Recovering of curves with involution by extended Prym data

Abstract

With every smooth, projective algebraic curve $\tilde{C}$ with involution $\sigma :\tilde{C}\longrightarrow \tilde{C}$ without fixed points is associated the Prym data which consists of the Prym variety $P:=(1-\sigma )J(\tilde{C})$ with principal polarization $\Xi$ such that $2\Xi$ is algebraically equivalent to the restriction on $P$ of the canonical polarization $\Theta $ of the Jacobian $J(\tilde{C})$. In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair $(\tilde{C},\sigma )$ up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We consider all symmetric theta divisors $\Theta $ of $J(\tilde{C})$ which have even multiplicity at every point of order 2 of $P$. It turns out that they form three $P_2$ orbits. The restrictions on $P$ of the divisors of one of the orbits form the orbit $\{ 2\Xi \} $, where $\Xi $ are the symmetric theta divisors of $P$. The other restrictions form two $P_2$-orbits $O_1,O_2\subset \mid 2\Xi \mid $. The extended Prym data consists of $(P,\Xi )$ together with $O_1,O_2$. We prove that it determines uniquely the pair $(\tilde{C} ,\sigma )$ up to isomorphism provided $g(\tilde{C})\geq 3$. The proof is analogous to Andreotti's proof of Torelli's theorem and uses the Gauss map for the divisors of $O_1,O_2$. The result is an analog in genus $>1$ of a classical theorem for elliptic curves.