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Resolution of Singularities of Embedded Algebraic Surfaces

  • Book
  • © 1998

Overview

Authors:
  1. Shreeram S. Abhyankar

    1. Department of Mathematics, Purdue University, West Lafayette, USA

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  • Description of the author's proof of desingularization of algebraic surfaces Self-contained introduction to birational algebraic geometry, based only on basic commutative algebra.
  • The unique place where desigularization for solids in characteristic p is done

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Shreeram S. Abhyankar
    Pages 1-6

  3. Local Theory

    • Shreeram S. Abhyankar
    Pages 7-154

  4. Global Theory

    • Shreeram S. Abhyankar
    Pages 155-237

  5. Some Cases of Three-Dimensional Birational Resolution

    • Shreeram S. Abhyankar
    Pages 238-284

  6. Back Matter

    Pages 285-311
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Keywords

About this book

The common solutions of a finite number of polynomial equations in a finite number of variables constitute an algebraic variety. The degrees of freedom of a moving point on the variety is the dimension of the variety. A one-dimensional variety is a curve and a two-dimensional variety is a surface. A three-dimensional variety may be called asolid. Most points of a variety are simple points. Singularities are special points, or points of multiplicity greater than one. Points of multiplicity two are double points, points of multiplicity three are tripie points, and so on. A nodal point of a curve is a double point where the curve crosses itself, such as the alpha curve. A cusp is a double point where the curve has a beak. The vertex of a cone provides an example of a surface singularity. A reversible change of variables gives abirational transformation of a variety. Singularities of a variety may be resolved by birational transformations.

Authors and Affiliations

  • Department of Mathematics, Purdue University, West Lafayette, USA

    Shreeram S. Abhyankar

Bibliographic Information

  • Book Title: Resolution of Singularities of Embedded Algebraic Surfaces

  • Authors: Shreeram S. Abhyankar

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-662-03580-1

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1998

  • Hardcover ISBN: 978-3-540-63719-6Published: 05 March 1998

  • Softcover ISBN: 978-3-642-08351-8Published: 04 December 2010

  • eBook ISBN: 978-3-662-03580-1Published: 06 December 2012

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 2

  • Number of Pages: XII, 312

  • Additional Information: Originally published by Academic Press, USA, 1966

  • Topics: Algebraic Geometry, Number Theory

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