# Brill-Noether theory of curves on abelian surfaces and applications

##
Margherita Lelli-Chiesa

created by risa on 09 Oct 2015

15 oct 2015
-- 14:00

Aula 211, Dip. Matematica, Università "Roma Tre", Roma

**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 show that a general curve in $

L

$ is Brill-Noether general. We will then study the locus $

L

^r_d$ of smooth curves in $

L

$ possessing a $g^r_d$ and prove that this is nonempty in some unexpected cases (with negative Brill-Noether number). 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 $M_g$. Time permitting, we will mention applications to enumerative geometry and hyperkähler manifolds. Most of this work is joint with A. L. Knutsen and G. Mongardi.