Constructions of Nontautological Classes on Moduli Spaces of Curves T. Graber and R. Pandharipande

0. Introduction The tautological rings R(M g,n) are natural subrings of the Chow rings of the Deligne-Mumford moduli spaces of pointed curves: (1) R(M g,n) ⊂ A (M g,n) (the Chow rings are taken with Q-coefficients). The system of tautological subrings (1) is defined to be the set of smallest Q-subalgebras satisfying the following three properties [FP]: (i) R(Mg,n) contains the cotangent line classes ψ1, . . . , ψn ∈ A (M g,n). (ii) The system is closed under push-forward via all maps forgetting markings: π∗ : R (M g,n) → R (M g,n−1). (iii) The system is closed under push-forward via all gluing maps: π∗ : R (M g1,n1∪{∗})⊗Q R (M g2,n2∪{•}) → R (M g1+g2,n1+n2), π∗ : R (Mg1,n1∪{∗,•}) → R (M g1+1,n1). The tautological rings possess remarkable algebraic and combinatorial structures with basic connections to topological gravity. A discussion of these properties together with a conjectural framework for the study of R(M g,n) can be found in [F], [FP]. In genus 0, the equality R(M 0,n) = A (M0,n), for n ≥ 3, is well-known from Keel’s study [K].