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# Mathematical Research Letters

## Volume 23 (2016)

### Number 6

### Theta-regularity of curves and Brill–Noether loci

Pages: 1761 – 1787

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n6.a9

#### Authors

#### 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.

#### 2010 Mathematics Subject Classification

14H45, 14H51, 14K12

Published 21 February 2017