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Dynkin Graphs and Quadrilateral Singularities

  • Book
  • © 1993

Overview

Authors:
  1. Tohsuke Urabe
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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1548)

Part of the book sub series: Mathematisches Institut der Universität und Max-Planck-Institut für Mathematik, Bonn (LNMMATH)

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Table of contents (5 chapters)

  1. Front Matter

    Pages I-VI
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  2. Introduction

    • Tohsuke Urabe
    Pages 1-16

  3. Quadrilateral singularities and elliptic K3 surfaces

    • Tohsuke Urabe
    Pages 17-59

  4. Theorems with the Ik-conditions for J 3,0, Z 1,0 and Q 2,0

    • Tohsuke Urabe
    Pages 60-97

  5. Obstruction components

    • Tohsuke Urabe
    Pages 98-184

  6. Concept of co-root modules

    • Tohsuke Urabe
    Pages 185-226

  7. Back Matter

    Pages 227-233
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Keywords

About this book

The study of hypersurface quadrilateral singularities can be reduced to the study of elliptic K3 surfaces with a singular fiber of type I * 0 (superscript *, subscript 0), and therefore these notes consider, besides the topics of the title, such K3 surfaces too. The combinations of rational double points that can occur on fibers in the semi-universal deformations of quadrilateral singularities are examined, to show that the possible combinations can be described by a certain law from the viewpoint of Dynkin graphs. This is equivalent to saying that the possible combinations of singular fibers in elliptic K3 surfaces with a singular fiber of type I * 0 (superscript *, subscript 0) can be described by a certain law using classical Dynkin graphs appearing in the theory of semi-simple Lie groups. Further, a similar description for thecombination of singularities on plane sextic curves is given. Standard knowledge of algebraic geometry at the level of graduate students is expected. A new method based on graphs will attract attention of researches.

Bibliographic Information

  • Book Title: Dynkin Graphs and Quadrilateral Singularities

  • Authors: Tohsuke Urabe

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/BFb0084369

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1993

  • Softcover ISBN: 978-3-540-56877-3Published: 29 July 1993

  • eBook ISBN: 978-3-540-47769-3Published: 15 November 2006

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: CCXLVIII, 242

  • Topics: Graph Theory, Algebraic Geometry, Analysis

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