Integrable Hamiltonian Systems Associated to Families of Curves and Their Bi-hamiltonian Structure

In this paper we show how there is associated an integrable Hamiltonian system to a certain set of algebraic-geometric data. Roughly speaking these data consist of a family of algebraic curves, parametrized by an aane algebraic variety B, a subalgebra C of O(B) and a polynomial '(x; y) in two variables. The phase space is constructed geometrically from the family of curves and has a natural projection onto B; the regular functions on B lead to an algebra of functions in involution and the level sets of the moment map are symmetric products of algebraic curves. While completely transparant from the geometrical point of view, a slight change of these integrable Hamiltonian systems is needed in order to explicitly realize these integrable Hamiltonian systems. Thus, we associate to the same data another integrable Hamiltonian system and show how they relate to the rst one: there is a birational map between them (which is regular in one direction) which is (in the regular direction) a morphism of integrable Hamiltonian systems. Both the Poisson structure and the functions in involution are found by performing an Euclidean division of two polynomials, so that when the data are explicitly given, all ingredients of the integrable Hamiltonian system can be easily computed from it in an explicit way. In the same spirit we also construct a large class of integrable bi-Hamiltonian systems. They depend on the extra datum of a polynomial (x; y) in two variables, which speciies a deformation of our family of curves. Our construction shows clearly how and why (certain) symmetries in the family of curves lead to a bi-Hamiltonian structure for the corresponding integrable Hamiltonian system.