On Hurwitz spaces of coverings with one special fiber

Francesca Vetro

Vol. 240 (2009), No. 2, 383–398

DOI: 10.2140/pjm.2009.240.383

Abstract

Let X X^′ Y be a covering of smooth, projective complex curves such that π is a degree 2 étale covering and f is a degree d covering, with monodromy group S_d, branched in n + 1 points one of which is a special point whose local monodromy has cycle type given by the partition e = (e₁,…,e_r) of d. We study such coverings whose monodromy group is either W(B_d) or wN(W(B_d))(G₁)w⁻¹ for some w ∈ W(B_d), where W(B_d) is the Weyl group of type B_d, G₁ is the subgroup of W(B_d) generated by reflections with respect to the long roots 𝜀_i − 𝜀_j and N(W(B_d))(G₁) is the normalizer of G₁. We prove that in both cases the corresponding Hurwitz spaces are not connected and hence are not irreducible. In fact, we show that if n + |e|≥ 2d, where |e| = ∑ _i=1^r(e_i − 1), they have 2^2g − 1 connected components.

Keywords

Hurwitz spaces, connected components, special fiber, Weyl groups of type B_d

Mathematical Subject Classification 2000

Primary: 14H30

Secondary: 14H10

Milestones

Received: 13 December 2007

Revised: 13 November 2008

Accepted: 14 November 2008

Published: 4 March 2009