Maps into projective spaces

We compute the cohomology of the Picard bundle on the desingularization of the compactified Jacobian of an irreducible nodal curve Y. We use it to compute the cohomology classes of the Brill-Noether loci in . We show that the moduli space M of morphisms of a fixed degree from Y to a projective space has a smooth compactification. As another application of the cohomology of the Picard bundle, we compute a top intersection number for the moduli space M confirming the Vafa-Intriligator formulae in the nodal case.