This page describes ongoing work on the birational model of \(M_7\) constructed as the GIT quotient of the first syzygy points following this paper.

Here is what we know so far about the stability of the first syzygy points of canonical genus 7 curves.

- A generic curve is
*stable*. Therefore, the GIT quotient is birational to \(M_7\). - A general tetragonal curve is stable.
- A Casnati–Ekedahl special tetragonal curve is strictly semistable. All such curves have the same syzygy point.

Here is a write up that includes the proofs of the results above (except the proof of the stability of a general tetragonal curve—that's coming soon.) It uses the following two pieces of Macaulay2 code: mcsyzygy.m2 (written by David Swinarski) and delPezzo.m2.

The complete description of the GIT quotient and its relation to \(\overline{M}_7\) will be very intrictate. I welcome any suggestions and contributions towards this goal. The next natural questions would be:

- Is a general tetragonal curve stable?
- What are the replacements of trigonal curves?
- What are the replacements of hyperelliptic curves?
- How is the syzygy model related to the Mukai model?