ON THE USE OF PARAMETER AND MODULI SPACES IN CURVE COUNTING

In order to solve problems in enumerative algebraic geometry, one works with various kinds of parameter or moduli spaces: Chow varieties, Hilbert schemes, Kontsevich spaces. In this note we give examples of such spaces. In particular we consider the case where the objects to be parametrized are algebraic curves lying on a given variety. The classical problem of enumerating curves of a given type and satisfying certain given conditions has recently received new attention in connection with string theories in theoretical physics. This interest has led to much new work—on the one hand, within the framework of more traditional algebraic geometry, on the other hand, with rather surprising results, using newmethods and ideas, such as the theory of quantum cohomology and generating functions.