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 $V_{|L|, δ} (S)$ be the Severi variety of irreducible $δ$-nodal curves in $|L|$. We denote by $π:X\to S$ 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.

Figures (2)

• Figure 1: $\eta _{\tilde{C}} = \nu_C^*(\eta_C) \neq 0$
• Figure 2: $\eta _{\tilde{C}} =\nu_C^*(\eta_C) = 0$

Theorems & Definitions (3)

• proposition 1
• proof
• corollary 1