A note on Severi varieties of nodal curves on Enriques surfaces

Let $\mathrm{\mid }L\mathrm{\mid }$ 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 $V_{\mathrm{\mid }L\mathrm{\mid },\delta }(S)$ be the Severi variety of irreducible $\delta$-nodal curves in $\mathrm{\mid }L\mathrm{\mid }$. We denote by $\pi :X\to S$ the universal covering of $S$. In this note we compute the dimensions of the irreducible components $V$ of $V_{\mathrm{\mid }L\mathrm{\mid },\delta }(S)$. In particular we prove that, if $C$ is the curve corresponding to a general element $[C]$ of $V$, then the codimension of $V$ in $\mathrm{\mid }L\mathrm{\mid }$ is $\delta$ if ${\pi }^{-1}(C)$ is irreducible in $X$ and it is $\delta -1$ if ${\pi }^{-1}(C)$ consists of two irreducible components.