Hurwitz spaces and liftings to the Valentiner group

TL;DR

This paper investigates the structure of Hurwitz spaces with monodromy in the alternating group A_6, using liftings to the Valentiner group to distinguish connected components, and proves the existence of two irreducible components for positive genus coverings.

Contribution

It introduces the use of Valentiner group liftings as an invariant to analyze Hurwitz scheme components, extending methods from spin structure studies to new monodromy cases.

Findings

Hurwitz scheme has two irreducible components for genus > 0

Lifting to Valentiner group detects connected components

Results align with previous asymptotic analyses

Abstract

We study the components of the Hurwitz scheme of ramified coverings of $\mathbb{P}^1$ with monodromy given by the alternating group $A_6$ and elements in the conjugacy class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, inspired by the case of the spin structures studied by Fried for the $3$-cycles, we use as invariant the lifting to the Valentiner group, triple covering of $A_6$. We prove that the Hurwitz scheme has two irreducible components when the genus of the covering is greater than zero, in accordance with the asymptotic solution found by Bogomolov and Kulikov.