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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
Abstract

We classify the Deligne–Mumford stacks compactifying the moduli space 1,n of smooth n-pointed curves of genus one under the condition that the points of represent Gorenstein curves with distinct smooth markings. This classification uncovers new moduli spaces ¯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 ¯1,n(Q)’s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model program for ¯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: 18 February 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