Moduli spaces of curves with polynomial point counts

2024/10/25 by Samir Canning, Canning, Samir, Hannah Larson +5 · 1 voice · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2410.19913

openalex publication_date 2024/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the number of curves of a fixed genus g over finite fields is a polynomial function of the size of the field if and only if g is at most 8. Furthermore, we determine for each positive genus g the smallest n such that the moduli space of curves of genus g with n marked points does not have polynomial point count. A key ingredient in the proofs, which is also a new result of independent interest, is the computation of the thirteenth cohomology group of the moduli spaces of stable curves of genus g with n marked points, for all g and n.

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• Getzler-Kapranov graph complex cohomology computations in weight 13

Discussions

• Really nice new paper by Canning, Larsen, Payne, and Willwacher which shows how far we've come in understanding unstable cohomology of M_g arxiv.org/abs/2410.19913 [bsky, 3 points, 0 comments]

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