[Paper Review] Cohomology of moduli spaces of curves of genus three via point counts

Jonas Bergström|arXiv (Cornell University)|Nov 27, 2006

Algebraic Geometry and Number Theory6 references4 citations

TL;DR

This paper computes the $σ$-equivariant cohomology of the moduli space of stable curves of genus 3 using point counts over finite fields. By combining $σ$-equivariant counts of non-hyperelliptic genus 3 curves (via plane quartics) with known counts for lower genus and hyperelliptic loci, it determines the $σ$-equivariant Galois and Hodge structures of the cohomology of $\overline{\mathcal{M}}_{3,n}$ and $\mathcal{M}_{3,n}$ for $n \leq 5$, and extends results to $n=6,7$ via Euler characteristics of local systems.

ABSTRACT

In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of genus 3 for $n \leq 5$ (to obtain $n \leq 7$ we would need counts of ``8-pointed curves of genus 2'').

Motivation & Objective

• To determine the $σ$-equivariant Galois and Hodge structures of the cohomology of the moduli space of stable curves of genus 3.
• To compute $σ$-equivariant point counts over finite fields for the moduli space of non-hyperelliptic genus 3 curves with $n \leq 7$ marked points.
• To use these counts, combined with known counts for genus 0, 1, 2, and hyperelliptic genus 3 curves, to deduce cohomological invariants of $\overline{\mathcal{M}}_{3,n}$.
• To extend results to $n=6$ and $n=7$ using Euler characteristics of local systems on $\mathcal{M}_3$, where full polynomial point counts are not yet available.

Proposed method

• Uses the sieve method to count non-singular plane quartics passing through $n$-tuples of points over $\mathbb{F}_q$, removing singular curves step-by-step.
• Applies the Lefschetz trace formula and properties of Frobenius eigenvalues to relate point counts on elliptic curves to cohomological data.
• Employs the stratification of $\overline{\mathcal{M}}_{g,n}$ to combine point counts from lower-genus and hyperelliptic loci.
• Uses the fact that polynomial point counts over $\mathbb{F}_q$ imply determination of $\sigma$-equivariant Galois and Hodge structures via theorems of van den Bogaart–Edixhoven and Faber–van der Geer.
• Computes dimensions of linear systems of quartics and cubics via combinatorial geometry over finite fields.
• Applies Lemma 15.3 to recursively compute $\sigma$-equivariant counts for $\mathcal{M}_{1,n}$ up to $n=10$, using the trace formula and automorphism group data.

Experimental results

Research questions

• RQ1 What is the $\sigma$-equivariant Galois and Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n \leq 5$?
• RQ2 Can $\sigma$-equivariant point counts of non-hyperelliptic genus 3 curves be computed for $n \leq 7$ using finite field methods?
• RQ3 How do the cohomological invariants of $\mathcal{M}_{3,n}$ relate to those of $\mathcal{M}_{1,n}$, $\mathcal{M}_{2,n}$, and $\mathcal{H}_{3,n}$?
• RQ4 What is the role of the moduli space of plane quartics in computing the cohomology of $\mathcal{M}_{3,n}$?
• RQ5 Can the cohomology of $\mathcal{M}_{3,n}$ for $n=6,7$ be determined even when full polynomial point counts are not available?

Key findings

• The $\sigma$-equivariant Galois and Hodge structures of the cohomology of $\overline{\mathcal{M}}_{3,n}$ and $\mathcal{M}_{3,n}$ are fully determined for $n \leq 5$ using polynomial point counts.
• For $n=6$ and $n=7$, the $\sigma$-equivariant Galois Euler characteristic of certain local systems on $\mathcal{M}_3$ is computed, extending cohomological information beyond the polynomial point count regime.
• The $\sigma$-equivariant count of $\mathcal{M}_{1,n}$ for $n \leq 10$ is computed via recursive application of Lemma 15.3, consistent with Getzler’s Hodge Euler characteristic results.
• The number of non-singular plane quartics through $n$-tuples of points is computed via a sieve method that removes singular curves with one, two, or more singularities, with careful correction terms.
• For $n=8$, new phenomena arise from conic and pair-of-lines configurations, requiring additional correction terms $c$ and $d$ in the point count formula.
• The method relies on the fact that if point counts over $\mathbb{F}_q$ are polynomial in $q$, then the $\sigma$-equivariant Galois and Hodge structures of the cohomology are determined by the polynomial.