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 $L^2>0$, and let $V_{|L|,\delta }(S)$ be the Severi variety of irreducible $\delta$-nodal curves in $|L|$. We denote by $\pi :X\to S$ 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.

Final version incorporating referee's corrections. To appear in the volume dedicated to the BGMS Conference 2018 of the Springer Indam Series

Cited by in corpus (4)

• Nonemptiness of Severi varieties on Enriques surfaces
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• Nodal rational curves on Enriques surfaces of base change type
• Severi varieties on blow--ups of the symmetric square of an elliptic curve