# Brill-Noether theory of curves on abelian surfaces

##
Margherita Lelli-Chiesa

created by daniele on 09 Mar 2015

18 mar 2015
-- 14:30

Aula Riunioni, DM, Pisa

Seminari di Geometria Algebrica, Pisa

**Abstract.**

Brill-Noether theory of curves on K3 surfaces is well understood. Quite little is known for curves lying on abelian surfaces. Given a general abelian surface S with polarization L of type $(1,n)$, we will first focus on the gonality of smooth curves $C$ in the linear system $

L

$ and show that this is not constant when moving $C$ in $

L

$.
We will then study linear series of type $g^r_d$ with $r\geq 2$. We will prove that, as soon as the Brill-Noether number is negative and some other inequalities are satisfied, the locus $

L

^r_d$ of smooth curves in $

L

$ possessing a $g^r_d$ is nonempty and has the expected dimension. As an application, we obtain the existence of a component of the Brill-Noether locus $M^r_{g,d}$ having the expected codimension in the moduli space of curves $M_g$. This is a joint work with A. L. Knutsen and G. Mongardi.