On the cohomology of moduli spaces of stable maps to Grassmannians

Let $overline{M}_{0,n}(G(r,V), d)$ be the coarse moduli space that parametrizes stable maps of class d from n-pointed genus 0 curves to a Grassmann variety G(r,V). We provide a recursive method for the computation of the Betti numbers and the Hodge numbers of $overline{M}_{0,n}(G(r,V), d)$ for all n and d. Our method is a generalization of Getzler and Pandharipande's work "The Betti numbers of $overline{M}_{0,n}(r,d)$". First, we reduce our problem to the calculation of the Hodge numbers of the open locus $M_{0,n}(G(r,V), d)$ corresponding to maps from smooth curves. Then, we show that those can be determined by considering a suitable compactification of the space of degree d morphisms from $mathbb{P}^1$ to G(r,V), combined with previous results on the configuration space of n distinct points on $mathbb{P}^1$.