Recovering of curves with involution by extended Prym data
type: Item
instance of: scholarly article
title: Recovering of curves with involution by extended Prym data
zbMATH Open document ID: 0803.14026
DOI: 10.1007/BF01459791
author: Vassil Kanev
published in: Mathematische Annalen
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/alg-geom/9304006
full work available at URL: https://eudml.org/doc/165215
review text: 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 \(\lambda (\tau) = - \theta_{01} (0, \tau)^ 4/ \theta_{10} (0, \tau)^ 4\).
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H42
Mathematics Subject Classification ID: 14K30
zbMATH DE Number: 683106
zbMATH Keywords: polarization of the Jacobian
zbMATH Keywords: Prym variety
zbMATH Keywords: Torelli theorem
zbMATH Keywords: extended Prym data
zbMATH Keywords: Gauss map
MaRDI profile type: Publication
arXiv ID: alg-geom/9304006
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description: scientific article