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Factorization semigroups and irreducible components of the Hurwitz space. II. (English. Russian original) Zbl 1251.14018

Izv. Math. 76, No. 2, 356-364 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 2, 151-160 (2012).
Summary: We continue the investigation started in [Izv. Math. 75, No. 4, 711–748 (2011); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 75, No. 4, 49–90 (2011; Zbl 1242.14026)]. Let \(\text{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)\) 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 \(f_C\geq 2\) elements of \( I_d\) fixed, then the Hurwitz space \(\text{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)\) is irreducible.

MSC:

14H30 Coverings of curves, fundamental group
20M50 Connections of semigroups with homological algebra and category theory
57M05 Fundamental group, presentations, free differential calculus

Citations:

Zbl 1242.14026
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