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:
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(i)
O Y,y is reduced.
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(ii)
f is universally open at the generic points of the components of Xy which contain x.
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(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.
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(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).
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GROTHENDIECK, A., DIEUDONNE, J.: Eléments de Géométrie algébrique chap. IV: Etude locale des schémas ... (cité EGA IV), Paris, P.U.F. (I.H.E.S., Publ. Math., no24 (1965), no28 (1966), no32 (1967).
GROTHENDIECK, A., SEYDI, H.: Morphismes universellement ouverts (à paraître).
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Grothendieck, A., Seydi, H. Platitude d'une adherence schematique et lemme de Hironaka generalise. Manuscripta Math 5, 323–339 (1971). https://doi.org/10.1007/BF01367768
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DOI: https://doi.org/10.1007/BF01367768