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

Abstract

Abstract We prove, for infinitely many values of g and n, the existence of non-tautological algebraic cohomology classes on the moduli space $$\mathcal {M}_{g,n}$$

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

$$\mathcal {M}_g$$

M g

for

$$g=12$$

g = 12

and all

$$g \ge 16$$

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

$$\mathcal {M}_g$$

M g

: the bielliptic cycle on

$$\mathcal {M}_{12}$$

M 12

. We extend their work by using the existence of holomorphic forms on certain moduli spaces

$$\overline{\mathcal {M}}_{g,n}$$

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.