Maps into projective spaces

Abstract

We compute the cohomology of the Picard bundle on the desingularization J^~d(Y) of the compactified Jacobian of an irreducible nodal curve Y. We use it to compute the cohomology classes of the Brill–Noether loci in J^~d(Y). 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.