Castelnuovo-Mumford Regularity and GV-Sheaves on Irregular Varieties

TL;DR

This paper investigates the relationship between Castelnuovo-Mumford regularity and GV-sheaves on irregular varieties, providing evidence that positivity of cycles influences this connection and confirming the hypothesis for certain surfaces and threefolds.

Contribution

It introduces a new perspective linking regularity and GV-sheaves, and proves the relationship holds for specific classes of irregular varieties.

Findings

Affirmative answer for natural polarizations on many irregular surfaces

Evidence that positivity of cycles governs the regularity-GV-sheaves relationship

Extension of results to some polarizations on ruled threefolds over a curve

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.