Recovering of curves with involution by extended Prym data
With every smooth, projective algebraic curve \(\widetilde C\) with involution \(\sigma : \widetilde C \to \widetilde C\) without fixed points is associated the Prym data which consists of the Prym variety \(P : = (1 - \sigma) J (\widetilde C)\) with principal polarization \(\Xi\) such that \(2 \Xi\) is algebraically equivalent to the restriction on \(P\) of the canonical polarization \(\Theta\) of the Jacobian \(J (\widetilde C)\). In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair \((\widetilde C, \sigma)\) up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We consider all symmetric theta divisors \(\Theta\) of \(J (\widetilde C)\) 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 \(O_ 1,O_ 2 \subset | 2 \Xi |\). The extended Prym data consists of \((P, \Xi)\) together with \(O_ 1,O_ 2\). We prove that it determines uniquely the pair \((\widetilde C, \sigma)\) up to isomorphism provided \(g (\widetilde C) \geq 3\). The proof is similar to Andreotti's proof of Torelli's theorem and uses the Gauss map for the divisors of \(O_ 1,O_ 2\). Separate treatment is necessary in the hyperelliptic case, the bi-elliptic case and the case of \(g (\widetilde C) = 5\). The result is an analogue in genus \(>1\) of the following classically known fact for elliptic curves: Any pair \((E = \mathbb{C}/ \mathbb{Z} \tau + \mathbb{Z}\), \(\mu = {1 \over 2} \tau + {1 \over 2} \pmod {\mathbb{Z} \tau + \mathbb{Z}})\) is determined uniquely up to isomorphism by \(k(\tau) = \lambda (\tau) + 1/ \lambda (\tau)\) where \(\lambda (\tau) = - \theta_{01} (0, \tau)^ 4/ \theta_{10} (0, \tau)^ 4\).
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