Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant

摘要

This article contains an elementary constructive proof of resolution of singularities in characteristic zero. Our proof applies in particular 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|,O_X) satisfying certain natural conditions. If a∈|X|, then inv_X(a) depends only on O_(X,a). More generally, inv_X is defined inductively after any sequence of blowings-up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of inv_X(.). The paper is self-contained and includes detailed examples. One of our goals is that the reader understand the desingularization theorem, rather than simply "know" it is true.