Inventiones mathematicae

, Volume 128, Issue 2, pp 207–302 | Cite as

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

  • Edward Bierstone


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 \({\rm inv}_X(a)\) whose maximum locus determines a smooth centre of blowing up, leading to desingularization. To define \({\rm inv}_X\), we need only to work with a category of local-ringed spaces \(X=(|X|,{\cal O}_X)\) satisfying certain natural conditions. If \(a\in |X|\), then \({\rm inv}_X(a)\) depends only on \(\widehat{{\cal O}}_{X,a}\). More generally, \({\rm inv}_X\) 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 \({\rm inv}_X(\cdot)\). 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.


Natural Condition Maximum Locus Characteristic Zero Finite Type Analytic Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Edward Bierstone
    • 1
  1. 1.Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 1A1 (e-mail:; milman@math.toronto.eduCA

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