VGIT presentation of the second flip of $\overline{M}_{2,1}$

Abstract

We perform a variation of geometric invariant theory stability analysis for 2nd Hilbert points of bi-log-canonically embedded pointed curves of genus two. As a result, we give a GIT construction of the last three non-trivial log canonical models of the moduli space of pointed genus two curves, and obtain a VGIT presentation of the second flip in its Hassett-Keel program.