CANONICAL DESINGULARIZATION IN CHARACTERISTIC ZERO BY BLOWING-UP THE MAXIMUM STRATA OF A LOCAL INVARIANT

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.