Fourier Transform and Prym varieties

Abstract

Let $P$ be the Prym variety attached to an unramified double covering $\tilde{C} \rightarrow C$. Let $X=X(\tilde{\boldsymbol{C}}, C)$ be the variety of special divisors which birationally parametrizes the theta divisor in $P$. We prove that $X$ is the projectivization of the Fourier-Mukai transform of a coherent sheaf $p_*(M)$, where $M$ is an invertible sheaf on $\tilde{C}$ and $p: \tilde{C} \rightarrow P$ is the natural embedding. We apply this fact to give an algebraic proof of the following Torelli type statement proved by Smith and Varley in the complex case: under some hypothesis the variety $X$ determines the covering $\tilde{C} \rightarrow C$.