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

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 $g\ge 16$. 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 ${\bar{\mathcal{M}}}_{g,n}$ 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.