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

TL;DR

This paper investigates the algebraic and geometric properties of torsion bundles on curves and their moduli spaces, proving new results about their structure, cohomology, and birational type, with computational and probabilistic insights.

Contribution

It formulates and verifies conjectures related to the resolutions of rings associated with torsion points on Jacobians, and analyzes the birational geometry of level l moduli spaces.

Findings

Proves that R_{g,3} is of general type for g>11

Determines the Kodaira dimension of R_{11,3} is at least 19

Computes the cohomology class of certain non-vanishing loci in moduli spaces

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.