# Tautological rings

The tautological ring is the subring of the the Chow ring which is meant to contain all of the natural geometric information. Faber and Pandharipande gave the following elegant formulation (which is equivalent to previous definitions). There are forgetful morphisms and gluing morphisms and . The system of tautological rings is then the smallest system of -subalgebras of the Chow rings which is closed under the gluing and pushforward maps and which contains all of the Witten classes . Tautological rings for the uncompactified moduli space and its partial compactifications are defined by restriction.

Jeffrey Herschel Giansiracusa 2005-06-27