Log canonical models and variation of GIT for genus four canonical curves

We discuss GIT for canonically embedded genus four curves and the connection to the Hassett-Keel program. A canonical genus four curve is a complete intersection of a quadric and a cubic, and, in contrast to the genus three case, there is a family of GIT quotients that depend on a choice of linearization. We discuss the corresponding VGIT problem and show that the resulting spaces give the final steps in the Hassett-Keel program for genus four curves.