Symmetries and intrinsic vs. extrinsic properties of $\overline{\mathcal{M}}_{0, n}$

TL;DR

This paper investigates the intrinsic combinatorial and geometric properties of the moduli space ar_{0,n} by analyzing the effects of the symmetric group action, revealing new insights into its cohomology, Chow ring, and connections to Hodge-Riemann relations.

Contribution

It demonstrates how the symmetric group action influences the structure of ar_{0,n} and relates degree 1 properties to geometric features, proposing a recursive structure for log concave sequences.

Findings

Most higher-degree cohomology terms reflect differences in face intersection patterns.

Degree 1 terms relate to geometric properties and satisfy recursive structures.

Sequences from Hodge-Riemann relations can be expressed via polynomials involving quantum Littlewood-Richardson coefficients.

Abstract

We consider the following question: How much of the combinatorial structure determining properties of $\overline{\mathcal{M}_{0, n}}$ is ``intrinsic'' and how much new information do we obtain from using properties specific to this space? Our approach is to study the effect of the $S_n$-action. Apart from being a natural action to consider, it is known that this action does not extend to other wonderful compactifications associated to the $A_{n - 2}$ hyperplane arrangement. We find the differences in intersection patterns of faces on associahedra and permutohedra which characterize the failure to extend to other compactifications and show that this is reflected by most terms of degree $\ge 2$ of the cohomology/Chow ring. Even from a combinatorial perspective, terms of degree 1 are more naturally related to geometric properties. In particular, imposing $S_n$-invariance implies that many of the log concave sequences obtained from degree 1 Hodge--Riemann relations (and all of them for $n \le 2000$) on the Chow ring of $\overline{\mathcal{M}_{0, n}}$ can be restricted to those with a special recursive structure. A conjectural result implies that this is true for all $n$. Elements of these sequences can be expressed as polynomials in quantum Littlewood--Richardson coefficients multiplied by terms such as partition components, factorials, and multinomial coefficients. After dividing by binomial coefficients, polynomials with these numbers as coefficients can be interepreted in terms of volumes or resultants. Finally, we find a connection between the geometry of $\overline{\mathcal{M}_{0, n}}$ and higher degree Hodge--Riemann relations of other rings via Toeplitz matrices.