Syzygies of torsion bundles and the geometry of the level ℓ modular variety over $\overline{\mathcal{M}}_{g}$

Abstract

ABSTRACT. We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of ℓ in its Jacobian. These statements can be viewed an analogues of Green's Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding nonvanishing locus in the moduli space R g,ℓ of twisted level ℓ curves of genus g and use this to derive results about the birational geometry of R g,ℓ . For instance, we prove that Rg,3 is a variety of general type when g > 11 and the Kodaira dimension of R11,3 is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.One way of proving that the moduli space of curves of odd genus g ≫ 0 has general type is via the divisor of curves whose canonical ring has "extra" syzygies. In this paper we apply the same philosophy to prove that, in certain cases, the moduli spaces of curves with torsion bundles-the modular varieties of the title-are also of general type. These modular varieties are natural generalizations, in higher genus, of the much studied modular curves X 1 (ℓ) := H/Γ 1 (ℓ) classifying elliptic curves together with an ℓ-torsion point in their Jacobian.To explain what we mean by "extra" syzygies, consider a finitely generated graded module M over a polynomial ring S = C[x 0 , . . . , x n ]. Such a module has a minimal free resolution of the form, called the graded Betti numbers of M , are uniquely defined; in fact b i,j is the dimension of the degree i + j component of Tor S i (M, C). We say that the resolution of M is natural if, for each j, the number b i,j−i (M ) is nonzero for at most one value of i, that is, at most one Betti number on each diagonal of the Betti diagram of M is non-zero. In an irreducible flat family of modules M λ the b i,j (M λ ) are semicontinuous, and simultaneously take on minimum values on an open set. We say that M λ has "extra" syzygies when one of the values b i,j (M λ ) is larger than this minimum. If the resolution of some M λ is natural in the sense above, then its Betti numbers have the minimum value.For example, given a curve C, a line bundle L ∈ Pic(C) and a sheaf F on C, we consider the S : Following notation introduced by Mark Green, the vector spacefor generic curves of genus g (proved in [V]), asserts that the resolution of the canonical ring of C or, equivalently, the resolution of Γ C (O C , K C ), is natural. For odd genus, it follows that the resolution is pure, which in Green's notation says Generalizing the case above, we study modules of the form Γ C (ξ, K C ⊗ η) where η and ξ are line bundles of degree 0 on a smooth curve C. As a first step, we prove:Theorem 0.1. Let C be a general curve of genus g, η ∈ Pic 0 (C)[ℓ] a torsion bundle of order ℓ ≥ 2 and ξ ∈ Pic 0 (C) a general line bundle of degree degree 0. Then the module Γ C (ξ, K C ⊗η) has nat...

Twists of Mukai bundles and the geometry of the level $3$ modular variety over $\overline {\mathcal {M}}_{8}$

“…Similarly, we also improve a divisor class found in [CEFS13]. Combining these results we can prove our Main Theorem 1.2.…”

“…The Kodaira dimension of R 11,3 is at least 19 (proved in [CEFS13]) but our theorem actually suggests that all three spaces should be of general type.…”

“…In fact it is more natural to study the question on the modular variety R g,ℓ parametrizing pairs [C, η] of smooth curves of genus g together with a nontrivial ℓ-torsion line bundle. These spaces have been constructed and compactified in [CEFS13]. On R g,ℓ we define the locus B g,ℓ = [C, η] ∈ R g,ℓ H 0 (C, E C ⊗ η) = 0 which is of codimension at most one in R g,ℓ and expected to be a divisor.…”

“…It is known (see [FL10] and [Bru16]) that R g,2 is of general type for g 14 while R g,3 is known to be of general type for g 12 (see [CEFS13]). We can use the divisor B 8,3 previously constructed to prove the following: Theorem 1.2.…”

“…The method of obtaining general type results is by constructing divisors with a divisor class in Pic Q (R g,ℓ ) satisfying certain numerical bounds. Here R g,ℓ is the modular compactification obtained by using quasi-stable level ℓ curves as described in [CEFS13]. Hence the first step is to calculate the divisor class of the closure B g,ℓ in the compactification R g,ℓ .…”