THE EFFECTIVE CONE OF \overline{M}_{g} AND THE SLOPE CONJECTURE

GAVRIL FARKAS

ABSTRACT: This is an shortened written version of my talk at the Mathematische Arbeitstagung 2003, reporting on joint work with M. Popa and mostly contained in the papers [FP1] and [FP2].

We denote by \overline{M}_{g} the Deligne-Mumford moduli space of stable curves of genus g. A fundamental problem which goes back to Mumford is to describe the cone Eff(\overline{M}_{g}) of effective divisors on \overline{M}_{g}. Knowledge of Eff(\overline{M}_{g}) will have wide ranging applications to the birational geometry of \overline{M}_{g}, in particular giving a description of all rational maps from \overline{M}_{g} to other projective varieties.

We start by recalling a few facts about \overline{M}_{g}. The boundary \overline{M}_{g} − M_{g} corresponding to singular stable curves, breaks up into a union of irreducible divisors Δ_{0} ∪ … ∪ Δ_{[g/2]}. We denote by δ_{i} := [Δ_{i}] ∈ Pic(\overline{M}_{g}) the associated class in the Picard group of the moduli stack, and by λ the Hodge class. It is known that for g ≥ 3, the group Pic(\overline{M}_{g}) is freely generated by the classes λ, δ_{0}, …, δ_{[g/2]}.

If D ∈ Eff(\overline{M}_{g}) is such that Δ_{i} is not contained in supp(D) for i = 0, …, [g/2], it is easy to show that we can write

D ≡ aλ − b_{0}δ_{0} − ⋯ − b_{[g/2]}δ_{[g/2]}, with a, b_{i} ≥ 0.

For such a divisor D one defines the slope by the formula

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

Since the class λ is big and nef on \overline{M}_{g}, it is very easy to write down effective divisors on \overline{M}_{g} of large slope. One is particularly interested in effective divisors of small slope and in particular the number inf_{D∈Eff(\overline{M}_{g})}s(D) is a very subtle invariant of \overline{M}_{g} encoding a lot of informations on the nature of \overline{M}_{g} as an algebraic variety. We have the following conjecture due to J. Harris and I. Morrison (cf. [HMo]):

Conjecture 0.1. For every effective divisor D on \overline{M}_{g} we have the inequality s(D) ≥ 6 + 12/(g + 1), with equality if and only if g + 1 is composite and D is essentially a combination of Brill-Noether divisors.

We recall that the Brill-Noether divisors on \overline{M}_{g} for g + 1 composite are defined as loci M_{g,d}^{r} := {[C] ∈ M_{g} : C has a 𝔤_{d}^{r}}, for all positive integers r and d such that

ρ(g, r, d) = g − (r + 1)(g − d + r) = −1.

Here 𝔤_{d}^{r}, as usual, is a linear system of degree d and projective dimension r.

In such a case M_{g,d}^{r} is an irreducible divisor, and the class of its compactification has been computed by Eisenbud and Harris (cf. [EH]):

\overline{M}_{g,d}^{r} = c((g + 3)λ − \frac{g + 1}{6}δ_{0} − ∑_{i≥1} i(g − i)δ_{i}).

Thus the Slope Conjecture would single out the Brill-Noether divisors as those minimizing the slope function. The Slope Conjecture holds for g ≤ 9 and g = 11. The main evidence for it comes from the large number of divisor class calculations on \overline{M}_{g} where in each case it turns out that the slope exceeds 6 + 12/(g + 1) which is the slope of the Brill-Noether divisors.

If true, the Slope Conjecture would imply that the Kodaira dimension of M_{g} is −∞ for g ≤ 22. This is known at the moment only for g ≤ 16. Recall that for g ≥ 24 we have the celebrated result of Harris, Mumford and Eisenbud stating that M_{g} is of general type (cf. [HM]), while for the intermediate case g = 23 we know that κ(M_{23}) ≥ 2 (cf. [Fa]). The Slope Conjecture would also provide myriads of new Schottky relations coming from Siegel modular forms of small slope.

It turns out that the Slope Conjecture is intimately related to curves sitting on K3 surfaces. If we denote by K_{g} := {[C] ∈ M_{g} : C sits on a K3 surface} we have the following:

Proposition 0.2. For every D ∈ Eff(\overline{M}_{g}) with s(D) < 6 + 12/(g + 1), we must have K_{g} ⊂ D.

Due to work of Mukai it is known that dim(K_{g}) = 19 + g for g ≥ 13. The naive dimension count breaks down for g = 10, where because of the existence of a homogeneous variety associated to the exceptional Lie group G_{2} we obtain that K = K_{10} is a divisor on M_{10}.

We prove the following result:

Theorem 0.3. The class of the divisor \overline{K} in Pic(\overline{M}_{10}) is given by

\overline{K} ≡ 7λ − δ_{0} − 5δ_{1} − 9δ_{2} − 12δ_{3} − 14δ_{4} − 15δ_{5},

where λ is the class of the Hodge bundle and δ_{0}, …, δ_{5} are the boundary classes corresponding to singular curves.

We compute the class of \overline{K} by reinterpreting K3 sections in a way that makes no reference to K3 surfaces:

Theorem 0.4. The divisor K has four incarnations as a geometric subvariety of M_{10}:

(1) (By definition) The locus of curves sitting on a K3 surface.

(2) The locus of curves C with a non-surjective Wahl map ψ_{K} : ∧^{2}H^{0}(K_{C}) → H^{0}(3K_{C}).

(3) The locus of curves C carrying a stable rank two vector bundle E with ∧^{2}(E) = K_{C} and h^{0}(E) ≥ 7.

(4) The locus of genus 10 curves sitting on a quadric in an embedding C ⊂ P^{4} with deg(C) = 12.

The equivalence of descriptions (1) and (2) was established before by Cukierman and Ulmer (cf. [CU]). Description (3) reveals the true nature of the divisor K as a higher rank Brill-Noether divisor. Moreover, characterizations (3) and (4) are generalizable to other genera to provide new counterexamples to the Slope Conjecture. For instance we can show that on M_{13} the locus D consisting of genus 13 curves C which lie on a quadric in an embedding in P^{5} with deg(C) = 16, is a divisor which also violates the Slope Conjecture.

There is a striking similarity between the class of \overline{K} and the class of the Brill-Noether divisors on \overline{M}_{11}. On this space there are two distinct Brill-Noether divisors, namely \overline{M}_{11,6}^{1} and \overline{M}_{11,9}^{2}, which both have classes proportional to the ℚ-class

BN = 7λ − δ_{0} − 5δ_{1} − 9δ_{2} − 12δ_{3} − 14δ_{4} − 15δ_{5}.

This similarity, especially surprising since K behaves very differently from all Brill-Noether divisors, might suggest the existence of a correspondence between genus 10 and genus 11 curves, that would associate to a K3-section of genus 10 a genus 11 curve with a 𝔤_{6}^{1}.

We also studied divisors on \overline{M}_{g} having minimal slope 6+12/(g+1) (for those g for which the original Slope Conjecture holds on M_{g}. We have the following:

Theorem 0.5. The Iitaka dimension of the Brill-Noether linear system on \overline{M}_{11} is equal to 19.

This seems to contradict the hypothesis formulated in [HMo] that the Brill-Noether divisors are essentially the only effective divisors on \overline{M}_{g} having slope 6 + 12/(g + 1).

Although the Slope Conjecture fails, a weakened version appears to be true:

Conjecture 0.6. There exists a constant ϵ > 0 such that that s(D) ≥ ϵ, for every D ∈ Eff(\overline{M}_{g}) and for every g.

If true this would highlight a fundamental difference between M_{g} and the moduli space of abelian varieties A_{g} on which there are Siegel modular forms (leading to effective divisors on A_{g}) and having arbitrarily small slope. We also believe that the following should hold:

Conjecture 0.7. For every g there exists ϵ_{g} > 0 such that for all D ∈ Eff(\overline{M}_{g}) with a/b_{0} ≤ ϵ_{g}, we have that b_{i} ≥ b_{0}, hence s(D) = a/b_{0}.

We proved that this holds on \overline{M}_{g} for all g ≤ 23 (cf. [FP1]).

References

[CU] F. Cukierman and D. Ulmer, Curves of genus 10 on K3 surfaces, Compositio Math. 89 (1993), 81–90.

[EH] D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ 23, Invent. Math. 90 (1987), 359-387.

[Fa] G. Farkas, The geometry of the moduli space of curves of genus 23, Math. Ann. 318 (2000), 43-65.

[FP1] G. Farkas and M. Popa, Effective divisors on \overline{M}_{g} and a counterexample to the Slope Conjecture, math.AG/0209171.

[FP2] G. Farkas and M. Popa, The geometry of the divisor of K3 sections, math.AG/0305112.

[HM] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23-86.

[HMo] J. Harris and I. Morrison, Slopes of effective divisors on the moduli space of stable curves, Invent. Math. 99 (1990), 321–355.

Department of Mathematics, University of Michigan, 525 East University, Ann Arbor, MI 48109

      E-mail address: gfarkas@umich.edu