The tautological ring of ℳ g,n is rarely Gorenstein

Canning, Samir

doi:10.2140/gt.2025.29.3905

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The tautological ring of ${}\mkern4mu\overline{\mkern-4mu\mathcal{M}\mkern-1mu}\mkern1mu{}_{g,n}$ is rarely Gorenstein

Samir Canning

Geometry & Topology 29 (2025) 3905–3919

DOI: 10.2140/gt.2025.29.3905

Abstract

We prove that the tautological rings $R^{*} (\bar{ℳ} b_{g, n})$ and ${R H}^{*} ({\bar{ℳ}}_{g, n})$ are not Gorenstein when $g \geq 2$ and $2 g + n \geq 24$ , extending results of Petersen and Tommasi in genus $2$ . The proof uses the intersection of tautological classes with nontautological bielliptic cycles. We conjecture the converse: the tautological rings should be Gorenstein when $g = 0, 1$ or $g \geq 2$ and $2 g + n < 24$ . The conjecture is known for $g = 0, 1$ by work of Keel and Petersen, and we prove several new cases of this conjecture for ${R H}^{*} ({\bar{ℳ}}_{g, n})$ when $g \geq 2$ .

Keywords

moduli of curves, tautological ring, Gorenstein conjecture, Pixton's conjecture

Mathematical Subject Classification

Primary: 14C15, 14C17

References

Publication

Received: 18 June 2024

Revised: 12 February 2025

Accepted: 12 March 2025

Published: 10 October 2025

Proposed: Sándor J Kovács

Seconded: Mark Gross, Arend Bayer

Authors

Samir Canning
Department of Mathematics ETH Zurich Zurich Switzerland

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Volume 29, issue 7 (2025)