Non-tautological Hurwitz cycles

TL;DR

This paper demonstrates that certain loci of stable curves with degree d covers of positive genus curves form non-tautological algebraic cycles on moduli spaces, assuming a modular form coefficient condition, advancing previous work on degree 2 covers.

Contribution

Introduces a new method to intersect these loci with boundary strata, establishing their non-tautological nature under specific modular form conditions.

Findings

Loci of stable curves with degree d covers are non-tautological cycles.

Method to intersect cycles with boundary strata is developed.

Results extend previous work on degree 2 covers of elliptic curves.

Abstract

We show that various loci of stable curves of sufficiently large genus admitting degree $d$ covers of positive genus curves define non-tautological algebraic cycles on $\overline{\mathcal{M}}_{g,N}$, assuming the non-vanishing of the $d$-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.