Brown's dihedral moduli space and freedom of the gravity operad

TL;DR

This paper proves that the gravity cooperad's structure is cofree with cogenerators from Brown's partial compactification of moduli space, revealing new insights into the cohomology and Hodge structures of these spaces.

Contribution

It establishes the cofree property of the gravity cooperad with explicit basis constructions and links cohomology injectivity to pure Hodge structures, providing new geometric and algebraic insights.

Findings

The gravity cooperad is cofree as a nonsymmetric anticyclic cooperad.

The cohomology groups of $M_{0,n}^ ext{δ}$ inject into those of $M_{0,n}$.

$H^k(M_{0,n}^ ext{δ})$ has a pure Hodge structure of weight $2k$.

Abstract

Francis Brown introduced a partial compactification $M_{0,n}^\delta$ of the moduli space $M_{0,n}$. We prove that the gravity cooperad, given by the degree-shifted cohomologies of the spaces $M_{0,n}$, is cofree as a nonsymmetric anticyclic cooperad; moreover, the cogenerators are given by the cohomology groups of $M_{0,n}^\delta$. This says in particular that $H^\bullet(M_{0,n}^\delta)$ injects into $H^\bullet(M_{0,n})$. As part of the proof we construct an explicit diagrammatically defined basis of $H^\bullet(M_{0,n})$ which is compatible with cooperadic cocomposition, and such that a subset forms a basis of $H^\bullet(M_{0,n}^\delta)$. We show that our results are equivalent to the claim that $H^k(M_{0,n}^\delta)$ has a pure Hodge structure of weight $2k$ for all $k$, and we conclude our paper by giving an independent and completely different proof of this fact. The latter proof uses a new and explicit iterative construction of $M_{0,n}^\delta$ from $\mathbb{A}^{n-3}$ by blow-ups and removing divisors, analogous to Kapranov's and Keel's constructions of $\overline M_{0,n}$ from $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$, respectively.