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15 - Absolute desingularization in characteristic zero

Published online by Cambridge University Press:  07 October 2011

By
Michael Temkin
Edited by
Raf Cluckers ,
Johannes Nicaise  and
Julien Sebag
Michael Temkin
Affiliation:
The Hebrew University
Raf Cluckers
Affiliation:
Université de Lille
Johannes Nicaise
Affiliation:
Katholieke Universiteit Leuven, Belgium
Julien Sebag
Affiliation:
Université de Rennes I, France
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Summary

Introduction

Preamble

This paper is an expository lecture notes originally based on a lecture on the results of [Tem1] given by the author at the workshop on Motivic Integration in May 2008, at ICMS, Edinburgh. Since a substantial progress was done since May 2008, it seemed natural to include the new results of [BMT], [Tem2] and [Tem3] in this exposition. We will mainly concentrate on the functorial non-embedded desingularization constructed in [Tem2] because it seems that the results of [Tem3] on the embedded case can be improved further. We pursue expository goals, so we will concentrate on explaining the results and the main ideas of our method and we will refer to the cited papers for proofs and technical details. Also, we try to include more examples and general remarks than in a pure research paper. Thus, this survey can serve as a companion to or a light version of [Tem1] and [Tem2]. I would like to warn the reader that the current situation described in the paper can change soon (similarly to the change since 2008), but this is always a danger with a survey on an active research area.

The history

In 1964 Hironaka proved many fundamental desingularization results including strong desingularization of algebraic varieties in characteristic zero. The latter means that any reduced variety of characteristic zero can be modified to a smooth one by successive blow ups along nowhere dense smooth centers.

Type
Chapter
Information
Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry , pp. 213 - 250
Publisher: Cambridge University Press
Print publication year: 2011

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References

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