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arXiv · 1205.0661

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

Alessandro Chiodo·David Eisenbud·Gavril Farkas·Frank-Olaf Schreyer
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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.

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Alessandro Chiodo, David Eisenbud, Gavril Farkas, Frank-Olaf Schreyer. 2014-02-24. Syzygies of torsion bundles and the geometry of the level l modular variety over M_g. https://doi.org/10.1007/s00222-012-0441-0

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