A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves

Bozlee, Sebastian; Kuo, Bob; Neff, Adrian

doi:10.2140/ant.2023.17.127

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A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves

Sebastian Bozlee, Bob Kuo and Adrian Neff

Vol. 17 (2023), No. 1, 127–163

DOI: 10.2140/ant.2023.17.127

Abstract

We classify the Deligne–Mumford stacks $ℳ$ compactifying the moduli space $ℳ_{1, n}$ of smooth $n$ -pointed curves of genus one under the condition that the points of $ℳ$ represent Gorenstein curves with distinct smooth markings. This classification uncovers new moduli spaces ${\bar{ℳ}}_{1, n} (Q)$ , which we may think of as coming from an enrichment of the notion of level used to define Smyth’s $m$ -stable spaces. Finally, we construct a cube complex of Artin stacks interpolating between the ${\bar{ℳ}}_{1, n} (Q)$ ’s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model program for ${\bar{ℳ}}_{g}$ .

Keywords

moduli of curves, tropical geometry, log geometry, Gorenstein singularities

Mathematical Subject Classification

Primary: 14D23, 14H10

Milestones

Received: 12 July 2021

Revised: 28 December 2021

Accepted: 4 March 2022

Published: 24 March 2023

Authors

Sebastian Bozlee
Department of Mathematics Tufts University Medford, MA United States

Bob Kuo
Department of Mathematics University of Colorado Boulder, CO United States

Adrian Neff
Department of Mathematics University of Colorado Boulder, CO United States

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Vol. 17, No. 1, 2023