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

TL;DR

This paper establishes a new relationship between the Poincaré polynomials of certain moduli spaces of genus 0 curves, revealing an inversion formula involving the cohomology of a specific smooth affine scheme.

Contribution

It proves that the inverse of the ordinary generating series for the Poincaré polynomial of ,n is given by the series for a new moduli space ,n^, extending previous results on exponential series.

Findings

Inverse of the ordinary generating series for ,n's Poincare9 polynomial matches the series for ,n^.

New smooth affine scheme ,n^ relates to ,n and ,n-bar in cohomological inversion.

Results generalize known exponential series inversion to ordinary series in moduli space cohomology.

Abstract

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