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

TL;DR

This paper classifies certain compactifications of the moduli space of pointed elliptic curves using Gorenstein curves, introduces new moduli spaces, and constructs an interpolating cube complex of stacks.

Contribution

It provides a comprehensive classification of modular compactifications of elliptic curves with Gorenstein conditions and introduces a novel cube complex structure connecting these spaces.

Findings

New moduli spaces ar{\u03bc}_{1,n}(Q) identified

Cube complex of Artin stacks constructed

Connections to the log minimal model program established

Abstract

We classify the Deligne-Mumford stacks M compactifying the moduli space of smooth $n$-pointed curves of genus one under the condition that the points of M represent Gorenstein curves with distinct markings. This classification uncovers new moduli spaces $\bar{\mathcal{M}}_{1,n}(Q)$, which we may think of 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 $\bar{\mathcal{M}}_{1,n}(Q)$'s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model program for $\bar{\mathcal{M}}_g$.