The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$

TL;DR

This paper investigates the structure of the 11th rational cohomology group of the moduli space of stable curves, revealing vanishing conditions, Hodge properties, and implications for point counts over finite fields.

Contribution

It establishes vanishing and purity results for cohomology groups of moduli spaces and constructs new examples of moduli spaces with nonvanishing odd cohomology and non-tautological classes.

Findings

$H^{11}(ar{ ext{M}}_{g,n})$ vanishes unless $g=1$ and $n \\geq 11$

$H^k(ar{ ext{M}}_{g,n})$ is pure Hodge-Tate for even $k \\leq 12$

Point counts over finite fields are well approximated by polynomials in $q$

Abstract

We prove that the rational cohomology group $H^{11}(\bar{\mathcal{M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$. We show furthermore that $H^k(\bar{\mathcal{M}}_{g,n})$ is pure Hodge-Tate for all even $k \leq 12$ and deduce that $\# \bar{\mathcal{M}}_{g,n}(\mathbb{F}_q)$ is surprisingly well approximated by a polynomial in $q$. In addition, we use $H^{11}(\bar{\mathcal{M}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and non-tautological algebraic cycle classes in Chow cohomology.