Modular compactifications of M_{2,n} with Gorenstein curves

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 0 < m < n, a stability condition — using one of the markings as a reference point, and thus not S_n-symmetric — defines proper Deligne–Mumford stacks M_{2, n}(m) with a dense open substack representing smooth curves.