Factorization semigroups and irreducible components of Hurwitz space

TL;DR

This paper introduces a semigroup structure on the irreducible components of Hurwitz spaces of coverings, proves its finite presentation, and characterizes when ramification types uniquely determine these components, with a complete description for three-sheeted cases.

Contribution

It establishes a semigroup structure on Hurwitz space components, proves its finite presentation, and characterizes uniqueness conditions for ramification types.

Findings

Semigroup structure on Hurwitz space components is finitely presented.

Complete description of irreducible components for three-sheeted coverings.

Conditions for ramification types to uniquely determine components.

Abstract

We introduce a natural structure of a semigroup (isomorphic to a factorization semigroup of the unity in the symmetric group) on the set of irreducible components of Hurwitz space of marked degree $d$ coverings of $\mathbb P^1$ of fixed ramification types. It is proved that this semigroup is finitely presented. The problem when collections of ramification types define uniquely the corresponding irreducible components of the Hurwitz space is investigated. In particular, the set of irreducible components of the Hurwitz space of three-sheeted coverings of the projective line is completely described.