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

Abstract

We prove, for infinitely many values of g and n, the existence of non-tautological algebraic cohomology classes on the moduli space Mg,n$$\mathcal {M}_{g,n}$$ of smooth, genus-g, n-pointed curves. In particular, when n=0$$n=0$$, our results show that there exist non-tautological algebraic cohomology classes on Mg$$\mathcal {M}g$$ for g=12$$g=12$$ and all g≥16$$g \ge 16$$. 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 Mg$$\mathcal {M}g$$: the bielliptic cycle on M12$$\mathcal {M}{12}$$. We extend their work by using the existence of holomorphic forms on certain moduli spaces M¯g,n$$\overline{\mathcal {M}}{g,n}$$ 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.