Intersections of loci of admissible covers with tautological classes

TL;DR

This paper develops combinatorial formulas to analyze the intersection of admissible cover cycles with tautological classes, enabling explicit computations of hyperelliptic and bielliptic loci classes in moduli spaces of curves.

Contribution

It introduces a method to compute intersections of admissible cover cycles with tautological classes using combinatorial formulas and algorithms.

Findings

Derived formulas for pullbacks of admissible cover cycles

Showed that certain push-pull maps preserve tautological classes

Computed explicit classes for hyperelliptic and bielliptic loci

Abstract

For a finite group $G$, let $\H_{g,G,\xi}$ be the stack of admissible $G$-covers $C\to D$ of stable curves with ramification data $\xi$, $g(C)=g$ and $g(D)=g'$. There are source and target morphisms $\phi\colon \H_{g,G,\xi}\to \M_{g,r}$ and $\delta\colon \H_{g,G,\xi}\to \M_{g',b}$, remembering the curves $C$ and $D$ together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form $\phi_* [\H_{g,G,\xi}]$. Examples include the fundamental classes of the loci of hyperelliptic or bielliptic curves $C$ with marked ramification points. The two main results of this paper are as follows: Firstly, for the gluing morphism $\xi_A\colon \M_A\to \M_{g,r}$ associated to to a stable graph $A$ we give a combinatorial formula for the pullback $\xi^*_A \phi_*[\H_{g,G,\xi}]$ in terms of spaces of admissible $G$-covers and $\psi$ classes. This allows us to describe the intersection of the cycles $\phi_*[\H_{g,G,\xi}]$ with tautological classes. Secondly, the pull-push $\delta_*\phi^*$ sends tautological classes to tautological classes and we also give a combinatorial description of this map in terms of standard generators of the tautological rings. We show how to use the pullbacks to algorithmically compute tautological expressions for cycles of the form $\phi_* [\H_{g,G,\xi}]$. In particular, we compute the classes $[\Hyp_5]$ and $[\Hyp_6]$ of the hyperelliptic loci in $\M_5$ and $\M_6$ and the class $[\B_4]$ of the bielliptic locus in $\M_4$.