Tautological Ring

The tautological ring of the moduli spaces of stable curves is the system of subrings of their rational Chow rings generated by natural marked-point and boundary classes and closed under forgetting and gluing pushforwards. The cotangent-line class is the first Chern class of the line bundle whose fiber is the cotangent line at the th marking. The kappa class is for the map forgetting the last marking and stabilizing.

The strata algebra is the algebra of formal decorated stable graphs, with decoration degrees truncated at the dimensions of vertex factors. Realization sends a decorated graph to its gluing pushforward in the rational Chow ring,

Its image is , the tautological ring, and its kernel consists of tautological relations. Pixton's graph formula gives a known system of such relations. Completeness asks whether its original span accounts for the entire kernel.