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- W2952842355 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 $2Xi$ 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 ${ 2Xi } $, where $Xi $ are the symmetric theta divisors of $P$. The other restrictions form two $P_2$-orbits $O_1,O_2subset mid 2Xi 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." @default.
- W2952842355 created "2019-06-27" @default.
- W2952842355 creator A5019608861 @default.
- W2952842355 date "1993-04-14" @default.
- W2952842355 modified "2023-09-27" @default.
- W2952842355 title "Recovering of curves with involution by extended Prym data" @default.
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- W2952842355 hasPublicationYear "1993" @default.
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