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

TL;DR

This paper uses variation of geometric invariant theory to analyze stability of genus two pointed curves, constructing certain moduli space models and presenting the second flip in the Hassett-Keel program.

Contribution

It provides a GIT construction of the last three non-trivial log canonical models of the moduli space of pointed genus two curves and presents the VGIT description of the second flip.

Findings

GIT stability analysis of 2nd Hilbert points of genus two curves.

Construction of specific log canonical models of the moduli space.

VGIT presentation of the second flip in the Hassett-Keel program.

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.