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

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 $overline{mathcal{M}}_{1,n}(Q)$, which we may think of coming from an enrichment of the notion of level used to define Smyths $m$-stable spaces. Finally, we construct a cube complex of Artin stacks interpolating between the $overline{mathcal{M}}_{1,n}(Q)$s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model program for $overline{mathcal{M}}_g$.