Intersections of Q-Divisors on Kontsevich's Moduli Space $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry

TL;DR

This paper develops a recursive algorithm for computing intersection numbers of Q-divisors on Kontsevich's moduli space, enabling the calculation of enumerative invariants like characteristic numbers of rational curves in projective space.

Contribution

It introduces a recursive method to compute all top intersection products of Q-divisors and applies it to determine enumerative invariants of rational curves.

Findings

Computed generators and Picard numbers of the moduli space

Developed an algorithm for all top intersection products of Q-Divisors

Explicitly evaluated the degree of the 1-cuspidal rational locus

Abstract

The theory of 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 Q-Divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in 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.