Second flip in the Hassett–Keel program: a local description

Abstract

This is the first of three papers in which we give a moduli interpretation of the second flip in the log minimal model program for $\overline{M}_{g}$, replacing the locus of curves with a genus $2$ Weierstrass tail by a locus of curves with a ramphoid cusp. In this paper, for $\unicode[STIX]{x1D6FC}\in (2/3-\unicode[STIX]{x1D716},2/3+\unicode[STIX]{x1D716})$, we introduce new $\unicode[STIX]{x1D6FC}$-stability conditions for curves and prove that they are deformation open. This yields algebraic stacks $\overline{{\mathcal{M}}}_{g}(\unicode[STIX]{x1D6FC})$ related by open immersions $\overline{{\mathcal{M}}}_{g}(2/3+\unicode[STIX]{x1D716}){\hookrightarrow}\overline{{\mathcal{M}}}_{g}(2/3){\hookleftarrow}\overline{{\mathcal{M}}}_{g}(2/3-\unicode[STIX]{x1D716})$. We prove that around a curve $C$ corresponding to a closed point in $\overline{{\mathcal{M}}}_{g}(2/3)$, these open immersions are locally modeled by variation of geometric invariant theory for the action of $\text{Aut}(C)$ on the first-order deformation space of $C$.