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

Abstract

Abstract 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 ) {(1,n)} , we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system | L | {|L|} for 0 ≤ δ ≤ n - 1 = p - 2 {0\leq\delta\leq n-1=p-2} (here p is the arithmetic genus of any curve in | L | {|L|} ). We also show that a general genus g curve having as nodal model a hyperplane section of some ( 1 , n ) {(1,n)} -polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many ( 1 , n ) {(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 equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in | L | {|L|} . It turns out that a general curve in | L | {|L|} is Brill–Noether general. However, as soon as the Brill–Noether number is negative and some other inequalities are satisfied, the locus | L | d r {|L|^{r}_{d}} of smooth curves in | L | {|L|} possessing a g d r {g^{r}_{d}} 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 , d r {{\mathcal{M}}^{r}_{p,d}} having the expected codimension in the moduli space of curves ℳ p {{\mathcal{M}}_{p}} . For r = 1 {r=1} , the results are generalized to nodal curves.