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Recovering the curve data from a general Prym variety. (English) Zbl 0639.14026

The author gives a new proof of the generic Torelli theorem for Prym varieties (of dimension \(\geq 16)\) by constructing original curves geometrically from the data of Prym varieties. The idea is analogous to the proof of the usual Torelli theorem for curves due to Andreotti.
Let \(\tilde C\) be a curve of genus \(2p+1\) and i be an involution on it inducing an etale covering \(\pi: \tilde C\to C\) (over \({\mathbb{C}})\), and denote by (P,\(\Xi)\) the associated Prym variety of dimensionp. The author introduces an intermediate surface \(F=\{[x-ix]+[y-iy]: x,y\in \tilde C\}\) in P. Then the projective tangent cone of F at the origin can be identified with the image of the rational map of the complete linear system on C defined by \(\omega_ C+\epsilon\) where \(\epsilon\) is the 2- torsion point in J(C) inducing \(\tilde C,\) while the latter rational map is an embedding generically for \(p\geq 7\). Now the main result by the author is that, if \(p\geq 16\), then this surface can be defined purely with the geometrical data on (P,\(\Xi)\) as \(F=\{\xi \in P; \xi +Sing(\Xi)\subset \Xi \}\). In proving this claim, the author reduces the proof to degenerate curves.
Reviewer: Y.Namikawa

MSC:

14K99 Abelian varieties and schemes
14H40 Jacobians, Prym varieties
14H10 Families, moduli of curves (algebraic)
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