Ravi Vakil, November 12th, 2013

We determine the Chow ring (with rational coefficients) of the moduli space of curves of

genus 6 by showing that all Chow classes are tautological. In particular, all algebraic

cohomology is tautological, and the natural map from Chow to cohomology is injective.

We rely on results of many others, notably Faber, Vistoli, and Mukai. To show the efficacy

of these methods, we give quick derivations of the Chow groups of moduli spaces of

curves of lower genus as well. What makes the genus 6 argument tractable (and even

short) is the particularly beautiful Brill-Noether theory in this case. This is joint work with

Nikola Penev.