Severi varieties and Brill-Noether theory of curves on abelian surfaces

Severi varieties and Brill-Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type .1; n/, we prove nonemptiness and regularity of the Severi variety parametrizing ı-nodal curves in the linear system jLj for 0 ı n 1 D p 2 (here p is the arithmetic genus of any curve in jLj). We also show that a general genus g curve having as nodal model a hyperplane section of some .1; n/-polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many .1; n/-polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigener-ically to a nodal curve. The rest of the paper deals with the Brill-Noether theory of curves in jLj. It turns out that a general curve in jLj is Brill-Noether general. However, as soon as the Brill-Noether number is negative and some other inequalities are satisfied, the locus jLj rd of smooth curves in jLj possessing a g dr is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill-Noether locus M p;dr having the expected codimension in the moduli space of curves M p . For r D 1, the results are generalized to nodal curves