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Prym varieties: Theory and applications. (English. Russian original) Zbl 0572.14025
Math. USSR, Izv. 23, 93-147 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 4, 785-855 (1983).
After the seminal work of Farkas and Rauch, the modern theory of Prym varieties has been developed notably by Mumford and Beauville. The present paper is at once a very readable and instructive introduction to this theory, and an exposition of the author’s main result:
The generalized Prym variety introduced by A. Beauville [Invent. Math. 41, 149-196 (1977; Zbl 0333.14013)] is isomorphic, as principally polarized abelian variety, to a sum of Jacobians of nonsingular curves if and only if the quotient curve is of one of the following types: hyperelliptic, trigonal, quasi-trigonal, plane quintic (and the Beauville pair is odd.) - As an application new components are found in the Andreotti-Mayer variety of principally polarized abelian varieties of dimension g whose theta-divisors have singular loci of dimension $$\geq g- 4$$, and a rationality criterion for conic bundles over a minimal rational surface is obtained in terms of the intermediate Jacobian.
Reviewer: H.H.Martens

##### MSC:
 14K30 Picard schemes, higher Jacobians 14H40 Jacobians, Prym varieties 14K25 Theta functions and abelian varieties 14M20 Rational and unirational varieties 14H10 Families, moduli of curves (algebraic)
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