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

2024/02/06 by Veronica Arena, Samir Canning, Arena, Veronica +11
Mathematics · #Meromorphic and Entire Functions #Algebraic and Geometric Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.03874

Abstract

We prove, for infinitely many values of g and n, the existence of non-tautological algebraic cohomology classes on the moduli space 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 M_g 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 M_g: the bielliptic cycle on M₁₂. We extend their work by using the existence of holomorphic forms on certain moduli spaces 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.

Related

• Non-tautological cycles on moduli spaces of smooth pointed curves
• Tautological and non-tautological cohomology of the moduli space of curves
• Constructions of nontautological classes on moduli spaces of curves
• On the tautological ring of M_g,n
• On the Chow and cohomology rings of moduli spaces of stable curves