Intersections of ℚ-divisors on Kontsevich’s moduli space \overline{𝕄}_{0,𝕟}(ℙ^{𝕣},𝕕) and enumerative geometry

Abstract

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