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.

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) |