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Integral Chow rings of modular compactifications of $\mathcal{M}_{1,n\leq 6}$

Luca Battistella, Andrea Di Lorenzo

TL;DR

The paper delivers a complete integral Chow-ring computation for all modular compactifications of $\\mathcal{M}_{1,n}$ with $n\\le 6$ by exploiting the stack $\\mathcal{G}_{1,n}$ of log-canonically polarised Gorenstein genus-one curves and a two-tier stratification by tails and elliptic singularities. Through Vistoli-style patching and an explicit calculation of top Chern classes on strata, the authors produce presentations of $A^*(\\overline{\\mathcal{M}}_{1,n}(Q))$ generated by the Hodge class $\\lambda$, boundary divisors $\\tau_B$ (and $\\nu$ in codimension 2 for $n=6$), subject to Keel-type and bespoke relations. They establish Chow‑Künneth generation and cycle-map isomorphisms for all such compactifications, and they analyze Getzler’s relation integrality, showing a corrected integral form for $n=4$ while proving the integral Getzler relation fails in general. The results yield both explicit polynomial expressions for fundamental loci and a uniform framework for computing cohomology and point counts across a large family of modular compactifications, with potential applications in intersection theory and enumerative geometry of genus-one moduli spaces.

Abstract

For $n\leq 6$, we compute the integral Chow ring of every modular compactification of $\mathcal{M}_{1,n}$ parametrising only Gorenstein curves with smooth, distinct markings. These include the Deligne--Mumford, Schubert, and Smyth compactifications, and many more. They can all be excised from the stack of log-canonically polarised Gorenstein curves. The Chow ring of the latter admits a simple, combinatorial description, which we compute by patching along a natural stratification by core level. We further deduce that all these modular compactifications satisfy the Chow-Künneth generation property, that the cycle class map is an isomorphism, and for $n=4$ we study whether the Getzler's relation hold integrally.

Integral Chow rings of modular compactifications of $\mathcal{M}_{1,n\leq 6}$

TL;DR

The paper delivers a complete integral Chow-ring computation for all modular compactifications of with by exploiting the stack of log-canonically polarised Gorenstein genus-one curves and a two-tier stratification by tails and elliptic singularities. Through Vistoli-style patching and an explicit calculation of top Chern classes on strata, the authors produce presentations of generated by the Hodge class , boundary divisors (and in codimension 2 for ), subject to Keel-type and bespoke relations. They establish Chow‑Künneth generation and cycle-map isomorphisms for all such compactifications, and they analyze Getzler’s relation integrality, showing a corrected integral form for while proving the integral Getzler relation fails in general. The results yield both explicit polynomial expressions for fundamental loci and a uniform framework for computing cohomology and point counts across a large family of modular compactifications, with potential applications in intersection theory and enumerative geometry of genus-one moduli spaces.

Abstract

For , we compute the integral Chow ring of every modular compactification of parametrising only Gorenstein curves with smooth, distinct markings. These include the Deligne--Mumford, Schubert, and Smyth compactifications, and many more. They can all be excised from the stack of log-canonically polarised Gorenstein curves. The Chow ring of the latter admits a simple, combinatorial description, which we compute by patching along a natural stratification by core level. We further deduce that all these modular compactifications satisfy the Chow-Künneth generation property, that the cycle class map is an isomorphism, and for we study whether the Getzler's relation hold integrally.
Paper Structure (25 sections, 38 theorems, 54 equations, 1 figure, 1 table)

This paper contains 25 sections, 38 theorems, 54 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Strategy of proof
  3. Relation to previous work
  4. Organization of the paper
  5. Acknowledgments
  6. The stack of log-canonically polarised Gorenstein curves
  7. Gorenstein curves of genus one
  8. Levels and strata
  9. Geometry of the open substack parametrising minimal curves
  10. $n=1$
  11. $n=2$
  12. $n=3$
  13. $n=4$
  14. $n=5$
  15. $n=6$
  16. ...and 10 more sections

Key Result

Theorem 1

For $n\leq 5$, the integral Chow ring of $\overline{\mathcal{M}}_{1,n}(Q)$ is generated by $\lambda$, the first Chern class of the Hodge line bundle, and by the boundary divisors $\tau_B$, $B\subset [n]$ of cardinality $\geq 2$, parametrising curves with a rational tail marked by $B$. The ideal of r For $n=6$, we need an extra generator $\nu$ in codimension $2$, the fundamental class of a locus of

Figures (1)

  • Figure 1: The dual graph of the curve in \ref{['exa1']}.

Theorems & Definitions (85)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.6
  • Definition 2.7
  • Example 2.8
  • Lemma 2.9
  • ...and 75 more