Hurwitz families and nonabelian cohomology
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 \({\mathcal H}\).
scientific article; zbMATH DE number 1409066
scientific article