Towards the Intersection Theory on Hurwitz Spaces

Abstract

Moduli spaces of algebraic curves and closely related to them Hurwitz spaces, that is, spaces of meromorphic functions on the curves, arise naturally in numerous problems of algebraic geometry and mathematical physics, especially in relationship with the string theory and Gromov--Witten invariants. In particular, the classical Hurwitz problem about enumeration of topologically distinct ramified coverings of the sphere with prescribed ramification type reduces to the study of geometry and topology of these spaces. The cohomology rings of such spaces are complicated even in the simplest cases of rational curves and functions. However, the cohomology classes that are the most important from the point of view of applications (namely, the classes Poincaré dual to the strata of functions with given singularities) can be expressed in terms of relatively simple ``basic'' classes (which are, in a sense, tautological). The aim of the present paper is to identify these basic classes, to describe relations among them, and to find expressions for the strata in terms of these classes. Our approach is based on R. Thom's theory of universal polynomials of singularities, which has been extended to the case of multisingularities by the first author. Although the general Hurwitz problem still remains open, our approach allows one to achieve a significant progress in its solution, as well as in the understanding of the geometry and topology of Hurwitz spaces.