The Chow rings of M_7, M_8, and M_9
The synchronous discussion for Sam Canning’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2021-04-16-sc (and will be deleted after ~3-7 days).
The rational Chow ring of the moduli space of smooth curves is known when the genus is at most 6 by work of Mumford (g=2), Faber (g=3,4), Izadi (g=5), and Penev-Vakil (g=6). In each case, it is generated by the tautological classes. On the other hand, van Zelm has shown that the bielliptic locus is not tautological when g=12. In recent joint work with Hannah Larson, we show that the Chow rings of M_7, M_8, and M_9 are generated by tautological classes, which determines the Chow ring by work of Faber. I will explain an overview of the proof with an emphasis on the special geometry of curves of low genus and low gonality.