Castelnuovo-Mumford Regularity and GV-Sheaves on Irregular Varieties

Abstract

Inspired by Beauville's recent construction of Ulrich sheaves on abelian surfaces, we pose the question of whether a torsion-free sheaf on a polarized smooth projective variety with Castelnuovo-Mumford regularity 1 is a GV (generic vanishing) sheaf, and present evidence that this question is governed by the positivity of cycles on generalized Brill-Noether loci. We prove that it has an affirmative answer for natural polarizations on many well-known irregular surfaces, as well as some polarizations on ruled threefolds over a curve.

Introduction revised, minor changes

References in corpus (2)

• Regularity on abelian varieties III: relationship with Generic Vanishing and applications
• Theta-regularity of curves and Brill-Noether loci