Some properties of second order theta functions on Prym varieties

Abstract

Let $P\cup P^{\prime }$ be the two component Prym variety associated to an étale double cover $\tilde{C}\to C$ of a non-hyperelliptic curve of genus $g\ge 6$ and let $|2{\Xi }_0|$ and $|2{\Xi }_0^{\prime }|$ be the linear systems of second order theta divisors on $P$ and $P^{\prime }$ respectively. The component $P^{\prime }$ contains canonically the Prym curve $\tilde{C}$. We show that the base locus of the subseries of divisors containing $\tilde{C}\subset P^{\prime }$ is scheme-theoretically the curve $\tilde{C}$. We also prove canonical isomorphisms between some subseries of $|2{\Xi }_0|$ and $|2{\Xi }_0^{\prime }|$ and some subseries of second order theta divisors on the Jacobian of $C$.

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