Theta-regularity of curves and Brill-Noether loci

We provide a bound on the $\mathrm{\Theta }$-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 $\mathrm{\Theta }$-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.