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Let M_{g,n}^rt be the moduli space of stable n-pointed curves of genus g>1with rational tails. We also consider the space C_g^n classifying smooth curvesof genus g with not necessarily distinct n ordered points. There is a naturalproper map from M_{g,n}^rt to C_g^n which contracts all rational components.Tautological classes on these spaces are natural algebraic cycles reflectingthe geometry of curves. In this short note we study the connection betweentautological classes on M_{g,n}^rt and C_g^n. We show that there is a naturalfiltration on the tautological ring of M_{g,n}^rt consisting of g-2+n steps. Aconjectural dictionary between tautological relations on M_{g,n}^rt and C_g^nis presented. Our conjecture predicts that the space of relations inR^*(M_{g,n}^rt) is generated by relations in R^*(C_g^n) together with a classof relations obtained from the geometry of blow-ups. This conjecture isequivalent to the the independence of certain tautological classes in Chow. Weprove the analogue version of our conjecture for the Gorenstein quotients oftautological rings.

A conjectural connection between R^*(C_g^n) and R^*(M_{g,n}^rt)

A conjectural connection between R^(C_g^n) and R^(M_{g,n}^rt)