Non-tautological cycles on moduli spaces of smooth pointed curves

TL;DR

This paper extends previous work to demonstrate the widespread existence of non-tautological algebraic cohomology classes on moduli spaces of smooth pointed curves across nearly all genera and markings, revealing new geometric structures.

Contribution

It generalizes earlier techniques to prove the existence of non-tautological classes on most moduli spaces of smooth pointed curves, covering nearly all cases.

Findings

Non-tautological classes exist for almost all genera and markings.

The technique applies to most remaining cases beyond initial results.

The work broadens understanding of the cohomology of moduli spaces.

Abstract

In recent work by Arena, Canning, Clader, Haburcak, Li, Mok, and Tamborini it was proven that for infinitely many values of $g$ and $n$, there exist non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth genus $g$, $n$-pointed curves. Here we show how a generalization of their technique allows to cover most of the remaining cases, proving the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ for all but finitely many values of $g$ and $n$.