Syzygies of torsion bundles and the geometry of the level l modular variety over M_g

Abstract

We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order 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 non-vanishing 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 R_{g,3} is a variety of general type when g>11 and the Kodaira dimension of R_{11,3} 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.

35 pages, appeared in Invent Math. We correct an inaccuracy in the statement of Prop 2.3

References in corpus (1)

• Generic Syzygy Schemes

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• The Prym-Green Conjecture for torsion bundles of high order
• Moduli of $G$-covers of curves: geometry and singularities
• A characterization of bielliptic curves via syzygy schemes
• Birational geometry of moduli of curves with an $S_3$-cover
• The resolution of paracanonical curves of odd genus
• Non-ordinary curves with a Prym variety of low $p$-rank