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

2011/01/28 by Faber, C., Pandharipande, R.
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1101.5489

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.

Related

• Non-tautological cycles on moduli spaces of smooth pointed curves
• Holomorphic forms and non-tautological cycles on moduli spaces of curves
• A calculus for the moduli space of curves
• The fourth tautological group of M_g,n and relations with the cohomology
• A conjectural description of the tautological ring of the moduli space of curves