Modular compactifications of $\mathcal M_{2,n}$ with Gorenstein singularities

Abstract: We study the geometry of Gorenstein curve singularities of genus two, and of their stable limits. These singularities come in two families, corresponding to either Weierstrass or conjugate points on a semistable tail. For every $1\leq m <n$, a stability condition - using one of the markings as a reference point, and therefore not $\mathfrak S_n$-symmetric - defines proper Deligne-Mumford stacks $\overline{\mathcal M}_{2,n}^{(m)}$ containing the locus of smooth curves as a dense open substack.