Modular compactifications of M2,n with Gorenstein curves

Abstract

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