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 \(\overline{\mathcal {M}}_{g,n}\) 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.
