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

TL;DR

This paper explores the structure of Severi varieties of curves on elliptic surfaces and their relation to Hurwitz spaces, providing recursive formulas and component classifications for these algebraic geometric objects.

Contribution

It offers a new description of hyperplane sections of Severi varieties on elliptic surfaces and characterizes components of Hurwitz spaces of genus one covers.

Findings

Derived a recursive-like formula for Severi degrees.

Determined components of the Hurwitz space of genus one covers.

Described components of Severi varieties in certain degree ranges.

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.