Rational cohomology of $\mathcal M_{4,1}$

TL;DR

The paper determines the rational cohomology of the moduli space ${\mathcal{M}}_{4,1}$ by applying Gorinov-Vassiliev's discriminant method to stratify genus $4$ curves with a marked point according to their ambient quadric, handling both non-singular quadrics and quadrics with cones. It computes the Hodge-Grothendieck polynomial $P_{{\mathcal{M}}_{4,1}}(t)$, verifies consistency with finite-field point counts, and uses a generalized Leray-Hirsch framework to pass from incidence varieties to quotients, ultimately obtaining the cohomology of the hyperelliptic loci and the non-hyperelliptic pieces $C_{0,1}$ and $C_{1,1}$ through intricate spectral-sequence analyses. The approach yields explicit mixed Hodge structures for the relevant cohomology groups, and establishes the first case where $H^ullet({\mathcal{M}}_{g};R^1 f_\*\mathbf{Q})$ is non-trivial for $g=4$. The results contribute rigorous cohomological descriptions and tautological-ring identifications for ${\mathcal{M}}_{4,1}$, aligned with the known polynomial counts and the broader structure of the moduli of curves with marked points.

Abstract

We compute the rational cohomology of the moduli space $\mathcal{M}_{4,1}$ of non-singular genus $4$ curves with $1$ marked point, using Gorinov-Vassiliev's method.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• proof
• Theorem 1.4
• Theorem 1.5
• proof : Proof of Theorem \ref{['thm:Main_thm']}
• Remark 1.6
• Remark 1.7
• Proposition 2.1: Gor