The Hurwitz space of covers of an elliptic curve $E$ and the Severi variety of curves in $E\times {\mathbb{P}}^1$

Abstract

We describe the hyperplane sections of the Severi variety of curves in $E\times {\mathbb{P}}^1$ in a similar fashion to Caporaso-Harris' seminal work. From this description we almost get a recursive formula for the Severi degrees (we get the terms, but not the coefficients). As an application, we determine the components of the Hurwitz space of simply branched covers of a genus one curve. In return, we use this characterization to describe the components of the Severi variety of curves in $E\times {\mathbb{P}}^1$, in a restricted range of degrees.

With 18 figures. Comments are very welcome!

References in corpus (2)

• A note on Hurwitz schemes of covers of a positive genus curve
• Irreducibility of Hurwitz spaces