Tautological and non-tautological cohomology of the moduli space of curves

Abstract

After a short exposition of the basic properties of the tautological ring of the moduli space of genus g Deligne-Mumford stable curves with n markings, we explain three methods of detecting non-tautological classes in cohomology. The first is via curve counting over finite fields. The second is by obtaining length bounds on the action of the symmetric group S_n on tautological classes. The third is via classical boundary geometry. Several new non-tautological classes are found.

arXiv comment40 pages

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Cited by in corpus (5)

• Holomorphic anomaly equations and the Igusa cusp form conjecture
• The Chern character of the Verlinde bundle over the moduli space of stable curves
• The quantum Witten-Kontsevich series and one-part double Hurwitz numbers
• Birational geometry of moduli of curves with an $S_3$-cover
• Some remarks on cyclic Galois coverings of the projective line over finite fields