Intersections of $\mathbb{Q}$-divisors on Kontsevich's moduli space $\overline M_{0,n}(\mathbb P^r,d)$ and enumerative geometry

The theory of $\mathbb Q$-Cartier divisors on the space of $n$-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 $\mathbb Q$-divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in $\mathbb P^r$ 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 $d$ plane curves is explicitly evaluated.