Covering semigroups

Abstract

International audienceWe introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As application, we give new conditions on the ramification type that are sufficient for irreducibil-ity of the Hurwitz spaces, suggest some bounds on the number of irreducibility components under certain more general conditions, and show that the number of irreducible components coincides with the number of topological classes of the coverings if the number of brunch points is big enough

Hurwitz spaces and liftings to the Valentiner group

“…, g r ). A description of the Hurwitz action, together with example of monodromies, can be found in [5] and a point of view on the study of the connected components of the Hurwitz schemes by means of semigroups over groups is carried out in [14]. Notice that if the group G is the alternating group A n , then N Sn (A n ) is the whole S n , because A n is unique in its conjugacy class.…”

“…The Valentiner group is described as the subgroup of S 18 generated by {v 1 , v 2 } = {(2, 6)(4, 11) (7,9) (8,13) (10,14) (12,16), (1,2,7,4) (3,8,6,10) (5,9,13,12) (11,15) (14,17) (16,18)…”

“…Cases [4] and [5] are equivalent to Case [2], this can achieved by conjugating (1, 4) (5,6), (2,5) (3,6) in [4] and (1, 6)(4, 5), (2, 3) (5,6) in [5]. Eventually, [3.2] is equivalent to [3.1] by using φ (,) 14 .…”

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

“…This result was extended in [55] to covers of a fixed curve Y of genus ≥ 1. The papers [37,38,6] are devoted to determining the number of connected components of the Hurwitz spaces when every n i of the branching data is large enough. The paper [31] extends the connectivity result of Clebsch and Hurwitz to Hurwitz spaces of Galois covers of P 1 with Galois group isomorphic to a Weyl group and branching data consisting of reflections.…”