Holomorphic forms and non-tautological cycles on moduli spaces of curves

TL;DR

This paper demonstrates the existence of non-tautological algebraic cohomology classes on moduli spaces of curves for infinitely many genera and points, expanding understanding of the cohomological structure of these spaces.

Contribution

It introduces new methods using holomorphic forms to produce non-tautological classes on moduli spaces, extending prior results beyond bielliptic loci.

Findings

Non-tautological classes exist for infinitely many g and n.

Constructs new non-tautological classes from double-cover loci.

Shows non-tautological classes have nontrivial interior restrictions.

Abstract

We prove, for infinitely many values of $g$ and $n$, the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth, genus-$g$, $n$-pointed curves. In particular, when $n=0$, our results show that there exist non-tautological algebraic cohomology classes on $\mathcal{M}_g$ for $g=12$ and all $g \geq 16$. These results generalize the work of Graber--Pandharipande and van Zelm, who proved that the classes of particular loci of bielliptic curves are non-tautological and thereby exhibited the only previously-known non-tautological class on any $\mathcal{M}_g$: the bielliptic cycle on $\mathcal{M}_{12}$. We extend their work by using the existence of holomorphic forms on certain moduli spaces $\overline{\mathcal{M}}_{g,n}$ to produce non-tautological classes with nontrivial restriction to the interior, via which we conclude that the classes of many new double-cover loci are non-tautological.