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The structure of the tautological ring in genus one. (English) Zbl 1291.14045

Let \(\overline{\mathcal M}_{g,n}\) be the moduli space of stable genus \(g\) curves with \(n\) ordered marked points. The main results of this paper show that for \(g=1\) the even-dimensional cohomology of \(\overline{\mathcal M}_{1,n}\) is generated additively by the classes of the boundary strata, and moreover all relations among these generators are given by the WDVV equation and E. Getzler’s relation [J. Am. Math. Soc. 10, No. 4, 973–998 (1997; Zbl 0909.14002)] (The results were announced previously in [loc. cit.] without proof). As a corollary, the tautological ring of \(\overline{\mathcal M}_{1,n}\) is isomorphic to its even cohomology and it is Gorenstein. The main tools in the proof are Deligne’s mixed Hodge theory and Eichler-Shimura theory for local systems.

MSC:

14H10 Families, moduli of curves (algebraic)
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)

Citations:

Zbl 0909.14002
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References:

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