Sur le problème de Torelli pour les variétés de Prym. (On the Torelli problem for Prym varieties). (French) Zbl 0699.14052

The Prym map associates to every étale double covering \(\pi\) of some curve of genus g a principally polarized abelian variety P, the Prym variety of \(\pi\). The Torelli problem for Prym varieties is to determine the preimages of the Prym map. Differently from the usual Torelli problem for curves the Prym map is not always injective. However it is shown by V. I. Kanev [Math. USSR, Izv. 20, 235-257 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.2, 244-268 (1982; Zbl 0566.14014)] for \(g\geq 9\) and R. Friedman and R. Smith [Invent. Math. 67, 473-490 (1982; Zbl 0506.14042)] for \(g\leq 7\) that a double covering of a general curve of genus g is determined by its Prym variety. A constructive proof of this result for \(g\geq 17\) was given by G. E. Welters [Am. J. Math. 109, 165-182 (1987; Zbl 0639.14026)]. The present paper gives a constructive proof for \(g\geq 8.\)
The proof is analogous to Green’s proof of the usual Torelli theorem for non-hyperelliptic and non-trigonal curves of genus \(\geq 7:\) Let P be a general Prym variety of dimension \(g-1\) and \(\Theta\) its theta divisor. The tangent cones of \(\Theta\) at singular points x of multiplicity 2 are quadrics of rank \(\leq 6\) in the projectivized tangent space of P at x which contain the base curve \(C\) of the double covering. The first step in the proof consists in showing that for a general covering these quadrics generate the vector space of all quadrics containing C. - The second step, analogue to Petri’s theorem is to show that the ideal of the curve C is generated by quadrics. As a byproduct one gets that generically the singular locus of \(\Theta\) is not empty of dimension \(g- 7\) for \(g\geq 7\), reduced for \(g\equiv 7\), and irreducible and reduced for \(g\geq 8\) with cohomology class \(16\frac{1}{6!}[\Theta]^ 6\).
Reviewer: H.Lange


14K10 Algebraic moduli of abelian varieties, classification
14K30 Picard schemes, higher Jacobians
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