AMS/IP Studies in Advanced Mathematics

Birational aspects of the geometry of \overline{M}_{g}

Gavril Farkas

1. Introduction

The study of the moduli space M_{g} begins of course with Riemann, who in 1857 was the first to consider a space whose points correspond to isomorphism classes of smooth curves of genus g. By viewing curves as branched covers of P^{1}, Riemann correctly computed the number of moduli, that is he showed that

dim(M_{g}) = 3g − 3

for all g ≥ 2. Riemann is also responsible for the term moduli, meaning essential parameters for varieties of certain kind: ”... es hängt also eine Klasse von Systemen gleichverzweigter 2p + 1 fach zusammenhangender Funktionen und die zu ihr gehörende Klassen algebraischer Gleichungen von 3p−3 stetig veränderlichen Grössen ab, welche die Moduln dieser Klasse werden sollen”. The best modern way of reproving Riemann’s result is via Kodaira-Spencer deformation theory. The first rigorous construction of M_{g} was carried out by Mumford in 1965, in the book [GIT]. By adapting Grothendieck’s ”functorial ideology”, Mumford, used Geometric Invariant Theory and developed a purely algebraic approach to study M_{g}. He indicated that one has to study the coarse moduli scheme that is as close as any scheme can be to the moduli stack of smooth curves: Although the coarse moduli scheme exists over Spec(ℤ), one has to pass to an algebraically closed field k to get a bijection between Hom(Spec(k), M_{g}) and isomorphism classes of smooth curves of genus g defined over k.

Despite the fact that the rigorous construction of M_{g} was achieved so late, various geometric properties of the space M_{g}, whose existence was somehow taken for granted, have been established. Hurwitz [Hu] following earlier work of Clebsch, proved in 1891 that M_{g} is irreducible by using Riemann’s existence theorem and showing that the space parameterizing branched covers of P^{1} having fixed degree and genus is connected. In 1915, Severi [S] used plane models of curves to prove that when g ≤ 10 the space M_{g} is unirational. For g ≤ 10 (and only in this range), it is possible to realize a general curve [C] ∈ M_{g} as a nodal plane curve Γ ⊂ P^{2} having minimal degree d = [(2g+8)/3], such that the nodes of Γ are general points in P^{2}. In the same paper Severi conjectures that M_{g} is unirational (or even

\begin{matrix}c \\ ◯ \end{matrix}2011 American Mathematical Society and International Press

Research partially supported by an Alfred P. Sloan Fellowship and the NSF Grant DMS-0500747.

Work on this paper was started during a stay at the Mittag-Leffler Institute in Djursholm in 2007.

rational!) for all g. This would correspond to being able to write down the general curve of genus g explicitly, in a family depending on 3g − 3 free parameters. Severi himself and later B. Segre made several attempts to prove the conjecture for g ≥ 11 using curves of minimal degree in P^{r} with r ≥ 3, cf. [Seg], [God].

Severi’s Conjecture seemed plausible and was widely believed until the 1980s: In [M3] Mumford declares ”How rational is the moduli space of curves” to be one of the main problems of present day mathematics. In ”Curves and their Jacobians”

[M2] Mumford elaborates: ”Whether more M_{g}’s g ≥ 11 are unirational or not is a very interesting problem, but one which looks very hard too, especially if g is quite large”. Probably thinking by analogy with the well-understood case of moduli of elliptic curves (with level structure), Oort formulates in his 1981 survey [O] a principle that naturally defined moduli spaces should be unirational: ”... generally speaking it seems that coarse moduli spaces tend to be close to rational varieties while high up in the tower of fine moduli spaces, these varieties possibly are of general type”.

It came as a major surprise when in 1982 Harris and Mumford [HM] showed that Severi’s Conjecture is maximally wrong in the sense that M_{g} itself rather than a higher level cover is almost always a variety of general type!

Theorem 1.1. For g ≥ 24, the moduli space of stable curves \overline{M}_{g} is a variety of general type.

An easy consequence of Theorem 1.1 is the following negative result:

Corollary 1.2. For g ≥ 24, if [C] ∈ M_{g} is a general curve and S is a surface containing C on which C moves in a non-trivial linear system, then S is birational to C × P^{1}. A general curve of genus g ≥ 24 does not occur in any non-trivial linear system on any non-ruled surface.

The proof of Theorem 1.1 uses in an essential way the Deligne-Mumford compactification \overline{M}_{g} by means of stable curves. The key idea is to reduce the problem of producing pluricanonical forms on \overline{M}_{g} to a divisor class calculation on \overline{M}_{g}. For instance, in the case g = 2k −1, Harris and Mumford consider the Hurwitz divisor

M_{g,k}^{1} := {[C] ∈ M_{g} : ∃ C →^{k:1} P^{1}}.

By computing the class of the closure \overline{M}_{g,k}^{1} of M_{g,k}^{1} inside \overline{M}_{g}, it follows that for g = 2k − 1 ≥ 25, the canonical class K_{\overline{M}_{g}} lies in the cone spanned inside

Pic(\overline{M}_{g})_{ℚ} by [\overline{M}_{g,k}^{1}], the Hodge class λ ∈ Pic(\overline{M}_{g}) and the irreducible components of the boundary \overline{M}_{g} − M_{g}. Since the class λ is big, that is, high multiples of λ have the maximal number of sections, it follows that high multiples of K_{\overline{M}_{g}} will also have the maximum number of sections, that is, \overline{M}_{g} is of general type. The main technical achievement of [HM] is the calculation of the class [\overline{M}_{g,k}^{1}] via the theory of admissible coverings. The case of even genus was initially settled in

[H1] for g = 2k − 2 ≥ 40 and later greatly simplified and improved by Eisenbud and Harris [EH3] via the theory of limit linear series. In this survey, apart from reviewing the work of Harris, Mumford and Eisenbud, we present a different proof of Theorem 1.1 by replacing the divisor M_{g,k}^{1} by a Koszul divisor on \overline{M}_{g} in the spirit of [F3]. It turns out that modulo Voisin’s proof [V2] of the generic Green Conjecture on syzygies of canonical curves, one obtains a very short proof of the Harris-Mumford Theorem 1.1, which does not resort to enumerative calculation on Hurwitz stacks of admissible coverings or to limit linear series.

BIRATIONAL ASPECTS OF THE GEOMETRY OF \overline{M}_{g} 3

After [HM] there has been a great deal of work trying to describe the geometry of M_{g} in the intermediate cases 11 ≤ g ≤ 23. Extending Severi’s result to genera g ≥ 11 requires subtle ideas and the use of powerful modern techniques, even though the idea of the proof is simple enough. Sernesi [Se1] was the first to go past the classical analysis of Severi by proving that M_{12} is unirational. A few years later, M. C. Chang and Z. Ran proved that M_{11} and M_{13} are also unirational (cf. [CR1]). In the process, they gave another proof for Sernesi’s theorem for M_{12}. The case g = 14 remained open for a long time, until Verra, using liaison techniques as well as Mukai’s work on models of canonical curves of genus at most 9, proved that M_{14} is unirational. Verra’s approach gives a much simpler proof of the unirationality of M_{g} in the cases g = 11, 12, 13 as well. We shall explain his main ideas following [Ve].

Chang and Ran showed that κ(\overline{M}_{g}) = −∞ for g = 15, 16, cf. [CR2], [CR3]. This was recently improved by Bruno and Verra [BV] who proved that M_{15} is rationally connected. Precisely, they proved that a general curve [C] ∈ M_{15} em-

bedded via a linear series C ⟶^{|L|} P^{6}, where L ∈ W_{19}^{6}(C), lies on a smooth complete intersection surface S ⊂ P^{6} of type (2, 2, 2, 2), in such a way that dim |O_{S}(C)| = 1. This last statement follows via a standard exact sequence argument because such a surface S is canonical.

Turning to genus 16, it is proved in [CR3] that K_{\overline{M}_{16}} is not a pseudo-effective class. It follows from [BDPP], that this actually implies that \overline{M}_{16} is uniruled.^{1} The question whether \overline{M}_{15} or \overline{M}_{16} are actually unirational remains open and seems difficult. Note that the above mentioned argument from [BV] actually implies that through a general point of \overline{M}_{15} there passes a rational surface.

Question 1.3. What is the Kodaira dimension of \overline{M}_{g} for 17 ≤ g ≤ 21?

A partial result for \overline{M}_{23} was obtained in [F1] where the inequality κ(\overline{M}_{23}) ≥ 2 is proved. Section 7 of this paper is devoted to the proof of the following result:

Theorem 1.4. The moduli space \overline{M}_{22} is of general type.

Similar questions about the birational type of other moduli spaces have been studied. Logan [Log] has proved that for all 4 ≤ g ≤ 22 there exists an explicitly known integer f(g) such that \overline{M}_{g,n} is of general type for n ≥ f(g). The bounds on the function f(g) have been significantly improved in [F3]. The moduli space A_{g} of principally polarized abelian varieties of dimension g is known to be of general type for g ≥ 7 due to results of Freitag [Fr], Mumford [M5] and Tai [T] (For a comprehensive recent review of developments on the global geometry of A_{g}, see [Gru]). Freitag was the first to go beyond the classical picture and show that for g ≥ 17, g ≡ 1 mod 8, the space A_{g} rather than one of its covers corresponding to ”moduli with level structure” is of general type. Freitag’s work seems to have been essential in making Mumford realize that Severi’s Conjecture might be fundamentally false, see the discussion in [HM] pg. 24. We mention that using e.g. the moduli space of Prym varieties, one can show that A_{g} is known to be unirational for g ≤ 5, cf. [Don], [Ve2]. The remaining question is certainly difficult and probably requires new ideas:

1More generally, it follows that \overline{M}_{g} is uniruled whenever one can show that K_{\overline{M}_{g}} is not a pseudoeffective class. I am grateful to J. McKernan for pointing this out to me.

Question 1.5. What is the Kodaira dimension of A_{6}?

Tai also discovered an important criterion (now called the Reid-Shepherd-Barron-Tai criterion) for canonical forms on the smooth locus of spaces with finite quotient singularities to extend to any resolution of singularities. He then showed that A_{g} satisfies the Reid-Shepherd-Barron-Tai criterion. A similar analysis of singularities (which is needed whenever one shows that a coarse moduli space is of general type), in the case of \overline{M}_{g}, has been carried out in [HM] Theorem 1.

A very interesting moduli space (also in light of Section 6 of this paper and the connection with the Slope Conjecture), is the moduli space F_{g} of polarized K3 surfaces [S, h] of degree h^{2} = 2g − 2. On F_{g} one considers the P^{g}-bundle

P_{g} := {([S, h], C) : [S, h] ∈ F_{g}, C ∈ |h|}

together with the projections p_{1} : P_{g} → F_{g} and p_{2} : P_{g} − − > M_{g}. The image K_{g} := p_{2}(P_{g}) is the locus of curves that can be abstractly embedded in a K3 surface. For g ≥ 13 the map p_{2} is generically finite (in fact, generically injective cf. [CLM]), hence dim(K_{g}) = 19 + g. This locus appears as an obstruction for an effective divisor on \overline{M}_{g} to have small slope, cf. Proposition 4.7. The geometry of F_{g} has been studied in low genus by Mukai and in general, using automorphic form techniques, initially by Kondo [K] and more recently, to great effect, by Gritsenko, Hulek and Sankaran [GHS]. Using Borcherds’s construction of automorphic forms on locally symmetric domains of type IV, they proved that (any suitable compactification of) F_{g} is of general type for g > 62 as well as for g = 47, 51, 55, 58, 59, 61. The largest g for which F_{g} is known to be unirational is equal to 20, cf. [M4].

Problem 1.6. Prove purely algebro-geometrically that F_{g} is of general type for g sufficiently large. Achieve this by computing the class of a geometric (Noether-Lefschetz, Koszul) divisor on F_{g} and comparing this calculation against the canonical class.

More generally, it is natural to ask whether the time is ripe for a systematic study of the birational invariants of the Alexeev-Kollár-Shepherd-Barron moduli spaces of higher dimensional varieties (see [AP], [H] for a few beautiful, yet isolated examples when the geometry of such spaces has been completely worked out).

We end this discussion by describing the birational geometry of the moduli space R_{g} classifying pairs [C, η] where [C] ∈ M_{g} and η ∈ Pic^{0}(C)[2] is a point of order 2 in its Jacobian. This moduli space provides an interesting correspondence between M_{g} and A_{g−1} via the natural projection π : R_{g} → M_{g} and the Prym map

𝔓𝔯_{g} : R_{g} → A_{g−1}.

For g ≤ 6 the Prym map is dominant, thus a study of the birational invariants of R_{g} gives detailed information about A_{g−1} as well. For g ≥ 7 the Prym map 𝔓𝔯_{g} is generically injective (though never injective) and we view R_{g} as a desingularization of the moduli space of Prym varieties 𝔓𝔯_{g}(R_{g}) ⊂ A_{g−1}. There is a good compactification \overline{R}_{g} of R_{g}, by taking \overline{R}_{g} to be the coarse moduli space associated to the moduli stack of stable maps \overline{M}_{g}(Bℤ_{2}). Note that the Galois covering π extends to a finite ramified covering π : \overline{R}_{g} → \overline{M}_{g}. We have the following result [FL]:

Theorem 1.7. The compact moduli space of Pryms \overline{R}_{g} is of general type for g > 13 and g ≠ 15. The Kodaira dimension of \overline{R}_{15} is at least 1.

Thus there are genera (e.g. g = 14) for which \overline{M}_{g} is unirational but \overline{R}_{g} is of general type. Note that \overline{R}_{g} is unirational for g ≤ 7 and it appears to be difficult to extend the range of g for which \overline{R}_{g} is unirational much further. An essential ingredient in the proof of Theorem 1.7 is the analysis of the singularities of \overline{R}_{g}. Kodaira-Spencer theory shows that singularities of \overline{R}_{g} correspond to automorphisms of Prym curves. A delicate local analysis shows that, even though the Reid-Shepherd-Barron-Tai criterion does not hold everywhere on \overline{R}_{g} (precisely, there is a codimension 2 locus of non-canonical singularities), for g ≥ 4 every pluricanonical form defined on the smooth part of \overline{R}_{g} extends to any desingularization. Equivalently, for any resolution of singularities ϵ : R̂_{g} → \overline{R}_{g} and l ≥ 0, there is an isomorphism of groups

ϵ^{∗} : H^{0}(\overline{R}_{g,reg}, K_{\overline{R}_{g}}^{⊗l}) ⟶^{≅} H^{0}(R̂_{g}, K⊗l ; R̂_{g}).

Since \overline{R}_{g} = \overline{M}_{g}(Bℤ_{2}), it makes sense to raise the following more general ques-

tion:

Problem 1.8. For a finite group G, study the birational invariants (Kodaira dimension and singularities, Picard groups, cones of ample and effective divisors) of the moduli spaces of twisted stable maps \overline{M}_{g}(BG).

We close by outlining the structure of the paper. In Section 2 we describe various attempts to prove that M_{g} is unirational, starting with Severi’s classical proof when g ≤ 10 and concluding with Verra’s recent work on M_{g} for g ≤ 14. While our presentation follows [Ve], several arguments have been streamlined, sometimes with the help of Macaulay 2. In Section 3 we present the structure of the Picard group of M_{g} while in Section 4 we recall Harris and Mumford’s spectacular application of the Grothendieck-Riemann-Roch theorem [HM] in order to compute the canonical class K_{\overline{M}_{g}} and then discuss Pandharipande’s recent lower bound on the slope of \overline{M}_{g}. In Section 5 we present a much shorter proof of the Harris-Mumford Theorem 1.1 using syzygies of canonical curves. Relying somewhat on Mukai’s earlier work, we highlight the importance of the locus K_{g} ⊂ M_{g} of curves lying on K3 surfaces in order to construct effective divisors on \overline{M}_{g} having small slope and produce a criterion which each divisor of small slope must satisfy (Section 6). We then explain how to construct and compute the class of certain effective divisors on \overline{M}_{g} defined in terms of Koszul cohomology of line bundles on curves (cf. [F2], [F3]). In Section 7 we prove that \overline{M}_{22} is of general type.

2. How rational is M_{g}?

As a matter of terminology, if M is a Deligne-Mumford stack, we denote by M its coarse moduli space. This is contrary to the convention set in [ACV] but for moduli spaces of curves it makes sense from a traditionalist point of view. Throughout the paper we denote by \overline{M}_{g} : Sch → Sets the contravariant functor (stack) of stable curves of genus g, which associates to every scheme S the set \overline{M}_{g}(S) of isomorphism classes of relative stable curves f : X → S of genus g.

The functor \overline{M}_{g} is not representable, for this would imply that each iso-trivial family of stable curves is actually trivial. This, of course, is not the case. To remedy this problem one looks for a compromise solution by retaining the requirement that the moduli space of curves be a scheme, but relaxing the condition that it represent \overline{M}_{g}. The result is the coarse moduli space of curves \overline{M}_{g} which is an irreducible projective variety of dimension 3g − 3 with finite quotient singularities, cf. [DM], [GIT], [M2]. For a family of stable curves [f : X → S] ∈ \overline{M}_{g}(S) we shall denote by m_{f} : S → \overline{M}_{g} the associated moduli map.

2.1. Brill-Noether theory.

We recall a few basic facts from Brill-Noether theory, cf. [ACGH]. For a smooth curve C of genus g and for integers d, r ≥ 0, one considers the cycle inside the Jacobian

W_{d}^{r}(C) := {L ∈ Pic^{d}(C) : h^{0}(C, L) ≥ r + 1}.

The variety of linear series of type 𝔤_{d}^{r} is defined as

G_{d}^{r}(C) := {(L, V) : L ∈ W_{d}^{r}(C), V ∈ G(r + 1, H^{0}(L))}.

There is an obvious forgetful map c : G_{d}^{r}(C) → W_{d}^{r}(C) given by c(L, V) := L.

We fix a point l = (L, V) ∈ G_{d}^{r}(C), and describe the tangent space T_{l}(G_{d}^{r}(C)). One has the standard identification T_{L}(Pic^{d}(C)) = H^{1}(C, O_{C}) = H^{0}(C, K_{C})^{∨} and we denote by

μ_{0}(L, V) : V ⊗ H^{0}(C, K_{C} ⊗ L^{∨}) → H^{0}(C, K_{C})

the Petri map given by multiplication of sections. The deformations of [L] ∈ Pic^{d}(C) preserving the space of sections V correspond precisely to those elements ϕ ∈ H^{0}(C, K_{C})^{∨} for which ϕ_{|Im μ_{0}(L,V)} = 0. One obtains an exact sequence

0 ⟶ Hom(V, H^{0}(C, L)/V) ⟶ T_{l}(G_{d}^{r}(C)) ⟶ Ker μ_{0}^{∨} ⟶ 0.

It follows that G_{d}^{r}(C) is smooth and of dimension

ρ(g, r, d) := g − (r + 1)(g − d + r)

at the point l if and only if μ_{0}(L, V) is injective.

The Gieseker-Petri Theorem states that if [C] ∈ M_{g} is general, then the Petri map

μ_{0}(L) : H^{0}(C, L) ⊗ H^{0}(C, K_{C} ⊗ L^{∨}) → H^{0}(C, K_{C})

is injective for every L ∈ Pic^{d}(C). In particular it implies that both W_{d}^{r}(C) and G_{d}^{r}(C) are irreducible varieties of dimension ρ(g, r, d). The variety G_{d}^{r}(C) is smooth while Sing W_{d}^{r}(C) = W_{d}^{r+1}(C). Furthermore, W_{d}^{r}(C) = ∅ if ρ(g, r, d) < 0.

The first rigorous proof of Petri’s theorem is due to Gieseker. The original proof has been greatly simplified by Eisenbud and Harris, cf. [EH2], using degeneration to curves of compact type and the theory of limit linear series. A very different proof, in which the degeneration argument is replaced by Hodge theory and the geometry of curves on K3 surfaces, has been found by Lazarsfeld [La1].

If [C, p] ∈ M_{g,1} and l = (L, V) ∈ G_{d}^{r}(C), we define the vanishing sequence of l at p

a^{l}(p) : 0 ≤ a_{0}^{l}(p) < … < a_{r}^{l}(p) ≤ d

by ordering the set {ord_{p}(σ)}_{σ∈V}. The ramification sequence of l and p

α^{l}(p) : 0 ≤ α_{0}^{l}(p) ≤ … ≤ α_{r}(p) ≤ d − r

is obtained from the vanishing sequence by setting α_{i}^{l}(p) := a_{i}^{l}(p) − i for i = 0 … r.

The theory of degenerations of linear series (in the case of curves of compact type) has been beautifully developed by Eisenbud and Harris [EH1]. The major successes of the theory include a simple proof of the Brill-Noether-Petri theorem cf. [EH2] and especially its essential use in the work on the Kodaira dimension of \overline{M}_{g} cf. [EH3].

Definition 2.1. If X is a stable curve whose dual graph is a tree, a limit linear series 𝔤_{d}^{r} on X, consists of a collection of linear series

l = {l_{Y} = (L_{Y} , V_{Y} ⊂ H^{0}(L_{Y})) ∈ G_{d}^{r}(Y) : Y component of X}

satisfying the following compatibility condition: If p ∈ Y ∩ Z is a node lying on two irreducible components Y and Z of X, then

a_{i}^{l_{Y}} (p) + a_{r−i}^{l_{Z}}(p) ≥ d, for i = 0 … r.

Limit linear series behave well in families: If M_{g}^{∗} ⊂ \overline{M}_{g} denotes the open substack of tree-like curves, then there exists an algebraic stack σ : 𝔊̃_{d}^{r} → M_{g}^{∗} classifying limit linear series. Each irreducible component of 𝔊̃_{d}^{r} has dimension at least 3g − 3 + ρ(g, r, d). In particular if l ∈ \overline{G}_{d}^{r}(C) = σ^{−1}(C) is a limit 𝔤_{d}^{r} on a curve [C] ∈ M_{g}^{∗} belonging to a component of \overline{G}_{d}^{r}(C) of the expected dimension ρ(g, r, d), then l can be smoothed to curves in an open set of M_{g} (cf. [EH1]).

2.2. Severi’s proof of the unirationality of M_{g} when g ≤ 10.

We outline Severi’s classical argument [S] showing that M_{g} is unirational for small genus (for a beautiful modern presentation see [AC1]). The idea is very simple: One tries to represent the general curve [C] ∈ M_{g} as a nodal plane curve Γ ⊂ P^{2} of minimal degree d such that ρ(g, 2, d) ≥ 0 and then show that the nodes are in general position. Since the varieties of plane curves with fixed nodes are linear spaces, hence rational varieties, this implies that M_{g} is unirational.

We fix d ≥ (2g + 8)/3 and set δ := \binom{d−1}{2} − g. We consider the Severi variety

U_{d,g} := {[Γ ↪ P^{2}] : deg(Γ) = d, Γ is a nodal irreducible plane curve , p_{g}(Γ) = g}.

It is well-known that U_{d,g} is an irreducible variety of dimension

dim U_{d,g} = dim M_{g} + ρ(g, 2, d) + dim P GL(3) = 3d + g − 1.

Furthermore, there exists a global desingularization map ν_{d,g} : U_{d,g} −−→ M_{g} which associates to each plane curve the class of its normalization. The Brill-Noether theorem guarantees that ν_{d,g} is surjective. (Indeed, since ρ(g, 2, d) ≥ 0 one has that G_{d}^{2}(C) ≠ ∅ and it is straightforward to prove that a general 𝔤_{d}^{2} corresponds to a nodal model of a general curve [C] ∈ M_{g}, see for instance [EH1]).

One defines the incidence correspondence between curves and their nodes

Σ := {([Γ ↪ P^{2}], p_{1} + ⋯ + p_{δ}) ∈ U_{d,g} × Sym^{δ}(P^{2}) : {p_{1}, …, p_{δ}} = Sing(Γ)}, together with the projection π_{2} : Σ → Sym^{δ}(P^{2}). The fibres of π_{2} being linear spaces, in order to conclude that Σ is rational (and hence M_{g} unirational), it suffices to prove that π_{2} is dominant. A necessary condition for π_{2} to be dominant is that

dim Σ = 3d + g − 1 ≥ 2δ.

This, together with the condition ρ(g, 2, d) ≥ 0, implies that g ≤ 10. We have the following result [S], [AC1] Theorem 3.2:

Theorem 2.2. We fix non-negative integers g, d, δ satisfying the inequalities

δ = \binom{d − 1}{2} − g, ρ(g, 2, d) ≥ 0 and 3d + g − 1 ≥ 2δ.

If p_{1}, …, p_{δ} ∈ P^{2} are general points and (n, δ) ≠ (6, 9), then there exists an irreducible plane curve Γ ⊂ P^{2} of degree d having nodes at p_{1}, …, p_{δ} and no other singularity. It follows that M_{g} is unirational for g ≤ 10.

Remark 2.3. As explained Severi’s argument cannot be extended to any M_{g} for g ≥ 11. In a similar direction, a classical result of B. Segre [Seg] shows that if S is any algebraic surface and Ξ ⊂ S × V is any algebraic system of smooth genus g curves contained in S, then whenever g > 6, the moduli map m_{Ξ} : V − − > M_{g} cannot be dominant, that is, no algebraic system of smooth curves of genus g > 6 with general moduli can lie on any given surface.

2.3. Verra’s proof of the unirationality of M_{g} for 11 ≤ g ≤ 14.

We fix an integer g ≥ 11 and aim to prove the unirationality of M_{g} by showing that a suitable component of a Hilbert scheme of curves

Hilb_{C} := {C ⊂ P^{r} : p_{a}(C) = g, deg(C) = d},

where ρ(g, r, d) ≥ 0 is unirational. The component Hilb_{C} must have the property that the forgetful rational map

m_{C} : Hilb_{C} −−→ M_{g}

is dominant (in particular, the general point of Hilb_{C} corresponds to a smooth curve C ⊂ P^{r}). To prove that Hilb_{C} is unirational we shall use an incidence correspondence which relates Hilb_{C} to another Hilbert scheme of curves Hilb_{D} parameterizing curves D ⊂ P^{r} such that

deg(D) = d′, g(D) = g′ and H^{1}(D, O_{D}(1)) = 0

(thus r = d′ − g′). If [D ↪ P^{r}] ∈ Hilb_{D} is a smooth curve with H^{1}(D, O_{D}(1)) = 0, then trivially H^{1}(D, N_{D/P^{r}}) = 0, which implies that Hilb_{D} is smooth at the point [D ↪ P^{r}] and of dimension

h^{0}(D, N_{D/P^{r}}) = χ(D, N_{D/P^{r}}) = (r + 1)d′ − (r − 3)(g′ − 1)

(see e.g. [Se2]). Moreover, there exists an open subvariety U_{D} ⊂ Hilb_{D} parameterizing smooth non-special curves D ⊂ P^{r} such that (i) the moduli map

m_{D} : U_{D} → M_{g′}

is dominant, and (ii) the restriction maps μ_{f} : Sym^{f}H^{0}(D, O_{D}(1)) → H^{0}(D, O_{D}(f)) are of maximal rank for all integers f.

The correspondence between Hilb_{C} and U_{D} is given by liaison with respect to hypersurfaces of a fixed degree f, that is, via the variety

Σ := {(D, V) : [D ↪ P^{r}] ∈ U_{D}, V ∈ G(r − 1, H^{0}(P^{r}, I_{D/P^{r}}(f)))}.

One has a projection map u : Σ → U_{D} given by u(D, V) := [D], and a residuation map

res : Σ → Hilb_{C}, res(D, V) := [C ↪ P^{r}],

where C ⊂ P^{r} is the scheme residual to D in the base locus of the linear system |V |. The morphism u : Σ → U_{D} has the structure of a Grassmann bundle corresponding to the vector bundle F over U_{D} with fibres F(D) = H^{0}(P^{r}, I_{D/P^{r}}(f)), thus clearly Σ is unirational provided that Hilb_{D} is unirational. Since U_{D} parameterizes non-special curves, proving its unirationality is equivalent to showing that the universal Picard variety 𝔓𝔦𝔠_{g′}^{d′} → M_{g′} is unirational.

In order to show that m_{C} : Hilb_{C} −−→ M_{g} is dominant (and thus, that the general curve [C] ∈ M_{g} is linked to a curve [D ↪ P^{r}] ∈ U_{D}), it suffices to exhibit a single nodal complete intersection

C ∪ D = X_{1} ∩ … ∩ X_{r−1}

with X_{i} ∈ |O_{P}^{r}(f)|, such that both C and D are smooth and the Petri map

μ_{0}(C) : H^{0}(C, O_{C}(1)) ⊗ H^{0}(C, K_{C} ⊗ O_{C}(−1)) → H^{0}(C, K_{C})

is injective. Indeed, it is well-known (see e.g. [Se2]) that via Kodaira-Spencer theory, the differential dm_{[C]} : T_{[C]}(Hilb_{C}) → T_{[C]}(M_{g}) is given by the coboundary map H^{0}(C, N_{C/P^{r}}) → H^{1}(C, T_{C}) obtained by taking cohomology in the exact sequence which defines the normal bundle of C:

(1)  0 ⟶ T_{C} ⟶ T_{P}^{r} ⊗ O_{C} ⟶ N_{C/P^{r}} ⟶ 0.

On the other hand, one has the pull-back of the Euler sequence from P^{r}

(2)  0 ⟶ O_{C} ⟶ H^{0}(C, O_{C}(1))^{∨} ⊗ O_{C}(1) ⟶ T_{P}^{r} ⊗ O_{C} ⟶ 0,

and after taking cohomology we identify H^{1}(T_{P}^{r} ⊗ O_{C}) with the dual of the Petri map μ_{0}(C). Thus if μ_{0}(C) is injective, then m_{C} is a dominant map around the point [C ↪ P^{r}].

The numerical invariants of C and D are related by well-known formulas for linked subschemes of P^{r}, see [Fu] Example 9.1.12: Suppose

C ∪ D = X_{1} ∩ … ∩ X_{r−1}

is a nodal complete intersection with X_{i} ∈ |O_{P}^{r}(f)| for 1 ≤ i ≤ r − 1. Then one has that the formulas:

(3)  deg(C) + deg(D) = f^{r−1},
(4)  2(g(C) − g(D)) = ((r − 1)f − r − 1)(deg(C) − deg(D)), and
(5)  #(C ∩ D) = deg(C) · ((r − 1)f − r − 1) + 2 − 2g(C).

We shall prove that if we choose

(6)  f = \frac{r + 2}{r − 2} ∈ ℤ,

the condition that O_{D}(1) be non-special is equivalent to h^{0}(P^{r}, I_{C/P^{r}}(f)) = r − 1. Furthermore, under the same assumption, μ_{0}(C) is injective if and only if I_{D/P^{r}}(f) is globally generated.

To summarize, we have reduced the problem of showing that M_{g} is unirational to showing (1) that the universal Picard variety 𝔓𝔦𝔠_{g′}^{d′} is unirational and (2) that one can find a non-special curve D ⊂ P^{d′−g′} whose ideal is cut out by hypersurfaces of degree f. This program can be carried out provided one can solve the equations (3), (4), (5) and (6) while keeping ρ(g, r, d) ≥ 0. To prove (1) Verra relies on Mukai’s work on the geometry of Fano 3-folds and on the existence of models of canonical curves of genus g ≤ 9 as linear sections of certain rational homogeneous varieties. We first explain Mukai’s work on existence of models of canonical curves of genus g ≤ 9. The standard references for this part are [Mu1], [Mu2], [Mu3]:

Theorem 2.4. We fix integers g ≤ 9, r ≥ 3 and d ≥ g + 3. Then the universal Picard variety 𝔓ic_{g}^{d} is unirational. Moreover, if H_{d,g,r} denotes the unique component of the Hilbert scheme of curves whose generic point corresponds to a smooth curve D ⊂ P^{r} with deg(D) = d, g(D) = d and H^{1}(D, O_{D}(1)) = 0, then H_{d,g,r} is unirational as well.

The theorem is easily proved for g ≤ 6 because, in this case, the general canonical curve of genus g is a complete intersection in some projective space: For instance, the canonical model of the general curve [C] ∈ M_{4} is a (2, 3) complete intersection in P^{4} while the canonical model of the general curve [C] ∈ M_{5} is a (2, 2, 2) complete intersection in P^{4}. In the cases g = 7, 8, 9, Mukai has found a rational homogeneous space

G_{g} ⊂ P^{dim(G_{g})+g−2},

such that K_{G_{g}} = O_{G_{g}}(−dim(G_{g})+2), and with the property that the general canonical curve of genus g appears as a curve section of G_{g}.

For g = 8, we consider a vector space V ≅ ℂ^{6} and we introduce the Grassmannian of lines G_{8} := G(2, V) together with the Plücker embedding

G(2, V) ↪ P(∧^{2}V).

Then K_{G_{8}} = O_{G_{8}}(−6). If H ∈ G(8, ∧^{2}V) is a general 7-dimensional projective subspace and C_{H} := G_{8} ∩ P(H) ↪ P(H), then by adjunction K_{C} = O_{C}(1). In other words, a transversal codimension 7 linear section of G_{8} is a canonical curve of genus 8. Mukai showed that any curve [C] ∈ M_{8} such that W_{4}^{1}(C) = ∅, can be recovered in this way (cf. [Mu1]).

The case g = 9 is described in [Mu3]: One takes G_{9} := SpG(3, 6) ⊂ P^{13} to be the symplectic Grassmannian, that is, the Grassmannian of Lagrangian subspaces of a 6-dimensional symplectic vector space V. Then dim(G_{9}) = 6 and K_{G_{9}} = O_{G_{9}}(−4). Codimension 5 linear sections G_{9} ∩ H_{1} ∩ … ∩ H_{5} ⊂ P^{8} are canonical curves of genus 9. A genus 9 curve is a transversal section of G_{9} if and only if W_{5}^{1}(C) = ∅. In particular a general [C] ∈ M_{9} is obtained through this construction. Finally, we mention Mukai’s construction for g = 7, cf. [Mu2]: For a vector space V ≅ ℂ^{10}, the subset of the Grassmannian G(5, V) consisting of totally isotropic quotient spaces has two connected components, one of which is the 10-dimensional spinor variety G_{7} ⊂ P^{15}.

Proof of Theorem 2.4. It is enough to deal with the cases g = 7, 8, 9. For each integer d ≥ g + 3, we fix non-zero integers n_{1}, …, n_{g} such that

2g − 2 + n_{1} + ⋯ + n_{g} = d

and note that for every [C] ∈ M_{g}, the map C^{g} → Pic^{d}(C) sending

(x_{1}, …, x_{g}) ↦ K_{C} ⊗ O_{C}(n_{1}x_{1} + ⋯ + n_{g} x_{g})

is surjective. Then the rational map ϕ : G_{g}^{g} −−→ 𝔓𝔦𝔠_{g}^{d} defined by

ϕ(x_{1}, …, x_{g}) :=(C_{x} = G_{g} ∩ P(⟨x_{1}, …, x_{g}⟩), K_{C_{x}} ⊗ O_{C_{x}}(n_{1} x_{1} + ⋯ + n_{g} x_{g})),

is dominant. Thus 𝔓𝔦𝔠_{g}^{d} is unirational.

To establish the unirationality of H_{d,g,r} when 3 ≤ r ≤ d − g, we consider the dominant map f : H_{d,g,r} → 𝔓𝔦𝔠_{g}^{d} given by f([C ⊂ P^{r}]) := [C, O_{C}(1)]. The fibres of f are obviously rational varieties. It follows that H_{d,g,r} is unirational too. □

Now we explain Verra’s work cf. [Ve], focusing on the cases g = 11, 14. Several (admittedly beautiful) arguments of classical geometric nature have been straightened or replaced by Macaulay 2 calculations in the spirit of [F2], Theorems 2.7, 2.10 or [ST].

Theorem 2.5. The moduli space of curves \overline{M}_{g} is unirational for 11 ≤ g ≤ 14.

Proof for g = 11, 14. We place ourselves in the situation when

f = \frac{r + 2}{r − 2} ∈ ℤ.

The relevance of this condition is that a surface complete intersection of type (f, …, f) in P^{r} is a canonical surface in P^{r}. We consider a nodal complete intersection

C ∪ D = X_{1} ∩ … ∩ X_{r−1}

with X_{i} ∈ |O_{P}^{r}(f)|, with C and D being smooth curves and g(C) = g. Assuming that I_{C∪D/P^{r}}(f) is globally generated (this will be the case in all the situations we consider), then C ∪D lies on a smooth complete intersection of r −2 hypersurfaces of degree f, say S := X_{1} ∩ … ∩ X_{r−2}. Thus S is a surface with K_{S} = O_{S}(1) and moreover h^{0}(P^{r}, I_{S/P^{r}}(f)) = r − 2 (use the Koszul resolution of I_{S/P^{r}}). From the exact sequence

0 ⟶ I_{C/P^{r}} ⟶ I_{S/P^{r}} ⟶ O_{S}(−C) ⟶ 0,

we find that

h^{0}(P^{r}, I_{C/P^{r}}(f)) = h^{0}(S, O_{S}(C)) + h^{0}(P^{r}, I_{S/P^{r}}(f)) = h^{0}(O_{S}(D)) + r − 2

(Note that C + D ∈ |O_{S}(f)|). Moreover, from the exact sequence

0 ⟶ O_{S}(1) ⊗ O_{S}(−D) ⟶ O_{S}(1) ⟶ O_{D}(1) ⟶ 0,

using also Serre duality, we obtain that

h^{0}(S, O_{S}(D)) = h^{2}(S, O_{S}(H − D)) − h^{2}(S, O_{S}(H)) = 1 + h^{1}(D, O_{D}(1)).

Therefore O_{D}(1) is non-special if and only if

(7)  h^{0}(P^{r}, I_{C/P^{r}}(f)) = r − 1.

Assume now that r = d′ − g′ and that g(D) = g′, deg(D) = d′ g(C) = g and deg(C) = d, where these invariants are related by the formulas (3)-(5). Using a simple argument involving diagram chasing, we claim that the Petri map

μ_{0}(C) : H^{0}(C, O_{C}(1)) ⊗ H^{0}(C, K_{C}(−1)) → H^{0}(C, K_{C})

is of maximal rank if and only if the multiplication map

(8)  ν_{D}(f) : H^{0}(P^{r}, I_{D/P^{r}}(f)) ⊗ H^{0}(D, O_{D}(1)) → H^{0}(P^{r}, I_{D/P^{r}}(f + 1))

is of maximal rank (see [Ve], Lemma 4.4). Indeed, since K_{C}(−1) = O_{C}(C), we find that

Ker(μ_{0}(C)) = Ker{μ_{S} : H^{0}(S, O_{S}(C))⊗H^{0}(S, O_{S}(C+D)) → H^{0}(S, O_{S}(D+2C))}.

Next, we note that I_{D/S}(f) = O_{S}(C) and then the claim follows by applying the Snake Lemma to the diagram obtained by taking cohomology in the sequence

0 → H^{0}(O_{P}^{r}(1))⊗I_{S/P^{r}}(f) → H^{0}(O_{P}^{r}(1))⊗I_{D/P^{r}}(f) → H^{0}(O_{P}^{r}(1))⊗O_{S}(C) → 0.

Thus to prove that the moduli map

m_{C} : Hilb_{C} −−→ M_{g}

is dominant, it suffices to exhibit a smooth curve [D] ∈ Hilb_{D} such that the map ν_{D}(f) is injective.

Having explained this general strategy, we start with the case g = 14 and suppose that [C] ∈ M_{14} is a curve satisfying Petri’s theorem, hence

dim W_{8}^{1}(C) = ρ(14, 1, 8) = 0.

For each pencil A ∈ W_{8}^{1} we have that L := K_{C} ⊗ A^{∨} ∈ W_{18}^{6}(C) and when [C] ∈ M_{14} is sufficiently general, each such linear series gives rise to an embedding

C ↪^{|L|} P^{6}. By Riemann-Roch we obtain that

dim Ker\{Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2})\} = \binom{8}{2} − (2 deg(C) + 1 − g(C)) = 5,

that is C lies on precisely 5 independent quadrics Q_{1}, …, Q_{5} ∈ |O_{P}^{6}(2)|. Writing

Q_{1} ∩ … ∩ Q_{5} = C ∪ D,

we find that g(D) = 8 and deg(D) = 14. In particular, we also find that

H^{1}(D, O_{D}(1)) = 0.

Thus we have reduced the problem of showing that M_{14} is unirational to two questions:

(1) 𝔓𝔦𝔠_{8}^{14} is unirational. This has already been achieved (cf. Theorem 2.4).

(2) If D ⊂ P^{6} is a general smooth curve with deg(D) = 14 and g(D) = 8, then the map

ν_{D}(2) : H^{0}(P^{6}, I_{D/P^{6}}(2)) ⊗ H^{0}(D, O_{D}(1)) → H^{0}(P^{6}, I_{D/P^{6}}(3))

is an isomorphism. This is proved using liaison and a few classical arguments (cf. [Ve], Propositions 5.5-5.16). We shall present a slightly more direct proof using Macaulay2.

When g = 11, we choose d = 14 and r = 4, hence f = 3. We find that if [C] ∈ M_{11} is general then dim W_{14}^{4}(C) = ρ(11, 4, 14) = 6 and h^{1}(C, L) = 1 for every L ∈ W_{14}^{4}(C). Moreover, for a general linear series L ∈ W_{11}^{4}(C),

dim Ker{Sym^{3}H^{0}(C, L) → H^{0}(C, L^{⊗3})} = 3,

(in particular condition (7) is satisfied). Hence there exist hypersurfaces

X_{1}, X_{2}, X_{3} ∈ |O_{P}^{4}(3)|

such that X_{1} ∩ X_{2} ∩ X_{3} = C ∪ D. Moreover, g(D) = 9 and deg(D) = 13, and the unirationality of \overline{M}_{11} has been reduced to showing that:

(1) 𝔓𝔦𝔠_{9}^{13} is unirational. This again follows from Theorem 2.4.

(2) If D ⊂ P^{4} is a general smooth curve with deg(D) = 13 and g(D) = 9, then the map

ν_{D}(3) : H^{0}(P^{4}, I_{D/P^{4}}(3)) ⊗ H^{0}(O_{D}(1)) → H^{0}(P^{4}, I_{D/P^{4}}(4))

is injective. □

We complete the proof of Theorem 2.5, and we focus on the case g = 14. A similar argument deals with the case g = 11:

Theorem 2.6. If D ↪^{|L|} P^{6} is the embedding corresponding to a general curve [D, L] ∈ 𝔓𝔦𝔠_{8}^{14}, then the multiplication map

H^{0}(P^{6}, I_{D/P^{6}}(2)) ⊗ H^{0}(P^{6}, O_{P}^{6}(1)) → H^{0}(P^{6}, I_{D/P^{6}}(3))

is an isomorphism.

Proof. We consider 11 general points in P^{2} denoted by p_{1}, …, p_{5} and q_{1}, …, q_{6} respectively, and define the linear system

H ≡ 6h − 2(E_{p_{1}} + ⋯ + E_{p_{5}}) − (E_{q_{1}} + ⋯ + E_{q_{6}})

on the blow-up S = Bl_{11}(P^{2}). Here h denotes the pullback of the line class from

P^{2}. Using the program Macaulay2 it is easy to check that S ↪^{|H|} P^{6} is an embedding and the graded Betti diagram of S is the following:

\begin{matrix}1 & − & − & − & − \\ − & 5 & − & − & − \\ − & − & 15 & 16 & 15 \end{matrix}

Thus S satisfies property (N_{1}). To carry out this calculation we chose the 11 points in P^{2} randomly using the Hilbert-Burch theorem so that they satisfy the Minimal Resolution Conjecture (see [ST] for details on how to pick random points in P^{2} using Macaulay). Next we consider a curve D ⊂ S in the linear system

(9)  D ≡ 10h − 3(E_{p_{1}} + E_{p_{2}}) − 4 ∑_{i=3}^{5} E_{p_{i}} − E_{q_{1}} − E_{q_{2}} − 2 ∑_{j=3}^{6} E_{q_{j}}.

By using Macaulay2, we pick D randomly in its linear system and then check that D is smooth, g(D) = 8 and deg(D) = 14. We can compute directly the Betti diagram of D:

\begin{matrix}1 & − & − & − & − \\ − & 7 & − & − & − \\ − & − & 35 & 56 & 35 \end{matrix}

Hence K_{1,1}(D, O_{D}(1)) = 0, which shows that ν_{2}(D) is an isomorphism. This last part also follows directly: Since S is cut out by quadrics, to conclude that D is also cut out by quadrics, it suffices to show that the map

ν_{S} : H^{0}(S, O_{S}(H)) ⊗ H^{0}(S, O_{S}(2H − D)) → H^{0}(S, O_{S}(3H − D))

is surjective (or equivalently injective). Since h^{0}(S, O_{S}(2H −D)) = 2, from the base point free pencil trick we get that Ker(ν(S)) = H^{0}(S, O_{S}(D − H)) = 0, because D − H is clearly not effective for a general choice of the 11 points in P^{2}. □

We end this section, by pointing out that already existing results in [CR3], coupled with recent advances in higher dimensional birational geometry, imply the following:

Theorem 2.7. (Chang-Ran) The moduli space \overline{M}_{16} is a uniruled variety.

Proof. Chang and Ran proved in [CR3] that κ(\overline{M}_{16}) = −∞, by exhibiting an explicit collection of curves \{F_{i}\}_{i=1}^{n} ⊂ \overline{M}_{16}, with the property that each F_{i} lies on a divisor D_{i} ⊂ \overline{M}_{16} such that F_{i} is nef as a curve on D_{i} with respect to ℚ-Cartier divisors, and moreover

F_{i} · ∑_{j=1}^{n} D_{j} > 0 for i = 1, …, n.

By explicit calculation they noted that F_{i} · K_{\overline{M}_{16}} < 0 for i = 1, …, n. This clearly implies that K_{\overline{M}_{16}} is not pseudo-effective. Since pseudo-effectiveness of the canonical bundle is a birational property, the canonical bundle of any smooth model of \overline{M}_{16} will lie outside the pseudo-effective cone as well. One can apply the the main result of [BDPP] to conclude that \overline{M}_{16} is uniruled. □

3. The Picard group of the moduli stack \overline{M}_{g}

For a stable curve [C] ∈ \overline{M}_{g} one can consider its dual graph with vertices corresponding to the irreducible components of C and edges corresponding to nodes joining two components. By specifying the dual graph, one obtains the topological stratification of \overline{M}_{g}, where the codimension a strata correspond to the irreducible components of the closure of the locus of curves [C] having precisely a nodes. The closure of the codimension 1 strata are precisely the boundary divisors of \overline{M}_{g}: For 1 ≤ i ≤ [g/2] we denote by Δ_{i} ⊂ \overline{M}_{g} the closure of the locus of stable curves [C_{1} ∪ C_{2}], where C_{1} and C_{2} are smooth curves of genera i and g − i respectively. Similarly, Δ_{0} ⊂ \overline{M}_{g} is the closure of the locus of irreducible 1-nodal stable curves. We have the decomposition

\overline{M}_{g} = M_{g} ∪ Δ_{0} ∪ … ∪ Δ_{[g/2]}.

Next we describe the Picard group of the moduli stack \overline{M}_{g}. The difference between the Picard group of the stack \overline{M}_{g} and that of the coarse moduli space \overline{M}_{g}, while subtle, is not tremendously important in describing the birational geometry of \overline{M}_{g}. Remarkably, one can define Pic(\overline{M}_{g}) without knowing exactly what a stack itself is! This approach at least respects the historical truth: In 1965 Mumford [M1] introduced the notion of a sheaf on the functor (stack) \overline{M}_{g}. One had to wait until 1969 for the definition of a Deligne-Mumford stack, cf. [DM].

Definition 3.1. A sheaf L on the stack \overline{M}_{g} is an assignment of a sheaf L(f) on S for every family [f : X → S] ∈ \overline{M}_{g}(S), such that for any morphism of schemes ϕ : T → S, if p_{2} : X_{T} := X ×_{S} T → T denotes the family obtained by pulling-back f, then there is an isomorphism of sheaves over T denoted by

L(ϕ, f) : L(p_{2}) → ϕ^{∗}(L(f)).

These isomorphisms should commute with composition of morphisms between the bases of the families. Precisely, if χ : W → T is another morphism and

σ_{2} : X_{W} := X_{T} ×_{T} W → W ∈ \overline{M}_{g}(W),

then L(ϕχ, f) = χ^{∗}L(ϕ, f) ∘ L(χ, p_{2}). If L and E are sheaves on \overline{M}_{g}, we define their tensor product by setting

(L ⊗ E)(f) := L(f) ⊗ E(f)

for each [f : X → S] ∈ \overline{M}_{g}(S).

A sheaf L on \overline{M}_{g} is a line bundle if L(f) ∈ Pic(S) for every [f : X → S] ∈ \overline{M}_{g}(S). We denote by Pic(\overline{M}_{g}) the group of isomorphism classes of line bundles on \overline{M}_{g}.

Similarly, for i ≥ 0, one defines a codimension i cycle class γ ∈ A^{i}(\overline{M}_{g}), to be a collection of assignments γ(f) ∈ A^{i}(S) for all [f : X → S] ∈ \overline{M}_{g}(S), satisfying an obvious compatibility condition like in Defintion 3.1

Example 3.2. For each n ≥ 1 the Hodge classes λ_{1}^{(n)} ∈ Pic(\overline{M}_{g}) are defined by taking λ_{1}^{(n)}(f) := c_{1}(𝔼_{n}(f)), where the assignment

\overline{M}_{g}(S) ∋ [f : X → S] ↦ 𝔼_{n}(f) := f_{∗}(ω_{f}^{⊗n}),

gives rise to a vector bundle 𝔼_{n} on \overline{M}_{g} for each n ≥ 1. Clearly rank(𝔼_{1}) = g and rank(𝔼_{n}) = (2n − 1)(g − 1) for n ≥ 2. One usually writes 𝔼 := 𝔼_{1}. Similarly, one can define the higher Hodge classes λ_{i}^{(n)} ∈ A^{i}(\overline{M}_{g}), by taking

λ_{i}^{(n)}(f) := c_{i}(𝔼_{n}(f)) ∈ A^{i}(S).

It is customary to write that λ_{i} := λ_{i}^{(1)} and sometimes, λ := λ_{1}.

There is an obvious group homomorphism ρ : Pic(\overline{M}_{g}) → Pic(\overline{M}_{g}) defined by ρ(L)(f) := m_{f}^{∗}(L) for every L ∈ Pic(\overline{M}_{g}) and [f : X → S] ∈ \overline{M}_{g}(S).

To get to grips with the group Pic(\overline{M}_{g}) one can also use the GIT realization of the moduli space and consider for each ν ≥ 3 the Hilbert scheme Hilb_{g,ν} of νcanonical stable embedded curves C ⊂ P^{(2ν−1)(g−1)−1}. One has an isomorphism of varieties cf. [GIT], [M2]

\overline{M}_{g} ≅ Hilb_{g,ν}//P GL((2ν − 1)(g − 1)).

Using this we can define an isomorphism of groups

β : Pic(\overline{M}_{g}) → Pic(Hilb_{g,ν})^{P GL((2ν−1)(g−1))}.

If σ : C_{g,ν} → Hilb_{g,ν} denotes the universal ν-canonically embedded curve, where we have that C_{g,ν} ⊂ Hilb_{g,ν} × P^{(2ν−1)(g−1)−1}, we set β(L) := L(σ) ∈ Pic(Hilb_{g,ν}).

To define β^{−1} we start with a line bundle L ∈ Pic(Hilb_{g,ν}) together with a fixed lifting of the P GL((2ν − 1)(g − 1))-action on Hilb_{g,ν} to L. For a family of stable curves f : X → S, we choose a local trivialization of the projective bundle P(f_{∗}(ω_{f}^{⊗ν})), that is, we fix isomorphisms over S_{α}

P((f_{α})_{∗}(ω_{f_{α}}^{⊗ν})) ≅ P^{(2ν−1)(g−1)−1} × S_{α},

where {S_{α}}_{α} is a cover of S and f_{α} = f_{|f^{−1}(S_{α})} : X_{α} → S_{α}. Since the Hilbert scheme is a fine moduli space, these trivializations induce morphisms

g_{α} : S_{α} → Hilb_{g,ν}

such that on intersections S_{α} ∩ S_{β}, the morphisms g_{α} and g_{β} differ by an element from P GL((2ν − 1)(g − 1)). The choice of the L-linearization ensures that the sheaves {g_{α}^{∗}(L)}_{α} can be glued to form a sheaf which we call β^{−1}(L)(f) ∈ Pic(S).

Example 3.3. If O_{Hilb_{g,ν}} (δ) = ⊗_{i=0}^{[g/2]}O_{Hilb_{g,ν}} (δ_{i}) is the divisor of all singular nodal curves on the universal curve σ : C_{g,ν} → Hilb_{g,ν}, then

ρ([Δ_{0}]) = β^{−1}(δ_{0}), ρ([Δ_{1}]) = 2β^{−1}(δ_{1}), ρ([Δ_{i}]) = β^{−1}(δ_{i}) for 2 ≤ i ≤ [g/2].

To put it briefly, we write that δ_{i} := [Δ_{i}] for i ≠ 1 and δ_{1} := \frac{1}{2}[Δ_{1}] in Pic(\overline{M}_{g})_{ℚ}.

Theorem 3.4. 1) The group homomorphism ρ : Pic(\overline{M}_{g}) → Pic(\overline{M}_{g}) is injective with torsion cokernel. Thus

ρ_{ℚ} : Pic(\overline{M}_{g})_{ℚ} ≅ Pic(\overline{M}_{g})_{ℚ}.

2) For g ≥ 3, the group Pic(\overline{M}_{g}) is freely generated by the classes λ, δ_{0}, …, δ_{[g/2]}.

From now on we shall identify Pic(\overline{M}_{g})_{ℚ} = Pic(\overline{M}_{g})_{ℚ}. The first part of Theorem 3.4 was established by Mumford in [M2] Lemma 5.8. The second part is due to Arbarello and Cornalba [AC2] and uses in an essential way Harer’s theorem that H^{2}(M_{g}, ℚ) ≅ ℚ. Unfortunately there is no purely algebraic proof of Harer’s result yet.

4. The canonical class of \overline{M}_{g}

In this section we explain the calculation of the canonical class of \overline{M}_{g} in terms of the generators of Pic(\overline{M}_{g}), cf. [HM]. This calculation has been one of the spectacular successes of the Grothendieck-Riemann-Roch theorem. In order to apply GRR one needs however a good modular interpretation of the cotangent bundle Ω_{\overline{M}_{g}}^{1}. This is provided by Kodaira-Spencer theory. We first compute the canonical

class of the stack \overline{M}_{g}, then we use the branched cover \overline{M}_{g} → \overline{M}_{g} to obtain the canonical class of the coarse moduli scheme \overline{M}_{g}.

For every stable curve [C] ∈ \overline{M}_{g}, we denote by Ω_{C}^{1} the sheaf of Kähler differentials and by ω_{C} the locally free dualizing sheaf (see [Ba] for a down-to-earth introduction to the deformation theory of stable curves). These sheaves sit in an exact sequence:

0 ⟶ Torsion(Ω_{C}^{1}) ⟶ Ω_{C}^{1} ⟶ ω_{C} ⟶^{res} ⨁_{p∈Sing(C)} ℂ_{p} ⟶ 0.

Kodaira-Spencer theory coupled with Serre duality provides an identification

T_{[C]}(\overline{M}_{g}) = Ext^{1}(Ω_{C}^{1}, O_{C}) = H^{0}(C, ω_{C} ⊗ Ω_{C}^{1})^{∨}.

One can globalize this observation and describe the cotangent bundle of \overline{M}_{g} as follows. We denote by π : \overline{M}_{g,1} → \overline{M}_{g} the universal curve and we denote by ω_{π} the relative dualizing sheaf and by Ω_{π}^{1} the sheaf of relative Kähler differentials, respectively. Then by Kodaira-Spencer theory we have the identification

Ω_{\overline{M}_{g}}^{1} = π_{∗}(Ω_{π}^{1} ⊗ ω_{π})

and call the class

K_{\overline{M}_{g}} = c_{1}(Ω_{\overline{M}_{g}}^{1}) ∈ Pic(\overline{M}_{g})

the canonical class of the moduli stack \overline{M}_{g}. To compute the first Chern class of this push-forward bundle we use the Grothendieck-Riemann-Roch theorem.

Suppose that we are given a proper map f : X → Y with smooth base Y and a sheaf F on X. Then the Grothendieck-Riemann-Roch (GRR) theorem reads

ch(f_{!}(F)) = f_{∗}(ch(F) · td(Ω_{f}^{1})) ∈ A^{∗}(Y), where
td(Ω_{f}^{1}) := 1 − \frac{c_{1}(Ω_{f}^{1})}{2} + \frac{c_{1}(Ω_{f}^{1})^{2} + c_{2}(Ω_{f}^{1})}{2} + (higher order terms )

denotes the Todd class.

Remark 4.1. One uses the GRR theorem to prove Mumford’s relation

κ_{1} := π_{∗}(c_{1}(ω_{π}^{2})) = 12λ − δ ∈ Pic(\overline{M}_{g}),

where δ := δ_{0} + ⋯ + δ_{[g/2]} is the total boundary (cf. [M1] pg. 101-103). Similarly, for n ≥ 2 we have the relation (to be used in Section 5), cf. [M2] Theorem 5.10:

λ_{1}^{(n)} = λ + \binom{n}{2}κ_{1} ∈ Pic(\overline{M}_{g}).

To compute K_{\overline{M}_{g}} we set f = π : \overline{M}_{g,1} → \overline{M}_{g}, F = Ω_{π}^{1} ⊗ ω_{π}, hence π_{∗}F = Ω_{\overline{M}_{g}}^{1}

and R^{i}π_{∗}F = 0 for i ≥ 1. Using Grothendieck-Riemann-Roch we can write: (10)

K_{\overline{M}_{g}} = π_{∗}[(1+c_{1}(F)+\frac{c_{1}^{2}(F) − 2c_{2}(F)}{2}+⋯)·(1−\frac{c_{1}(Ω_{π}^{1})}{2}+\frac{c_{1}(Ω_{π}^{1})^{2} + c_{2}(Ω_{π}^{1})}{12}+⋯)]_{1}.

Next we determine the Chern classes of F. Suppose [f : X → S] ∈ \overline{M}_{g}(S) is a family of stable curves such that both X and S are smooth projective varieties. Then codim(Sing(f), X) = 2 and the sheaf homomorphism Ω_{f}^{1} → ω_{f} induces an isomorphism Ω_{f}^{1} = ω_{f} ⊗ I_{Sing(f)} (in particular, Ω_{f}^{1} is not locally free). This claim follows from a local analysis around each point p ∈ Sing(f). Since the versal deformation space of a node is 1-dimensional, there exist affine coordinates x, y on the fibres of f and an affine coordinate t on S, such that locally around p, the variety X is given by the equation xy = t^{n} for some integer n ≥ 1. By direct calculation in a neighbourhood of p,

Ω_{f}^{1} = (f^{∗}O_{C} · dx + f^{∗}O_{C} · dy)/(xdy + ydx) · O_{C}.

Similarly, the dualizing sheaf ω_{f} is the free O_{X} module generated by the meromorphic differential η given by dx/x on the locus x ≠ 0 and by −dy/y on the locus y ≠ 0, hence we find that locally Ω_{f}^{1} = I_{x=y=t=0} · ω_{f}, which proves our claim.

The sheaves ω_{f} and Ω_{f}^{1} agree in codimension 1, thus c_{1}(Ω_{f}^{1}) = c_{1}(ω_{f}). An

application of Grothendieck-Riemann-Roch for the inclusion Sing(f) ↪ X, shows that c_{2}(Ω_{f}^{1}) = [Sing(f)]. Then by the Whitney formula we obtain that c_{1}(F) = 2c_{1}(ω_{f}) and c_{2}(F) = [Sing(f)]. Since this analysis holds for an arbitrary family of stable curves, the same relation must hold for the universal curve over \overline{M}_{g}. Returning to (10), we find the following formula:

K_{\overline{M}_{g}} = \frac{13}{12}π_{∗}(c_{1}(ω_{π})^{2}) − \frac{11}{12}π_{∗}[Sing(π)] = \frac{13}{12}κ_{1} − \frac{11}{12}δ = 13λ − 2δ ∈ Pic(\overline{M}_{g}).

Theorem 4.2. For g ≥ 4, the canonical class of the coarse moduli space \overline{M}_{g} is given by the formula

K_{\overline{M}_{g}} ≡ 13λ − 2δ_{0} − 3δ_{1} − 2δ_{2} − ⋯ − 2δ_{[g/2]} ∈ Pic(\overline{M}_{g}).

Proof. We consider the morphism ϵ : \overline{M}_{g} → \overline{M}_{g} which is simply branched

along the divisor is the divisor Δ_{1} ⊂ \overline{M}_{g}, hence ϵ^{∗}([Δ_{1}]) = 2δ_{1} ∈ Pic(\overline{M}_{g}). The Riemann-Hurwitz formula gives that K_{\overline{M}_{g}} = ϵ^{∗}(K_{\overline{M}_{g}}) + δ_{1} which finishes the proof. □

Remark 4.3. A slight difference occurs in the case g = 3. The morphism ϵ : \overline{M}_{3} → \overline{M}_{3} is simply branched along both the boundary Δ_{1} and the closure of the hyperelliptic locus

M_{3,2}^{1} := {[C] ∈ M_{3} : W_{2}^{1}(C) ≠ ∅}.

It follows that K_{\overline{M}_{3}} = ϵ^{∗}K_{\overline{M}_{3}} + δ_{1} + ρ([\overline{M}_{3,2}^{1}]), hence K_{\overline{M}_{3}} = 4λ − δ_{0}.

Using Theorem 4.2, we reformulate the problem of determining the Kodaira dimension of \overline{M}_{g} in terms of effective divisors: A sufficient condition for \overline{M}_{g} to be of general type is the existence of an effective divisor

D ≡ aλ − b_{0}δ_{0} − ⋯ − b_{[g/2]}δ_{[g/2]} ∈ Pic(\overline{M}_{g}),

with coefficients satisfying the following inequalities

(11)  \frac{a}{b_{0}} < \frac{13}{2}, \frac{a}{b_{1}} ≤ \frac{13}{3} and \frac{a}{b_{i}} ≤ \frac{13}{2} for 2 ≤ i ≤ [g/2].

This formulation using (11) clearly suggests the definition of the following numerical invariant of the moduli space cf. [HMo ]: If δ := δ_{0} + ⋯ + δ_{[g/2]} is the class of the total boundary and Eff(\overline{M}_{g}) ⊂ Pic(\overline{M}_{g})_{ℝ} denotes the cone of effective divisors, then we can define the slope function s : Eff(\overline{M}_{g}) → ℝ ∪ \{∞\} by the formula

s(D) := inf \{\frac{a}{b} : a, b > 0 such that aλ − bδ − D ≡ ∑_{j=0}^{[g/2]} c_{j}δ_{j}, where c_{j} ≥ 0\}.

From the definition it follows that s(D) = ∞ unless D ≡ aλ − ∑_{j=0}^{[g/2]} b_{j}δ_{j} with a, b_{j} ≥ 0 for all j. It is also well-known that s(D) < ∞ for any D which is the closure of an effective divisor on M_{g}. In this case, one has that

s(D) = \frac{a}{min_{j=0}^{[g/2]}b_{j}} .

We denote by s(\overline{M}_{g}) the slope of the moduli space \overline{M}_{g}, defined as

s(\overline{M}_{g}) := inf \{s(D) : D ∈ Eff(\overline{M}_{g})\}.

Proposition 4.4. We fix a moduli space \overline{M}_{g} with g ≥ 4. If s(\overline{M}_{g}) < 13/2 then \overline{M}_{g} is of general type. If s(\overline{M}_{g}) > 13/2 then the Kodaira dimension of \overline{M}_{g} is negative.

Proof. If there exists D ∈ Eff(\overline{M}_{g}) with s(D) < s(K_{\overline{M}_{g}}), it follows that one

can write K_{\overline{M}_{g}} ≡ α · λ + β · D + ∑_{j=1}^{[g/2]} c_{j}δ_{j}, where α, β > 0 and c_{j} ≥ 0 for 1 ≤ j ≤ [g/2]. Since the class λ ∈ Eff(\overline{M}_{g}) is big, we obtain that K_{\overline{M}_{g}} ∈ int Eff(\overline{M}_{g}), hence by definition \overline{M}_{g} is a variety of general type. □

Any explicit calculation of a divisor class on \overline{M}_{g} provides an upper bound for s(\overline{M}_{g}). Estimating how small slopes of effective divisors on \overline{M}_{g} can be, is the subject of the Harris-Morrison Slope ”Conjecture” [HMo ]:

Conjecture 4.5.

s(\overline{M}_{g}) ≥ 6 + \frac{12}{g + 1}.

The conjecture would obviously imply that κ(\overline{M}_{g}) = −∞ for g ≤ 22. However Conjecture 4.5 is false and counterexamples have been found in [FP], [F2], [F3], [Kh]-see also Section 6.1 of this paper.

There is a somewhat surprising connection between the Slope Conjecture and curves sitting on K3 surfaces. This has been first observed in [FP]: Given g ≥ 1 we consider a Lefschetz pencil of curves of genus g lying on a general K3 surface of degree 2g − 2 in P^{g}. This gives rise to a curve B in the moduli space \overline{M}_{g}. These pencils B fill up the entire moduli space \overline{M}_{g} for g ≤ 9 or g = 11 (see [Mu1]), and the divisor \overline{K}_{10} of curves lying on a K3 surface for g = 10. When g ≥ 13, the pencils B fill up the locus \overline{K}_{g} ⊂ \overline{M}_{g} of K3 sections of genus g and dim(\overline{K}_{g}) = 19 + g.

Lemma 4.6. We have the formulas B · λ = g + 1, B · δ_{0} = 6g + 18 and B · δ_{j} = 0 for j ≠ 0.

It will turn out that the locus K_{g} becomes an obstruction for an effective divisor on \overline{M}_{g} to have small slope. The next result shows that in order to construct geometric divisors on \overline{M}_{g} having small slope, one must search for geometric conditions which have codimension 1 in moduli, and which are a relaxation of the condition that a curve be a section of a K3 surface. This philosophy governs the construction of Koszul divisors on \overline{M}_{g} carried out in [F2], [F3].

Proposition 4.7. Let D be the closure in \overline{M}_{g} of an effective divisor on M_{g}. If the inequality s(D) < 6 + 12/(g + 1) holds, then D contains the locus \overline{K}_{g} of curves lying on K3 surfaces.

Proof. We consider as above the curve B ⊂ \overline{M}_{g} corresponding to a Lefschetz pencil of curves of genus g on a general K3 surface S. From Lemma 4.6 we obtain that

\frac{B · δ}{B · λ} = 6 + \frac{12}{g + 1} > s(D),

which implies that B · D < 0 hence B ⊂ D. By varying both B and S inside the moduli space of polarized K3 surfaces, we obtain the inclusion \overline{K}_{g} ⊂ D. □

Bounding s(\overline{M}_{g}) from below, remains one of the main open problems in the field. There is a straightforward (probably far from optimal) way of obtaining a bound on s(\overline{M}_{g}) by writing down any moving curve R ⊂ \overline{M}_{g}, that is, a curve which moves in an algebraic family \{R_{t}\}_{t∈T} of curves on \overline{M}_{g} such that the set ⋃_{t∈T} R_{t} is dense in \overline{M}_{g}. One instance of a moving curve is a complete intersection curve R = H_{1} ∩ ⋯ ∩ H_{3g−4}, where H_{i} are numerically effective divisors on \overline{M}_{g}.

If R ⊂ \overline{M}_{g} is a moving curve, then R · D ≥ 0, for any D ∈ Eff(\overline{M}_{g}), hence

s(\overline{M}_{g}) ≥ \frac{R · δ}{R · λ}.

Obviously writing down and then computing the invariants of a moving curve in \overline{M}_{g} can be difficult. An experimental bound s(\overline{M}_{g}) ≥ O(1/g) was initially obtained in [HMo ] using Hurwitz schemes of covers of P^{1}. A similar (but nevertheless different) bound is obtained by D. Chen [C] using covers of elliptic curves.

4.1. Pandharipande’s lower bound on s(\overline{M}_{g}).

Recently, Pandharipande [P] has found a short way of proving the inequality

s(\overline{M}_{g}) ≥ O(\frac{1}{g}),

in a way that uses only descendent integrals over \overline{M}_{g,n} as well as some calculations on Hodge integrals that appeared in [FaP]. We explain the main idea of his proof.

One constructs a covering curve for \overline{M}_{g} by pushing forward products of nef tautological classes on moduli spaces \overline{M}_{g,n} via the morphisms forgetting the marked points. In the simplest incarnation of this method, one considers the universal curve π : \overline{M}_{g,1} → \overline{M}_{g} curve and the nef tautological class ψ_{1} = c_{1}(ω_{π}) ∈ A^{1}(\overline{M}_{g}). Then π_{∗}(ψ_{1}^{3g−4}) ∈ A_{1}(\overline{M}_{g}) is a covering curve class, in particular for every divisor

D ≡ aλ − ∑_{i=0}^{[g/2]} b_{i}δ_{i} ∈ Pic(\overline{M}_{g})

which does not contain boundary components, we have that π_{∗}(ψ_{1}^{3g−4}) · D ≥ 0, hence,

s(D) ≥ \frac{a}{b_{0}} ≥ \frac{π_{∗}(ψ_{1}^{3g−3}) · δ_{0}}{π_{∗}(ψ_{1}^{3g−3}) · λ} = \frac{∫_{\overline{M}_{g,1}} δ_{0} · ψ_{1}^{3g−3}}{∫_{\overline{M}_{g,1}} λ · ψ_{1}^{3g−3}} .

We outline the calculation of the numerator appearing in this fraction. For the degree 2 natural map

ϵ : \overline{M}_{g−1,3} → \overline{M}_{g,1}, ϵ([C, p, x, y]) := [\frac{C}{x ∼ y} , p],

one has that ϵ_{∗}([\overline{M}_{g−1,3}] = 2δ_{0} ∈ A^{1}(\overline{M}_{g,1}), hence via the push-pull formula we find,

∫_{\overline{M}_{g,1}} δ_{0} · ψ_{1}^{3g−3} = \frac{1}{2} ∫_{\overline{M}_{g−1,3}} ψ_{1}^{3g−3} = \frac{1}{2} ∫_{\overline{M}_{g−1,1}} ψ_{1}^{3g−5}.

The last equality here is an easy consequence of the string equation [W]

∫_{\overline{M}_{g,n+1}} ψ_{1}^{a_{1}} ⋯ ψ_{n}^{a_{n}} = ∑_{i=1}^{n} ∫_{\overline{M}_{g,n}} ψ_{1}^{a_{1}} ⋯ ψ_{i}^{a_{i}−1} ⋯ ψ_{n}^{a_{n}},

where a_{1}, …, a_{n} ≥ 0 such that ∑_{i=1}^{n} a_{i} = 3g − 2 + n.

The following evaluation follows by putting together [M6] Section 6 and [FaP] Section 3. For the sake of completeness, we outline a proof:

Lemma 4.8.

∫_{\overline{M}_{g,1}} ψ_{1}^{3g−2} = \frac{1}{24^{g} · g!}.

Proof. The cokernel of the sheaf morphism π^{∗}(𝔼) → ω_{π} on \overline{M}_{g,1} given by multiplication of global sections, is supported on the locus

X_{2} ∪ … ∪ X_{g},

where X_{j} ⊂ \overline{M}_{g,1} is the closure of the subvariety of pointed curves

[R ∪ C_{1} ∪ … ∪ C_{j}, p],

where R is a smooth rational curve, p ∈ R and the curves C_{i} are smooth with the property that #(R ∩ C_{i}) = 1, for 1 ≤ i ≤ j and ∑_{i=1}^{j} g(C_{i}) = g. Clearly dim(X_{j}) = 3g − 2 − j, and there is a natural map

f_{j} : X_{j} → \overline{M}_{0,j+1}

forgetting the tails C_{1}, …, C_{j}, while retaining the intersection points R ∩ C_{i} for 1 ≤ i ≤ j. One has that ψ_{1| X_{j}} = f_{j}^{∗}(ψ_{p}), where ψ_{p} ∈ A^{1}(\overline{M}_{0,j+1}) denotes the cotangent line class on \overline{M}_{0,j+1} corresponding to the marked point labeled by p ∈ R. For dimension reasons it follows that ψ_{1| X_{2}∪...∪X_{g}}^{g−1} = 0, whereas ψ_{1| X_{2}∪...∪X_{g}}^{g−2} must be entirely supported on the locus X_{g}. Putting these observations together, one finds that

(12)  (\frac{π^{∗}c(𝔼)}{1 + ψ_{1}})_{2g−2} = ψ_{1}^{g−2}[X_{g}]_{ℚ}.

To estimate ∫_{\overline{M}_{g,1}} ψ_{1}^{3g−2}, one uses Mumford’s relation c(𝔼) · c(𝔼^{∨}) = 1, cf. [M6]. This comes from the exact sequence which globalizes Serre duality

0 ⟶ 𝔼 → R^{1}π_{∗}Ω_{π}^{1} → 𝔼^{∨} → 0,

where the rank 2g vector bundle in the middle possesses a Gauss-Manin connection. Accordingly, we can write that

∫_{\overline{M}_{g,1}} ψ_{1}^{3g−2} = (\frac{π^{∗}(𝔼)}{1 + ψ_{1}} )_{2g−2} · (λ_{g} + λ_{g−1}ψ_{1} + ⋯ + ψ_{1}^{g}) = λ_{g}ψ_{1}^{g−2} · [X_{g}]_{ℚ}.

This last intersection number can be evaluated via the map of degree g!,

ϕ : (\overline{M}_{1,1})^{g} × \overline{M}_{0,g+1} → X_{g},

which attaches g elliptic tails at the first g marked points of a rational (g + 1)- pointed stable curve. Clearly ϕ^{∗}(𝔼_{|X_{g}}) = 𝔼_{1} ⊠ ⋯ ⊠ 𝔼_{1}, where 𝔼_{1} is the Hodge bundle on \overline{M}_{1,1}. Since ∫_{\overline{M}_{1,1}} λ_{1} = 1/24, one finds that,

λ_{g}ψ_{1}^{g−2} · [X_{g}]_{ℚ} = \frac{1}{g!}(∫_{\overline{M}_{1,1}} λ_{1})^{g} · ∫_{\overline{M}_{0,g+1}} ψ_{1}^{g−2} = \frac{1}{24^{g} · g!}.

□

To evaluate the integral ∫_{\overline{M}_{g,1}} λ · ψ_{1}^{3g−3}, first one uses the GRR calculation of ch(𝔼) applied to the universal curve π : \overline{M}_{g,1} → \overline{M}_{g}. One finds that

∫_{\overline{M}_{g,1}} λ · ψ_{1}^{3g−3} = \frac{1}{12} ∫_{\overline{M}_{g,2}} ψ_{1}^{3g−3}ψ_{2}^{2} − \frac{1}{12} ∫_{\overline{M}_{g,1}} ψ_{1}^{3g−2} + \frac{1}{24} ∫_{\overline{M}_{g−1,3}} ψ_{1}^{3g−3}.

The last integral is evaluated using again the string equation, for the middle one we use Lemma 4.8. The first integral is evaluated using [W] and one finally proves:

Theorem 4.9.

s(\overline{M}_{g}) ≥ \frac{∫_{\overline{M}_{g,1}} δ_{0} · ψ_{1}^{3g−3}}{∫_{\overline{M}_{g,1}} λ · ψ_{1}^{3g−3}} = \frac{60}{g + 4}.

Note that the bound O(1/g) obtained in this theorem is quite similar to the experimental bound \frac{576}{5g} obtained in [HMo ] using Hurwitz covers.

Remark 4.10. Another very natural covering curve for \overline{M}_{g}, which potentially could produce a much better lower bound for s(\overline{M}_{g}) than the one in [P], has been recently proposed by Coskun, Harris and Starr [CHS]: If Hilb_{g,1} denotes the Hilbert scheme of canonically embedded curves C ⊂ P^{g−1}, then dim Hilb_{g,1} = g^{2} + 3g − 4. We denote by r(g) the largest number r, such that through r general points in P^{g−1} there passes a canonical curve [C ↪ P^{g−1}] ∈ Hilb_{g,1}. It has been determined in [St] that

r(g) = g + 5 + [\frac{6}{g − 2}].

This number is, the smallest solution of the necessary inequality

dim Hilb_{g,1} + r ≥ (g − 1)r.

In particular r(3) = 14 (as it should be!) and r(5) = 12. For g ≥ 9, one fixes general points p_{1}, …, p_{g+5} ∈ P^{g−1} as well as a general linear space P^{g−7} ⊂ P^{g−1}. The family X_{g} ⊂ \overline{M}_{g} consisting of canonical curves [C ↪ P^{g−1}] ∈ Hilb_{g,1} passing through p_{1}, …, p_{g+5}, and such that C ∩ P^{g−7} ≠ ∅ is a covering curve for \overline{M}_{g}. It is an interesting problem to determine the slope X_{g} · δ/X_{g} · λ.

5. The Harris-Mumford theorem revisited: An alternative proof via syzygies

In this section we present a different proof of the main result from [HM] by replacing the calculation of the class of the Hurwitz divisor \overline{M}_{g,k}^{1} of k-gonal curves of genus g = 2k − 1 by the calculation of the class of a certain Koszul divisor \overline{Z}_{g,k−2}, consisting of canonical curves [C] ∈ M_{g} with extra syzygies at the (k − 2)-nd step in its minimal graded resolution. The advantage of this approach is that the proof that \overline{M}_{g} is of general type becomes shorter since one can completely avoid having to develop the theory of admissible covers and do without the enumerative calculations that occupy a large part of [HM], precisely pg. 53-86, or alternatively, develop the theory of limit linear series [EH1]. The proof becomes also more direct and logical, since it uses solely the geometry of canonical curves of genus g and that of the corresponding Hodge bundles on \overline{M}_{g} , rather than the geometry of an auxiliary Hurwitz stack. The disadvantage of this approach, is that the statement that the locus \overline{Z}_{g,k−2} is a divisor on \overline{M}_{g} is highly non-trivial and it is equivalent to Green’s Conjecture for generic curves of odd genus (Voisin’s theorem [V1], [V2]). This situation is somewhat similar to that encountered in the streamlined proof of Theorem 1.1 presented by Eisenbud and Harris in [EH3] (and which is comparable in length to our proof): Showing that the a priori virtual Brill-Noether locus is an actual divisor in \overline{M}_{g}, requires the full force of the Brill-Noether theory and is arguably more difficult than computing the class of the Brill-Noether divisor on \overline{M}_{g}.

We start by recalling a few basic facts on syzygies. For a smooth curve C and a globally generated line bundle L ∈ Pic^{d}(C), we denote by K_{i,j}(C, L) the Koszul cohomology group obtained from the complex

∧^{i+1}H^{0}(L)⊗H^{0}(L^{⊗(j−1)}) ⟶^{d_{i+1,j−1}} ∧^{i}H^{0}(L)⊗H^{0}(L^{⊗j}) ⟶^{d_{i,j}} ∧^{i−1}H^{0}(L)⊗H^{0}(L^{⊗(j+1)}),

where the maps d_{i,j} are the Koszul differentials defined by (cf. [La2], [PR])

d_{i,j}(f_{1} ∧ … ∧ f_{i} ⊗ u) := ∑_{l=0}^{i}(−1)^{l}(f_{1} ∧ … ∧ f̌_{l} … ∧ f_{i}) ⊗ (uf_{l}),

with f_{l} ∈ H^{0}(C, L) and u ∈ H^{0}(C, L^{⊗j}). If R(C, L) := ⊕_{n≥0}H^{0}(C, L^{⊗n}) denotes the graded module over the polynomial ring S := Sym H^{0}(C, L), then

K_{i,j}(C, L) = Tor_{i}^{S}(ℂ, R(C, L))_{i+j}.

There is a standard geometric way of computing Koszul cohomology groups using Lazarsfeld bundles. Since L is assumed to be globally generated, we can define the vector bundle M_{L} on C through the following exact sequence on C:

0 → M_{L} → H^{0}(L) ⊗ O_{C} → L → 0.

A diagram chasing argument using the exact sequences

0 ⟶ ∧^{a}M_{L} ⊗ L^{⊗b} → ∧^{a}H^{0}(L) ⊗ L^{⊗b} ⟶ ∧^{a−1}M_{L} ⊗ L^{⊗(b+1)} ⟶ 0

for various a and b, shows that there is an identification cf. [La2]

(13)  K_{a,b}(C, L) = \frac{H^{0}(C, ∧^{a}M_{L} ⊗ L^{⊗b})}{Image\{∧^{a+1}H^{0}(C, L) ⊗ H^{0}(C, L^{⊗(b−1)})\}} .

Example 5.1. From (13) we find that K_{0,2}(C, L) = 0 if and only if the multiplication map Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2}) is surjective. Assuming L is normally

generated, we have that K_{1,2}(C, L) = 0 if and only if C →^{|L|} P(H^{0}(C, L)^{∨}) is cut out by quadrics. More generally, one says that L satisfies the Green-Lazarsfeld property (N_{p}) for p ≥ 0, if the vanishing

K_{i,2}(C, L) = 0

holds for all 0 ≤ i ≤ p. This corresponds intuitively to the situation that the first p

syzygies of the image curve C →^{|L|} P(H^{0}(C, L)^{∨}) are as simple as possible, that is, linear.

From now on we specialize to the case L = K_{C} ∈ W_{2g−2}^{g−1}(C) and we consider

the canonical map C ⟶^{|K_{C}|} P^{g−1}. If C is non-hyperreliptic, we set I_{C/P^{g−1}} to be the ideal of the canonically embedded curve.

Proposition 5.2. For any non-hyperrelliptic curve [C] ∈ M_{g} and any integer 0 ≤ i ≤ (g − 1)/2 we have the following equivalence:

K_{i,2}(C, K_{C}) ≠ 0 ⟺

⟺ h^{0}(P^{g−1}, Ω_{P^{g−1}}^{i}(i + 2) ⊗ I_{C/P^{g−1}}) ≥ \binom{g − 1}{i + 2}\frac{(g − 2i − 3)(i + 1)}{g − i − 1} + 1.

Proof. We start with a canonically embedded curve C ↪^{|K_{C}|} P^{g−1}. Throughout the proof we use the identification M_{P}^{g−1} = Ω_{P}^{g−1}(1) coming from the Euler sequence on P^{g−1}. Since the vector bundle M_{K_{C}} is stable (cf. [PR] Corollary 3.5), we have the vanishing

H^{1}(C, ∧^{i} ⊗ K_{C}^{⊗2}) = 0

because μ(∧^{i}M_{K_{C}} ⊗ K_{C}^{⊗2}) > 2g − 1. It follows from (13) that K_{i,2}(C, K_{C}) ≠ 0 if and only if the map

H^{1}(C, ∧^{i+1}M_{K_{C}} ⊗ K_{C}) → ∧^{i+1}H^{0}(C, K_{C}) ⊗ H^{1}(C, K_{C})

is an isomorphism, or equivalently h^{1}(C, ∧^{i+1}M_{K_{C}} ⊗K_{C}) = \binom{g}{i+1}. We write down the following commutative diagram, where by abusing notation, we shall denote by the same letter a sheaf morphism and the group morphism it induces at the level of global sections:

0  0  0
↓  ↓  ↓
∧^{i+1}M_{P}^{g−1} ⊗ I_{C/P^{g−1}}(1) ⟶ ∧^{i+1}H^{0}(O_{P}^{g−1}(1)) ⊗ I_{C/P^{g−1}}(1) ⟶ ∧^{i}M_{P}^{g−1} ⊗ I_{C/P^{g−1}}(2)
↓  ↓  ↓
∧^{i+1}M_{P}^{g−1}(1)  ⟶ ∧^{i+1}H^{0}(O_{P}^{g−1}(1)) ⊗ O_{P}^{g−1}(1) ⟶  ∧^{i}M_{P}^{g−1}(2)
↓^{α}  ↓^{β}  ↓^{γ}
∧^{i+1}M_{K_{C}} ⊗ K_{C}  ⟶  ∧^{i+1}H^{0}(K_{C}) ⊗ K_{C}  ⟶  ∧^{i}M_{K_{C}} ⊗ K_{C}^{⊗2}
↓  ↓  ↓
0  0  0

Applying the Snake Lemma, we find that H^{0}(P^{g−1}, ∧^{i}M_{P}^{g−1}⊗I_{C/P^{g−1}}(2)) = Coker(α). We also note that h^{0}(P^{g−1}, ∧^{i+1}M_{P}^{g−1}(1)) = \binom{g}{i+2} (use for instance Bott’s vanishing theorem). Thus the condition K_{i,2}(C, K_{C}) = 0 is satisfied if and only if

dim Coker(α) = h^{0}(C, ∧^{i+1}M_{K_{C}} ⊗ K_{C}) − h^{0}(P^{g−1}, ∧^{i+1}M_{P}^{g−1}(1)) =

= \binom{g − 1}{i + 1}(g − 2i − 3) + h^{1}(C, ∧^{i+1}M_{K_{C}} ⊗ K_{C}) − \binom{g}{i + 2} ≤

≤ \binom{g − 1}{i + 2}\frac{(g − 2i − 3)(i + 1)}{g − i − 1}.

□
For g = 2i + 3, we find that K_{i,2}(C, K_{C}) ≠ 0 if and only if the map
(14)  H^{0}(P^{g−1}, ∧^{i}M_{P}^{g−1}(2)) ⟶^{γ} H^{0}(C, ∧^{i}M_{K_{C}} ⊗ K_{C}^{⊗2})

is not an isomorphism. We note that γ is a map between vector spaces of the same dimensions:

h^{0}(P^{g−2}, ∧^{i}M_{P}^{g−1}(2)) = (i+1)\binom{g + 1}{i + 2} = χ(C, ∧^{i}M_{K_{C}} ⊗K_{C}^{⊗2}) = h^{0}(C, ∧^{i}M_{K_{C}} ⊗K_{C}^{⊗2})

(for the left hand side use Bott vanishing, for the right hand-side the Riemann-Roch theorem.) This shows that the locus

Z_{g,i} := {[C] ∈ M_{g} : K_{i,2}(C, K_{C}) ≠ 0},

being the degeneracy locus of a morphism between two vector bundles of the same rank over M_{g}, is a virtual divisor on the moduli space of curves.

Example 5.3. By specializing to the case g = 3, we find the following interpretation

Z_{3,0} := {[C] ∈ M_{3} : Sym^{2}H^{0}(C, K_{C}) → H^{0}(C, K_{C}^{⊗2}) is not an isomrphism }.

Using M. Noether’s theorem [ACGH], it follows that Z_{3,0} consists precisely of hyperelliptic curves, that is,

supp(Z_{3,0}) = supp(M_{3,2}^{1}).

In the next case g = 5, we use Petri’s theorem stating that a non-hyperelliptic canonical curve [C] ∈ M_{5} is cut out by quadrics unless it has a 𝔤_{3}^{1}. We obtain that supp(Z_{5,1}) = supp(M_{5,3}^{1}).

In order to describe the closure \overline{Z}_{g,i} of Z_{g,i} inside \overline{M}_{g}, we shall extend the determinantal description of Z_{g,i} over a partial compactification of M_{g}. Our methods seem well-suited for divisor class calculations but harder to implement in the case of Koszul cycles on \overline{M}_{g} of higher codimension.

We denote by M_{g}^{∗} := M_{g} ∪ (∪_{j=0}^{i+1}Δ_{j}^{0}) the locally closed substack of \overline{M}_{g} defined as the union M_{g} and the open substacks Δ_{j}^{0} ⊂ Δ_{j} for 1 ≤ j ≤ i + 1 consisting of 1-nodal genus g curves [C ∪_{y} D], with [C, y] ∈ M_{g−j,1} and [D, y] ∈ M_{j,1}, that is, Δ_{j}^{0} is the intersection of Δ_{j} with the codimension 1 stratum in the topological stratification of \overline{M}_{g}. The substack Δ_{0}^{0} ⊂ Δ_{0} classifies 1-nodal irreducible genus g curves

[C_{yq} := \frac{C}{q ∼ y} ] ∈ \overline{M}_{g},

where [C, q, y] ∈ M_{g−1,2} together with their degenerations consisting of unions of a smooth genus g − 1 curve and a nodal rational curve. We set

M̃_{g} := M_{g} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0} ⊂ M_{g}^{∗}.

For integers 0 ≤ a ≤ i and b ≥ 2 we define vector bundles G_{a,b} over M̃_{g} with fibre

G_{a,b}[C] = H^{0}(C, ∧^{a}M_{K_{C}} ⊗ K_{C}^{⊗b})

over every point [C] ∈ M_{g}. The question is of course how the extend this description of G_{a,b} over the locus of stable curves. In this paper we shall only describe how to construct the bundles G_{a,b} over M̃_{g}, which will suffice in order to compute the slope of \overline{Z}_{g,i} and prove Theorem 1.1 for g = 2i + 3. For full details on how to extend the vector bundles G_{a,b} over M̃_{g} (that is, outside codimension 2 over all the boundary divisors on \overline{M}_{g}), we refer to [F2] p. 75-86. We start by constructing the bundles G_{0,b}:

Proposition 5.4. For each b ≥ 2 there exists a vector bundle G_{0,b} over M̃_{g} of rank (2b − 1)(g − 1) with fibres admitting the following description:

• For [C] ∈ M_{g} we have that G_{0,b}[C] = H^{0}(C, K_{C}^{⊗b}).

• For [C ∪_{y} E]) ∈ Δ_{1}^{0}, where E is an elliptic curve, if

u ∈ H^{0}(C, K_{C} ⊗ O_{C}(2y)) − H^{0}(C, K_{C})

denotes any meromorphic 1-form with non-zero residue at y, then

G_{0,b}[C ∪_{y} E] = H^{0}(C, K_{C}^{⊗b} ⊗ O_{C}((2b − 2) · y)) + ℂ · u^{b} ⊂

⊂ H^{0}(C, K_{C}^{⊗b} ⊗ O_{C}(2b · y)).

• For [C_{yq} = C/y ∼ q] ∈ Δ_{0}^{0}, where q, y ∈ C and

u ∈ H^{0}(C, K_{C} ⊗ O_{C}(y + q)) − H^{0}(C, K_{C})

is a meromorphic 1-form with non-zero residues at y and q, we have that

G_{0,b}[C_{yq}] = H^{0}(C, K_{C}^{⊗b} ⊗ O_{C}((b − 1)y + (b − 1)q)) ⊕ ℂ · u^{b} ⊂

⊂ H^{0}(C, K_{C}^{⊗b} ⊗ O_{C}(by + bq)).

The idea to define the vector bundles G_{0,b} as suitable twists by boundary divisors of powers of the relative dualizing sheaf over the universal curve, that is,

G_{0,b} = π_{∗}(ω_{π}^{⊗b} ⊗ ∑_{j=1}^{[g/2]} π^{∗}(O_{\overline{M}_{g}}(c_{j}^{b} δ_{j})))

for precisely determined constants c_{j}^{b} ≥ 0, comes of course from the theory of limit linear series. Recalling that σ : 𝔊_{2g−2}^{g−1} → M̃_{g} denotes the stack of limit 𝔤_{2g−2}^{g−1}’s, then for a curve [C ∪_{y} E] ∈ Δ_{1}^{0}, the fibre σ^{−1}[C ∪_{y} E] consists of a single limit linear series

(l_{C} = (ω_{C}(2y), H^{0}(ω_{C}(2y)) ∈ G_{2g−2}^{g−1}(C),
l_{E} = (O_{E}(2gy), (g − 1)y + |(g − 1)y|) ∈ G_{2g−2}^{g−1}(E))).

The bundle G_{0,1} retains the aspect of this limit 𝔤_{2g−2}^{g−1} corresponding to the component of genus g − 1, while dropping the information coming from the elliptic tail. Similarly, for b ≥ 2, it is an easy exercise in limit linear series to show the fibre G_{0,b}[C ∪_{y} E] is precisely the C-aspect of the limit 𝔤_{(2b−1)(g−1)−1}^{2b(g−1)} induced from ω_{C∪E}^{⊗b}. The situation becomes more complicated when extending G_{0,b} over the entire stack M_{g}^{∗}. As explained in [F2] Theorem 3.13 in the case of the Hurwitz stack (and the same holds true for M_{g}^{∗} itself), the twisting coefficients c_{j}^{b} are chosen in a unique way such that the resulting bundles G_{a,b} fit in exact sequences of type (15).

Having defined G_{0,b} we now define inductively all vector bundles G_{a,b}. First we define G_{1,b} as the kernel of the multiplication map G_{0,1} ⊗ G_{0,b} → G_{0,b+1}, that is, by the exact sequence

0 ⟶ G_{1,b} ⟶ G_{0,1} ⊗ G_{0,b} ⟶ G_{0,b+1} ⟶ 0.

Having defined G_{l,b} for all l ≤ a − 1, the vector bundle G_{a,b} is defined through the following exact sequence over M̃_{g}:

(15)  0 ⟶ G_{a,b} ⟶ ∧^{a}G_{0,1} ⊗ G_{0,b} ⟶^{ϕ_{a,b}} G_{a−1,b+1} ⟶ 0.

Proposition 5.5. The Koszul maps ϕ_{a,b} : ∧^{a}G_{0,1} ⊗ G_{0,b} → G_{a−1,b+1} are welldefined and surjective for all integers b ≥ 2 and 0 ≤ a ≤ b. In particular the exact sequences (15) make sense and the vector bundles G_{a,b} are well-defined.

Proof. This proof is similar to [F2] Proposition 3.10. We use that the vector bundle M_{K_{C}⊗O_{C}(y+q)} is semi-stable for [C, y, q] ∈ M_{g−1,2}, in particular

H^{1}(C, ∧^{a}M_{K_{C}⊗O_{C}(y+q)} ⊗ K_{C}^{⊗b}((b − 1) · (y + q))) = 0,

that is, the map

∧^{a}H^{0}(C, K_{C}(y + q)) ⊗ H^{0}(C, K_{C}^{⊗b}((b − 1)(y + q))) ⟶
⟶ H^{0}(C, ∧^{a−1}M_{K_{C}⊗O_{C}(y+q)} ⊗ K_{C}^{⊗(b+1)}(b(y + q)))

is surjective. The rest now follows from the description of the fibres of the bundles G_{a,b} provided in Proposition 5.4. □

For 0 ≤ a ≤ i and b ≥ 1 we define vector bundles H_{a,b} over M̃_{g} having fibre

H_{a,b}[C] = H^{0}(P^{g−1}, ∧^{a}M_{P}^{g−1} ⊗ O_{P}^{g−1}(b))

over each point corresponding to a smooth curve [C] ∈ M_{g} with the canonical

map C ⟶^{|K_{C}|} P^{g−1}. First we set H_{0,b} := Sym^{b}(𝔼 ⊗ O_{M̃_{g}}(δ_{1})) for b ≥ 1. Having already defined H_{a−1,b} for all b ≥ 1, we define H_{a,b} via the exact sequence

(16)  0 ⟶ H_{a,b} ⟶ ∧^{a}H_{0,1} ⊗ Sym^{b}H_{0,1} ⟶ H_{a−1,b+1} ⟶ 0.

Note that the bundles H_{a,b} are defined entirely in terms of the Hodge bundle 𝔼. There is a natural vector bundle morphism over M̃_{g}

γ_{a,b} : H_{a,b} → G_{a,b}.

When g = 2i + 3 then rank(H_{i,2}) = rank(G_{i,2}) and the degeneracy locus Z(γ_{i,2}) of the morphism γ_{i,2} is a codimension 1 compactification in M̃_{g} of the locus Z_{g,i}.

We shall determine the class c_{1}(G_{i,2} − H_{i,2}) ∈ Pic(M̃_{g}) by computing its intersection it with the following test curves lying in the boundary of \overline{M}_{g}: We fix a pointed curve [C, q] ∈ M_{g−1,1} and a general elliptic curve [E, y] ∈ M_{1,1}. We define two 1-parameter families (17)

C^{0} := \{\frac{C}{y ∼ q} : y ∈ C\} ⊂ Δ_{0} ⊂ \overline{M}_{g} and C^{1} := \{C ∪_{y} E : y ∈ C\} ⊂ Δ_{1} ⊂ \overline{M}_{g}.

These families intersect the generators of Pic(\overline{M}_{g}) as follows (cf. [HM] pg. 83-85):

C^{0} · λ = 0, C^{0} · δ_{0} = −2g + 2, C^{0} · δ_{1} = 1 and C^{0} · δ_{a} = 0 for a ≥ 2, and
C^{1} · λ = 0, C^{1} · δ_{0} = 0, C^{1} · δ_{1} = −2g + 4, C^{1} · δ_{a} = 0 for a ≥ 2.

Lemma 5.6. We fix [C, q] ∈ M_{g−1} and we consider the test curves C^{0}, C^{1} ⊂ \overline{M}_{g}. Then for all integers j ≥ 1 the following formulas:

(1) C^{1} · c_{1}(G_{0,j}) = −2g + 4.
(2) C^{0} · c_{1}(G_{0,j}) = (j − 1)(j(g − 1) + j − 1) + j.

Proof. We denote by p_{1}, p_{2} : C × C → C the two projections and Δ ⊂ C × C is the diagonal. We give details only for the first calculation the remaining one being similar. We have the identification G_{0,1|C^{1}} = (π_{1})_{∗}(π_{2}^{∗}(K_{C}) ⊗ O(2Δ)), from which we obtain that c_{1}(G_{0,1|C^{1}}) = −2g + 4. For j ≥ 2 we use the following exact sequences of bundles on C:

0 → (π_{1})_{∗}(π_{2}^{∗}(K_{C}^{⊗j})⊗O((2j−2)Δ)) ⟶ G_{0,j|C^{1}} ⟶ (π_{1})_{∗}(π_{2}^{∗}(K_{C}^{⊗j})⊗O_{Δ}(2jΔ)) → 0.

An immediate application of Grothendieck-Riemann-Roch for the projection morphism p_{1} : C × C → C gives that

c_{1}(π_{1})_{∗}(π_{2}^{∗}(K_{C}^{⊗j}) ⊗ O_{C×C}((2j − 2)Δ)) = 2(g − 1)(j − 1),

which finishes the proof. □

Theorem 5.7. The class of the virtual divisor \overline{Z}_{2i+3,i} in Pic(M̃_{2i+3}) equals

[\overline{Z}_{2i+3,i}]^{virt} = c_{1}(G_{i,2} − H_{i,2}) = \frac{1}{i + 2}\binom{2i}{i}(6(i + 3)λ − (i + 2)δ_{0} − 6(i + 1)δ_{1}).

Proof. We have constructed the vector bundle morphism γ_{i,2} : H_{i,2} → G_{i,2} over the stack M̃_{g}. For g = 2i + 3 we know that rank(H_{i,2}) = rank(G_{i,2}) and the virtual Koszul class [\overline{Z}_{g,i}]^{virt} equals c_{1}(G_{i,2} − H_{i,2}). We recall that for a rank e vector bundle E over a stack X and for i ≥ 1, we have formulas

c_{1}(∧^{i}E) = \binom{e − 1}{i − 1}c_{1}(E) and c_{1}(Sym^{i}E) = \binom{e + i − 1}{e}c_{1}(E).

We write c_{1}(G_{i,2} − H_{i,2}) = 𝔞λ − 𝔟_{0}δ_{0} − 𝔟_{1}δ_{1}. Using the exact sequences (15) we find that

c_{1}(G_{i,2}) = ∑_{l=0}^{i}(−1)^{l}c_{1}(∧^{i−l}G_{0,1} ⊗ G_{0,l+2}) = ∑_{l=0}^{i}(−1)^{l}\binom{g}{i − l}c_{1}(G_{0,l+2})+

+ ∑_{l=0}^{i}(−1)^{l}((g − 1)(2l + 3))\binom{g − 1}{i − l − 1}c_{1}(G_{0,1}).

Using repeatedly the exact sequence (16) we find that

c_{1}(H_{i,2}) = ∑_{l=0}^{i}(−1)^{l}c_{1}(∧^{i−l}H_{0,1} ⊗ Sym^{l+2}H_{0,1}) =

= ∑_{l=0}^{i}(−1)^{l}\binom{g}{i − l}c_{1}(Sym^{l+2}(H_{0,1})) + ∑_{l=0}^{i}(−1)^{l}\binom{g + l + 1}{l + 2}c_{1}(∧^{i−l}H_{0,1})

= ∑_{l=0}^{i}(−1)^{l}(\binom{g}{i − l}\binom{g + l + 1}{g} + \binom{g + l + 1}{l + 2}\binom{g − 1}{i − l − 1})c_{1}(H_{0,1}) =

= 4(2i + 1)\binom{2i}{i}c_{1}(H_{0,1}),

with G_{0,1} = H_{0,1} = 𝔼 ⊗ O_{M̃_{g}}(δ_{1}). We intersect both these formulas with the test curves C^{0} and C^{1} and write that

(2g − 2)𝔟_{0} − 𝔟_{1} = C^{0} · [\overline{Z}_{g,i}]^{virt} = (i + 1)\binom{2i + 2}{i} and

(2g − 4)𝔟_{1} = C^{1} · [\overline{Z}_{g,i}]^{virt} = 6(i + 1)\binom{2i + 2}{i}.

These relations determine 𝔟_{0} and 𝔟_{1}. Finally we claim that we also have the relation 𝔞 − 12𝔟_{0} + 𝔟_{1} = 0 which finishes the proof. Indeed, we consider q the curve R ⊂ \overline{M}_{g} obtained by attaching to a fixed point q ∈ C of a curve of genus g − 1 a Lefschetz pencil of plane cubics. Then R · λ = 1, R · δ_{0} = 12, R · δ_{1} = −1 and R · δ_{j} = 0 for j ≥ 2. Then

𝔞 − 12𝔟_{0} + 𝔟_{1} = 0 = R · c_{1}(G_{i,2} − H_{i,2}) = 0,

and this follows because G_{a,b|R} are numerically trivial (It is clear that G_{0,b|R} are trivial for b ≥ 1 and then one uses (15) and (16)). □

Example 5.8. For i = 0 hence g = 3, Theorem 5.7 reads like

[\overline{Z}_{3,0}]^{virt} = c_{1}(G_{0,2} − Sym^{2}G_{0,1}) = 9λ − δ_{0} − 3δ_{1} ≡ \overline{M}_{3,2}^{1} ∈ Pic(\overline{M}_{3}).

Thus our calculation yields a computation of the compactified divisor \overline{M}_{3,2}^{1} on \overline{M}_{3} of hyperelliptic curves. Thus we have the relation Z(γ_{0,2}) = \overline{Z}_{3,0} and the vector bundle morphism γ_{0,2} : H_{0,2} → G_{0,2} provides the ”correct” determinantal structure of the compactification of the hyperelliptic divisor. A different compactification of M_{3,2}^{1} is provided by the vector bundle morphism between Hodge bundles

χ_{3} : Sym^{2}(𝔼_{1}) → 𝔼_{2}, χ_{3}[X] : Sym^{2}H^{0}(X, ω_{X}) → H^{0}(X, ω_{X}^{⊗2})

for [X] ∈ \overline{M}_{3}. The class of its degeneration locus is c_{1}(𝔼_{2} −Sym^{2}𝔼_{1}) = 9λ−δ_{0} −δ_{1} (use Remark 4.1). It follows that there is an equality of cycles

Z(χ_{3}) = Z(γ_{0,2}) + 2δ_{1} ∈ A^{1}(\overline{M}_{3}),

that is, χ_{3} is an everywhere degenerate morphism along the divisor Δ_{1}. This discussion in low genus, already indicates that the determinantal structure induced by the morphism γ_{i,2} : H_{i,2} → G_{i,2} provides the right compactification of Z_{g,i} over M̃_{g}.

In a way analogous to [F2], one can extend the vector bundles G_{a,b} and H_{a,b} as well as the vector bundle morphism γ_{a,b} : H_{a,b} → G_{a,b} over the larger codimension 1 compactification M_{g}^{∗}, in a way that the exact sequence (15) and (16) extend to M_{g}^{∗}. Using these sequences, we can compute the class c_{1}(G_{i,2} − H_{i,2}) ∈ Pic(M_{g}^{∗}) = Pic(\overline{M}_{g}). One finds a formula

c_{1}(G_{i,2} − H_{i,2}) = 𝔞λ − 𝔟_{0}δ_{0} − ⋯ − 𝔟_{[g/2]}δ_{[g/2]},

where 𝔟_{j} ≥ 𝔟_{0} for j ≥ 1. It follows that

s([\overline{Z}_{g,i}]^{virt}) = \frac{𝔞}{𝔟_{0}} = 6 + \frac{12}{g + 1}.

This finishes the proof of Theorem 1.1 provided we can show that Z_{g,i} is an ”honest” divisor on M_{2i+3}, that is, γ_{i,2} is generically nondegenerate. This is the subject of Voisin’s theorem [V2] which gives an affirmative answer to Green’s Conjecture for generic curves of odd genus (see e.g. [GL] for more background):

Theorem 5.9. For a general curve [C] ∈ M_{2i+3} we have the vanishing

K_{i,2}(C, K_{C}) = 0.

It follows that Z_{2i+3,i} is a divisor on M_{2i+3}.

Remark 5.10. For g = 23 Theorem 5.7 shows that s(\overline{Z}_{23,10}) = s(K_{\overline{M}_{23}}) = 13/2. This implies that κ(\overline{M}_{23}) ≥ 0, in particular \overline{M}_{23} is not uniruled. A finer analysis using Brill-Noether divisors on \overline{M}_{23} proves the stronger inequality κ(\overline{M}_{23}) ≥ 2, cf. [F1].

We finish this section by briefly discussing the proof of Theorem 1.1 in even genus. This is achieved in [EH3] and it relies on the calculation of class of the Gieseker-Petri divisor on \overline{M}_{g}. We fix integers r, s ≥ 1 and set g := s(r + 1) and d := r(s + 1). Note that ρ(g, r, d) = 0 and every even genus appears in this way. A general curve [C] ∈ M_{g} has a finite number of linear series L ∈ W_{d}^{r}(C) and for each of them, the multiplication map

μ_{0}(L) : H^{0}(C, L) ⊗ H^{0}(C, K_{C} ⊗ L^{∨}) → H^{0}(C, K_{C})

is an isomorphism. We define the Gieseker-Petri locus

GP_{g,d}^{r} := {[C] ∈ M_{g} : ∃L ∈ W_{d}^{r}(C) such that μ_{0}(L) is not injective }.

The following result is proved in [F3] Theorem 1.6. The case s = 2, g = 2r + 2, which is the most important and the one used in the proof of Theorem 1.1, has been originally settled in [EH3]. The proof given in [F3] which uses the techniques of Koszul cohomology, is however substantially shorter.

Theorem 5.11. For d = rs + r and g = rs + s, the locus GP_{g,d}^{r} has at least one divisorial component. The slope of the divisorial part of its compactification \overline{GP}_{g,d}^{r} in \overline{M}_{g} is given by the formula:

s(\overline{GP}_{g,d}^{r}) = 6 + \frac{12}{g + 1} + \frac{6(s + r + 1)(rs + s − 2)(rs + s − 1)}{s(s + 1)(r + 1)(r + 2)(rs + s + 4)(rs + s + 1)}.

6. The locus of K3 sections in the moduli space

To extend Theorem 1.1 and show that \overline{M}_{g} is of general for a genus g ≤ 23, one needs to construct effective divisors D ∈ Eff(\overline{M}_{g}) of slope

s(D) < \frac{13}{2} ≤ 6 + \frac{12}{g + 1}.

One is lead to consider geometric conditions for curves [C] ∈ M_{g}, which are divisorial in moduli but are satisfied by all curves lying on K3 surfaces. Thus it makes sense to study more systematically the geometry of curves of arbitrary genus on K3 surfaces.

Let S be a K3 surface and C ⊂ S a smooth curve of genus g. We choose a linear series A ∈ W_{d}^{r}(C) with d ≤ g − 1, satisfying the following properties:

• Both linear series A ∈ W_{d}^{r}(C) and K_{C} ⊗ A^{∨} ∈ W_{2g−2−d}^{g−d+r−1}(C) are base point free.

• Both multiplication maps

H^{0}(C, A) ⊗ H^{0}(C, K_{C}) → H^{0}(C, A ⊗ K_{C})

and

H^{0}(C, A) ⊗ H^{0}(C, K_{C}^{⊗2} ⊗ A^{∨}) → H^{0}(C, K_{C}^{⊗2})

are surjective.

We recall that the Lazarsfeld bundle M_{A} on C comes from the exact sequence

0 → M_{A} → H^{0}(A) ⊗ O_{C} →^{ev_{C}} A → 0

and we set Q_{A} := M_{A}^{∨}, hence rank(Q_{A}) = r and det(Q_{A}) = A. Following an idea due to Mukai [Mu3], we show that C possesses many higher rank vector bundles with unexpectedly many global sections. These bundles are restrictions of vector bundles on S and their existence will ultimately single out the K3 locus K_{g} in M_{g}:

Theorem 6.1. Given a smooth curve C ⊂ S and A ∈ W_{d}^{r}(C) as above, there exists a rank r + 1 vector bundle E_{A} on C with det(E_{A}) = K_{C}, sitting in an exact sequence

0 ⟶ Q_{A} → E_{A} ⟶ K_{C} ⊗ A^{∨} ⟶ 0,

and satisfying the condition h^{0}(C, E_{A}) = h^{0}(C, A)+h^{0}(C, K_{C} ⊗A^{∨}) = g −d+2r +1.

Proof. Viewing A as a sheaf on S, we define the sheaf F̃_{A} through the exact sequence

0 → F̃_{A} → H^{0}(A) ⊗ O_{S} →^{ev_{S}} A → 0.

Since A is a base point free line bundle, F̃_{A} is a vector bundle on S. We consider the vector bundle Ẽ_{A} := F̃_{A}^{∨} on S, which sits in an exact sequence

(18)  0 ⟶ H^{0}(A)^{∨} ⊗ O_{S} ⟶ Ẽ_{A} ⟶ K_{C} ⊗ A^{∨} ⟶ 0.

We write down the following commutative diagram

0  0
↓  ↓
H^{0}(A) ⊗ O_{S}(−C) ⟶^{=} H^{0}(A) ⊗ O_{S}(−C)
↓  ↓
0 ⟶  F̃_{A}  ⟶  H^{0}(A) ⊗ O_{S}  ⟶ A ⟶ 0
↓  ↓  ↓^{=}
0 ⟶  M_{A}  ⟶  H^{0}(A) ⊗ O_{C}  ⟶ A ⟶ 0
↓  ↓
0  0

from which, if we set F_{A} := F̃_{A} ⊗ O_{C} and E_{A} := Ẽ_{A} ⊗ O_{C}, we obtain the exact sequence

0 ⟶ M_{A} ⊗ K_{C}^{∨} ⟶ H^{0}(A) ⊗ K_{C}^{∨} ⟶ F_{A} ⟶ M_{A} ⟶ 0

(use that Tor_{O_{S}}^{1}(M_{A}, O_{C}) = M_{A} ⊗ K_{C}^{∨}). Taking duals, we find the exact sequence

(19)  0 ⟶ Q_{A} ⟶ E_{A} ⟶ K_{C} ⊗ A^{∨} ⟶ 0.

Since S is regular, from (18) we obtain that h^{0}(S, Ẽ_{A}) = h^{0}(C, A) + h^{0}(C, K_{C} ⊗ A^{∨}) while H^{0}(S, Ẽ_{A} ⊗ O_{S}(−C)) = 0, that is,

h^{0}(S, Ẽ_{A}) ≤ h^{0}(C, E_{A}) ≤ h^{0}(C, A) + h^{0}(C, K_{C} ⊗ A^{∨}).

This shows that the sequence (19) is exact on global sections and completes the proof. □

Corollary 6.2. Let C ⊂ S and A ∈ W_{d}^{r}(C) be as above. Then the multiplication map

H^{0}(C, K_{C} ⊗ A^{∨}) ⊗ H^{0}(C, K_{C} ⊗ M_{A}) → H^{0}(C, K_{C}^{⊗2} ⊗ A^{∨} ⊗ M_{A})

is not surjective. In particular, for every base point free pencil A ∈ W_{d}^{1}(C) with d ≤ g−1, the multiplication map

Sym^{2}H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, K_{C}^{⊗2} ⊗ A^{⊗(−2)})

is not surjective.

Proof. The existence of the bundle

E_{A} ∈ Ext^{1}(K_{C} ⊗ A^{∨}, Q_{A}) = H^{0}(C, K_{C}^{⊗2} ⊗ A^{∨} ⊗ M_{A})^{∨}

satisfying h^{0}(C, E_{A}) = h^{0}(C, Q_{A}) + h^{0}(C, K_{C} ⊗ A^{∨}) implies that the coboundary map

Ext^{1}(K_{C} ⊗ A^{∨}, Q_{A}) → Hom(H^{0}(C, K_{C} ⊗ A^{∨}), H^{1}(C, Q_{A}))

given by E ↦ δ_{E}, is not injective. We finish the proof by applying Serre duality.

□

Corollary 6.3. For C ⊂ S and A ∈ W_{d}^{r}(C) as above, we have that

h^{0}(C, Q_{A} ⊗ Q_{K_{C}⊗A^{∨}}) ≥ h^{0}(C, A)h^{0}(C, K_{C} ⊗ A^{∨}) + 1.

Proof. We tensor the exact sequence

0 ⟶ M_{K_{C}⊗A^{∨}} ⟶ H^{0}(K_{C} ⊗ A^{∨}) ⊗ O_{C} → K_{C} ⊗ A^{∨} ⟶ 0

by the vector bundle M_{A} ⊗ K_{C}, then apply Corollary 6.2. The conclusion follows because by assumption H^{1}(C, K_{C}^{⊗2} ⊗ A^{∨} ⊗ M_{A}) = 0. □

Corollary 6.2 can be simplified in the case of linear series of dimension ≥ 2. For instance we have the following characterization which will be used in Section 6:

Proposition 6.4. Given C ⊂ S a Brill-Noether general curve and A ∈ W_{d}^{2}(C) a complete linear series as above, the multiplication map

Sym^{2}H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, K_{C}^{⊗2} ⊗ A^{⊗(−2)})

is not surjective.

Proof. We start by choosing points p, q ∈ C such that A ⊗ O_{C}(−p − q) ∈ W_{d−2}^{1}(C). We can write the following exact sequence

0 ⟶ O_{C}(p + q) ⟶ Q_{A} ⟶ A ⊗ O_{C}(−p − q) ⟶ 0,

which we use together with Corollary 6.3 to write the inequalities

h^{0}(C, A) h^{0}(C, K_{C} ⊗ A^{∨}) + 1 ≤ h^{0}(C, Q_{A} ⊗ Q_{K_{C}⊗A^{∨}}) ≤
≤ h^{0}(C, Q_{K_{C}⊗A^{∨}} ⊗ O_{C}(p + q)) + h^{0}(C, Q_{K_{C}⊗A^{∨}} ⊗ A ⊗ O_{C}(−p − q)).

We apply the Base point free pencil trick to note that the multiplication map

H^{0}(C, K_{C}(−p − q)) ⊗ H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, K_{C}^{⊗2} ⊗ A^{∨}(−p − q))

is surjective, hence h^{0}(C, Q_{K_{C}⊗A^{∨}}(p + q)) = h^{0}(C, K_{C} ⊗ A^{∨}). Then one must have

h^{0}(C, Q_{K_{C}⊗A^{∨}} ⊗ A(−p − q)) > 2h^{0}(C, K_{C} ⊗ A^{∨}),

which implies that the multiplication map

H^{0}(C, K_{C} ⊗ A^{∨}) ⊗ H^{0}(C, K_{C} ⊗ A^{∨}(p + q)) → H^{0}(C, K_{C}^{⊗2} ⊗ A^{⊗(−2)}(p + q))

is not surjective. Since h^{0}(C, K_{C} ⊗ A^{∨}(p + q)) = h^{0}(C, K_{C} ⊗ A^{∨}) + 1, this implies that the map

Sym^{2}H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, K_{C}^{⊗2} ⊗ A^{⊗(−2)})

is not surjective either. □

Example 6.5. As an illustration, a general curve [C] ∈ M_{21} carries a finite

number of linear series A ∈ W_{16}^{2}(C) and C ↪^{|K_{C}⊗A^{∨}|} P^{6} is an embedding for all A ∈ W_{16}^{2}(C). The locus

Z_{21} := {[C] ∈ M_{21} : ∃A ∈ W_{16}^{2}(C) with
Sym^{2}H^{0}(C, K_{C} ⊗ A^{∨}) ≇ H^{0}(C, K_{C}^{⊗2} ⊗ A^{⊗(−2)})}

contains the locus K_{21} of sections of K3 surfaces. Since

rank Sym^{2}H^{0}(C, K_{C} ⊗ A^{∨}) = rank H^{0}(C, K_{C}^{⊗2} ⊗ A^{⊗(−2)}),

clearly Z_{21} is a virtual divisor on M_{21}. In fact \overline{Z}_{21} is an ”honest” divisor on \overline{M}_{21} of slope s(\overline{Z}_{21}) < 6 + 12/22 (cf. [F3], [Kh]). Unfortunately, s(\overline{Z}_{21}) > 6.5, so one cannot conclude that \overline{M}_{21} is of general type.

To summarize, the existence of the vector bundles E_{A} shows that curves C on K3 surfaces carry line bundles of the form K_{C} ⊗A^{∨} having very special geometric properties (Corollary 6.3). The vector bundles E_{A} are produced starting from any linear series A ∈ W_{d}^{r}(C) satisfying suitable genericity condition. This leads to the construction of Koszul divisors on \overline{M}_{g} as being push-forwards of degeneracy loci defined on stacks 𝔊̃_{d}^{r} of limit linear series, cf. [F2], [F3].

6.1. Koszul divisors on \overline{M}_{g}. We can rewrite Corollary 6.2 in terms of Koszul cohomology groups. A curve [C] ∈ K_{g} enjoys the property that K_{0,2}(C, K_{C} ⊗ A^{∨}) ≠ 0 for every pencil A ∈ W_{d}^{1}(C) with d ≤ g − 1 such that K_{C} ⊗ A^{∨} is globally generated. This suggests an obvious ways of constructing geometric divisors on M_{g} which contain the K3 locus K_{g} by looking at the higher (rather than 0-th order) Koszul cohomology groups K_{i,2}(C, K_{C} ⊗ A^{∨}). From a technical point of view the simplest case is when one considers syzygies of linear series residual to a pencil of minimal degree in the case when the general curve [C] ∈ M_{g} has a finite number of such pencils. The situation when the Brill-Noether number is positive will be considered in the forthcoming paper [F4]. A special case of that new construction can be found in Section 7 of this paper.

We fix an integer i ≥ 0 and set

g := 6i + 10, d := 9i + 12, and r := 3i + 4

hence ρ(g, r, d) = 0. We consider the open substack M_{g}^{0} ⊂ M_{g} consisting of curves

[C] ∈ M_{g} such that W_{d−1}^{r}(C) = ∅ and W_{d}^{r+1}(C) = ∅. Note that for a curve [C] ∈ M_{g}^{0}, each L ∈ W_{9i+12}^{3i+4}(C) is complete and base point free. From Riemann-Roch, the residual linear series K_{C} ⊗L^{∨} ∈ W_{3i+6}^{1}(C) is a pencil of minimal degree. We would like to study the locus of curves [C] ∈ M_{g} carrying a linear series L ∈ W_{d}^{r}(C) with extra syzygies of order i. Our numerical choices for g, r and d imply that this locus is a (virtual) divisor on \overline{M}_{g}. Whenever it is a divisor, it is guaranteed to contain K_{g}. The next theorem comes from [F2]:

Theorem 6.6. There exists a partial compactification M_{g}^{0} ⊂ M̃_{g} ⊂ \overline{M}_{g} of the stack of smooth curves with codim(\overline{M}_{g} − M̃_{g}) ≥ 2, such that if

σ : 𝔊̃_{3i+6}^{1} → M̃_{g}

denotes the stack of limit linear series, then there exist vector bundles A and B of the same rank together with a vector bundle morphism ϕ_{i} : A → B over 𝔊̃_{3i+6}^{1} such that the degeneracy locus of ϕ_{i} over σ^{−1}(M_{g}^{0}) equals

Z_{g,i} := {[C, A] ∈ 𝔊_{3i+6}^{1} : K_{i,2}(C, K_{C} ⊗ A^{∨}) ≠ 0}.

The slope of the virtual class of \overline{Z}_{g,i} is equal to

s([\overline{Z}_{g,i}]^{virt}) = s(σ_{∗}c_{1}(B − A)) = \frac{3(4i + 7)(6i^{2} + 19i + 12)}{(i + 2)(12i^{2} + 31i + 18)} < 6 + \frac{12}{g + 1}.

The question of generic non-degeneracy of the morphism ϕ_{i} is addressed in [F2]. It is proved that ϕ_{i} is generically non-degenerate for i = 0, 1, 2. In particular, the locus \overline{Z}_{22,2} is an effective divisor on \overline{M}_{22} of slope s(\overline{Z}_{22,2}) = 1665/256 = 6.5032.... This barely fails to make \overline{M}_{22} of general type!

It is conjectured in [F2] that \overline{Z}_{g,i} is an actual divisor on \overline{M}_{6i+10} for all i ≥ 0. To

show that Z_{22,2} is a divisor on M_{22} (rather than the entire space M_{22}), we use that (i) the Hurwitz stack 𝔊_{12}^{1} is irreducible and (ii) one can find a smooth embedded

genus 22 curve C ↪^{𝔤_{30}^{10}} P^{10} of genus 22, such that K_{2,2}(C, 𝔤_{30}^{10}) = 0. In other words, C ⊂ P^{10} is cut out by quadrics and all the syzygies among the quadrics are linear.

Because 𝔊_{12}^{1} is irreducible, this implies that if [C] ∈ M_{22} is a general curve, then K_{2,2}(C, K_{C} ⊗ A^{∨}) = 0, for all A ∈ W_{12}^{1}(C). The irreducibility of the Hurwitz stack 𝔊_{12}^{1} makes it possible to derive information about all 𝔤_{12}^{1}’s on a general curve, even though we can only see one 𝔤_{12}^{1} at a time. This trick (which has been used again in [F3] to prove the Maximal Rank Conjecture), only works in the case ρ(g, r, d) = 0. Proving transversality statements for Koszul divisors in the case ρ(g, r, d) ≥ 1 requires different ideas.

7. The Kodaira dimension of \overline{M}_{22}

In this section we outline the calculation of the class [\overline{𝔇}_{22}] of an effective divisor on \overline{M}_{22} of slope less than 13/2. Complete details of a more general construction (of which Theorem 7.1 is a particular case) will appear in [F4]. Precisely, we shall present in [F4] a way of computing the class of all Koszul divisors on \overline{M}_{g} defined in terms of linear series 𝔤_{d}^{r} in the case ρ(g, r, d) = 1. (The case ρ(g, r, d) = 0 has been dealt with in [F3]). Specializing (g, r, d) = (22, 6, 25) we obtain our result on the Kodaira dimension of \overline{M}_{22}.

Theorem 7.1. The following locus of smooth curves of genus 22

𝔇_{22} := {[C] ∈ M_{22} : ∃L ∈ W_{25}^{6}(C) withSym^{2}H^{0}(L) → H^{0}(L^{⊗2}) not injective}

is a divisor on M_{22}. The class of its compactification on \overline{M}_{22} is given by the formula:

\overline{𝔇}_{22} ≡ 132822768(\frac{17121}{2636}λ − δ_{0} − \frac{14511}{2636}δ_{0} − ∑_{j=2}^{11} b_{j}δ_{j}),

where b_{j} > 1 for 2 ≤ j ≤ 11. It follows that s(\overline{𝔇}_{22}) = 17121/2636 = 6.49506 …, therefore \overline{M}_{22} is of general type.

We discuss the calculation of the class of \overline{D}_{22} viewed as a virtual degeneracy locus on a partial compactification of \overline{M}_{22}. The proof that \overline{D}_{22} is indeed a divisor on \overline{M}_{22}, that is, that for a general curve [C] ∈ M_{22} we have that

Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2})

is injective for all L ∈ W_{25}^{6}(C) will be presented in [F4] as part of a more general version of the Maximal Rank Conjecture (see again [F3] Theorem 1.5 for the corresponding statement when ρ(g, r, d) = 0).

The idea is to construct two tautological vector bundles over the Severi variety 𝔊_{17}^{2} of curves [C] ∈ M_{22} with a plane model 𝔤_{17}^{2} and then define the divisor 𝔇_{22} as the image of the first degeneration locus of a natural map between these bundles.

We denote by M_{22}^{p} the open substack of M_{22} consisting of curves [C] ∈ M_{22} such that W_{24}^{6}(C) = ∅ and W_{25}^{7}(C) = ∅. Standard results in Brill-Noether theory guarantee that codim(M_{22} − M_{22}^{p}, M_{22}) ≥ 2. If 𝔓𝔦𝔠_{22}^{25} denotes the Picard stack of degree 25 over M_{22}^{p}, then we consider the substack 𝔊_{25}^{6} ⊂ 𝔓𝔦𝔠_{22}^{25} parameterizing pairs [C, L] where [C] ∈ M_{22}^{p} and L ∈ W_{25}^{6}(C). We denote by

σ : 𝔊_{25}^{6} → M_{22}^{p}

the forgetful morphism. For a general [C] ∈ M_{22}^{p}, the fibre σ^{−1}([C]) = W_{25}^{6}(C) is a smooth curve and 𝔊_{25}^{6} is an irreducible stack of dimension dim 𝔊_{25}^{6} = dim M_{22} + 1.

Let π : M_{22,1}^{p} → M_{22}^{p} be the universal curve and then
p_{2} : M_{22,1}^{p} ×_{M}_{22}^{p} 𝔊_{25}^{6} → 𝔊_{25}^{6}

denotes the natural projection. If L is a Poincaré bundle over M_{22,1}^{p} ×_{M}_{22}^{p} 𝔊_{25}^{6}, then by Grauert’s Theorem E := (p_{2})_{∗}(L) and F := (p_{2})_{∗}(L^{⊗2}) are vector bundles over 𝔊_{25}^{6} with rank(E) = 7 and rank(F) = 29. There is a natural vector bundle morphism over 𝔊_{25}^{6}

ϕ : Sym^{2}(E) → F

and we denote by U_{22} ⊂ 𝔊_{25}^{6} its first degeneracy locus. We set 𝔇_{22} := σ_{∗}(U_{22}) and clearly U_{22} has expected codimension 2 inside 𝔊_{25}^{6} hence 𝔇_{22} is a virtual divisor on M_{22}^{p}.

Using Proposition 6.4, we are guaranteed that 𝔇_{22} contains the K3 locus K_{22}, in particular it is a good candidate for a divisor on \overline{M}_{22} of exceptionally small slope. We shall extend the vector bundles E and F over a partial compactification of 𝔊_{25}^{6}. We denote by Δ_{1}^{p} ⊂ Δ_{1}^{0} ⊂ \overline{M}_{g} the locus of curves [C ∪_{y} E], where E is an arbitrary elliptic curve, [C] ∈ M_{g−1} is a Brill-Noether general curve and y ∈ C is an arbitrary point. We also denote by Δ_{0}^{p} ⊂ Δ_{0}^{0} ⊂ \overline{M}_{g} the locus consisting of curves [C_{yq}] ∈ Δ_{0}^{0}, where [C, q] ∈ M_{g−1,1} is Brill-Noether general and y ∈ C is arbitrary, as well as their degenerations [C ∪_{q} E_{∞}] where E_{∞} is a rational nodal curve (that is, j(E_{∞}) = ∞). Once we set

\overline{M}_{g}^{p} := M_{g}^{p} ∪ Δ_{0}^{p} ∪ Δ_{1}^{p} ⊂ M̃_{g},

we can extend the map σ to a proper morphism σ : 𝔊̃_{25}^{6} → \overline{M}_{22}^{p} from the stack 𝔊̃_{25}^{6} of limit linear series 𝔤_{25}^{6} over the partial compactification \overline{M}_{22}^{p} of M_{22}.

Like in to [F2], [F3] or in Section 5 of this paper, we intersect the (virtual) divisor \overline{𝔇}_{22} with the test curves C^{0} ⊂ Δ_{0}^{p} and C^{1} ⊂ Δ_{1}^{p} obtained from a general pointed curve [C, q] ∈ M_{21,1} and a general elliptic curve [E, y] ∈ M_{1,1}. We explicitly describe the pull-back 2-cycles under σ of the test curves C^{0} and C^{1}:

Proposition 7.2. Fix general curves [C] ∈ M_{21} and [E, y] ∈ M_{1,1} and consider the associated test curve C^{1} ⊂ Δ_{1} ⊂ \overline{M}_{22}. Then we have the following equality of 2-cycles in 𝔊̃_{25}^{6}:

σ^{∗}(C^{1}) = X + X_{1} × X_{2} + Γ_{0} × Z_{0} + n_{1} · Z_{1} + n_{2} · Z_{2} + n_{3} · Z_{3},

where

X := {(y, L) ∈ C × W_{25}^{6}(C) : h^{0}(C, L ⊗ O_{C}(−2y)) = 6},
X_{1} := {(y, L) ∈ C × W_{25}^{6}(C) : a^{L}(y) = (0, 2, 3, 4, 5, 6, 8)},

X_{2} := \{l_{E} ∈ G_{8}^{6}(E) : a_{1}^{l_{E}}(y) ≥ 2, a_{6}^{l_{E}}(y) = 8\} ≅ P(\frac{H^{0}(O_{E}(8y))}{H^{0}(O_{E}(6y))})

Γ_{0} := {(y, A ⊗ O_{C}(y)) : y ∈ C, A ∈ W_{24}^{6}(C)}, Z_{0} = G_{7}^{6}(E) ≅ E,

Z_{1} := \{l_{E} ∈ G_{9}^{6}(E) : a_{1}^{l_{E}}(y) ≥ 3, a_{6}^{l_{E}}(y) = 9\} ≅ P(\frac{H^{0}(O_{E}(9y))}{H^{0}(O_{E}(6y))}),

Z_{2} := \{l_{E} ∈ G_{8}^{6}(E) : a_{2}^{l_{E}}(y) ≥ 3, a_{6}^{l_{E}}(y) = 8\} ≅ P(\frac{H^{0}(O_{E}(8y))}{H^{0}(O_{E}(5y))}),

Z_{3} := \{l_{E} ∈ G_{8}^{6}(E) : a^{l_{E}}(y) ≥ (0, 2, 3, 4, 5, 6, 7)\} ≅ ⋃_{z∈E} P(\frac{H^{0}(O_{E}(7y + z))}{H^{0}(O_{E}(5y + z))}),

where the constants n_{1}, n_{2} and n_{3} are explicitly known positive integers.

Remark 7.3. The constants n_{i}, 1 ≤ i ≤ 3 have the following enumerative interpretation. First n_{1} is the number of linear series L ∈ W_{25}^{6}(C) such that there exists an unspecified point y ∈ C with a^{L}(y) = (0, 2, 3, 4, 5, 6, 9). Similarly, n_{2} is the number of those L ∈ W_{25}^{6}(C) for which there exists y ∈ C with a^{L}(y) = (0, 2, 3, 4, 5, 7, 8). Finally n_{3} is the number of points y ∈ C such that there exists L ∈ W_{24}^{6}(C) which is ramified at y. If n_{0} is the number of 𝔤_{24}^{6}’s on C, then Γ_{0} consists of n_{0} disjoint copies of the curve C.

Before describing σ^{∗}(C^{0}), we set some more notation. For a general pointed curve [C, q] ∈ M_{21,1} we denote by Y the surface

Y := {(y, L) ∈ C × W_{25}^{6}(C) : h^{0}(C, L ⊗ O_{C}(−y − q)) = 6}

and by π_{1} : Y → C the first projection. Inside Y we consider two curves corresponding to 𝔤_{25}^{6}’s with a base point at q:

Γ_{1} := {(y, A ⊗ O_{C}(y)) : y ∈ C, A ∈ W_{24}^{6}(C)} and
Γ_{2} := {(y, A ⊗ O_{C}(q)) : y ∈ C, A ∈ W_{24}^{6}(C)}

intersecting transversally in n_{0} = #(W_{24}^{6}(C)) points. Note that since [C] ∈ M_{21} is Brill-Noether general, W_{24}^{6}(C) is a reduced 0-dimensional scheme consisting of n_{0} very ample (in particular, base point free) 𝔤_{24}^{6}’s. We denote by Y′ the blow-up of Y at these n_{0} points and at the points (q, B) ∈ Y where B ∈ W_{25}^{6}(C) is a linear series with the property that h^{0}(C, B ⊗ O_{C}(−8q)) ≥ 1. We denote by E_{A}, E_{B} ⊂ Y′ the exceptional divisors corresponding to (q, A ⊗ O_{C}(q)) and (q, B) respectively, by ϵ : Y′ → Y the projection and by Γ̃_{1}, Γ̃_{2} ⊂ Y′ the strict transforms of Γ_{1} and Γ_{2} respectively.

Proposition 7.4. Fix a general curve [C, q] ∈ M_{21,1} and consider the associated test curve C^{0} ⊂ Δ_{0} ⊂ \overline{M}_{22}. Then we have the following equality of 2-cycles in 𝔊̃_{25}^{6}:

σ^{∗}(C^{0}) = Y′/Γ̃_{1} ≅ Γ̃_{2},

that is, σ^{∗}(C^{0}) can be naturally identified with the surface obtained from Y′ by identifying the disjoint curves Γ̃_{1} and Γ̃_{2} over each pair (y, A) ∈ C × W_{24}^{6}(C).

Proof. We fix a point y ∈ C − \{q\}, denote by [C_{yq} := C/y ∼ q] ∈ Δ_{0}^{p} ⊂ \overline{M}_{22} and by ν : C → C_{yq} the normalization map. We describe the variety

\overline{W} _{25}^{6}(C_{yq}) ⊂ \overline{Pic}^{25}(C_{yq})

of torsion-free sheaves L on the curve C_{yq}, with deg(L) = 25 and h^{0}(C_{yq}, L) ≥ 7.

If L ∈ W_{25}^{6}(C_{yq}) ⊂ \overline{W} _{25}^{6}(C_{yq}), that is, L is a locally free sheaf, then L is completely determined by ν^{∗}(L) ∈ W_{25}^{6}(C) which has the property that

h^{0}(C, ν^{∗}L ⊗ O_{C}(−y − q)) = 6.

However, the line bundles of type A ⊗ O_{C}(y) or A ⊗ O_{C}(q) with A ∈ W_{24}^{6}(C), do not appear in this association even though they have this property. In fact, they correspond to the situation when L ∈ \overline{W} _{25}^{6}(C_{yq}) is not locally free, in which case necessarily one has that L = ν_{∗}(A), for some A ∈ W_{24}^{6}(C). Thus Y ∩ π_{1}^{−1}(y) is the partial normalization of \overline{W} _{25}^{6}(C_{yq}) at the n_{0} points of the form ν_{∗}(A) with A ∈ W_{24}^{6}(C). A special analysis is required when y = q, that is, when C_{y}^{0} degenerates to C∪_{q}E_{∞}, where E_{∞} is a rational nodal cubic. If {l_{C}, l_{E_{∞}}} ∈ σ^{−1}([C∪_{q}E_{∞}]), then an argument along the lines of Theorem 7.2 shows that ρ(l_{C}, q) ≥ 0 and ρ(l_{E_{∞}}, q) ≤ 1. Then either l_{C} has a base point at q and then the underlying line bundle of l_{C} is of type A ⊗ O_{C}(q) while l_{E_{∞}}(−18q) ∈ \overline{W} _{7}^{6}(E_{∞}), or else, a^{l_{C}} (q) = (0, 2, 3, 4, 5, 6, 8) and then l_{E_{∞}}(−17q) ∈ P(H^{0}(E_{∞}(8q))/H^{0}(E_{∞}(6q))) ≅ E_{B}, where B ∈ W_{25}^{6}(C) is the underlying line bundle of l_{C}. □

We extend the vector bundles E and F over the stack 𝔊̃_{25}^{6} of limit linear series. The proof of the following result proceeds along the lines of the proof of Proposition 3.9 in [F2]:

Proposition 7.5. There exist two vector bundles E and F defined over 𝔊̃_{25}^{6} with rank(E) = 7 and rank(F) = 29 together with a vector bundle morphism ϕ : Sym^{2}(E) → F, such that the following statements hold:

• For (C, L) ∈ 𝔊_{25}^{6}, with [C] ∈ M_{22}^{p}, we have that E(L) = H^{0}(C, L) and F(L) = H^{0}(C, L^{⊗2}).

• For t = (C ∪_{y} E, l_{C}, l_{E}) ∈ σ^{−1}(Δ_{1}^{p}), where g(C) = 21, g(E) = 1 and l_{C} = |L_{C}| is such that L_{C} ∈ W_{25}^{6}(C) has a cusp at y ∈ C, then E(t) = H^{0}(C, L_{C}) and

F(t) = H^{0}(C, L_{C}^{⊗2}(−2y)) ⊕ ℂ · u^{2},

where u ∈ H^{0}(C, L_{C}) is any section such that ord_{y}(u) = 0. If L_{C} has a base point at y, then

E(t) = H^{0}(C, L_{C}) = H^{0}(C, L_{C} ⊗ O_{C}(−y))

and the image of a natural map F(t) → H^{0}(C, L_{C}^{⊗2}) is the subspace

H^{0}(C, L_{C}^{⊗2} ⊗ O_{C}(−2y)).

• Fix t = (C_{yq} := C/y ∼ q, L) ∈ σ^{−1}(Δ_{0}^{p}), with q, y ∈ C and L ∈ \overline{W} _{25}^{6}(C_{yq}) such that h^{0}(C, ν^{∗}L ⊗ O_{C}(−y − q)) = 6, where ν : C → C_{yq} is the normalization map.

In the case when L is locally free we have that

E(t) = H^{0}(C, ν^{∗}L) and
F(t) = H^{0}(C, ν^{∗}L^{⊗2} ⊗ O_{C}(−y − q)) ⊕ ℂ · u^{2},

where u ∈ H^{0}(C, ν^{∗}L) is any section not vanishing at y and q. In the case when L is not locally free, that is, L ∈ \overline{W} _{25}^{6}(C_{yq})−W_{25}^{6}(C_{yq}), then L = ν_{∗}(A), where A ∈ W_{24}^{6}(C) and the image of the natural map F(t) → H^{0}(C, ν^{∗}L^{⊗2}) is the subspace

H^{0}(C, A^{⊗2}).

We determine the cohomology classes of the surfaces X and Y introduced in Propositions 7.2 and 7.4 respectively. Our result are expressible in terms of standard cohomology classes on Jacobians (cf. [ACGH], [F5]), which we now recall.

If [C] ∈ M_{g} is a curve satisfying the Brill-Noether theorem, we denote by P a Poincaré bundle on C × Pic^{d}(C) and by

π_{1} : C × Pic^{d}(C) → C and π_{2} : C × Pic^{d}(C) → Pic^{d}(C)

the projections. We define the cohomology class η = π_{1}^{∗}([point]) ∈ H^{2}(C×Pic^{d}(C)), and if δ_{1}, …, δ_{2g} ∈ H^{1}(C, ℤ) ≅ H^{1}(Pic^{d}(C), ℤ) is a symplectic basis, then we set

γ := − ∑_{α=1}^{g}(π_{1}^{∗}(δ_{α})π_{2}^{∗}(δ_{g+α}) − π_{1}^{∗}(δ_{g+α})π_{2}^{∗}(δ_{α})).

We have the formula c_{1}(P) = dη + γ, corresponding to the Hodge decomposition of c_{1}(P). We also record that γ^{3} = γη = 0, η^{2} = 0 and γ^{2} = −2ηπ_{2}^{∗}(θ). On W_{d}^{r}(C) we have the tautological rank r + 1 vector bundle M := (π_{2})_{∗}(P_{|C×W_{d}^{r}(C)}). The Chern numbers of M can be computed using the Harris-Tu formula. By repeatedly applying it, we get all intersection numbers on W_{d}^{r}(C) which we need:

Lemma 7.6. If [C] ∈ M_{21} is Brill-Noether general and c_{i} := c_{i}(M^{∨}) are the Chern classes of the dual of the tautological bundle on W_{17}^{2}(C), we have the following identities in H^{∗}(W_{17}^{2}(C), ℤ):

[W_{17}^{2}(C)] = \frac{θ^{18}}{73156608000}.

x_{1} · ξ = \frac{θ^{19} · ξ}{219469824000},

x_{2} · ξ = x_{3} · ξ = 0, for anyξ ∈ H^{4}(Pic^{21}(C)).

x_{1}x_{2} · ξ = \frac{θ^{20}}{1755758592000} · ξ,

x_{1}x_{3} · ξ = x_{2}x_{3} · ξ = 0, for anyξ ∈ H^{2}(Pic^{21}(C)),

x_{1}^{2} · ξ = \frac{θ^{20}}{1097349120000} · ξ,

x_{2}^{2} · ξ = −x_{1}x_{2} · ξ, x_{3}^{2} · ξ = 0, for anyξ ∈ H^{2}(Pic^{21}(C)),

x_{1}^{3} = \frac{θ^{21}}{7242504192000}, x_{2}^{3} = −\frac{t^{21}}{6584094720000},

x_{3}^{3} = x_{1}x_{2}x_{3} = \frac{θ^{21}}{36870930432000},

x_{1}^{2}x_{2} = −x_{2}^{3}, x_{1}x_{2}^{2} = x_{1}^{2}x_{3} = x_{2}x_{3}^{2} = 0, x_{1}x_{3}^{2} = x_{2}^{2}x_{3} = −x_{1}x_{2}x_{3}.

The next calculation is a particular case of [F5] Proposition 2.7:

Proposition 7.7. Let [C] ∈ M_{21} be a Brill-Noether general curve and q ∈ C a general point. If M denotes the tautological rank 3 vector bundle over W_{17}^{2}(C) and c_{i} := c_{i}(M^{∨}), then one has the following relations:

(1) [X] = π_{2}^{∗}(c_{2}) − 6ηθ + (74η + 2γ)π_{2}^{∗}(c_{1}) ∈ H^{4}(C × W_{17}^{2}(C)).

(2) [Y] = π_{2}^{∗}(c_{2}) − 2ηθ + (16η + γ)π_{2}^{∗}(c_{1}) ∈ H^{4}(C × W_{17}^{2}(C)).

Proof. By Riemann-Roch, if (y, L) ∈ X, then the line bundle

M := K_{C} ⊗ L^{∨} ⊗ O_{C}(2y) ∈ W_{17}^{2}(C)

has a cusp at y. We realize X as the degeneracy locus of a vector bundle map over C × W_{17}^{2}(C). For each pair (y, M) ∈ C × W_{17}^{2}(C), there is a natural map

H^{0}(C, M ⊗ O_{2y})^{∨} → H^{0}(C, M)^{∨}

which globalizes to a vector bundle morphism

ζ : J_{1}(P)^{∨} → π_{2}^{∗}(M)^{∨}

over C × W_{17}^{2}(C) (Note that W_{17}^{2}(C) is a smooth 3-fold). Then we have the identification X = Z_{1}(ζ) and the Thom-Porteous formula gives that

[X] = c_{2}(π_{2}^{∗}(M) − J_{1}(P^{∨})).

From the usual exact sequence over C × Pic^{17}(C)

0 ⟶ π_{1}^{∗}(K_{C}) ⊗ P ⟶ J_{1}(P) ⟶ P ⟶ 0,

we can compute the total Chern class of the jet bundle

c_{t}(J_{1}(P)^{∨})^{−1} = (∑_{j≥0}(17η + γ)^{j}) · (∑_{j≥0}(57η + γ)^{j}) = 1 − 6ηθ + 74η + 2γ,

which quickly leads to the formula for [X]. To compute [Y] we proceed in a similar way. We denote by p_{1}, p_{2} : C × C × Pic^{17}(C) → C × Pic^{17}(C) the two projections, by Δ ⊂ C × C × Pic^{17}(C) the diagonal and we set Γ_{q} := {q} × Pic^{17}(C). We introduce the rank 2 vector bundle B := (p_{1})_{∗}(p_{2}^{∗}(P) ⊗ O_{Δ+p_{2}^{∗}(Γ_{q})}) defined over C × W_{17}^{2}(C) and we note that there is a bundle morphism χ : B^{∨} → (π_{2})^{∗}(M)^{∨} such that Y = Z_{1}(χ). Since we also have that

c_{t}(B^{∨})^{−1} = (1 + (17η + γ) + (17η + γ)^{2} + ⋯)(1 − η),

we immediately obtained the desired expression for [Y]. □

The next results are simple applications of Grothendieck-Riemann-Roch for the projection morphism p_{2} : C × C × Pic^{17}(C) → C × Pic^{17}(C):

Proposition 7.8. Let [C] ∈ M_{21} and denote by p_{1}, p_{2} : C × C × Pic^{17}(C) →

C ×Pic^{17}(C) the natural projections. We denote by A_{2} the vector bundle on C ×Pic^{17}(C) with fibre at each point A_{2}(y, M) = H^{0}(C, K_{C}^{⊗2} ⊗ M^{⊗(−2)} ⊗ O_{C}(2y)). We have the following formulas:

c_{1}(A_{2}) = −4θ − 4γ − 28η andc_{2}(A_{2}) = 8θ^{2} + 104ηθ + 16γθ.

Proposition 7.9. Let [C, q] ∈ M_{21,1} be a general pointed curve an we denote by B_{2} the vector bundle on C × Pic^{17}(C) having fibre

B_{2}(y, M) = H^{0}(C, K_{C}^{⊗2} ⊗ M^{⊗(−2)} ⊗ O_{C}(y + q))

at each point (y, M) ∈ C × Pic^{17}(C). Then we have that:

c_{1}(B_{2}) = −4θ + 7η − 2γ andc_{2}(B_{2}) = 8θ^{2} − 28ηθ + 8θγ.

As a first step towards computing [\overline{𝔇}_{22}] we determine the δ_{1} coefficient in its expression:

Theorem 7.10. Let [C] ∈ M_{21} be Brill-Noether general and denote by C^{1} ⊂ Δ_{1} the associated test curve. Then

σ^{∗}(C^{1}) · c_{2}(F − Sym^{2}(E)) = 4847375988.

It follows that the coefficient of δ_{1} in the expansion of \overline{𝔇}_{22} is equal to b_{1} = 731180268.

Proof. We intersect the degeneracy locus of the map Sym^{2}(E) → F with the surface σ^{∗}(C^{1}) and use that the vector bundles E and F were defined by retaining the sections of the genus 21 aspect of each limit linear series and dropping the information coming from the elliptic curve. It follows that Z_{i}·c_{2}(F −Sym^{2}(E)) = 0 for 1 ≤ i ≤ 3 (since F and Sym^{2}(E)) are both trivial along the surfaces Z_{i}), and [X_{1} × X_{2}] · c_{2}(F − Sym^{2}(E)) = 0 (because c_{2}(F − Sym^{2}(E))_{|X_{1}×X_{2}} is in fact the pull-back of a codimension 2 class from the 1-dimensional cycle X_{1}, therefore the intersection number is 0 for dimensional reasons). We are left with estimating the contribution coming from X and write that

σ^{∗}(C^{1}) · c_{2}(F − Sym^{2}(E)) = c_{2}(F_{|X}) − c_{1}(F_{|X})c_{1}(Sym^{2}E_{|X})+
+c_{1}^{2}(Sym^{2}E_{|X}) − c_{2}(Sym^{2}E_{|X}).

We are going to compute separately each term in the right-hand-side of this expression.

The surface X appears as the first degeneracy locus of a vector bundle morphism ζ : J_{1}(P)^{∨} → π_{2}^{∗}(M)^{∨} which globalizes the maps

H^{0}(C, M ⊗ O_{2y})^{∨} → H^{0}(C, M)^{∨}

for all (y, M) ∈ C × W_{17}^{2}(C). We denote by U := Ker(ζ). In other words, U is a line bundle on X with fibre

U(y, M) = \frac{H^{1}(C, M ⊗ O_{C}(−2y))^{∨}}{H^{1}(C, M)^{∨}} = \frac{H^{0}(C, L)}{H^{0}(C, L ⊗ O_{C}(−2y))}

over a point (y, M) ∈ X. The Chern class of U can be computed from the Harris-Tu formula:

c_{1}(U)·ξ_{|X} = −c_{3}(π_{2}^{∗}(M)^{∨}−J_{1}(P)^{∨})·ξ_{|X} = −(π_{2}^{∗}(c_{3})−6ηθπ_{2}^{∗}(c_{1})+(74η+2γ)π_{2}^{∗}(c_{2}))·ξ_{|X},

for any class ξ ∈ H^{2}(C × W_{17}^{2}(C)), and

c_{1}^{2}(U) = c_{4}(π_{2}^{∗}(M)^{∨} − J_{1}(P)^{∨}) = π_{2}^{∗}(c_{3})(74η + 2γ) − 6π_{2}^{∗}(c_{2})ηθ.

If A_{3} denotes the rank 30 vector bundle on X having fibres

A_{3}(y, M) = H^{0}(C, L^{⊗2}) = H^{0}(C, K_{C}^{⊗2} ⊗ M^{⊗(−2)} ⊗ O_{C}(4y)),

then there is an injective bundle morphism U^{⊗2} ↪ A_{3}/A_{2} and we consider the quotient sheaf

G := \frac{A_{3}/A_{2}}{U^{⊗2}}

We note that since the morphism U^{⊗2} → A_{3}/A_{2} vanishes along the curve Γ_{0} corresponding to pairs (y, M) where M has a base point, G has torsion along Γ_{0}. A straightforward local analysis now shows that F_{|X} can be identified as a subsheaf of A_{3} with the kernel of the map A_{3} → G. Therefore, there is an exact sequence of vector bundles on X

0 ⟶ A_{2|X} ⟶ F_{|X} ⟶ U^{⊗2} ⟶ 0,

which over a generic point of X corresponds to the decomposition

F(y, M) = H^{0}(C, L^{⊗2} ⊗ O_{C}(−2y)) ⊕ ℂ · u^{2},

where u ∈ H^{0}(C, L) is such that ord_{y}(u) = 1 (The analysis above, shows that the sequence stays exact over Γ_{0} as well). Hence

c_{1}(F_{|X}) = c_{1}(A_{2|X}) + 2c_{1}(U)

and c_{2}(F_{|X}) = c_{2}(A_{2|X}) + 2c_{1}(A_{2|X})c_{1}(U). Furthermore, we note that the vector bundle π_{2}^{∗}(R^{1}π_{2∗}(P))_{|X}^{∨} is a subbundle of E_{|X} and we have an exact sequence

0 ⟶ π_{2}^{∗}(R^{1}π_{2∗}(P))_{|X}^{∨} ⟶ E_{|X} ⟶ U ⟶ 0

from which we find that c_{1}(E_{|X}) = −θ + π_{2}^{∗}(c_{1}) + c_{1}(U). Similarly, we have that

(20)  c_{2}(E_{|X}) = \frac{θ^{2}}{2} + π_{2}^{∗}(c_{2}) − θπ_{2}^{∗}(c_{1}) − c_{1}(U)π_{2}^{∗}(c_{1}) − θc_{1}(U).

It is elementary to check that c_{1}(Sym^{2}E_{|X}) = 8 c_{1}(E_{|X}) and that

c_{2}(Sym^{2}E_{|X}) = 27 c_{1}^{2}(E_{|X}) + 9 c_{2}(E_{|X}),

therefore we obtain that

σ^{∗}(C^{1}) · c_{2}(F − Sym^{2}(E)) = c_{2}(A_{2|X}) + c_{1}(A_{2|X})c_{1}(U^{⊗2})−
−8c_{1}(A_{2|X})c_{1}(E_{|X}) − 8c_{1}(E_{|X})c_{1}(U^{⊗2}) + 37c_{1}^{2}(E_{|X}) − 9c_{2}(E_{|X}) =

= (−120 ηθ+ \frac{17}{2}θ^{2} −16 θγ −9 π_{2}^{∗}(c_{2})+(224 η+32 γ −33 θ)π_{2}^{∗}(c_{1})+37π_{2}^{∗}(c_{1}^{2}))·[X]+

+(168 η + 24 γ − 25 θ + 49 π_{2}^{∗}(c_{1})) · c_{1}(U) + 21c_{1}^{2}(U) =

= 1754 ηθπ_{2}^{∗}(c_{2})+1386 ηπ_{2}^{∗}(c_{3})−2498 ηθπ_{2}^{∗}(c_{1}^{2})+741 ηθ^{2}π_{2}^{∗}(c_{1})−4068 ηπ_{2}^{∗}(c_{1})π_{2}^{∗}(c_{2})−

−51 ηθ^{3} + 2738 ηπ_{2}^{∗}(c_{1}^{3}),

where the last expression lives inside H^{4}(C × W_{17}^{2}(C)). Using [F5] Propositions 2.6, each term in this sum is evaluated and we find that

σ^{∗}(C^{1}) · c_{2}(F − Sym^{2}(E)) = 691 θ^{21}/1207084032000,

which implies the stated formula for b_{1}. □

Theorem 7.11. Let [C, q] ∈ M_{21,1} be a suitably general pointed curve and L ∈ W_{25}^{6}(C) a linear series with a cusp at q. Then the multiplication map

Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2})

is injective. It follows that we have the relation a − 12b_{0} + b_{1} = 0.

Proof. We consider the pencil R ⊂ \overline{M}_{g} obtained by attaching to C at the point q a pencil of plane cubics. It is well-known that R · λ = 1, R · δ_{0} = 12 and R · δ_{1} = −1, thus the relation a − 12b_{0} + b_{1} = 0 would be immediate once we show that R · c_{2}(F − Sym^{2}(E)) = 0. This follows because of the way the vector bundles E and F are defined over the boundary divisor Δ_{1}^{0} of M̃_{22}, by retaining the aspect of the limit linear series of the component of genus 21 and dropping the aspect of the elliptic component. □

Theorem 7.12. Let [C, q] ∈ M_{21,1} be a Brill-Noether general pointed curve and denote by C^{0} ⊂ Δ_{0} the associated test curve. Then

σ^{∗}(C^{0}) · c_{2}(F − Sym^{2}(E)) = 42b_{0} − b_{1} = 4847375988.

It follows that b_{0} = 132822768.

Proof. This time we look at the virtual degeneracy locus of the morphism Sym^{2}(E) → F along the surface σ^{∗}(C^{0}). The first thing to note is that the vector bundles E_{|σ^{∗}(C^{0})} and F_{|σ^{∗}(C^{0})} are both pull-backs of vector bundles on Y. For convenience we denote this vector bundles also by E and F, hence to use the notation of Proposition 7.4, E_{|σ^{∗}(C^{0})}) = ϵ^{∗}(E_{|Y}) and F_{|σ^{∗}(C^{0})} = ϵ^{∗}(F_{|Y}). We find that

σ^{∗}(C^{0}) · c_{2}(F − Sym^{2}(E)) = c_{2}(F_{|Y}) − c_{1}(F_{|Y}) · c_{1}(E_{|Y}) + c_{1}^{2}(E_{|Y}) − c_{2}(E_{|Y})

and like in the proof of Theorem 7.10, we are going to compute each term in this expression. We denote by V := Ker(χ), where χ : B^{∨} → π_{2}^{∗}(M)^{∨} is the bundle morphism on C × W_{17}^{2}(C) whose degeneracy locus is Y and which globalizes all the maps H^{0}(C, O_{y+q}(M))^{∨} → H^{0}(C, M)^{∨}. Thus the kernel bundle V is a line bundle on Y with fibre

V (y, M) = \frac{H^{0}(C, L)}{H^{0}(C, L ⊗ O_{C}(−y − q))},

over each point (y, M) ∈ Y, and where L := K_{C} ⊗ M^{∨} ⊗ O_{C}(y + q) ∈ W_{25}^{6}(C). By using again the Harris-Tu Theorem, we find the following formulas for the Chern numbers of V:

c_{1}(V)·ξ_{|Y} = −(c_{3}(π_{2}^{∗}(M)^{∨}−B^{∨})·ξ_{|Y}) = (π_{2}^{∗}(c_{3})+π_{2}^{∗}(c_{2})(16η+γ)−2π_{2}^{∗}(c_{1})ηθ)·ξ_{|Y} ,

for any class ξ ∈ H^{2}(C × W_{17}^{2}(C)), and

c_{1}^{2}(V) = c_{4}(π_{2}^{∗}(M)^{∨} − B^{∨}) = π_{2}^{∗}(c_{3})(16η + γ) − 2π_{2}^{∗}(c_{2})ηθ.

Recall that we have introduced in Proposition 7.9 the rank 28 vector bundle B_{2} over C × W_{17}^{2}(C) with fibre B_{2}(y, M) = H^{0}(C, L^{⊗2} ⊗ O_{C}(−y − q)). We claim that one has an exact sequence of bundles over Y

(21)  0 ⟶ B_{2|Y} ⟶ F_{|Y} ⟶ V^{⊗2} ⟶ 0.

If B_{3} is the rank 30 vector bundle on Y with fibres

B_{3}(y, M) = H^{0}(C, L^{⊗2}) = H^{0}(C, K_{C}^{⊗2} ⊗ M^{⊗(−2)} ⊗ O_{C}(2y + 2q)),

we have an injective morphism of sheaves V^{⊗2} ↪ B_{3}/B_{2} locally given by

v^{⊗2} ↦ v^{2} mod H^{0}(C, L^{⊗2} ⊗ O_{C}(−y − q)),

where v ∈ H^{0}(C, L) is any section not vanishing at q and y. Then F_{|Y} is canonically identified with the kernel of the projection morphism

B_{3} → \frac{B_{3}/B_{2}}{V^{⊗2}}

and the exact sequence (21) now becomes clear. Therefore c_{1}(F_{|Y}) = c_{1}(B_{2|Y}) + 2c_{1}(V) and c_{2}(F_{|Y}) = c_{2}(B_{2|Y}) + 2c_{1}(B_{2|Y})c_{1}(V). Reasoning along the lines of Theorem 7.10, we also have an exact sequence

0 ⟶ π_{2}^{∗}(R^{1}π_{2∗}(P))_{|Y}^{∨} ⟶ E_{|Y} ⟶ V ⟶ 0

and from this we obtain that

c_{1}(E_{|Y}) = −θ + π_{2}^{∗}(c_{1}) + c_{1}(V)

and

c_{2}(E_{|Y}) = \frac{θ^{2}}{2} + π_{2}^{∗}(c_{2}) − θπ_{2}^{∗}(c_{1}) − θc_{1}(V) + c_{1}(V)π_{2}^{∗}(c_{1}).

All in all, we can write the following expression for the total intersection number:

σ^{∗}(C^{0}) · c_{2}(F − Sym^{2}(E)) = c_{2}(B_{2|Y}) + c_{1}(B_{2|Y})c_{1}(V^{⊗2})−
−8c_{1}(B_{2|Y})c_{1}(E_{|Y}) − 8c_{1}(E_{|Y})c_{1}(V^{⊗2}) + 37c_{1}^{2}(E_{|Y}) − 9c_{2}(E_{|Y}) =

= (\frac{17}{2}θ^{2} + 28ηθ − 8θγ − 9π_{2}^{∗}(c_{2}) + (16γ − 33θ − 56η)π_{2}^{∗}(c_{1}) + 37π_{2}^{∗}(c_{1}^{2})) · [Y]+

+(49 π_{2}^{∗}(c_{1}) − 25 θ − 42 η + 12 γ)c_{1}(V) + 21c_{1}^{2}(V) =
= 428 ηθπ_{2}^{∗}(c_{2}) − 536 ηθπ_{2}^{∗}(c_{1}^{2}) + 168 ηθ^{2}π_{2}^{∗}(c_{1}) − 984 ηπ_{2}^{∗}(c_{1})π_{2}^{∗}(c_{2})+
+378ηπ_{2}^{∗}(c_{3}) − 17 ηθ^{3} + 592ηπ_{2}^{∗}(c_{1}^{3}),

and using once more [F5] Proposition 2.6, we get that

42b_{0} − b_{1} = \frac{509θ^{21}}{5364817920000}.

Since we already know the value of b_{1} and a − 12b_{0} + b_{1} = 0, this allows us to calculate a and b_{0}. □

End of the proof of Theorem 7.1. We write \overline{𝔇}_{22} ≡ aλ − ∑_{j=0}^{11} b_{j}δ_{j}. Since

\frac{a}{b_{0}} = \frac{17121}{2636} ≤ \frac{71}{10},

we are in a position to apply Corollary 1.2 from [FP] which gives the inequalities b_{j} ≥ b_{0} for 1 ≤ j ≤ 11, hence s(\overline{𝔇}_{22}) = a/b_{0} < 13/2. □

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Humboldt Universität zu Berlin, Institut für Mathematik, 10099 Berlin

    E-mail address: farkas@math.hu-berlin.de