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

TL;DR

This paper explores the structure of the cohomology of the moduli space of curves, introducing methods to identify non-tautological classes and presenting new findings in the field.

Contribution

It presents three novel methods for detecting non-tautological classes in the cohomology of the moduli space of curves, and reports new non-tautological classes discovered.

Findings

Three methods for detecting non-tautological classes are explained.

Several new non-tautological classes are identified.

Insights into the tautological ring of the moduli space are provided.

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.