A note on Severi varieties of nodal curves on Enriques surfaces

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 $L^ 2>0$, and let $V_{|L|, delta} (S)$ be the Severi variety of irreducible $delta$-nodal curves in $|L|$. We denote by $pi:X o S$ the universal covering of $S$. In this note we compute the dimensions of the irreducible components $V$ of $V_{|L|, delta} (S)$. 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 $delta$ if $pi^{-1}(C)$ is irreducible in $X$ and it is $delta-1$ if $pi^ {-1}(C)$ consists of two irreducible components.