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Desingularization: Invariants and Strategy

Application to Dimension 2

  • Book
  • © 2020

Overview

Authors:
  1. Vincent Cossart

    1. Université Paris-Saclay, UVSQ, LMV (UMR 8100) CNRS, Versailles Cedex, France

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  2. Uwe Jannsen

    1. Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany

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  3. Shuji Saito

    1. Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Japan

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  • Provides a complete proof of desingularization of surfaces, and several other well-known results not previously published in the literature
  • Briefly summarizes the history of the topic, with numerous readable references
  • Written in an accessible style, ideal for non-specialists
  • Features numerous useful computations, serving as a source of inspiration for experts exploring bigger dimensions

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2270)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-viii
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  2. Introduction

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 1-14

  3. Basic Invariants for Singularities

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 15-36

  4. Permissible Blow-Ups

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 37-48

  5. \(\mathcal {B}\)-Permissible Blow-Ups: The Embedded Case

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 49-63

  6. \( \mathcal {B}\)-Permissible Blow-Ups: The Non-embedded Case

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 65-77

  7. Main Theorems and Strategy for Their Proofs

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 79-104

  8. (u)-standard Bases

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 105-116

  9. Characteristic Polyhedra of J ⊂ R

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 117-131

  10. Transformation of Standard Bases Under Blow-Ups

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 133-144

  11. Termination of the Fundamental Sequences of \( \mathcal {B}\)-Permissible Blow-Ups, and the Case e x(X) = 1

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 145-153

  12. Additional Invariants in the Case e x(X) = 2

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 155-160

  13. Proof in the Case e x(X) = es x(X) = 2 , I: Some Key Lemmas

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 161-166

  14. Proof in the Case \(e_x(X)=\overline {e}_x(X)=2\) , II: Separable Residue Extensions

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 167-173

  15. Proof in the Case e x(X) = e x(X) = 2 , III: Inseparable Residue Extensions

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 175-190

  16. Non-existence of Maximal Contact in Dimension 2

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 191-200

  17. An Alternative Proof of Theorem 6.17

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 201-204

  18. Functoriality, Locally Noetherian Schemes, Algebraic Spaces and Stacks

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 205-209

  19. Appendix. Hironaka’s Characteristic Polyhedron. Notes for Novices(B.Schober)

    • Vincent Cossart, Uwe Jannsen, Shuji Saito
    Pages 211-247

  20. Back Matter

    Pages 249-258
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Keywords

About this book

This book provides a rigorous and self-contained review of desingularization theory. Focusing on arbitrary dimensional schemes, it discusses the important concepts in full generality, complete with proofs, and includes an introduction to the basis of Hironaka’s Theory.

The core of the book is a complete proof of desingularization of surfaces; despite being well-known, this result was no more than folklore for many years, with no existing references. 

Throughout the book there are numerous computations on standard bases, blowing ups and characteristic polyhedra, which will be a source of inspiration for experts exploring bigger dimensions. Beginners will also benefit from a section which presents some easily overlooked pathologies.

Authors and Affiliations

  • Université Paris-Saclay, UVSQ, LMV (UMR 8100) CNRS, Versailles Cedex, France

    Vincent Cossart

  • Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany

    Uwe Jannsen

  • Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Japan

    Shuji Saito

Bibliographic Information

  • Book Title: Desingularization: Invariants and Strategy

  • Book Subtitle: Application to Dimension 2

  • Authors: Vincent Cossart, Uwe Jannsen, Shuji Saito

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-52640-5

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

  • Softcover ISBN: 978-3-030-52639-9Published: 28 August 2020

  • eBook ISBN: 978-3-030-52640-5Published: 27 August 2020

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: VIII, 258

  • Number of Illustrations: 41 b/w illustrations

  • Topics: Algebra, Algebraic Geometry

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