The Chow ring of the moduli space of curves of genus six

Abstract

We determine the Chow ring (with Q-coecients) of M6 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 stratify the moduli space into locally closed strata which are group quotients, and use a theorem of Vistoli to show that their Chow rings are generated by Chern classes of natural vector bundles. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus six case relies in addition on the particularly beautiful Brill{Noether theory in this case, and in particular on a rank ve vector bundle \relativizing a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the rank six vector bundle of quadrics cutting out the canonical curve.