Hurwitz families and nonabelian cohomology
type: Item
instance of: scholarly article
title: Hurwitz families and nonabelian cohomology
zbMATH Open document ID: 0971.18011
author: Pierre Dèbes
author: Jean-Claude Douai
author: Michel Emsalem
publication date: 2000-03-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75410
review text: Nonabelian cohomology, more precisely the Grothendieck-Giraud theory of gerbes, is used by \textit{P. Dèbes} and \textit{J.-C. Douai} [Commun. Algebra 27, No. 2, 577-594 (1999; Zbl 0917.18008)] to study some questions related to fields of definition of Galois covers. The paper under review continues the same ideas, by studying the existence of Hurwitz families (i.e., of families of coverings of the Riemann sphere) parametrized by a (possibly coarse) moduli space of such coverings (i.e., a Hurwitz space). Given an irreducible component \({\mathcal H}\) of a Hurwitz space, the authors construct a gerbe \({\mathcal G}\) above the étale site \({\mathcal H}_{\text{ét}}\) of \({\mathcal H}\). The class of \({\mathcal G}\) in \(H^2({\mathcal H}_{\text{ét}},-)\), of equivalence classes of gerbes (of given band) encodes the obstruction to the existence of Hurwitz families above \({\mathcal H}\) (\S 3). Applications are given in \S 4, e.g. concrete criteria for the existence of Hurwitz families or determination of the function field of \({\mathcal H}\).
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14D22
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 1409066
zbMATH Keywords: Hurwitz families
zbMATH Keywords: covers
zbMATH Keywords: nonabelian cohomology
zbMATH Keywords: coarse moduli space
zbMATH Keywords: gerbes
zbMATH Keywords: Hurwitz space
zbMATH Keywords: étale site
reviewed by: Marc Reversat
MaRDI profile type: Publication
OpenAlex ID: W2327356071
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DOI: 10.5802/AIF.1749
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1409066