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 $1\le m<n$, 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.

38 pages, 8 figures, comments are welcome! v2: exposition improved largely due to the referee's comments: - simpler proof of the classification of isolated Gorenstein singularities of genus two using differentials, - description of semistable tails in the language of tropical geometry, - material on crimping spaces moved to an appendix. Revised version to appear in Algebra & Number Theory

References in corpus (2)

• A geometric invariant theory construction of moduli spaces of stable maps
• On some modular contractions of the moduli space of stable pointed curves

Cited by in corpus (4)

• A smooth compactification of the space of genus two curves in projective space via logarithmic geometry and Gorenstein curves
• A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves
• Hyperelliptic Gorenstein curves and logarithmic differentials
• Gorenstein curve singularities of genus three