The Chow rings of the moduli spaces of curves of genus 7, 8, and 9

Canning, Samir; Larson, Hannah

doi:10.1090/jag/818

Authors: Samir Canning and Hannah Larson
Journal: J. Algebraic Geom. 33 (2024), 55-116
DOI: https://doi.org/10.1090/jag/818
Published electronically: May 16, 2023
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Abstract | References | Additional Information

Abstract: The rational Chow ring of the moduli space $\mathcal {M}_g$ of curves of genus $g$ is known for $g \leq 6$. Here, we determine the rational Chow rings of $\mathcal {M}_7, \mathcal {M}_8$, and $\mathcal {M}_9$ by showing they are tautological. The key ingredient is intersection theory on Hurwitz spaces of degree $4$ and $5$ covers of $\mathbb {P}^1$ via their associated vector bundles. The main focus of this paper is a detailed geometric analysis of special tetragonal and pentagonal covers whose associated vector bundles on $\mathbb {P}^1$ are highly unbalanced, expanding upon previous work of the authors in the more balanced case. In genus $9$, we use work of Mukai to present the locus of hexagonal curves as a global quotient stack, and, using equivariant intersection theory, we show its Chow ring is generated by restrictions of tautological classes.

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Additional Information

Samir Canning
Affiliation: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
MR Author ID: 1445211
Email: samir.canning@math.ethz.ch

Hannah Larson
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
MR Author ID: 1071917
Email: hlarson@math.harvard.edu

Received by editor(s): June 26, 2021
Received by editor(s) in revised form: January 6, 2023
Published electronically: May 16, 2023
Additional Notes: During the preparation of this article, the first author was partially supported by NSF RTG grant DMS-1502651. The second author was supported by the Hertz Foundation and NSF GRFP under grant DGE-1656518.
Article copyright: © Copyright 2023 University Press, Inc.