Log canonical models and variation of git for genus 4 canonical curves

We discuss geometric invariant theory (GIT) for canonically embedded genus 4 curves and the connection to the Hassett-Keel program. A canonical genus 4 curve is a complete intersection of a quadric and a cubic, and, in contrast to the genus 3 case, there is a family of GIT quotients that depend on a choice of linearization. We discuss the corresponding variation of GIT (VGIT) problem and show that the resulting spaces give the final steps in the Hassett-Keel program for genus 4 curves.