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 them by the 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.
The talk will start with a pre-seminar at 2pm:
Title: Moduli spaces of trigonal curves
Abstract: I'll explain how to use the geometric Riemann-Roch theorem to give a simple presentation for the moduli space of trigonal curves. We'll then use this presentation to understand intersection-theoretic aspects of the geometry of the moduli space of trigonal curves.