Theta-regularity of curves and Brill-Noether loci

TL;DR

This paper establishes bounds on the Theta-regularity of curves and Brill-Noether loci in abelian varieties, extending classical Castelnuovo bounds and providing new insights into the geometry of these subvarieties.

Contribution

It introduces a new bound on Theta-regularity for curves in abelian varieties and extends results to higher-dimensional Brill-Noether loci, generalizing classical Castelnuovo bounds.

Findings

Bound on Theta-regularity in terms of degree and codimension

Castelnuovo type genus bounds for curves in abelian varieties

Bounds on Theta-regularity of Brill-Noether loci in Jacobians

Abstract

We provide a bound on the $\Theta$-regularity of an arbitrary reduced and irreducible curve embedded in a polarized abelian variety in terms of its degree and codimension. This is an "abelian" version of Gruson-Lazarsfeld-Peskine's bound on the Castelnuovo--Mumford regularity of a non-degenerate curve embedded in a projective space. As an application, we provide a Castelnuovo type bound for the genus of a curve in a (non necessarily principally) polarized abelian variety. Finally, we bound the $\Theta$-regularity of a class of higher dimensional subvarieties in Jacobian varieties, i.e. the Brill-Noether loci associated to a Petri general curve, extending earlier work of Pareschi-Popa.