Functoriality in resolution of singularities
type: Item
instance of: scholarly article
title: Functoriality in resolution of singularities
zbMATH Open document ID: 1151.14012
author: Edward Bierstone
author: Pierre Milman
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2008-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0702375
review text: The authors investigate a procedure to resolve singularities that has reasonable ``functorial'' properties. The main result is expressed in terms of ``marked ideals'', that is a 5-tuple \(\mathcal I = (M,N,E,I,d)\) where \(N\) is a subvariety of \(M\), both smooth over a field of characteristic zero, \(I\) is a coherent sheaf of ideals of \(N\), \(E\) is a normal crossings divisor of \(M\), transversal to \(N\) and \(d\) is a positive integer. The singular locus, or cosupport, of \(\mathcal I\) is the set of points \(x\) of \(N\) such that the order of the stalk \(I_x\) is \(\geq d\). One may define the transformation of such a marked ideal with center a suitable smooth subvariety of \(N\) (the \textit{admissible} transformations), the result is a new marked ideal. The objective is to obtain, by means of a finite sequence of admissible transformations, a marked ideal whose cosupport is empty. Such a sequence is called a resolution sequence. If this can be done in a reasonable constructive (or algorithmic) way other more classical desingularization theorems follow rather easily. (This is explained in the present article). The authors define a notion of equivalence of marked ideals as follows. A test transformation is either an admissible one, or one determined by the blowing-up of a center which is the intersection of two components of \(E\), or one induced by a projection \(M \times {\mathbb{A}}^1 \to M\). Two marked ideals \(\mathcal I = (M,N,E,I,d)\) and \(\mathcal J = (M,N',E,J,d')\) are equivalent if they have the same sequences of of test transformations. Then they prove that there is way to associate to each marked ideal a resolution sequence such that if \(\mathcal I\) and \(\mathcal J\) are equivalent, then the resolution sequence associated to \(\mathcal I\) is obtained by using the same centers as were needed for the sequence of \(\mathcal J\). Moreover, this procedure is compatible with smooth morphisms \(M' \to M\). This is what is meant by functoriality…
reviewed by: Augusto Nobile
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14E99
Mathematics Subject Classification ID: 32S45
Mathematics Subject Classification ID: 32S15
zbMATH DE Number: 5310672
zbMATH Keywords: resolution
zbMATH Keywords: blowing-up
zbMATH Keywords: marked ideal
zbMATH Keywords: admissible transformation
zbMATH Keywords: derivative ideals
zbMATH Keywords: test transformation
MaRDI profile type: Publication
OpenAlex ID: W2018490596
arXiv ID: math/0702375
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DOI: 10.2977/PRIMS/1210167338
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