Factorization semigroups and irreducible components of Hurwitz space. II

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 $f_C\ge 2$ elements of $I_d$, then the Hurwitz space {\rm ${\text{HUR}}_{d,t}^{{\mathcal{S}}_d}({\mathbb{P}}^1)$} is irreducible.

arXiv comment9 pages; the assertion of Theorem 2 is weakened and more detailed proof of Theorem 1 is given. Accepted in Izv. Math