Mathematical Research Letters

Volume 23 (2016)

Number 6

Theta-regularity of curves and Brill–Noether loci

Pages: 1761 – 1787



Luigi Lombardi (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Wenbo Niu (Department of Mathematical Sciences, University of Arkansas, Fayetteville, Ark., U.S.A.)


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

Accepted 25 April 2015

Published 21 February 2017