Prym Varieties I

Abstract

This chapter presents the first steps in a purely algebraic version (in all characteristics except two) of the Riemann–Prym–Wirtinger–Schottky–Jung theory of double coverings of one curve (or compact Riemann surface) over another. It also tries to incorporate some of the interesting generalizations of this theory in the thesis of Fay. The basic idea is this: π: ▪→C is a double covering, where C and ▪ are nonsingular complete curves with Jacobians J and ▪. The involution ι : ▪→C interchanging sheets extends to ι : ▪→J, and up to some points of order two, ▪ splits into an even part J and an odd part P, the Prym variety. The Prym P has a natural polarization on it, but only in two cases—where π has zero or two branch points—there is a unique principal polarization on P and, hence, a theta divisor Ξ ⊂ P.