write Ωi for the subspace of H (P(W ), OP(W )(m)) ˆ spanned by {M ∈ Bm|wλ(M) ≤ i}. Let

ˆ m,mˆ m 0 r Ωi := Ωi ⊗ H (P , OPr (m ˆ )) (25) m,mˆ ¯ˆ ˆ m,mˆ 0 m mˆ Ωi :=ρ ˆm,mˆ (Ωi ) ⊂ H (C,LW ⊗ Lr ) (26)

0 0 0 m mˆ whereρ ˆm,mˆ : H (P(W ), OP(W )(m)) ⊗ H (P(r), OP(r)(m ˆ )) → H (C,LW ⊗ Lr ) is the map induced by restriction. m,mˆ ¯ˆ 0 ¯ ¯m ¯mˆ 0 ¯ ¯m ¯mˆ We show that πm,mˆ ∗(Ωi ) ⊆ H (C, LW ⊗ Lr ⊗ OC¯((−3m + i)P )) ⊂ H (C, LW ⊗ Lr ) for 0 ≤ i ≤ 3m. When i = 0 this follows from the deﬁnitions. For 1 ≤ i ≤ 3m it is enough ˆ ¯ˆ m,mˆ ¯ˆ m,mˆ to show that for any monomial M ∈ Ωi − Ωi−1 we have ˆ 0 ¯ ¯m ¯mˆ ˆ πm,mˆ ∗(M) ∈ H (C, LW ⊗Lr 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 ˆ 0 ¯ ¯m ¯mˆ and i1 + 3i3 = i. By deﬁnition πm,mˆ ∗(M) ∈ H (C, LW ⊗ Lr ⊗ OC¯((−3i0 − 2i1)P )). But

3i0 + 2i1 = 3(i0 + i1 + i3) − (i1 + 3i3) = 3m − i so we have ˆ 0 ¯ ¯m ¯mˆ πm,mˆ ∗(M) ∈ H (C, LW ⊗ Lr OC¯((−3m + i)P )) as required. By the claim and Riemann-Roch,

m,mˆ ˆ ¯ˆ 0 ¯ ¯m ¯mˆ βi := dim Ωi ≤ h (C, LW ⊗ Lr ⊗ OC¯((−3m + i)P )) + dim ker πm,mˆ ∗ 1 ¯ ¯m ¯mˆ = em+dmˆ −3m+i−gC¯ +1+h (C, LW ⊗ Lr ⊗ OC¯((−3m+i)P ))+dim ker πm,mˆ ∗

57 We have the following three estimates :

I. dim ker πm,mˆ ∗ < q2.

Proof. Let IC denote the ideal sheaf of nilpotents in OC . We can choose an integer q2 0 such that h (C, IC ) < q2. Then we also have dim ker πm,mˆ ∗ < q2. To see this, recall that the 0 normalization morphism factors through Cred. We have π = ι ◦ π , where ι : Cred ,→ C. Then 0 0 we have πm,mˆ ∗ = πm,mˆ ∗ ◦ ιm,mˆ ∗. Now, πm,mˆ ∗ is injective, so ker πm,mˆ ∗ = ker ιm,mˆ ∗. We have shown that C is generically reduced, so IC has ﬁnite support. We have an exact sequence m mˆ m mˆ m mˆ of sheaves 0 → IC ⊗ LW ⊗ Lr → LW ⊗ Lr → LW red ⊗ Lrred → 0. This gives rise to an exact ∼ 0 m mˆ sequence in cohomology. It follows that ker πm,mˆ ∗ = ker ιm,mˆ ∗ = H (C, IC ⊗ LW ⊗ Lr ) and 0 m mˆ 0 hence dim ker πm,mˆ ∗ = h (C, IC ⊗ LW ⊗ Lr ) = h (C, IC ) < q2.

1 ¯ ¯m ¯mˆ II. h (C, LW ⊗ Lr ⊗ 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 ¯ ¯m ¯mˆ sheaves on C. Since LW ⊗ Lr is an invertible sheaf, it is locally free and in particular ﬂat. Tensoring, we obtain a second short exact sequence

¯m ¯mˆ ¯m ¯mˆ 0 → LW ⊗ Lr ⊗ OC¯((−3m + i)P ) → LW ⊗ Lr → O(−3m+i)P → 0

of sheaves on C¯. In the corresponding long exact sequence in cohomology we have 1 ¯ ¯m ¯mˆ 2 ¯ ¯m ¯mˆ H (C, LW ⊗Lr )=0 since m andm ˆ are large. Also H (C, LW ⊗Lr ⊗OC¯((−3m+i)P )) = 0. It 0 ¯ 1 ¯ ¯m ¯mˆ follows that the map H (C, O(−3m+i)P ) → H (C, LW ⊗Lr ⊗OC¯(−3m+i)P )) is a surjection. 1 ¯ ¯m ¯mˆ 0 ¯ Therefore h (C, LW ⊗ Lr ⊗ OC¯(−3m + i)P )) ≤ h (C, O(−3m+i)P ). But O(−3m+i)P is just a 0 ¯ skyscraper sheaf supported on P and we calculate h (C, O(−3m+i)P ) = 3m − i.

1 ¯ ¯m ¯mˆ III. h (C, LW ⊗ Lr ⊗ OC¯((−3m + i)P )) = 0 if 2g − 1 ≤ i ≤ 3m − 1.

¯ ¯m ¯mˆ Proof. C is integral and if i ≥ 2g − 1 then deg LW ⊗ Lr ⊗ OC¯((−3m + i)P ) ≥ 2g − 1. 1 ¯ ¯m ¯mˆ Therefore h (C, LW ⊗ Lr ⊗ OC¯((−3m + i)P )) = 0. Combining these inequalities we have ˆ q2 + em + dmˆ − 3m + i − gC¯ + 1 + 3m − i, 0 ≤ p ≤ 2g − 2 βi ≤ q2 + em + dmˆ − 3m + i − gC¯ + 1, 2g − 1 ≤ p ≤ m

ˆ 0 Now suppose that Hm,mˆ (C) is λ -semistable. Then there exist monomials ˆ ˆ ˆ ˆ ˆ Mi1 , ..., MiP (m)+dmˆ in Bm,mˆ such that {ρˆm,mˆ (Mi1 ), ..., ρˆm,mˆ (MiP (m)+dmˆ )} is a basis of

58 0 m mˆ H (C,LW ⊗ Lr ) and P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) j=1 ≤ i=0 . m(P (m) + dmˆ ) e − g + 1

P (m)+dmˆ 3m X ˆ X ˆ ˆ But wλ(Mij ) must be larger than i(βi − βi−1). We calculate: j=1 i=1 3m 3m−1 X ˆ ˆ ˆ X ˆ i(βi − βi−1) = 3mβ3m − βi i=1 i=0

3m−1 2g−2 X X ≥ 3m(em + dmˆ + 1 − g) − q2 + em + dmˆ − 3m + i − gC¯ + 1 − 3m − i i=0 i=0 9 = m2 − S m + (g − 1)(2g − 1) 2 7 9 ≥ m2 − S m 2 7 9 where S7 = 9g + 3q2 − 3gC¯ − 2 . N X Next we show that wλ(wi) ≤ 4. Note that the image of W0 under π∗ is contained i=0 0 ¯ ¯ 0 ¯ ¯ in H (C, LW (−3P )), and the image of W1 under π∗ is contained in H (C, LW (−2P )). We have two exact sequences ¯ ¯ 0 → LW (−P ) → LW → k(P ) → 0 ¯ ¯ 0 → LW (−3P ) → LW (−2P ) → k(P ) → 0. which give rise to long exact sequences in cohomology 0 ¯ 0 ¯ 0 0 → H (C, LW (−P )) → H (C, LW ) → H (C, k(P )) → · · · 0 ¯ 0 ¯ 0 0 → H (C, LW (−3P )) → H (C, LW (−2P )) → H (C, k(P ) → · · ·

¯ ∗ The second long exact sequence implies that dim W1/W0 ≤ 1. Recall that LW := π (LW ) and π is ramified at P . The ramification index must be at least two, so we have 0 ¯ 0 ¯ H (C, LW (−P )) = H (C, LW (−2P )). Then the first long exact sequence implies that N X dim W3/W1 ≤ 1. Thus wλ(wi) ≤ 1 + 3 = 4. i=0 1 ((3n+2)a−1)− 9 − 1 mˆ 2 2 g−1 By hypothesis m > S7(e−g+1) and m < 4n . Combining these inequalities leads to a contradiction as follows:

P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) 9 2 m − S m j=1 4 2 7 ≤ ≤ i=0 ≤ m(em + dmˆ + 1 − g) m(P (m) + dmˆ ) e − g + 1 e − g + 1

59 9 − S7 4 2 m ≤ dmˆ e − g + 1 e + m 9 S (e − g + 1) 4dmˆ (e − g + 1) − 4e ≤ 7 + 2 m m 1 ((3n + 2)a − 1) − 9 − 1 mˆ 2 2 g−1 ≤ . 4n m

ˆ 0 ˆ The contradiction implies that Hm,mˆ (C) is not λ -semistable, and therefore that Hm,mˆ (C) is not SL(W )-semistable.

In particular, Lemma 16.7 implies that

Proposition 16.8 (cf. [Gies] 1.0.5) Suppose that m, mˆ > m000 and

3 00 (g − 2 + e(q1 + 1) + q3 + µ1m )(e − g + 1), m > max 9 (9g + 3q2 − 3¯g − 2 )(e − g + 1) and ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 mˆ 1 ((3n + 2)a − 1) − 9 − 1 2 m < < 2 2 g−1 . (3n + 2)a − 1 − n m 4n ˆ Then if Hm,mˆ (C) is SL(W )-semistable, Cred does not have a cusp singularity.

−1 Proof. Suppose Q ∈ Cred is a cusp. Write π (Q) = P . Then π is ramiﬁed at P .

Proposition 16.9 (cf. [Gies] 1.0.4) Suppose that m, mˆ > m000 and

 3 00  (g − 2 + e(q1 + 1) + q3 + µ1m )(e − g + 1),  9  m > max (9g + 3q2 − 3¯g − 2 )(e − g + 1),  15  (7g − g¯ + q2 − 2 )(e − g + 1) and ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 mˆ 1 ((3n + 2)a − 1) − 9 − 1 2 m < < 2 2 g−1 . (3n + 2)a − 1 − n m 4n ˆ For all curves C such that Hm,mˆ (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 0 ev : H (P(W ), OP(W )(1)) → k(P ) be the evaluation map. Let W0 = ker ev. We have

60 0 N0 := dim W0 = e−g. Choose a basis of W0 and extend it to a basis of H (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 − 1 ∗ λ(t)we−g = twe−g, t ∈ C

0 ˆ and let λ be the associated 1-PS of SL(W ). Choose m andm ˆ suﬃciently large. Let Bm,mˆ be 0 r a basis of H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) consisting of monomials of bidegree (m, mˆ ). 0 m mˆ As in the previous proof, construct a ﬁltration of H (C,LW ⊗ Lr ) as follows: For m−p p 0 0 ≤ p ≤ m let W0 W1 be the subspace of H (P(W ), OP(W )(m)) spanned by elements of the following type:

{x1 ··· xm|xi ∈ W0} if p = 0 {x1 ··· xm−py1 ··· yp|xi ∈ W0, yi ∈ W1} if 0 < p < m {y1 ··· ym|yi ∈ W1} if p = m.

\m−p p m−p p 0 r Set W0 W1 = W0 W1 ⊗ H (P , OPr (m ˆ )), and let

¯\m−p ¯ p \m−p p 0 m mˆ W0 W1 :=ρ ˆm,mˆ (W0 W1 ) ⊂ H (C,LW ⊗ Lr ) (27) whereρ ˆm,mˆ is the homomorphism induced by restriction (see page 32). Note that a monomial ˆ ¯\m−p ¯ p ¯ m−\p+1 ¯ p−1 M ∈ W0 W1 − W0 W1 has weight p. Therefore we have a ﬁltration of 0 m mˆ H (C,LW ⊗ Lr ) in order of increasing weight:

¯ m ¯ 0 ¯\m−1 ¯ 1 ¯ 0 ¯ m 0 m mˆ 0 ⊆ W\0 W1 ⊆ W0 W1 ⊆ · · · ⊆ W\0 W1 = H (C,LW ⊗ Lr )

ˆ ¯\m−p ¯ p Write βp = dim W0 W1 .

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 −1 ¯ Pi ∈ π (P ) ∩ Ci for i = 1, 2, 3. In case 2, at least one of the components, say C1, satisﬁes −1 ¯ deg pW (C1) ≥ 3, and P is a singular point of C1. Choose P1,P2 ∈ π (P ) ∩ C1 and P3 ∈ −1 ¯ −1 π (P ) ∩ C2. In case 1, choose three distinct points P1,P2,P3 from the ﬁber π (P ). The normalization morphism induces a homomorphism

0 m mˆ 0 ¯ ¯m ¯mˆ πm,mˆ ∗ : H (C,LW ⊗ Lr ) → H (C, LW ⊗ Lr ).

¯\m−p ¯ p 0 ¯ ¯m ¯mˆ Note that πm,mˆ ∗(W0 W1 ) ⊆ H (C, LW ⊗ Lr ⊗ OC¯((p − m)D)). We have

ˆ ¯\m−p ¯ p 0 ¯ ¯m ¯mˆ βp := dim W0 W1 ≤ h (C, LW ⊗ Lr ⊗ OC¯((p − m)D)) + dim ker πm,mˆ ∗ 1 ¯ ¯m ¯mˆ = em + dmˆ + 3(p−m) − gC¯ + 1 + h (C, LW ⊗ Lr ⊗ OC¯((p−m)D)) + dim ker πm,mˆ ∗

The following estimates may be established by arguments entirely analogous to the proofs given on page 58:

0 I. h (C, IC ) < q2. 1 ¯ ¯m ¯mˆ II. h (C, LW ((p − m)D) ⊗ Lr ) ≤ 3(m − p). 1 ¯ ¯m ¯mˆ III. h (C, LW ((p − m)D) ⊗ Lr ) = 0 if p ≥ 2g − 1.

Entering these quantities into the previous formulae, we have ˆ q2 + em + dmˆ + 3(p − m) − gC¯ + 1 + 3(m − p), 0 ≤ p ≤ 2g − 2 βp ≤ q2 + em + dmˆ + 3(p − m) − gC¯ + 1, 2g − 1 ≤ p ≤ m

ˆ 0 Now suppose that Hm,mˆ (C) is λ -semistable. Then there exist monomials ˆ ˆ ˆ ˆ ˆ Mi1 , ..., MiP (m)+dmˆ in Bm,mˆ such that {ρˆm,mˆ (Mi1 ), ..., ρˆm,mˆ (MiP (m)+dmˆ )} is a basis of 0 m mˆ H (C,LW ⊗ Lr ) and P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) j=1 ≤ i=0 . m(P (m) + dmˆ ) e − g + 1

P (m)+dmˆ m X ˆ X ˆ ˆ But wλ(Mij ) must be larger than p(βp − βp−1). We calculate: j=1 p=1

m m−1 X ˆ ˆ ˆ X ˆ p(βp − βp−1) = mβm − βp p=1 p=0 m−1 2g−2 X X ≥ m(em+dmˆ +1−g)− em+dmˆ +3(p−m)−gC¯ +1+q2 − 3(m−p) p=0 p=0 3 3 = m2 − S m + (g − 1)(2g − 1) ≥ m2 − S m 2 9 2 9

62 15 where S9 = 7g − gC¯ + q2 − 2 . N X Note also that wλ(wi) = 1 since wλ(w0) = 1 and wλ(wi) = 0, i > 0. i=0 By hypothesis m > S9(e − g + 1). Combining these inequalities leads to a contradiction as follows:

P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) 3 2 m − S m j=1 1 2 9 ≤ ≤ i=0 ≤ m(em + dmˆ + 1 − g) m(P (m) + dmˆ ) e − g + 1 e − g + 1 3 − S9 1 2 m ≤ dmˆ e − g + 1 e + m 3 S (e − g + 1) dmˆ (e − g + 1) − e ≤ 9 + 2 m m 1 e − 3 (g − 1) − 1 mˆ 2 2 < d m 1 (3n + 2)a − 3 − 1 mˆ 2 2 g−1 < (28) n m 1 ((3n+2)a−1)− 9 − 1 mˆ 2 2 g−1 But the last line contradicts the hypothesis that m < 4n . The contradic- ˆ 0 ˆ tion implies that Hm,mˆ (C) is not λ -semistable, and therefore that Hm,mˆ (C) is not SL(W )- semistable.

1 (3n+2)a− 3 − 1 mˆ 2 2 g−1 Remark. To contradict (28) it suﬃces that m ≤ n . The stronger hypothesis 1 ((3n+2)a−1)− 9 − 1 mˆ 2 2 g−1 that m < 4n was imposed so that Lemma 16.7 could be used on page 61.

Proposition 16.10 (cf. [Gies] 1.0.6) Suppose that m, mˆ > m000 and  (g − 3 + e(q + 1) + q + µ m00)(e − g + 1),   2 1 3 1   9   (9g + 3q2 − 3¯g − 2 )(e − g + 1),   15  m > max (7g − g¯ + q2 − 2 )(e − g + 1),    10g − 2¯g + 2q2 − 6,   11  10g + 2q2 − 2 − g¯ and ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 mˆ 1 ((3n + 2)a − 1) − 9 − 1 2 m < < 2 2 g−1 . (3n + 2)a − 1 − n m 4n ˆ For all curves C such that Hm,mˆ (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 . 0 0 Once again we think of H (P(W ), OP(W )(1)) as a subspace of H (pW (C), OpW (C)(1)) and deﬁne subspaces

0 W0 = {s ∈ H (P(W ), OP(W )(1))|π∗pW ∗s vanishes to order ≥ 2 at Q1 and Q2} 0 W1 = {s ∈ H (P(W ), OP(W )(1))|π∗pW ∗s vanishes to order ≥ 1 at Q1 and Q2} and write N0 := dim W0 and N1 := dim W1. Choose a basis

w0, ..., wN0 , wN0+1, ..., wN1 , wN1+1, ..., wN+1

0 of W2 := H (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 2 ∗ λ(t)wi = t wi, t ∈ C ,N1 + 1 ≤ i ≤ N + 1

0 ˆ and let λ be the associated 1-PS of SL(W ). Choose m andm ˆ suﬃciently large. Let Bm,mˆ be 0 r a basis of H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) consisting of monomials of bidegree (m, mˆ ). 0 m mˆ As in the previous proof, construct a ﬁltration of H (C,LW ⊗ Lr ) as follows: For all m−p p 0 0 ≤ p ≤ m let W0 W1 be the subspace of H (P(W ), OP(W )(m)) spanned by elements of the following type:

{x1 ··· xm|xi ∈ W0} if p = 0 {x1 ··· xm−py1 ··· yp|xi ∈ W0, yi ∈ W1} if 0 < p < m {y1 ··· ym|yi ∈ W1} if p = m.

m−p p 0 Similarly for 0 ≤ p ≤ m let W1 W2 be the subspace of H (P(W ), OP(W )(m)) spanned by elements of the following type:

{x1 ··· xm|xi ∈ W1} if p = 0 {x1 ··· xm−py1 ··· yp|xi ∈ W1, yi ∈ W2} if 0 < p < m {y1 ··· ym|yi ∈ W2} if p = m.

\m−p p m−p p 0 r \m−p p m−p p 0 r Set W0 W1 = W0 W1 ⊗ H (P , OPr (m ˆ )) and W1 W2 = W1 W2 ⊗ H (P , OPr (m ˆ )), and let

¯\m−p ¯ p \m−p p 0 m mˆ W0 W1 :=ρ ˆm,mˆ (W0 W1 ) ⊂ H (C,LW ⊗ Lr ) ¯\m−p ¯ p \m−p p 0 m mˆ W1 W2 :=ρ ˆm,mˆ (W1 W2 ) ⊂ H (C,LW ⊗ Lr ).

64 whereρ ˆm,mˆ is the homomorphism induced by restriction (see page 32). Note that a monomial ˆ ¯\m−p ¯ p ¯ m−\p+1 ¯ p−1 ˆ ¯\m−p ¯ p ¯ m−\p+1 ¯ p−1 M ∈ W0 W1 −W0 W1 has weight p and a monomial M ∈ W1 W2 −W1 W2 0 m mˆ has weight m + p. Therefore we have a ﬁltration of H (C,LW ⊗ Lr ) in order of increasing weight:

0 ⊆ W¯\mW¯ 0 ⊆ W¯\m−1W¯ 1 ⊆ · · · ⊆ W¯\0W¯ m 0 1 0 1 0 1 (29) ¯ m ¯ 0 ¯\m−1 ¯ 1 ¯ 0 ¯ m 0 m mˆ = W\1 W2 ⊆ W1 W2 ⊆ · · · ⊆ W\1 W2 = H (C,LW ⊗ Lr )

¯\m−p ¯ p ˆ ¯\m−p ¯ p Writeγ ˆ := dim W0 W1 and βp = dim W1 W2 . We shall divide the proof into the following cases:

1. deg L ≥ 2 and deg L ≥ 2. C1 red W C2 red W 2. deg L = 1 and deg L ≥ 2. C1 red W C2 red W 3. deg L ≥ 2 and deg L = 1. C1 red W C2 red W

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 0 m mˆ 0 ¯ ¯m ¯mˆ πm,mˆ ∗ : H (C,LW ⊗ Lr ) → H (C, LW ⊗ Lr ). By deﬁnition,

¯\m−p ¯ p 0 ¯ ¯m ¯mˆ πm,mˆ ∗(W0 W1 ) ⊂ H (C, LW ⊗ Lr ⊗ OC¯((p − 2m)(Q1 + Q2))) ¯\m−p ¯ p 0 ¯ ¯m ¯mˆ πm,mˆ ∗(W1 W2 ) ⊂ H (C, LW ⊗ Lr ⊗ OC¯((p − m)(Q1 + Q2))).

Then we use Riemann-Roch to calculate

¯\m−p ¯ p 0 ¯ ¯m ¯mˆ γˆp := dim W0 W1 ≤ h (C, LW ⊗ Lr ⊗ OC¯((p − 2m)(Q1 + Q2))) + dim ker πm,mˆ ∗

= em + dmˆ + 2p − 4m − gC¯ + 1 1 ¯ ¯m ¯mˆ +h (C, LW ⊗ Lr ⊗ OC¯((p − 2m)(Q1 + Q2))) + dim ker πm,mˆ ∗. ˆ ¯\m−p ¯ p 0 ¯ ¯m ¯mˆ βp := dim W1 W2 ≤ h (C, LW ⊗ Lr ⊗ OC¯((p − m)(Q1 + Q2))) + dim ker πm,mˆ ∗

= em + dmˆ + 2p − 2m − gC¯ + 1 1 ¯ ¯m ¯mˆ +h (C, LW ⊗ Lr ⊗ OC¯((p − m)(Q1 + Q2))) + dim ker πm,mˆ ∗.

The following estimates may be established by arguments entirely analogous to the proofs given on page 58:

I. dim ker πm,mˆ ∗ < q2.

65 1 ¯ ¯m ¯mˆ II. h (C, LW ⊗ Lr ⊗ OC¯((p − 2m)(Q1 + Q2))) ≤ 4m − 2p if 0 ≤ p ≤ 2g − 2. 1 ¯ ¯m ¯mˆ III. h (C, LW ⊗ Lr ⊗ OC¯((p − 2m)(Q1 + Q2))) = 0 if 2g − 1 ≤ p ≤ m − 1. 1 ¯ ¯m ¯mˆ IV. h (C, LW ⊗ Lr ⊗ OC¯((p − m)(Q1 + Q2))) = 0 for all 0 ≤ p ≤ m − 1.

We have q2 + em + dmˆ + 2p − 4m − gC¯ + 1 + 4m − 2p, 0 ≤ p ≤ 2g − 2 γˆp ≤ q2 + em + dmˆ + 2p − 4m − gC¯ + 1, 2g − 1 ≤ p ≤ m − 1 ˆ βp ≤ q2 + em + dmˆ + 2p − 2m − gC¯ + 1, 0 ≤ p ≤ m − 1.

P (m)+dmˆ m m X ˆ X ˆ ˆ X But wλ(Mij ) must be larger than (m + p)(βp − βp−1) + p(ˆγp − γˆp−1). We j=1 p=1 p=1 calculate: m m X ˆ ˆ X (m + p)(βp − βp−1) + p(ˆγp − γˆp−1) p=1 p=1 m−1 m−1 ˆ X ˆ X = 2mβm − βp − γˆp p=0 p=0 m−1 X ≥ 2m(em + dmˆ + 1 − g) − q2 + em + dmˆ + 2p − 2m − gC¯ + 1 p=0 m−1 2g−2 X X − q2 + em + dmˆ + 2p − 4m − gC¯ + 1 − 4m − 2p p=0 p=0 2 2 = 4m − S10m + (2g − 2)(2g − 1) ≥ 4m − S10m where S10 = 10g − 2gC¯ + 2q2 − 6. N X Note also that wλ(wi) ≤ 3. To see this note that the image of W0 under π∗ is contained i=0 0 ¯ ¯ 0 ¯ ¯ in H (C, LW (−Q1)), and the image of W1 under π∗ is contained in H (C, LW (−Q1)). We have two exact sequences ¯ ¯ 0 → LW (−Q1) → LW → k(Q1) → 0 ¯ ¯ 0 → LW (−2Q1) → LW (−Q1) → k(Q1) → 0. which give rise to long exact sequences in cohomology

66 0 ¯ 0 ¯ 0 0 → H (C, LW (−Q1)) → H (C, LW ) → H (C, k(Q1)) → · · · 0 ¯ 0 ¯ 0 0 → H (C, LW (−2Q1)) → H (C, LW (−Q1)) → H (C, k(Q1) → · · ·

The ﬁrst long exact sequence implies that dim W2/W1 ≤ 1, and the second long exact N X sequence implies that dim W1/W0 ≤ 1. Thus wλ(wi) ≤ 1 + 2 = 3. i=0 By hypothesis m > S10(e − g + 1). Combining these inequalities leads to a contradiction as follows:

P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) 2 4m − S m j=1 3 10 ≤ ≤ i=0 ≤ m(em + dmˆ + 1 − g) m(P (m) + dmˆ ) e − g + 1 e − g + 1 4 − S10 3 m ≤ dmˆ e − g + 1 e + m S (e − g + 1) 3dmˆ 4(e − g + 1) − 3e ≤ 10 + m m e − 4(g − 1) − 1 mˆ ≤ 3d m (3n + 2)a − 4 − 1 mˆ g−1 ≤ n m But the last line contradicts the hypothesis that

mˆ 1 ((3n + 2)a − 1) − 9 − 1 < 2 2 g−1 . m 4n ˆ 0 ˆ The contradiction implies that Hm,mˆ (C) is not λ -semistable, and therefore that Hm,mˆ (C) is not SL(W )-semistable. Proof of Case 2. Recall that Case 2. means that deg L = 1 and deg L ≥ 2. C1 red W C2 red W 0 0 We continue to think of H (P(W ), OP(W )(1)) as a subspace of H (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 ¯\m−p ¯ p 0 ¯ ¯m ¯mˆ πm,mˆ ∗(W W ) ⊂ H (Yred, L ¯ ⊗ L ¯ ⊗ O ¯ ((p − 2m)Q2)). Therefore 0 1 W Yred rYred Y

¯ m−p ¯ p 0 ¯ ¯m ¯mˆ γˆ := dim W\W ≤ h (Yred, L ¯ ⊗ L ¯ ⊗ O ¯ ((p − 2m)Q )) + dim ker π p 0 1 W Yred rYred Y 2 m,mˆ ∗

= (e − 1)m + (d − d1)m ˆ + p − 2m − gY¯ + 1 1 ¯ ¯m ¯mˆ + h (Yred, L ¯ ⊗ L ¯ ⊗ O ¯ ((p − 2m)Q )) + dim ker π . W Yred rYred Y 2 m,mˆ ∗ ˆ ¯\m−p ¯ p 0 ¯ ¯m ¯mˆ βp := dim W1 W2 ≤ h (C, LW ⊗ Lr ⊗ OY¯ ((p − m)(Q1 + Q2)) + dim ker πm,mˆ ∗

67 = em + dmˆ + 2p − 2m − gC¯ + 1 1 ¯ ¯m ¯mˆ + h (C, LW ⊗ Lr ⊗ OY¯ ((p − m)(Q1 + Q2))) + dim ker πm,mˆ ∗.

The following estimates may be established by arguments entirely analogous to the proofs given on page 58:

I. dim ker πm,mˆ ∗ < q2. 1 ¯ ¯m ¯mˆ II. h (Yred, L ¯ ⊗ L ¯ ⊗ O ¯ ((p − 2m)Q2)) ≤ 2m − p if 0 ≤ p ≤ 2g − 2. W Yred rYred Y 1 ¯ ¯m ¯mˆ III. h (Yred, L ¯ ⊗ L ¯ ⊗ O ¯ ((p − 2m)Q2)) = 0 if 2g − 1 ≤ p ≤ m − 1. W Yred rYred Y 1 ¯ ¯m ¯mˆ IV. h (C, LW ⊗ Lr ⊗ OY¯ ((p − m)(Q1 + Q2))) ≤ 2m − 2p if 0 ≤ p ≤ 2g − 2. 1 ¯ ¯m ¯mˆ V. h (C, LW ⊗ Lr OY¯ ((p − m)(Q1 + Q2))) = 0 for all 2g − 1 ≤ p ≤ m − 1.

Therefore q2 + (e − 1)m + (d − d1)m ˆ + p − 2m − gY¯ + 1 + 2m − p, 0 ≤ p ≤ 2g − 2 γˆp ≤ q2 + (e − 1)m + (d − d1)m ˆ + p − 2m − gY¯ + 1, 2g − 1 ≤ p ≤ m − 1 ˆ q2 + em + dmˆ + 2p − 2m − gC¯ + 1 + 2m − 2p, 0 ≤ p ≤ 2g − 2 βp ≤ q2 + em + dmˆ + 2p − 2m − gC¯ + 1, 2g − 1 ≤ p ≤ m − 1

P (m)+dmˆ m m X ˆ X ˆ ˆ X But wλ(Mij ) must be larger than (m + p)(βp − βp−1) + p(ˆγp − γˆp−1). We j=1 p=1 p=1 calculate:

m m X ˆ ˆ X (m + p)(βp − βp−1) + p(ˆγp − γˆp−1) p=1 p=1 m−1 m−1 ˆ X ˆ X = 2mβm − βp − γˆp p=0 p=0 m−1 2g−2 X X ≥ 2m(em + dmˆ + 1 − g) − q2 + em + dmˆ + 2p − 2m − gC¯ + 1 − 2m − 2p p=0 p=0 m−1 2g−2 X X − q2 + (e − 1)m + (d − d1)m ˆ + p − 2m − gY¯ + 1 − 2m − p p=0 p=0 7 2 7 2 0 0 = 2 m + d1mmˆ − S10 m + (3g − 3)(2g − 1) ≥ 2 m − S10 m

68 11 0 where S10 = 10g + 2q2 − 2 − gC¯ − gY¯ . N X Note also that wλ(wi) ≤ 3 (cf. proof of case 1, page 66). i=0 By hypothesis m > S100 (e − g + 1). Combining these inequalities leads to a contradiction as follows:

P (m)+dmˆ N X ˆ X wλ(Mij ) wλ(wi) 7 2 m − S 0 m j=1 3 2 10 ≤ ≤ i=0 ≤ m(em + dmˆ + 1 − g) m(P (m) + dmˆ ) e − g + 1 e − g + 1 7 − S100 3 2 m ≤ dmˆ e − g + 1 e + m 7 S 0 (e − g + 1) 3dmˆ (e − g + 1) − 3e ≤ 10 + 2 m m 1 e − 7 (g − 1) − 1 mˆ 2 2 ≤ 3d m 1 (3n + 2)a − 7 − 1 mˆ 2 2 g−1 ≤ 3n m

1 ((3n+2)a−1)− 9 − 1 mˆ 2 2 g−1 But the last line contradicts the hypothesis that m < 4n . The contradic- ˆ 0 ˆ tion implies that Hm,mˆ (C) is not λ -semistable, and therefore that Hm,mˆ (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 C0 ⊂ C such that

deg 0 (L) h0(C0,L) ≤ C + 1. (30) 2 Furthermore C0 ∼= P1 only if L is trivial on C0.

Proof. Gieseker proves nearly all of this. It remains only to show that C0 6∼= P1 if L is not trivial on C0. So suppose that C0 ∼= 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(C0,L) = m + 1 when m ≥ 0. Combining m this with the inequality (30) we have m + 1 ≤ 2 + 1 which implies m = 0. Then L is trivial on C0.

Proposition 16.12 Suppose that m, mˆ > m000 and

 (g − 3 + e(q + 1) + q + µ m00)(e − g + 1),   2 1 3 1   (9g + 3q − 3¯g − 9 )(e − g + 1),   2 2   (7g − g¯ + q − 15 )(e − g + 1),  m > max 2 2 10g − 2¯g + 2q2 − 6,  11   10g + 2q2 − − g¯   2   12g + 4q2 − 4¯g  and ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 mˆ 1 ((3n + 2)a − 1) − 9 − 1 2 m < < 2 2 g−1 . (3n + 2)a − 1 − n m 4n ˆ 1 If Hm,mˆ (C) is SL(W )-semistable, then H (Cred,LW red) = 0.

1 Proof. Since Cred is nodal, it has a dualizing sheaf ω. If H (Cred,LW red) 6= 0 then by 0 −1 ∼ 1 duality H (Cred, ω ⊗ LW red) = H (Cred,LW red) 6= 0. By Proposition 16.5 LW red is not trivial 0 ∼ 1 on any component of Cred. Then by Lemma 16.11 there is a subcurve C 6= P of Cred for 0 0 e0 0 which e > 1 and h (C,LWC ) − 1 ≤ 2 . We apply Lemma 16.4 with k = 1 to obtain:

0 0 0 0 (d( e +1+)−d0(e−g+1))m ˆ e0 + k − S h0 + (dh −d (e−g+1))m ˆ e + 1 + 2 2 m ≤ em ≤ 2 em e e − g + 1 e − g + 1 k S e0 e0 mˆ (e − g + 1) e0 + − ≤ e + 1 + d + 1 − d0(e − g + 1) 2 m 2 2 m e dmˆ S k mˆ e0 e − g + 1 − − ≤ − (e − g + 1) + e + (d − d0(e − g + 1)) 2 2m m 2 m

1 ((3n+2)a−1)− 9 − 1 mˆ 2 2 g−1 e dmˆ Since m < 4n the quantity e − g + 1 − 2 − 2m > 0 and we proceed:

mˆ 2S 0 mˆ 0 2e + 2d m + (e − g + 1)( m − k − 2d m ) e ≤ mˆ e − 2(g − 1) − d m mˆ 2S 0 mˆ 2(3n + 2)a + 2n m + ((3n + 2)a − 1)( m − k − 2d m ) = mˆ (3n + 2)a − 2 − n m mˆ 2S 0 mˆ mˆ 2S 0 mˆ mˆ ((3n + 2)a − 2 − n m )( m − k − 2d m + 2) + (1 + n m )( m − k − 2d m ) + 4n m + 4 = mˆ (3n + 2)a − 2 − n m

70 2S mˆ (n mˆ + 1)(4 + 2S − k − 2d0 mˆ ) = − k − 2d0 + 2 + m m m m m mˆ (3n + 2)a − 2 − n m 2S (n mˆ + 1)(4 − k + 2S ) ≤ 2 − k + + m m . (31) m mˆ (3n + 2)a − 2 − n m Note that k = 1 and e0 ≥ 2. Furthermore the hypothesis

mˆ 1 ((3n + 2)a − 1) − 9 − 1 (3n + 2)a − 10 − 2 ≤ 2 2 g−1 = g−1 m 4n 8n implies that 1 1 mˆ ≤ 1 2 (3n + 2)a − 2 − n m (3n + 2)a − 2 − 8 ((3n + 2)a − 10 − g−1 ) 2S mˆ (3 + 2S )( 1 ((3n + 2)a − 10 − 2 ) + 1) (3 + m )(n m + 1) m 8 g−1 mˆ ≤ 1 2 (3n + 2)a − 2 − n m (3n + 2)a − 2 − 8 ((3n + 2)a − 10 − g−1 ) 2S 6 2 64 16 (3 + m )((3n + 2)a − 7 + 7(g−1) − 7 − 7(g−1) + 8) = 6 2 7((3n + 2)a − 7 + g−1 ) 3 + 2S ≤ m . 7 Combining these facts with the inequality (31) we obtain 2S 3 2S 2 ≤ 1 + + + . m 7 7m

Since by hypothesis m > 4S this is a contradiction.

Proposition 16.13 Suppose that m, mˆ > m000 and

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

1 1 1 · · · → H (C, IC ⊗ LW ) → H (C,LW ) → H (C, ι∗LW red) → 0.

1 Since C is generically reduced, IC has ﬁnite support, hence H (C, IC ⊗ LW ) = 0. By 1 1 the previous lemma, H (C, ι∗LW red) = H (Cred,LW red) = 0. The exact sequence implies 1 H (C,LW ) = 0 as well. Next, the map

0 0 0 H (P(W ), OP(W )(1)) → H (pW (C)red, OpW (C)red (1)) → H (Cred,LW red)

0 0 is injective by Proposition 16.1. Then e − g + 1 = h (P(W ), OP(W )(1)) ≤ h (Cred,LW red) = 0 0 0 0 h (C,LW ) − h (C, IC ⊗ LW ) = e − g + 1 − h (C, IC ⊗ LW ). Therefore h (C, IC ⊗ LW ) = 0.

Since IC ⊗ LW has ﬁnite support, IC = 0, so C is reduced.

ˆ Summary. We have shown that if C is a connected curve such that Hm,mˆ (C) is SL(W )- 1 semistable, then C is reduced, has at worst nodes as singularities, and H (C,LW ) = 0. ˆ Next, we improve Lemma 16.4. If C is nodal and Hm,mˆ (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, mˆ > m000 and

 (g − 3 + e(q + 1) + q + µ m00)(e − g + 1),   2 1 3 1   (9g + 3q − 3¯g − 9 )(e − g + 1),   2 2   (7g − g¯ + q − 15 )(e − g + 1),  m > max 2 2 10g − 2¯g + 2q2 − 6,  11   10g + 2q2 − − g¯   2   12g + 4q2 − 4¯g  and ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 mˆ 1 ((3n + 2)a − 1) − 9 − 1 2 m < < 2 2 g−1 . (3n + 2)a − 1 − n m 4n ˆ Let C be a curve such that Hm,mˆ (C) is SL(W )-semistable; then C has at worst nodes as singularities. Suppose C has at least two irreducible components. Let C0 6= C be a reduced, complete subcurve of C and let Y be the closure of C − C0 in C with the reduced structure.

72 ¯ 0 Suppose there exist points P1, ..., Pk on Y satisfying π(Pi) ∈ Y ∩ C for all 1 ≤ i ≤ k. Write 0 0 0 0 h (pW (C ), OpW (C )(1)) =: h . Then

0 0 (e0 + k ) h0 + (dh −d (e−g+1))m ˆ S 2 < em + , e e − g + 1 em

k where S = g + k(2g − 1) + q2 − gY¯ + 2 .

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, mˆ > m000 and  (g − 3 + e(q + 1) + q + µ m00)(e − g + 1),   2 1 3 1   (9g + 3q − 3¯g − 9 )(e − g + 1),   2 2   (7g − g¯ + q − 15 )(e − g + 1),  m > max 2 2 10g − 2¯g + 2q2 − 6,  11   10g + 2q2 − − g¯   2   12g + 4q2 − 4¯g  and ( 1 + 3g+q2 )((3n + 2)a − 1) − 1 mˆ 1 ((3n + 2)a − 1) − 9 − 1 2 m < < 2 2 g−1 . (3n + 2)a − 1 − n m 4n ˆ 0 0 0 If Hm,mˆ (C) is SL(W )-semistable, and C is a subcurve of C such that g = 0 and d = 0, then #C0 ∩ Y = k ≥ 2. Furthermore if g0 = d0 = 0 and k = 2 then e0 = 1.

0 Proof. We apply Ampliﬁcation 16.14 to C . Since C is nodal, we have gY = g − k + 1. 0 Also dY = d − d = d − 0 = d.

k mˆ S (e + )(e − g + 1) ≤ e(e − g + 1) + (d(e − g + 1) − d (e − g + 1)) + Y 2 Y Y Y Y Y m em k mˆ S 0 ≤ (e − e + )(g − 1) + d (e − e + k − 1) + (32) Y 2 m Y em ˆ 0 Since Hm,mˆ (C) is SL(W )-semistable, by Proposition 16.5 e ≥ 1. Thus eY − e ≤ −1, so k mˆ if k = 1, then the terms (eY − e + 2 )(g − 1) and d m (eY − e + k − 1) on the right hand side of k mˆ 1 equation (32) are negative. In fact (eY − e + 2 )(g − 1) + d m (eY − e + k − 1) ≤ − 2 whereas S 1 the hypothesis on m imply that em < 2 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 0 e = 1.

The corollary implies in particular that if C is SL(W )-semistable and C0 is a subcurve which is a chain of rational components and satisﬁes d0 = 0 and k = 2, then C0 ∼= P1. For 0 0 if C were irreducible, then we must have ei > 0 for each irreducible component of C by 0 X 0 Proposition 16.5. But we have 1 = e = ei so C must consist of exactly one irreducible 0 Ci⊂C 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 dim W = 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 0 1 and nonspecial (i.e. h (C, OC (1)) = e − g and h (C, OC (1)) = 0) and any complete subcurve Y ⊂ C satisﬁes the inequality

e + k h0(C, O (1)) Y 2 ≤ C . (33) e e − g + 1

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 destabilizing 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:

r fh r Deﬁnition 16.17 Let h ∈ Hilb(P(W )× P ). The map Ch → P 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

0 0 H (P(W ), OP(W )(1)) → H (Cred,LW red)

1 iii. h (C,LW ) = 0 iv. any complete subcurve C0 ⊂ C with C0 6= C satisﬁes the inequality

0 0 (e0 + k ) h0 + (dh −d (e−g+1))m ˆ S 2 < em + e e − g + 1 em of Ampliﬁcation 16.14.

Theorem 16.18 Suppose that m, mˆ > m000 and

r ss fh r For all h ∈ Hilb(P(W )× P ) the map Ch → P is potentially stable.

17 The Construction Finished

In the previous section we studied Hilb(P(W )× Pr)ss. In this section we focus on J ss. 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.

r iii. The multidegree of (OP(W )(1) ⊗ OP (1))|Ch equals the multidegree of ⊗a (ω ⊗ O r (3a + 1))| . Ch P Ch and J is the closed subscheme of U where the sheaves in line iii. above are isomorphic.

75 Proposition 17.1 Suppose that m, mˆ > m000 and

ss fh r For all h ∈ J the map Ch → P is stable.

Proof. By Theorem 16.18 in the previous section, under these hypotheses, for all h ∈ r ss fh r ss Hilb(P(W )× P ) , the map Ch → P is potentially stable. Suppose that h ∈ J and that fh is a potentially stable map but not a stable map. Then there is at least one irreducible 0 0 0 component C of Ch for which d = g = 0 and k ≤ 2. Then

⊗a (O (1) ⊗ O r (1))| ∼= (ω ⊗ O r (3a + 1))| P(W ) P Ch Ch P Ch since h ∈ J. But

r degC0 (OP(W )(1) ⊗ OP (1))|Ch = 1 + 0 = 1 while ⊗a deg 0 ω ⊗ O r (3a + 1))| = 2 · 0 − 2 + k + 0 ≤ 0. C Ch P Ch

The contradiction implies that fh is a stable map.

r mˆ 3a Now we shall construct Mg(P , d). We will assume m = 2a−1 because the proof is most transparent for this choice. Note that

( 1 + 3g+q2 )((3n + 2)a − 1) − 1 3a 1 ((3n + 2)a − 1) − 9 − 1 2 m < < 2 2 g−1 . (3n + 2)a − 1 − n 2a − 1 4n so Theorem 16.18 and Proposition 17.1 apply. After completing the construction with this mˆ mˆ value of m we shall make some remarks on extending the range of m .

76 Proposition 17.2 Suppose that m, mˆ > m000 and

 (g − 3 + e(q + 1) + q + µ m00)(e − g + 1),   2 1 3 1   (9g + 3q − 3¯g − 9 )(e − g + 1),   2 2   (7g − g¯ + q − 15 )(e − g + 1),  m > max 2 2 10g − 2¯g + 2q2 − 6,  11   10g + 2q2 − − g¯   2   12g + 4q2 − 4¯g 

mˆ 3a ss r ss and m = 2a−1 . Then J is closed in Hilb(P(W )× P ) .

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 J ss is a locally closed subset of Hilb(P(W )× Pr)ss. We will use the valuative criterion of properness to show that J ss 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) ⊂ J ss. Then we will show that α(0) ∈ J ss. 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

∼ ⊗a (OP(W )(1) ⊗ OPr (1))|D = (ω ⊗ OPr (3a + 1))|D . η D/SpecR η S We will write D0 =: C. Decompose D0 = C = Ci into its irreducible components. Then we can write

∼ ⊗a X (OPW (1) ⊗ OPr (1))|D = ω ⊗ OPr (3a + 1)) ⊗ OD(− aiCi) (34) D/SpecR ∼ 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 = Ci

ai=0 0 [ 0 and C = Ci. Since at least one of the ai is zero, we have Y 6= ∅ and C 6= C. Suppose for

ai>0 purposes of contradiction that C0 6= ∅ (hence Y 6= C); then the hypotheses of Ampliﬁcation 16.14 are satisﬁed for C0 and k. P Any local equation for the divisor OD(− aiCi) must vanish identically on every component of C0 and on no component of Y . Such an equation is zero therefore at each of the

77 k nodes in Y ∩ C0. Thus we obtain the inequality X k ≤ degY (OD(− aiCi) ⊗−a = degY (OP(W )(1) ⊗ ω ⊗ OPr (−3a)) ⊗−a r = degY (OP(W )(1)|D0 ⊗ (ω ⊗ OP (−3a))|D0

r = degY (OP(W )(1)|D0 − a degY (ω ⊗ OP (−3))|D0

= eY − a(2gY − 2 + k) − a(3dY ) (35)

0 0 0 Substituting e = e − eY , d = d − dY , g = g − gY − k + 1 and e = a(2g − 2 + 3d), inequality (35) is equivalent to e0 − 3ad0 − 2a(g0 − 1) ≤ ak − k. (36)

We show below that Ampliﬁcation 16.14 applied to C0 yields the inequality

eY − a(2gY − 2 + k) − a(3dY ) < k.

Together with line (35) this implies k < k. The contradiction implies that C0 = ∅ and ss Y = C. This implies that all the coeﬃcients ai are zero and α(0) ∈ J , 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)

0 0 0 Substituting e = e − eY , d = d − dY , g = g − gY − k + 1 and e = a(2g − 2 + 3d), we get

⇐⇒ (e − e0) − a(2(g − g0 − k + 1) − 2) − a(3(d − d0)) < k(a + 1) ⇐⇒ e − 2a(g − 1) − 3ad − e0 + 2a(g0 − 1) + 3ad0 + 2ak < k(a + 1)

⇐⇒ ak − k < e0 − 2a(g0 − 1) − 3ad0 k e0 3 k ⇐⇒ < − (g0 − 1) − d0 + (37) 2 2a 2 2a We apply Ampliﬁcation 16.14 to C0 to establish inequality (37).

0 0 0 e0 + k e0 − g0 + 1 + (d(e −g +1)−d (e−g+1))m ˆ S 2 < em + e e − g + 1 em k dmˆ dmˆ mˆ S (e − g + 1) ≤ e0( + g − 1) + (e + )(1 − g0) − d0(e − g + 1) + 2 m m m m

78 0 k e0( dmˆ + g − 1) ( d mˆ + g − 1)(1 − g0) d0mˆ S ≤ m + 1 − g0 + m − + 2 e − g + 1 e − g + 1 m (e − g + 1)m k e0(1 + nmˆ ) ( nmˆ + 1)(1 − g0) d0mˆ S ≤ m + 1 − g0 + m − + 2 (3n + 2)a − 1 (3n + 2)a − 1 m (e − g + 1)m k (e0 − g0 + 1)(1 + nmˆ ) d0mˆ S ≤ m + 1 − g0 − + 2 (3n + 2)a − 1 m (e − g + 1)m 0 0 m 0 k e − g + 1 mˆ 1 + nmˆ d mˆ 0 S ≤ ( 2 1 ) − + 1 − g + (38) 2 3a m 1 + 3n − 3na m (e − g + 1)m We showed in line (36) that e0 − 3ad0 − 2a(g0 − 1) ≤ ak − k. This implies that S S e0 − 3ad0 − 2a(g0 − 1) + ≤ ak − k + . (e − g + 1)m (e − g + 1)m

S S The hypotheses on m imply that (e−g+1)m < 1 so ak − k + (e−g+1)m < ak − k + 1 < (2a − 1)k. Thus S e0 − 3ad0 − 2a(g0 − 1) + < (2a − 1)k (e − g + 1)m 0 0 2a−1 0 0 e − g + 1 3a 1 + 3an 0 3a S e 3d k 2 1 − d + < − + . (39) 3a 2a − 1 1 + 3n − 3an 2a − 1 (e − g + 1)m 2a 2 2a

mˆ 3a Then if m = 2a−1 the inequalities (38) and (39) imply that

0 0 m 0 k e − g + 1 mˆ 1 + nmˆ d mˆ 0 S ≤ 2 1 − + 1 − g + 2 3a m 1 + 3n − 3na m (e − g + 1)m 0 0 2a−1 0 e − g + 1 3a 1 + 3an 3ad 0 S = 2 1 − + 1 − g + 3a 2a − 1 1 + 3n − 3na 2a − 1 (e − g + 1)m e0 3d0 k < − + + 1 − g0. 2a 2 2a

0 This is what we needed to show, for as remarked earlier substituting e = e − eY , 0 0 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 C0 and Y as described. Thus all the coeﬃcients ai must be zero, and we have an isomorphism

∼ ⊗a (OP(W )(1) ⊗ OPr (1))|D = (ω ⊗ OPr (3a + 1))|D. D/SpecR

79 ss ss r ss In particular, D0 ∈ J , so J is closed in Hilb(P(W )× P ) .

Corollary 17.3 Suppose that m, mˆ > m000 and

mˆ 3a ¯ and m = 2a−1 . Then J//SL(W ) = J//SL(W ). In particular the quotient J//SL(W ) is projective.

Proof. By Proposition 17.2, J ss = 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 quotient to the Kontsevich-Manin moduli space of maps.

Theorem 17.4 Suppose that m, mˆ > m000 and    (p + 1)n   (g − 3 + e(q + 1) + q + µ m00)(e − g + 1),   2 1 3 1   9   (9g + 3q2 − 3¯g − 2 )(e − g + 1),   15  m > max (7g − g¯ + q2 − 2 )(e − g + 1),  10g − 2¯g + 2q2 − 6,   11   10g + 2q2 − − g¯   2   12g + 4q2 − 4¯g 

mˆ 3a ¯ ∼ r where this p and n are those deﬁned on page 41. If m = 2a−1 , then J//SL(W ) = Mg(P , d).

mˆ Proof. The hypotheses on m,m ˆ , a, and m 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 ) r is Mg(P , d). Graber and Pandharipande note that the action of P GL(W ) on J is proper

80 r ([GP] Appendix A). Then the SL(W ) action is proper as well. Therefore j : J → Mg(P , d) is a closed morphism. J ss is an open subset of J, so J \J ss is closed in J. Therefore j(J \J ss) r ss ss ss is closed in Mg(P , d). But J is also SL(W )-invariant, so j(J )∩j(J \J ) = ∅. Therefore ss ss r j r j(J) \ j(J ) = j(J) \ j(J ) is closed in Mg(P , d). But J → Mg(P , d) is surjective, so ss r ss ss r j(J)\j(J ) = Mg(P , d)\j(J ) is closed, and we conclude that j(J ) is open in Mg(P , d). ss ss The property of being a categorical quotient is local, i.e. j|Jss : J → j(J ) is a categorical quotient of J ss for the SL(W ) action. The categorical quotient is unique up to isomorphism, so J//SL(W ) ∼= j(J ss). We have that J//SL(W ) is projective so j(J ss) is also projective, hence closed. Then j(J ss) is open and closed, and by Theorem 15.2 it is nonempty, r so it must be a connected component of Mg(P , d). But Pandharipande and Kim have shown r ∼ r (main theorem, [KP]) that Mg(P , d) is connected, so we have J//SL(W ) = Mg(P , d). ¯ ∼ r Applying the previous corollary we have J//SL(W ) = Mg(P , 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 K˜ ss 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 r Pandharipande that a projective scheme Mg(P , d) exists, is a coarse moduli space for this r moduli problem, is connected, and is the quotient of a proper action J → Mg(P , d); in the literature these facts are only presented over C. It seems that Alexeev proves the existence of r Mg(P , 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 ¯ r any base. However our argument comparing the quotient J//SL(W ) to Mg(P , 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.

mˆ Extending the Range of m

mˆ 3a To prove Theorem 17.2, Corollary 17.3, and Theorem 17.4 we ﬁxed the value m = 2a−1 . Recall that this was used in the proof to establish the inequality

0 0 m 0 0 0 e − g + 1 mˆ 1 + nmˆ d mˆ e 3d k 2 1 − < − + (40) 3a m 1 + 3n − 3na m 2a 2 2a mˆ of page 79. So we now ask: can we ﬁnd a wider range of m for which (40) is veriﬁed so that the proof of Theorem 17.2 goes through as written? The inequality (40) is equivalent to

mˆ (e0 − g0 + 1)n e0 3d0 k 3a − d0 ≤ − + − . (41) m 3an + 2a − 1 2a 2 2a 3an + 2a − 1 It is tempting to take 0 0 mˆ e − 3d + k − 3a ≤ 2a 2 2a 3an+2a−1 (42) m (e0−g0+1)n 0 3an+2a−1 − d 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 e0, g0, d0 of the quantity

e0 3d0 k 3a 2a − 2 + 2a − 3an+2a−1 (e0−g0+1)n 0 3an+2a−1 − d

mˆ 3a will be. Hence for the time being we are left with m = 2a−1 as the only linearization for ∼ r which we claim the quotient J//SL(W ) = Mg(P , d).

18 Another Linearization

We mentioned (page 36) that G.I.T quotients depend on a choice of linearization. In r mˆ our construction of Mg(P , d) we continually reﬁned our choice of m ; recall that this value determines the linearization used. The following argument suggests that for a diﬀerent range mˆ of values of m , namely when m >> mˆ , 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) ˆ r and recall our definition of the Hilbert point Hm,mˆ (C) from page 32: Let h ∈ Hilb(P(W )×P ). r ˆ The Hilbert polynomial of Ch ⊂ P(W )× P is em + dmˆ + 1 − g. We define Hm,mˆ (Ch), the

(m, mˆ )−th Hilbert point of Ch, as follows:

1 r Deﬁnition 18.1 If m and mˆ are suﬃciently large, then H (Ch, OP(W )(m)⊗OP (m ˆ )|Ch ) = 0 and the restriction map

0 r 0 r r ρˆm,mˆ : H (P(W )× P , OP(W )(m) ⊗ OP (m ˆ )) → H (Ch, OP(W )(m) ⊗ OP (m ˆ )|Ch ) is surjective. VP (m)+dmˆ VP (m)+dmˆ 0 r Then ρˆm,mˆ is a point of P( H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ ))). ˆ VP (m)+dmˆ We set Hm,mˆ (Ch) := ρˆm,mˆ for all m > m0.

Recall that there is an induced SL(W ) action on VP (m)+dmˆ 0 r P( H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )). We will make this more explicit than in 0 Section 14. If w0, ..., wN is a basis of H (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 PN g.wp = j=0 apjwj. γ0,γ1,...,γN γ0 γ1 γN Write w0,1,...,N := w0 w1 ··· wN where γ0 + ··· + γN = m. Let Bm be the resulting 0 monomial basis of H (P(W ), OP(W )(m)). The SL(W ) action is as follows:

γ0,γ1,...,γN γ0 γN g.w0,1,...,N . = (g.w0) ··· (g.wN ) .

r+1 Γ0,Γ1,...,Γr Γ0 Γ1 Γr Pick a basis f0, ..., fr of C . Set f0,1,...,r := f0 f1 ··· fr where Γ0+···+Γr =m ˆ . Let 0 r Bmˆ be the resulting monomial basis of H (P , OPr (m ˆ )). Tensor Bm with Bmˆ to get a basis ˆ 0 0 r ∼ 0 r Bm,mˆ of H (P(W ), OP(W )(m)) ⊗ H (P , OPr (m ˆ )) = H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) consisting of monomials having bidegree (m, mˆ ). The SL(W ) action may be written

γ0,γ1,...,γN Γ0,Γ1,...,Γr γ0 γN Γ0 Γr g.w0,1,...,N f0,1,...,r = (g.w0) ··· (g.wN ) f0 ··· fr .

VP (m)+dmˆ 0 r A basis for H (P(W )× P , OP(W )(m) ⊗ OPr (m ˆ )) is given by elements of the form

γ0,1,...,γN,1 Γ0,1,...,Γr,1 γ0,P (m)+dmˆ ,...,γN,P (m)+dmˆ Γ0,P (m)+dmˆ ,...,Γr,P (m)+dmˆ w0,...,N f0,...,r ∧ · · · ∧ w0,...,N f0,...,r (43)

83 such that for all j, γ0,j + ··· γN,j = m and Γ0,j + ··· + Γr,j =m ˆ 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)+dmˆ  P (m)+dmˆ  X X  γ0,j α0 + ··· +  γN,j αN (44) j=1 j=1

Letw ¯0,..., w¯N be the images of w0, . . . , wN under the map

0 0 0 H (P(W ), OP(W )(1)) → H (pW (C), OpW (C)(1)) → H (C,LW ).

0 Choose a section s ∈ H (C,Lr) which does not vanish identically on any component of C. Then the map

0 m 0 m mˆ H (C,LW ) → H (C,LW ⊗ Lr ) σ 7→ σ ⊗ smˆ

0 m mˆ 0 m mˆ is injective. Let S = H (C,LW ) ⊗ s ⊂ H (C,LW ⊗ Lr ). 0 m mˆ Complete S to a basis of H (C,LW ⊗ Lr ) by choosing a set of vectors

˜ ˜ ˜ γ0˜,j ,...,γ˜N,j Γ0,j ,...,Γr,j S = {w¯0,...,N f0,...,r |1 ≤ j ≤ dmˆ }

˜ 0 m mˆ P (m) so that S ∪ S is a basis of H (C,LW ⊗ Lr ). Now if Hm(C) := ∧ ϕm(C) (see page 25 P (m) X and [Gies] pages 6-9) has a nonzero coeﬃcient corresponding to weight ( γ0,j)α0 + ··· + j=1 P (m) X 0 0 m ( γN,j)αN there is a basis of H (P(W ), OP(W )(1)) which induces a basis of H (C,LW ) j=1 γ0˜,j ,...,γ˜N,j of the form {w¯0,...,N |1 ≤ j ≤ P (m)}. Then, writing S with this basis, we obtain a basis ˜ 0 m m S ∪ S of H (C,LW ⊗ Lr ) which may be written

γ0˜,j ,...,γ˜N,j γ0˜,j ,...,γ˜N,j Γ0˜,j ,...,Γ˜r,j {w¯0,...,N |1 ≤ j ≤ P (m)} ∪ {w¯0,...,N f0,...,r |1 ≤ j ≤ dmˆ }.

ˆ That this is a basis implies that Hm,mˆ (C) has a nonzero coeﬃcient corresponding to P (m) P (m) dmˆ dmˆ X X X X weight ( γ0,j)α0 + ··· + ( γN,j)αN + ( γ˜0,j)α0 + ··· + ( γ˜N,j)αN . j=1 j=1 j=1 j=1

84 Let CH denote the convex hull of the set of weights such that the corresponding coeﬃcient

of Hm(C) is nonzero, and let CHd denote the convex hull of the set of weights such that the ˆ corresponding coefficient of Hm,mˆ (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. ˆ By the argument above, the set of weights such that the coefficient of Hm,mˆ (C) is nonzero contains the set P (m) P (m) dmˆ dmˆ X X X X ( γ0,j)α0 + ··· + ( γN,j)αN + ( γ˜0,j)α0 + ··· + ( γÑ,j)αN (45) j=1 j=1 j=1 j=1 PN Since the γi,j and theγ ˜i,j are nonnegative integers and i=1 γi,j = m for all 1 ≤ j ≤ P (m) PN 1 while i=1 γi,j =m ˆ for all 1 ≤ j ≤ dmˆ we find that if m >> mˆ then mP (m) CHd is a 1 small perturbation of mP (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 CHd as well. Deligne-Mumford stable curves have SL(W )-stable Hilbert points, so this suggests that if ˆ m >> mˆ and C is a Deligne-Mumford stable curve then Hm,mˆ (C) is SL(W )-stable. Then for such linearizations J//SL¯ (W ) 6= ∅. However, if m >> mˆ , the inequality

0 0 (e0 + k ) h0 + (dh −d (e−g+1))m ˆ 2 < em , e e − g + 1 fails for all maps whose domains are not potentially stable curves in the sense of Definition ¯ ∼ r 16.16. This suggests that for such linearizations the quotient J//SL(W ) 6= Mg(P , d). Thaddeus [Th1] exploits the existence of different linearizations with nonisomorphic quotients 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 r results on the cohomology of Mg and Mg(P , d).

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

r One of the primary motivations for constructing Mg(P , d) as a G.I.T. quotient was to help us study other moduli spaces. We turn our thoughts in this direction before closing. r f r Let Mg(P , d) be the moduli space of isomorphism classes of morphisms C → P r from nonsingular curves C of genus g to P such that f∗([C]) = d. There is an inclusion

85 r r Mg(P , d) ,→ Mg(P , d) into the Kontsevich-Manin moduli space of isomorphism classes of r stable maps. Deﬁne H ⊂ Mg(P , d) as the locus of nondegenerate 10-canonical embedding r maps. Identify H with its image under inclusion into Mg(P , d). Let H be the closure in r Mg(P , d) of H. As stated in Section 10 we would like to study H and H. j r Let J → Mg(P , d) be the quotient map as before. It is natural to wonder whether we 1 ⊗10 −1 −1 can compare the closed subscheme V (Fittg−1R ϕ∗(ωC ⊗ ((OPr (1)|C) ) of J to j (H) or −1 1 ⊗10 −1 j (H) or else its image j(V (Fittg−1R ϕ∗(ωC ⊗ ((OPr (1)|C) )) to H or H. The proof of Proposition 12.3 shows that

1 ⊗10 −1 10 V (Fitt R ϕ (ω ⊗ ((O r (1)| ) ) ⊃ {h ∈ J|ω ∼ O r (1)| }, g−1 ∗ C P C Ch = P Ch

which allows us to conclude that every 10-canonical stable map C → Pr with C nonsingular 1 ⊗10 −1 has a model in V (Fittg−1R ϕ∗(ωC ⊗ ((OPr (1)|C) ), and by Theorem 15.2 such points are SL(W )-stable so their behavior under the quotient map is straightforward. Thus we can 1 ⊗10 −1 r conclude that j(V (Fittg−1R ϕ∗(ωC ⊗((OPr (1)|C) )) is a closed set of Mg(P , d) containing H. A word of caution: to use [FP] Proposition 1 to show

1 ⊗10 −1 10 V (Fitt R ϕ (ω ⊗ ((O r (1)| ) ) = {h ∈ J|ω ∼ O r (1)| } (46) g−1 ∗ C P C Ch = P Ch

10 we must have that the multidegrees of ω and O r (1)| are equal for all h ∈ J. But this Ch P Ch multidegree condition is only satisﬁed on an open set of J. So the stronger statement (46) r may not be true. So we have produced a closed set of Mg(P , 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 in Mg(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 HG(X), is the ordinary coho- ∗ mology H (EG ×G X). Here EG ×G X 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 HG(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 HG(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

r r ˙ r−q e(E) r+1 · · · → HG(B) → HG(E) → HG (B) → HG (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:

87 Theorem 20.2 Let T be a torus acting on a manifold M, let F index the components F of T the ﬁxed point set M , and let eF be the equivariant euler class of the normal bundle to F ∗ in M. Let η ∈ HT (M). Then

Z X Z η(X) η(X) = . eF (X) M F ∈F F

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, Poincaréduality, 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

ker ∂ : ICi(X) → ICi−1(X) IHi(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).

89 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 IHi(B; Q) is a direct summand of IHi(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).

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93 22 Index of Terms and Notation categorical quotient, 5 coarse moduli space, 4 Deligne-Mumford (semi)stable curve, 16 dualizing sheaf, 17 ﬁne moduli space, 4 Fitting ideal, 27 Hilbert functor, 24 Hilbert point of a curve, 25, Hilbert scheme, 24 linearization, 7 local universal property, 5 orbit space, 5 potentially stable curve, 74 potentially stable map, 74 prestable curve, 16 prestable map, 20 semistability, G.I.T. 7 stable map, 20 universal family, 4 Xss(L), the locus of G.I.T. semistable points with respect to a linearization L, 7 ss X//LG, the categorical quotient of X (L), 7 Mg, the moduli space of nonsingular curves of genus g, 16 Mg, the moduli space of Deligne-Mumford stable curves of genus g, 18 Mg(X, β), the moduli space of stable maps, 20 HilbP,X/T , the Hilbert scheme of subschemes of X with Hilbert polynomial P , 24 N HilbP,N , the Hilbert scheme of subschemes of P with Hilbert polynomial P , 24 Hm(C), the Hilbert point of C in Gieseker’s construction, 25 ˆ r Hm,mˆ (C), the Hilbert point of a map C → P , 32 K˜, the locus of n-canonically embedded curves in the Hilbert scheme, 26 Fittg(G), the gth Fitting ideal of an ideal sheaf G, 27 V (I), the closed subscheme corresponding to the quasicoherent sheaf of ideals I, 27 Φi(), the cohomology and base change theorem map, 28 i R ϕ∗, higher direct image functor, 28 ϕ C → Hilb, the universal family over the Hilbert scheme, 25, 31 Hilb(P(W )× Pr), the Hilbert scheme of subschemes of P(W )× Pr having bidegree (e, d) e, 29 d, the degree of the stable map to Pr, 29 d d n, denotes 2(g−1) on pages 21-29, and g−1 on pages 29-86 N, frequently denotes e − g a, constant relating e and d, 29 J, 29 ˆ 0 ˆ wλ0 (Mi), the λ -weight of a monomial Mi, 33 Pj ˆ 0 i=1 wλ0 (Mi), the total λ -weight of a collection of j monomials, 33 LW , Lr, 35 0 00 000 m , m , m , q1, q2, q3, µ1, µ2, constants, 35 C1, 47 g¯, the minimum value of gC¯ taken over all curves C of genus g, 56

94 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.