In algebraic geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes obtained from 1 by pushforward along various morphisms described below. The tautological cohomology ring is the image of the tautological ring under the cycle map (from the Chow ring to the cohomology ring).
Definition Let M ¯ g , n {\displaystyle {\overline {\mathcal {M}}}_{g,n}} be the moduli stack of stable marked curves ( C ; x 1 , … , x n ) {\displaystyle (C;x_{1},\ldots ,x_{n})} , such that
C is a complex curve of arithmetic genus g whose only singularities are nodes, the n points x1, ..., xn are distinct smooth points of C, the marked curve is stable, namely its automorphism group (leaving marked points invariant) is finite. The last condition requires 2 g − 2 + n > 0 {\displaystyle 2g-2+n>0} in other words (g,n) is not among (0,0), (0,1), (0,2), (1,0). The stack M ¯ g , n {\displaystyle {\overline {\mathcal {M}}}_{g,n}} then has dimension 3 g − 3 + n {\displaystyle 3g-3+n} . Besides permutations of the marked points, the following morphisms between these moduli stacks play an important role in defining tautological classes:
Forgetful maps M ¯ g , n → M ¯ g , n − 1 {\displaystyle {\overline {\mathcal {M}}}_{g,n}\to {\overline {\mathcal {M}}}_{g,n-1}} which act by removing a given point xk from the set of marked points, then restabilizing the marked curved if it is not stable anymore. Gluing maps M ¯ g , n + 1 × M ¯ g ′ , n ′ + 1 → M ¯ g + g ′ , n + n ′ {\displaystyle {\overline {\mathcal {M}}}_{g,n+1}\times {\overline {\mathcal {M}}}_{g',n'+1}\to {\overline {\mathcal {M}}}_{g+g',n+n'}} that identify the k-th marked point of a curve to the l-th marked point of the other. Another set of gluing maps is M ¯ g , n + 2 → M ¯ g + 1 , n {\displaystyle {\overline {\mathcal {M}}}_{g,n+2}\to {\overline {\mathcal {M}}}_{g+1,n}} that identify the k-th and l-th marked points, thus increasing the genus by creating a closed loop. The tautological rings R ∙ ( M ¯ g , n ) {\displaystyle R^{\bullet }({\overline {\mathcal {M}}}_{g,n})} are simultaneously defined as the smallest subrings of the Chow rings closed under pushforward by forgetful and gluing maps. The tautological cohomology ring R H ∙ ( M ¯ g , n ) {\displaystyle RH^{\bullet }({\overline {\mathcal {M}}}_{g,n})} is the image of R ∙ ( M ¯ g , n ) {\displaystyle R^{\bullet }({\overline {\mathcal {M}}}_{g,n})} under the cycle map. As of 2016, it is not known whether the tautological and tautological cohomology rings are isomorphic.
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