Syzygies of torsion bundles and the geometry of the level ℓ modular variety over /line{mathcal{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 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 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 PrymGreen conjecture in genus 8 and level 2.
 Publication:

Inventiones Mathematicae
 Pub Date:
 October 2013
 DOI:
 10.1007/s0022201204410
 arXiv:
 arXiv:1205.0661
 Bibcode:
 2013InMat.194...73C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Commutative Algebra
 EPrint:
 35 pages, appeared in Invent Math. We correct an inaccuracy in the statement of Prop 2.3