E. Bierstone et Pd. Milman, CANONICAL DESINGULARIZATION IN CHARACTERISTIC ZERO BY BLOWING-UP THE MAXIMUM STRATA OF A LOCAL INVARIANT, Inventiones Mathematicae, 128(2), 1997, pp. 207-302
This article contains an elementary constructive proof of resolution o
f singularities in characteristic zero, Our proof al,plies in particul
ar to schemes of finite type and to analytic spaces (so we recover the
great theorems of Hironaka), We introduce a discrete local invariant
inv(X)(a) whose maximum locus determines a smooth centre of blowing up
, leading to desingularization, To define inv(X), we need only to work
with a category of local-ringed spaces X = (/X/, C-X) satisfying cert
ain natural conditions, If a epsilon /X/, then inv(X)(a) depends only
on (C) over cap(X,a). More generally, inv(X) is defined inductively af
ter any sequence of blowings-up whose centres have only normal crossin
gs with respect to the exceptional divisors and lie in the constant lo
ci of inv(X)(.). The paper is self-contained and includes detailed exa
mples, One of our goals is that the reader understand the desingulariz
ation theorem, rather than simply ''know'' it is true.