Moduli of Curves

Bold entries in the main body index point to the defining occurrence of the cited term. The type is a very slightly modified version of the Lucida font family designed by Chuck Bigelow and Kristin Holmes.

Parameter spaces

A Parameters and moduli

Parameters and moduli 3 Unfortunately, few natural moduli functors are representable by

Exercise (1.4) Show that, if one exists, a rough moduli scheme (M,Ψ) for F is determined up to canonical isomorphism by condition 2) above. Exercise (1.5) Show that the cuspid curveM defined above is not a rough modulus space for lines inC2.

Construction of the Hilbert scheme 5 4) Show that the pair (Spec(C), Ψ) has the universal property in 2)

B Construction of the Hilbert scheme

For large m, then Q(m) is the dimension of the degree I(X)m of the ideal X in Pr. A key step in the proof of the lemma is to show that the supports are idealisK.

Construction of the Hilbert scheme 9 only if J is actually the degree m piece of the ideal of a variety X

This is the source of the universal properties that we will rely heavily on later in this book and one reflection is that the Hilbert scheme captures much finer structure. Construction of the Hilbert scheme 11There are a number of useful variants of the Hilbert scheme whose.

Construction of the Hilbert scheme 11 There are a number of useful variants of the Hilbert scheme whose

The first family is parameterized by the Hilbert scheme H, which we saw in the second exercise above is simply a PN projective space. A way out is to first define the module space Vd,g to be reduced by the subscheme eWd, the support of which is the set of reduced and irreducible plane curves of degrees and geometric genus, and then consider only the families of such curves with base B that are equipped with a mapB✲Vd , g.

C Tangent space to the Hilbert scheme

Tangent space to the Hilbert scheme 13 same point p) give a morphism from the ﬁbered sum of I with it-

The universal property of the Hilbert scheme implies that such a tangent vector to G will lie in the Zariski tangent space to the subscheme H if and only if X is just above I. The tangent space to the Hilbert scheme 15 is a cubic plane and an isolated point: this component has dimension 15.

Tangent space to the Hilbert scheme 15 plane cubic and an isolated point: this component has dimension 15

Components of the Hilbert scheme whose general term is not connected (let alone irreducible) are in fact the rule rather than the exception. Tangent space for the Hilbert scheme 17larger than that of the components parameterizing honest curves.

Tangent space to the Hilbert scheme 17 greater than that of the components that parameterize honest curves

There is little convincing evidence for or against the connection of the bounded Hilbert variety or its closure R: the examples known so far have provided neither a counterexample nor a reliable replacement for the class of fans used in Hartshorne's proof.

D Extrinsic pathologies

Extrinsic pathologies 19 construction of the Hilbert scheme. Most of the work there was de-

These phenomena are usually caused by the constraints imposed by the particular fiber models parameterized by the Hilbert scheme in question. One of the motivations for studying the space of intrinsic modules is the possibility of eliminating such extrinsic pathologies.

Mumford’s example

Extrinsic pathologies 21

Indeed, there exist first-order deformations of C that do not lie on any cubic, and these account for the extra dimension in the tangent space of H. In this case, the class of C in the Neron-Severi group of NS(S) will not be equal to 4H+2L.

Extrinsic pathologies 23 C is residual in this intersection to a quartic B of the form 2L + D

Other examples

Since the binomial coefficient on the right is the dimension of the space of degree surfaces containing the quarticS, C continues to lie in the complete intersection of S and a surface T of degree n. In the singular roll S,H∼(r−1)F and these coincide but are generally distinct; it follows (at least as long as r ≥4) that there are two components of the Hilbert scheme of curves of certain degree and genus, whose general members are the Castelnuovo curves.

E Dimension of the Hilbert scheme

Therefore, the choice of the linear subsystem of H0(C,L) with dimension (r +1) determines a point in a Grassmannian G(r, d−g) whose dimension (r +1)(d−g+r) is. The general question of what the dimensions of the components of H might be remains very open.

Severi varieties 29 2) For any d, g and r , what is the smallest dimension of a component

F Severi varieties

Furthermore, we see that the projection map π :Σ✲PN is an immersion in (C, p), with the tangent space to Σ1at (C, p) mapping isomorphically to the space of vanishing degree polynomials atp (i.e. i.e., having00=0). The tangent space with Vd,g at C is therefore the linear space of degree polynomials that vanish at the points pi.

G Hurwitz schemes

Hurwitz schemes 33 up to rescaling, that has B as its cycle of zeros, this might seem almost

The connectedness depends on an analysis (first carried out by Klein, Clebsch, L¨uroth and Hurwitz) of the braid monodromy of Hd,g. When the mapC✲P1 has non-standard branching (i.e. the branch divisor B has multiple points), then the number of possible combinatorial forms for the monodromy drops.

Basic facts about moduli spaces of curves

A Why do ﬁne moduli spaces of curves not exist?

Why do ﬁne moduli spaces of curves not exist? 37 There are a number of approaches to dealing with the obstructions

This will, in general, be a fine modulo space for the open subfunction of the "automorphism-free curve" Fof. The drawback of this approach is that each marked point increases the dimension of the module space by 1.

Why do ﬁne moduli spaces of curves not exist? 39 2) Show that j expresses U as a Galois cover of the j-line, with Galois

Categories of algebraic stacks are other extensions of the category schemes that have been widely used to study moduli problems. We leave it to you to formulate or find the correct definition of a morphism between stacks (a task that begins to bring out the flavor of the subject).

B Moduli spaces we’ll be concerned with

Again, the fiber of Pd,g over a point[C]∈ Mg corresponding to a curve C without automorphisms is the connected component Picd(C) of the Picard variety of C; in particular, P0,gis sometimes called the bundleover Jacobian modulus. For example, it follows from the Harer-Mestrano proof of Franchetta's conjecture (discussed later in this chapter; or see [112]) that P0,g Pd,g if and only if(2g−2)|d.

The Teichm¨ uller approach

We should also mention that this approach provides Mg with a natural metric, called the Weil-Petersson metric, whose positivity properties were used by Wolpert to construct an embedding of Mgin, a projective variety with many of the good properties of Deligne- Consistent Mumford compactification, which we introduce later in this section. An excellent survey of what is known along these lines can be found in the paper by Hain and Looijenga [70].

The Hodge theory approach

Getting the closure of Mg in Ag gives a compaction of Mg. which we will denote by Mg and also refer to as the Satake compaction. Exercise (2.10) Assuming the facts quoted above about the Satake compactness of Mg, show that through every finite collection of points of Mg there passes a complete curve extending to Mg. Hint: blow up the points and use the fact that the attraction in this swelling of a quantity - Large multiple of an abundant linear series in Mgminus the sum of the extraordinary divisors of the explosion is very abundant.

The geometric invariant theory (G.I.T.) approach

Moreover, a local calculation in deformation theory that we will perform in the next chapter shows that the locus of curves with more thanδnodes lies in the closure of the locus of those with exactlyδ. For each homeomorphism type, find the dimension of the locus of the corresponding curves in Mg and say which of these loci are in the closure of which others.

D Geometric and topological properties

Problem (2.24) Let B be a smooth curve of genusg−1,p ∈B each point, and for anyq∈B\ {p}letXqbe the stable curve obtained by identifying pandqonB. Problem (2.25) [prepared by Jean-Francois Burnol] Let ∆(α) ⊂ Mg be the locus of stable curves withαor more nodes.

Basic properties

Exercise (2.26) It is a classical fact that the automorphism group of a smooth genus curve can have order at most 84 (g−1).

Local properties

Geometric and topological properties 53 To begin with, the moduli space M g is smooth at a point [C] cor-
Geometric and topological properties 55
Geometric and topological properties 57 in this family will be a complete family of curves of genus 4g 0 − 1 with
Geometric and topological properties 59 that the low-dimensional cohomology ring of M g is independent of

At the same time, it is clear that the dimension of the families produced is only logarithmic in the genus g. The answers are striking: for example, it shows that the orbifold is the Euler characteristic of the universal curve.

Cohomology of the universal curve

Geometric and topological properties 63 One hypothesis that would guarantee the existence of the desired

A theorem due to Mestrano and Ramanan [113] asserts that there is a Poincar'e line bundle onPd,g if and only ifd−g+1 is relatively prime to 2g−2. This, combined with Harer's results, implies that any rationally definite line bundle over moduli is a rational multiple of the canonical bundle.

Cohomology of Hilbert schemes

Exercise (2.51) Compute the codimension in Mgof the component image of the Hilbert scheme shown in exercise (2.44); in particular, note that it is larger seg. Finally, we should say that there are analogous conjectures about the dimension and irreducibility of the Hilbert scheme.

Structure of the tautological ring

Geometric and topological properties 69 The other parts of Faber’s conjecture deal with the nature of the

On the other hand, pushing down to Mg any monomial in the classes Dd,ij andωd,i involved in the definition of the Fds turns out to yield an element of the tautological ring on Mg, so this relation pushes down to an inR∗(Mg). Conjecture (2.55) (Faber's conjecture, second part) The ideal of relations in the tautological ring R∗(Mg) is generated by that of the form.

Geometric and topological properties 71

Witten’s conjectures and Kontsevich’s theorem

Geometric and topological properties 75

As with the KdV form, it is clear that the system of equations (2.72) also uniquely determines F. More precisely, Witten devised a generalization from moduli spaces of stable curves to moduli spaces of stable maps to a fixed target manifold (moduli spaces of curves are those in which the target of the map was a point) and saw the resulting intersection numbers as a way of finding invariants of the target to produce.

E Moduli spaces of stable maps

Now, however, it is necessary to keep track not only of the decomposition of the curve, but also of the set of marked points and the class γ. In the case of the module space M0,4 P1, there are three such divisors corresponding to the three points 0, 1 and ∞ in this space that parameterize the reducible curves - see exercise (2.19) - and the linear equivalence is simply that of points in P1.

Modulus spaces of stable maps 79and{3,4} ∈B;In choosing for the image eCAandNd−over the image. cf. [135]) which has been used to prove associativity in several cases not yet handled by the types of methods discussed here.

Techniques

A Basic facts about nodal and stable curves

Dualizing sheaves

Basic facts about nodal and stable curves 83

The next two exercises show that the dualizing sheaf of a stable curve has sufficient properties only very slightly weaker than that of the canonical sheaf on a smooth curve, and in these terms give a characterization of moduli-stable curves under all nodal . Given this, the relative dualizing sheaf of a family ϕ : C✲B of nodal curves whose common fiber is smooth can be characterized as the unique line bundle extending the relative cotangent bundle on the locus C∗ of smooth points of ϕ.

Automorphisms

If not, there would be a non-zero-valued point of Aut(C) lying above the identity, or equivalently a non-zero regular vector field on C. Hint: Such a vector field would correspond to a regular vector field on the normalization C, which vanishes at all points above the nodes in C.

B Deformation theory

To define the automorphism group of a stable curve C, we just take D=Cin above, that is to identify Aut(C) with the stabilizer in PGL(r+1) of [C]∈ H. We conclude noting that similar arguments give a relative version: given two stable curves C ✲S and D✲S, there exists a scheme IsomS(C, D) which is finite and unbounded overSand represents the function of S-isomorphisms between the curves.

Overview

Deformation theory 87 deformations of smooth varieties, as our model. A deformation of a

Second, and usually much more important in practice, no uniqueness properties are required for :U✲Y mappings that realize a given deformation as a local indentation of the verso. In particular, the existence of a verso deformation does not imply the representability, even in the rough sense, of a deformation functor.

Deformation theory 89

Deformations of smooth curves

Generalizing the above exercise, we see that the annihilator of the image in Def1(C) is the tangent space to such a deformation. Instead, we give an alternative construction that has the advantage of working for all stable curves at the end of the next section.

Variations on the basic deformation theory plan

Deformation theory 93
Deformation theory 97 contained in a ﬁrst-order deformation Y ⊂ Z × I of Y : we look at the
Deformation theory 99 There are, of course, some cases whose spaces of ﬁrst-order defor-

Pr is the space H0(X, NX/Pr) of global sections of the normal sheafNX/Pr of X inPr. In the case of a curveC, we can dualize this to see that the cotangent space of the subschemaWr(C) is the annihilator of the image of the multiplication map. 4) The space of deformations of a pair (X, L) with X a smooth variety and La line bundle of X, X, which is not taken as being fixed, is the space H1(X,ΣL), whereΣLis the chain of first-order differential operators iL .

Universal deformations of stable curves

Deformation theory 103 morphisms and, after going through the construction, indicate what

Let J be the subscheme of Hconsisting of the Hilbert points of the subcurves C ⊂Pr that are abstractly isomorphic to C (ie that correspond to different choices of line bundles and bases for sections on C). By the universal property of the Hilbert scheme, this is induced by a unique mapχ:Z✲H, which by construction has image inKand that fits into a commutative square.

Deformations of maps

Deformation theory 109 Lemma (3.41) Let X ✲ B be a ﬂat smooth proper family, Y a smooth
Deformation theory 113 Now let t and ε be local coordinates on X, with ε a local coordinate
Deformation theory 115 and
Stable reduction 117 show there are no triple points, it’s enough to show that for any three

We letNbbe be the normal sheaf ofψb, that is, the cokernel of the morphismψb of sheaves onXb. Show that the Kodaira-Spencer map gives an isomorphism of the space of first-order deformations of ψ with H0(X, N).

C Stable reduction

Results

Stable reduction 119

This can be done by making analogous modifications to X (either puffs or base changes and puffs) without reintroducing multiple components of the special fiber. On the contrary: stable reduction is a key ingredient in proving the existence of Mg.

Stable reduction 121

However, before we do this, we need to solve the problem a little by removing the lump in the special fiber. Starting with the family in figure (3.57) and expanding one time, we arrive at the family in figure (3.58) whose special fiber consists of the normalizationCof.

Stable reduction 125 a single base change of composite order, but doing so makes it much

In short, the pullback toX of the divisor (t)onX is simply the sum of the components of the inverse image of the special fiber in that are not included in the branch divider. and with a doubling of the multiplicity for components in the branch divider. Again we can apply the same principles to a base change of any primary order p, except that the multiples of components of the special fiber Xt in the branch divider are then multiplied by the inverse image of X and the new special fiber is1.

Stable reduction 127

The intersection number of each of any component E of the special fiber with the whole special fiber is 0. If we blow down the five curves of this type, we then arrive at the family in Figure (3.67) whose special fiber consists of the association of.

Stable reduction 129 Exercise (3.68) At each stage of this process, calculate the self-

In fact, arranging this map requires a sequence of three augmentations exactly analogous to those required to desingularize the entire space X in the previous example.

Stable reduction 131

What conditions must the ray of an arc in the deformation space of the triple point satisfy to give a family of curves of this type. 3) (Harder) Describe the regularization of the rational map to moduli from the bottom of the triple point deformation space.

Stable reduction 133

One is that it illustrates that the ugliness of the nonsemistationary fiber in a nonsemistationary family has little to do with how singular or reducible the semistationary limit will be. If we think of the family as a family of abstract curves of genus 3, however, the natural limiting object is a smooth, hyperelliptic curve of genus 3—an object that has a large structure.

Exercise (3.82) 1) If {Ct} is a general pencil of curves, smooth for t≠0, specialized for a curve C with an ordinary fourfold point, show that the stable limit of the family will be the union of the normalization C with a curveBof genus 3 meetingCat fourpointsp1, . . . 3, obtained by identifying pi with qi. Problem (3.85) Is it always the case that the geometric place in the Mh,nof curves B, which are formed in this way as boundaries, has dimension 1 smaller than the dimension of the versal deformation space of the inflection point at that time.

Stable reduction 137

Next, at the end, we make further inflations and basis changes as necessary to ensure that closures of the inverse images of the nodes of the common fiber are disjoint parts of X. Aluﬁ and Faber [2] show that the set of possible limits that can arise in isotrivial families, such as the examples above, can depend on the intrinsic geometry of the general fiber D1.

Interlude: calculations on the moduli stack 139

D Interlude: calculations on the moduli stack

First, we don't need to define a stack; and given that we don't have to define them, we won't use any definition or result that depends on the definition of a stack. We have therefore chosen to make purely local and ad hoc definitions of the objects we need to use.

Divisor classes on the moduli stack

Of the various divisors and classes discussed so far, some seem to be naturally rational classes of divisors on the stack of modules: the class λ, for example. Interlude: Computations on Stack of Modules 143a Morphism of Stacks; there is a natural morphismπ from mod-.

Interlude: calculations on the moduli stack 143 a morphism of stacks; there is a natural morphism π from the mod-

Q to arrive at the rational divisor class on the module stack; or we can define the class of the rational divisor σ on the stack of modules as in proposition (3.91). Proposition (3.92) Let Σ⊂ Mg be an irreducible subvariety of codimension 1 and σ ∈Picfun(Mg) be the class of divisors on the stack of modules connected to Σ as in Proposition (3.91).

Interlude: calculations on the moduli stack 145 The answer is a practical one. As we will see very amply illustrated in

In short, we see that the retreat to B of the defining equation of the discriminant hypersurface in the vertical deformation space X0 vanishes in the order k1+k2+ · · · +knat 0; or, in other words, a manifold. Similar descriptions of the classes of divisors δi can be given on the stack of modules connected to the divisors ∆i⊂ Mg.

Existence of tautological families

Interlude: calculations on the moduli stack 149 and

It is worth noting that each step of this construction is already necessary in the case of the module space M1 of curves of genus 1: that is, the affine line with coordinate. Corollary (3.96) (Stable reduction with respect to general bases) For any morphism: X✲B of integral varieties whose general ﬁber is a smooth curve of genus g ≥ 2, there exists a generic ﬁnite map B ✲B, a family of stable curves X ✲ B and a birational isomorphism of X with the essential withdrawal to B in the family X✲B.

E Grothendieck-Riemann-Roch and Porteous

In this case, we will simply ignore the components of the fiber product that do not dominate B: we define the essential retraction X ✲B of our family as the unique irreducible component X of the fiber product X×BB dominatingB , equipped with the restriction of the projection map to B. We should note (although it is not really within the scope of this book) that an analogue of the basic stable reduction theorem Proposition (3.47) holds for families X ✲B of higher-dimensional manifolds over a one-dimensional basis B : after basis change and birational modifications, we can arrive at a family whose fibers are all schema-theoretic normal intersections.

Grothendieck-Riemann-Roch

Grothendieck-Riemann-Roch and Porteous 151 a topological invariant, and which is expressed by the Riemann-Roch

Beyond that, however, we would like to understand the twist of the π∗E sea, as measured by its Chern classes. Indeed, we can think of the rank of a bundle only as part of the degree 0 of its Chern character, an object we would like to fully understand.).

Grothendieck-Riemann-Roch and Porteous 153 in which σ i denotes the i th elementary symmetric function and n i the

Exercise Show that when B is a point, π!(E)=χ(E)and td(B)=1 and use this to reduce the above formula to the Hirzebruch-Riemann-Roch formula. Of course, when B has a higher dimension, the interpretation of the Grothendieck-Riemann-Roch formula is not so straightforward.

Chern classes of the Hodge bundle

Grothendieck-Riemann-Roch and Porteous 155 of cohomology classes on moduli spaces in Chapter 2: the expression

To begin with, notice that the conormal bundle of the mapπ — that is the difference TC−TM inK(Cg). Exercise Show that the relative dualizing sheaf of the new family is trivial on the exceptional divisor of the map µ, and hence that it is simply the retraction of the relative dualizing sheaf of ρ :X ✲B, i.e.

Grothendieck-Riemann-Roch and Porteous 157

We can use this to compute the Todd class of the relative cotangent groupΩ in terms of ω. To begin, to calculate the Chern character of the ideal sea of Z, we apply Grothendieck-Riemann-Roch to the inclusion:Z✲Y.

Grothendieck-Riemann-Roch and Porteous 159

Chern class of the tangent bundle

Porteous’ formula

Problem (3.117) Find the class in Pic(M3) of the locus of curves C with point p such that 4p ∼ KC, or the same union of the locus of plane quartics with hyperflex and the hyperelliptic locus. Forg=3 and 4, determine when this locus is reduced and has the expected codimension, and when it is, compute its class in A(M3). A discussion of these loci in moduli spaces of general genus curves can be found in Section 5.D.).

Relations amongst standard cohomology classes

We can generalize this further by defining Em to be the retraction to the Cgof bundle of canonical differentials and Jkm to be the bundle of kth-order jets of such differentials. The previous example is only casemen=1.) If it is a smooth curve, a point is on Candm≥2, then.

Divisor classes on Hilbert schemes

Hint: Reapply exercise (3.129).) What does this have to do with calculating the class of the hyperelliptic locus in M3. Problem (3.133) Consider two larger open subsets of the Hilbert scheme containing H: the location of stable curves embedded in Pr, and even larger, the location of nodal curves.

F Test curves: the hyperelliptic locus in M 3 begun

Test curves: the hyperelliptic locus in M 3 begun 169 We will refer to H as the locus of hyperelliptic stable curves of genus
Test curves: the hyperelliptic locus in M 3 begun 171 To ﬁnd the degree of λ, we observe that if π : X ✲ B is a family of
Test curves: the hyperelliptic locus in M 3 begun 173 diagram on the right and ﬁnally identify the now disjoint sections
Admissible covers 175

We have to describe the degree, on the basis B=P1 of this family, of the different divisor classes,λ,δ0 andδ1. It is tempting to jump straight to the conclusion that the image of the pencil in M3 also meets the cross 27 times.

G Admissible covers

Admissible covers 181 2) π −1 (B sing ) = C sing , and for every node q of B and every node r of
Admissible covers 183 The key fact to recall is that the connected components of the curve
Admissible covers 185 Deﬁnition (3.158) Let C be a stable curve. We say that a nodal curve
The hyperelliptic locus in M 3 completed 187 h on this family, and clearly the place to start is to identify which

If the smaller of the two sexes isi >0, then the stable model will lie in∆i. An example of the use of these covers (and their moduli) will be given in Diaz' proof.