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

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.