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

Veronica Arena, Samir Canning, Emily Clader, Richard Haburcak, Q. Li, Siao Chi Mok, Carolina Tamborini

• We prove, for infinitely many values of g and n, the existence of non-tautological algebraic cohomology classes on the moduli space Mg,n of smooth, genus-g, npointed curves. In particular, when n = 0, our results show that there exist nontautological algebraic cohomology classes on Mg for g = 12 and all g ≥ 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: the bielliptic cycle on M12. We extend their work by using the existence of holomorphic forms on certain moduli spaces Mg,n to produce non-tautological classes with nontrivial restriction to the interior, via which we conclude that the classes of many new doublecover loci are non-tautological.