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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 $\mathsc{ℳ}$ compactifying the moduli space ${\mathsc{ℳ}}_{1,n}$ of smooth $n$-pointed curves of genus one under the condition that the points of $\mathsc{ℳ}$ represent Gorenstein curves with distinct smooth markings. This classification uncovers new moduli spaces ${\overline{\mathsc{ℳ}}}_{1,n}\left(Q\right)$, 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 ${\overline{\mathsc{ℳ}}}_{1,n}\left(Q\right)$’s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model program for ${\overline{\mathsc{ℳ}}}_{g}$.

##### Keywords
moduli of curves, tropical geometry, log geometry, Gorenstein singularities
##### Mathematical Subject Classification
Primary: 14D23, 14H10