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

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 ${\bar{\mathcal{M}}}_g$.

35 pages; author's accepted manuscript for Algebra and Number Theory

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