Betti numbers of stable map spaces to Grassmannians

Let $\bar{M}_{0,n}(G(r,V), d)$ be the coarse moduli space of stable degree $d$ maps from $n$-pointed genus $0$ curves to a Grassmann variety $G(r,V)$. We provide a recursive method for the computation of the Hodge numbers and the Betti numbers of $\bar{M}_{0,n}(G(r,V), d)$ for all $n$ and $d$. Our method is a generalization of Getzler and Pandharipande's work for maps to projective spaces.