The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves

TL;DR

This paper investigates specific motivic structures in the cohomology of moduli spaces of curves, providing new nonvanishing results and supporting conjectures about the growth of cohomology dimensions.

Contribution

It identifies the presence of motivic structures $ extsf{LS}_{12}$ and $ extsf{S}_{16}$ in the cohomology, and proves new nonvanishing results for certain moduli spaces.

Findings

Nonvanishing of middle cohomology groups for $ ext{M}_9$ and $ ext{M}_{11}$

Evidence for exponential growth of cohomology dimensions with genus

Identification of motivic structures in cohomology

Abstract

We study the appearances of $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the weight-graded compactly supported cohomology of moduli spaces of curves. As applications, we prove new nonvanishing results for the middle cohomology groups of $\mathcal{M}_9$ and $\mathcal{M}_{11}$ and give evidence to support the conjecture that the dimension fo $H^{2g + k}_c(\mathcal{M}_g)$ grows at least exponentially with $g$ for almost all $k$.