VGIT Presentation of the Second Flip of 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 $2$. As a result, we give a GIT construction of the log canonical models ${\overline{M}}_{2,1}\left(\alpha \right)$ for $\alpha =2/3\pm ϵ$ and obtain a VGIT presentation of the second flip in the Hassett–Keel program for the moduli space of pointed genus $2$ curves.