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For $n\geq 3$, let $\mathcal{M}_{0,n}$ denote the moduli space of genus 0curves with $n$ marked points, and $\overline{\mathcal{M}}_{0,n}$ its smoothcompactification. A theorem due to Ginzburg, Kapranov and Getzler states thatthe inverse of the exponential generating series for the Poincar\'e polynomialof $H^{\bullet}(\mathcal{M}_{0,n})$ is given by the corresponding series for$H^{\bullet}(\overline{\mathcal{M}}_{0,n})$. In this paper, we prove that theinverse of the ordinary generating series for the Poincar\'e polynomial of$H^{\bullet}(\mathcal{M}_{0,n})$ is given by the corresponding series for$H^{\bullet}(\mathcal{M}^{\delta}_{0,n})$, where $\mathcal{M}_{0,n}\subset\mathcal{M}^{\delta}_{0,n} \subset \overline{\mathcal{M}}_{0,n}$ is a certainsmooth affine scheme.

Inversion of series and the cohomology of the moduli spaces $\mathcal{M}^{\delta}_{0,n}$