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The first higher Chow groups of $\mathcal{M}_{1,n}$ for $n\leq 4$

William C. Newman

TL;DR

The paper determines the indecomposable first higher Chow groups $\overline{\mathrm{CH}}(\mathcal{M}_{1,n},1)$ for $n\le 4$ and uses these computations to recover integral presentations of the Chow rings $\mathrm{CH}(\overline{\mathcal{M}}_{1,n})$, including explicit boundary-classes descriptions. The approach combines equivariant methods, universal separable homeomorphisms, and the motivic Künneth property to control both interior and boundary contributions via localization and stratification by stable graphs. A cohesive framework is developed to relate $\overline{\mathrm{CH}}(X,1)$ to $\mathrm{CH}(\partial X)$ and to propagate MKP-based vanishing and module-structure results across the moduli of genus 1 curves and their compactifications. The genus 0 case is handled in parallel, establishing MKP for $\mathcal{M}_{0,n}$ and related stacks and computing their Chow rings via hyperplane-complement arguments. Collectively, the results provide a detailed, integral, and computable description of indecomposable higher Chow groups and boundary phenomena in low genus, with implications for related moduli spaces and their tautological structures.

Abstract

For $n\leq 4$, we compute the indecomposible higher Chow groups $\overline{\operatorname{CH}}(\mathcal{M}_{1,n},1)$ with integer coefficients. As an application, we give new proofs of presentations of the integral Chow rings $\operatorname{CH}(\overline{\mathcal{M}}_{1,n})$ for $n\leq 4$ and determine formulas for the classes of boundary strata in these rings.

The first higher Chow groups of $\mathcal{M}_{1,n}$ for $n\leq 4$

TL;DR

The paper determines the indecomposable first higher Chow groups for and uses these computations to recover integral presentations of the Chow rings , including explicit boundary-classes descriptions. The approach combines equivariant methods, universal separable homeomorphisms, and the motivic Künneth property to control both interior and boundary contributions via localization and stratification by stable graphs. A cohesive framework is developed to relate to and to propagate MKP-based vanishing and module-structure results across the moduli of genus 1 curves and their compactifications. The genus 0 case is handled in parallel, establishing MKP for and related stacks and computing their Chow rings via hyperplane-complement arguments. Collectively, the results provide a detailed, integral, and computable description of indecomposable higher Chow groups and boundary phenomena in low genus, with implications for related moduli spaces and their tautological structures.

Abstract

For , we compute the indecomposible higher Chow groups with integer coefficients. As an application, we give new proofs of presentations of the integral Chow rings for and determine formulas for the classes of boundary strata in these rings.

Paper Structure

This paper contains 28 sections, 81 theorems, 669 equations.

Table of Contents

  1. Introduction
  2. Indecomposable Higher Chow Groups
  3. Higher Chow Groups
  4. Indecomposable Higher Chow
  5. Universal Separable Homeomorphisms
  6. The Motivic Künneth Property
  7. Comparison With $\ell$-adic Higher Chow Groups
  8. Equivariant Calculations
  9. $\mathcal{M}_{1,1}$
  10. $\mathcal{M}_{1,2}$
  11. $\mathcal{M}_{1,2}^0$
  12. $\mathcal{M}_{1,3}^0$
  13. $\mathcal{M}_{1,4}^0$
  14. Toward $\overline\mathop{\mathrm{CH}}\nolimits(\mathcal{M}_{1,3},1)$ and $\overline\mathop{\mathrm{CH}}\nolimits(\mathcal{M}_{1,4},1)$
  15. $\mathcal{M}\hbox{$\overline{\space}$}_{1,1}$
  16. ...and 13 more sections

Key Result

Theorem 1.1

For $n\leq 4$, one has $\overline\mathop{\mathrm{CH}}\nolimits^1(\mathcal{M}_{1,n},1)=0$, and for $i\geq 2$, we have

Theorems & Definitions (181)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 171 more