Prym varieties and applications

The classical definition of Prym varieties deals with the unramified covers of curves. The aim of this article is to give explicit algebraic descriptions of the Prym varieties associated with ramified double covers of algebraic curves. We make a careful study of the connection with the concept of algebraic completely integrable systems and we apply the methods to some problems such as the Hénon–Heiles system, the Kowalewski rigid body motion and Kirchhoff’s equations of motion of a solid in an ideal fluid.