Relations in the tautological ring of $\mathcal{M}_g$

Abstract

Using a simple geometric argument, we obtain an infinite family of nontrivial relations in the tautological ring of $\mathcal{M}_g$ (coming, in fact, from relations in the Chow ring of $\overline{\mathcal{M}}_{g,2}$). One immediate consequence of these relations is that the classes $\kappa_1,\ldots,\kappa_{[g/3]}$ generate the tautological ring of $\mathcal{M}_g$, which was conjectured by Faber in [F] and recently proven at the level of cohomology by Morita in [M].