Platitude d'une adherence schematique et lemme de Hironaka generalise

Abstract

The main aim of this article is to prove the following:Theorem (Generalized Hironaka's lemma). Let X→Y be a morphism of schemes, locally of finite presentation, x a point of X and y=f(x). Assume that the following conditions are satisfied: (i) OY,y is reduced. (ii) f is universally open at the generic points of the components of Xy which contain x. (iii) For every maximal generisation y′ of y in Y and every maximal generisation x′ of x in X which belongs to Xy, we have dimx, (Xy')=dimx(Xy)=d. (iv) Xy is reduced at the generic points of the components of Xy which contain x and (Xy)red is geometrically normal over K(y) in x. Then there exist an open neighbourhood U of x in X and a subscheme U0 of U which have the same underlying space as U such that f0:U0\arY is normal (i.e. f0 is a flat morphism whose geometric fibers are normal).