Sam Canning (April 16, 2021):  The Chow rings of M_7, M_8, and M_9
Channel: Algebraic Geometry at Stanford
Date: 20/04/2021

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.