Prym varieties of double coverings of elliptic curves

Abstract

We prove the generic injectivity of the Prym map $\mathscr{P}: \mathcal{R}_{1, r} \rightarrow \mathcal{A}_{\frac{r}{2}}^\delta$ sending a double covering of an elliptic curve ramified at $r \geq 6$ points to its polarized Prym variety. For $r=6$ the map is birational and both $\mathcal{R}_{1,6}$ and $\mathcal{A}_3^\delta$ are unirational.