Geodesic
Factorization semigroups and irreducible components of the Hurwitz space.~II
Izvestiya. Mathematics , Tome 76 (2012) no. 2, pp. 356-364.

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We continue the investigation started in [1]. Let $\mathrm{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)$ be the Hurwitz space of coverings of degree $d$ of the projective line $\mathbb P^1$ with Galois group $\mathcal S_d$ and monodromy type $t$. The monodromy type is a set of local monodromy types, which are defined as conjugacy classes of permutations $\sigma$ in the symmetric group $\mathcal S_d$ acting on the set $I_d=\{1,\dots,d\}$. We prove that if the type $t$ contains sufficiently many local monodromies belonging to the conjugacy class $C$ of an odd permutation $\sigma$ which leaves $f_C\geqslant 2$ elements of $I_d$ fixed, then the Hurwitz space $\mathrm{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)$ is irreducible.
Keywords: semigroup, factorizations of an element of a group, irreducible components of the Hurwitz space. 
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Vik. S. Kulikov. Factorization semigroups and irreducible components of the Hurwitz space.~II. Izvestiya. Mathematics , Tome 76 (2012) no. 2, pp. 356-364. http://geodesic.mathdoc.fr/item/IM2_2012_76_2_a5/

[1] V. S. Kulikov, “Factorization semigroups and irreducible components of the Hurwitz space”, Izv. Math., 75:4 (2011), 711–748 | DOI | Zbl

[2] B. Wajnryb, “Orbits of Hurwitz action for coverings of a sphere with two special fibers”, Indag. Math. (N.S.), 7:4 (1996), 549–558 | DOI | MR | Zbl