A note on Severi varieties of nodal curves on Enriques surfaces

Abstract

Let |L| be a linear system on a smooth complex Enriques surface S whose general member is a smooth and irreducible curve of genus p, with L2 > 0, and let V|L|,δ(S) be the Severi variety of irreducible δ-nodal curves in |L|. We denote by π : X → S the universal covering of S. In this note we compute the dimensions of the irreducible components V of V|L|,δ(S). In particular we prove that, if C is the curve corresponding to a general element [C] of V , then the codimension of V in |L| is δ if π−1(C) is irreducible in X and it is δ − 1 if π−1(C) consists of two irreducible components.