Intersections of and enumerative geometry

The theory of -Cartier divisors on the space of -pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of -divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree plane curves is explicitly evaluated.