Towards an Enumerative Geometry of the Moduli Space of Curves

Abstract

The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduli space M _g of curves of genus g and its compactification M _g, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci in M _g in terns of these classes. We take as a model for this the enumerative geometry of the Grassmannians. Here the basic classes are the Chern classes of the tautological or universal bundle that lives over the Grassmannian, and the most basic cycles are the loci of linear spaces satisfying various Schubert conditions: the so-called Schubert cycles. However, since Harris and I have shown that for g large, M _g is not unir.ational [H-M] it is not possible to expect that M _g has a decomposition into elementary cells or that the Chow ring of M _g is as simple as that of the Grassmannian. But in the other direction, J. Harer [Ha] and P. Miller [Mi] have strong results indicating that at least the low dimensional homology groups of M _g behave nicely. Moreover, it appers that many geometrically natural cycles are all expressible in terms of a small number of basic classes.