On the symmetric products of a curve

Both authors are supported by MURST and CNR of Italy.     

A note on the Petri loci

Let ${\mathcal {M}_g}$ be the coarse moduli space of complex projective nonsingular curves of genus g. We prove that when the Brill?CNoether number ??(g, r, n) is non-negative every component of the Petri locus ${P^r_{g,n} \subset \mathcal {M}_g}$ whose general member is a curve C such that ${W^{r+1}_n(C) = \emptyset}$ , has codimension one in ${\mathcal {M}_g}$ .     

Nodal Curves with General Moduli on K3 Surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p ? 2, for 2 ≤ g = p ? δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves ? _g that associates to X the isomorphism class [C] of its normalization.     

Curves of genus g on an abelian variety of dimension g

In this paper we prove a general theorem concerning the number of translation classes of curves of genus g belonging to a fixed cohomology class in a polarized abelian variety of dimension g. For g = 2 we recover results of Göttsche and Bryan-Leung. For g = 3 we deduce explicit numbers for these classes.     

Gaussian maps of plane curves with nine singular points

We consider nonsingular curves which are the normalization of plane curves with nine ordinary singular points, viewing them as embedded in the blow-up X of the projective plane along their singular points. For a large class of such curves we show that the gaussian map relative to the canonical line bundle has corank one. The proof makes essential use of the geometry of X.