Kulikov, V. S. Factorization semigroups and irreducible components of the Hurwitz space. (English. Russian original) Zbl 1242.14026 Izv. Math. 75, No. 4, 711-748 (2011); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 75, No. 4, 49-90 (2011). Given two covers of \(\mathbb P^1\) of the same degree \(d\) with specific local numbering of \(d\) sheets over chosen base points, one can form new such a cover by topologically cutting out sheets over large regions of \( \mathbb P^1(\mathbb C)\) and pasting them along boundaries that carry compatible numbering over the patched base point.Based on this operation, in this paper introduced “a natural structure of a semigroup (isomorphic to the factorization semigroup of the identity in the symmetric group) on the set of irreducible components of the Hurwitz space of coverings of marked degree \( d\) of \( \mathbb P^1\) of fixed ramification types”. Author’s abstract further continues to read: “We shall prove that this semigroup is finitely presented. We study the problem of when collections of ramification types uniquely determine the corresponding irreducible components of the Hurwitz space. In particular, we give a complete description of the set of irreducible components of the Hurwitz space of three-sheeted coverings of the projective line.” Reviewer: Hiroaki Nakamura (Okayama) Cited in 3 ReviewsCited in 3 Documents MSC: 14H30 Coverings of curves, fundamental group 20M50 Connections of semigroups with homological algebra and category theory 57M05 Fundamental group, presentations, free differential calculus Keywords:semigroups over groups; Hurwitz space; monodromy type of covers; factorization of an element of a group PDFBibTeX XMLCite \textit{V. S. Kulikov}, Izv. Math. 75, No. 4, 711--748 (2011; Zbl 1242.14026); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 75, No. 4, 49--90 (2011) Full Text: DOI arXiv