Factorization semigroups and irreducible components of Hurwitz space. II (1011.3619v2)

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.