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Monomialization of Morphisms from 3-Folds to Surfaces

  • Book
  • © 2002

Overview

Authors:
  1. Steven Dale Cutkosky
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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1786)

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Table of contents (21 chapters)

  1. Front Matter

    Pages N2-V
    Download chapter PDF

  2. 1. Introduction

    • Steven Dale Cutkosky
    Pages 1-8

  3. 2. Local Monomialization

    • Steven Dale Cutkosky
    Pages 9-10

  4. 3. Monomialization of Morphisms in Low Dimensions

    • Steven Dale Cutkosky
    Pages 11-13

  5. 4. An Overview of the Proof of Monomialization of Morphisms From 3 Folds to Surfaces

    • Steven Dale Cutkosky
    Pages 14-18

  6. 5. Notations

    • Steven Dale Cutkosky
    Pages 19-19

  7. 6. The Invariant \(\nu\)

    • Steven Dale Cutkosky
    Pages 20-55

  8. 7. The Invariant \(\nu\) Under Quadratic Transforms

    • Steven Dale Cutkosky
    Pages 56-76

  9. 8. Permissible Monoidal Transforms Centered at Curves

    • Steven Dale Cutkosky
    Pages 77-92

  10. 9. Power Series in 2 Variables

    • Steven Dale Cutkosky
    Pages 93-108

  11. 10. \(\bf {A_r(X)}\)

    • Steven Dale Cutkosky
    Pages 109-109

  12. 11. Reduction of \(\nu\) in a Special Case

    • Steven Dale Cutkosky
    Pages 110-130

  13. 12. Reduction of \(\nu\) in a Second Special Case

    • Steven Dale Cutkosky
    Pages 131-149

  14. 13. Resolution 1

    • Steven Dale Cutkosky
    Pages 150-162

  15. 14. Resolution 2

    • Steven Dale Cutkosky
    Pages 163-175

  16. 15. Resolution 3

    • Steven Dale Cutkosky
    Pages 176-184

  17. 16. Resolution 4

    • Steven Dale Cutkosky
    Pages 185-187

  18. 17. Proof of the Main Theorem

    • Steven Dale Cutkosky
    Pages 188-188

  19. 18. Monomialization

    • Steven Dale Cutkosky
    Pages 189-223

  20. 19. Toroidalization

    • Steven Dale Cutkosky
    Pages 224-231
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Keywords

About this book

A morphism of algebraic varieties (over a field characteristic 0) is monomial if it can locally be represented in e'tale neighborhoods by a pure monomial mappings. The book gives proof that a dominant morphism from a nonsingular 3-fold X to a surface S can be monomialized by performing sequences of blowups of nonsingular subvarieties of X and S.
The construction is very explicit and uses techniques from resolution of singularities. A research monograph in algebraic geometry, it addresses researchers and graduate students.

Bibliographic Information

  • Book Title: Monomialization of Morphisms from 3-Folds to Surfaces

  • Authors: Steven Dale Cutkosky

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/b83848

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2002

  • Softcover ISBN: 978-3-540-43780-2Published: 06 August 2002

  • eBook ISBN: 978-3-540-48030-3Published: 13 October 2004

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: VIII, 240

  • Topics: Algebraic Geometry

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