Factorization semigroups and irreducible components of Hurwitz space. II

TL;DR

This paper proves the irreducibility of certain Hurwitz spaces of degree d coverings of the projective line with symmetric group Galois group, under conditions involving the number of local monodromies of a specific type.

Contribution

It establishes conditions under which Hurwitz spaces are irreducible based on the distribution of local monodromies, extending previous results in the field.

Findings

Hurwitz space irreducibility when many local monodromies are in a specific conjugacy class

Conditions involving fixed points of permutations in the monodromy type

Extension of previous irreducibility results for Hurwitz spaces

Abstract

This article is a continuation of the article with the same title (see arXiv:1003.2953v1). Let {\rm $\text{HUR}_{d,t}^{G}(\mathbb P^1)$} be the Hurwitz space of degree $d$ coverings of the projective line $\mathbb P^1$ with Galois group $\mathcal S_d$ and having fixed monodromy type $t$ consisting of a collection of local monodromy types (that is, a collection of conjugacy classes of permutations $\sigma$ of the symmetric group $\mathcal S_d$ acting on the set $I_d=\{1,...,d\}$). We prove that if the type $t$ contains big enough number of local monodromies belonging to the conjugacy class $C$ of an odd permutation $\sigma$ which leaves fixed $f_C\geq 2$ elements of $I_d$, then the Hurwitz space {\rm $\text{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)$} is irreducible.