International Congress of Mathematicians

504 R. Pandharipandemay restrict inquiry to the system of tautological rings, R*(M g^n). The tautologicalsystem is defined to be the set of smallest Q-subalgebras of the Chow rings,F*(M S ,„)CA*(M S; „),satisfying the following three properties:(i) R*(M g: „) contains the cotangent line classes fii,... ,fi n wherefii=ci(Li),the first Chern class of the ith cotangent line bundle.(ii) The system is closed under push-forward via all maps forgetting markings:Tr,:R*(M g , n )^R*(M g^i).(iii) The system is closed under push-forward via all gluing maps:7T» : F*(Af Slj „ lU {»}) ®Q F*(Af ff2j „ 2U {»}) —t R*(M gi+g2:ni+n2 ),7T» : F*(Af Slj „ lU {» j »}) —¥ R*(Mg 1+ i^ni ).Natural algebraic constructions typically yield Chow classes lying in the tautologicalring. See [7], [18] for further discussion.Consider the following basic filtration of the moduli space of pointed curves:Mg tn D M£ n D M r g' n D Cg^n.Here, M c denotes the moduli of pointed curves of compact type, M g * n denotesthe moduli of pointed curves with rational tails, and C g>n denotes the moduli ofpointed curves with a fixed stabilized complex structure C g . The choice of C g willplay a role below.The tautological rings for the elements of the filtration are defined by theimages of R*(M g: „) in the associated quotient sequence:R*(M g ,n) -> R*(Ml n ) -+ R*(M r g%) -+ R"(C g , n ) ^ 0. (2.1)Remarkably, the tautological rings of the strata are conjectured to resemble cohomologyrings of compact manifolds.A finite dimension graded algebra R is Gorenstein with socle in degree s ifthere exists an evaluation isomorphism,cj>:R s^Q,for which the bilinear pairings induced by the ring product,R r x R s - r-ìF4Q,are nondegenerate. The cohomology rings of compact manifolds are Gorensteinalgebras.