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The Gorenstein conjecture fails for the tautological ring of \(\mathcal{\overline{M}}_{2,n}\)

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Abstract

We prove that for N equal to at least one of the integers 8, 12, 16, 20 the tautological ring \(R^{\bullet}(\overline {\mathcal {M}}_{2,N})\) is not Gorenstein. In fact, our N equals the smallest integer such that there is a non-tautological cohomology class of even degree on \(\overline {\mathcal {M}}_{2,N}\). By work of Graber and Pandharipande, such a class exists on \(\overline {\mathcal {M}}_{2,20}\), and we present some evidence indicating that N is in fact 20.

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Authors and Affiliations

  1. Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden

    Dan Petersen

  2. Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Orsola Tommasi

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  1. Dan Petersen
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  2. Orsola Tommasi
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Correspondence to Dan Petersen.

Additional information

D.P. is supported by the Göran Gustafsson foundation for scientific and medical research. O.T. is supported by the Deutsche Forschungsgemeinschaft under grant Hu337/6-2.

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Petersen, D., Tommasi, O. The Gorenstein conjecture fails for the tautological ring of \(\mathcal{\overline{M}}_{2,n}\) . Invent. math. 196, 139–161 (2014). https://doi.org/10.1007/s00222-013-0466-z

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  • DOI: https://doi.org/10.1007/s00222-013-0466-z

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