Modifications et cycles évanescents sur une base de dimension supérieure à un

Abstract

For a given morphism of schemes f:X->S, a sheaf F on X, a geometric point x on X, and s=f(x), the morphism f\_x : X(x) -> S(s) between the strict henselizations doesn't necessarily behave (with respect to F) like a proper morphism. However, we know it is so (assuming constructibility of F etc.) if S is the spectrum of a dvr (P. Deligne, SGA 4 1/2, [Th. finitude]). In this article, we prove it becomes so after an appropriate modification of the base S. The main ingredient is a theorem by A.J. de Jong on plurinodal fibrations. An application of this formalism to Lefschetz pencils is given.