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 \delta-nodal curves in the linear system |L| for 0<=\delta<= n-1=p-2 (here p is the arithmetic genus of any curve in |L|). 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 equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in |L|. It turns out that a general curve in |L| is Brill–Noether general. However, 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 a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus M^r _{p,d} having the expected codimension in the moduli space of curves M_p. For r=1, the results are generalized to nodal curves.
Severi varieties and Brill-Noether theory of curves on abelian surfaces
LELLI-CHIESA, MARGHERITA;
2016-01-01
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), we prove nonemptiness and regularity of the Severi variety parametrizing \delta-nodal curves in the linear system |L| for 0<=\delta<= n-1=p-2 (here p is the arithmetic genus of any curve in |L|). 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 equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in |L|. It turns out that a general curve in |L| is Brill–Noether general. However, 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 a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus M^r _{p,d} having the expected codimension in the moduli space of curves M_p. For r=1, the results are generalized to nodal curves.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.