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The integral chow ring of $M_2^{ct}$

Joseph Helfer, Eric Jovinelly, Eric Larson, Anda Tenie, Chengxi Wang

TL;DR

The paper determines the integral Chow ring of the moduli stack $\mathcal{M}_2^{\mathrm{ct}}$ of stable genus 2 curves of compact type by excising boundary strata from $\overline{\mathcal{M}}_2$ and computing the Chow rings of the resulting open strata with $\mathbb{Z}[1/2]$-coefficients. It combines an explicit equivariant framework with a boundary-excision strategy, using test bielliptic families and Grothendieck–Riemann–Roch to resolve pushforward ambiguities. The main result provides a concrete presentation

Abstract

This paper computes the integral Chow ring of the moduli space $M_2^{ct}$ of stable genus 2 curves of compact type. This is done by excising boundary strata from $\bar M_2$ one-by-one. During this process, we determine the Chow rings of all other open strata in $\bar M_2$ with $Z[1/2]$-coefficients.

The integral chow ring of $M_2^{ct}$

TL;DR

The paper determines the integral Chow ring of the moduli stack of stable genus 2 curves of compact type by excising boundary strata from and computing the Chow rings of the resulting open strata with -coefficients. It combines an explicit equivariant framework with a boundary-excision strategy, using test bielliptic families and Grothendieck–Riemann–Roch to resolve pushforward ambiguities. The main result provides a concrete presentation

Abstract

This paper computes the integral Chow ring of the moduli space of stable genus 2 curves of compact type. This is done by excising boundary strata from one-by-one. During this process, we determine the Chow rings of all other open strata in with -coefficients.

Paper Structure

This paper contains 27 sections, 20 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Boundary Stratification
  3. Methods
  4. Preliminaries
  5. Assumptions and conventions
  6. Equivariant Computations
  7. Common groups and classes
  8. Push pull
  9. Chow rings of M₂ and Δ₁
  10. Excising Δ₀₀₁ and Δ₀₁
  11. Quotient presentations of Δ₀₁ and Δ₀₀₁
  12. Chow ring of Δ₀₁ Δ₀₀₁
  13. Chow ring of Δ₁ Δ₀₁
  14. Chow rings of M₂ Δ₀₀₁ and M₂ Δ₀₁
  15. Bielliptic Curves
  16. ...and 12 more sections

Key Result

Theorem 1.1

Over a field of characteristic $\neq 2, 3$, we have

Figures (1)

  • Figure 1: The boundary stratification of $\overline{\mathcal{M}}_2$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: totaro-group-cohomology-and-algebraic-cycles [Theorem 5.2 and Lemma 7.1]larson-chow-m2
  • Theorem 2.5: Theorem 1.1, larson-chow-m2
  • Definition 2.6
  • Theorem 2.7: larson-chow-m2
  • Proposition 3.1
  • Proposition 4.1
  • ...and 22 more