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Joins and Intersections

  • Hubert Flenner
  • Liam O’Carroll
  • Wolfgang Vogel

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

Table of contents

  1. Front Matter
    Pages I-VI
  2. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 1-5
  3. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 7-42
  4. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 43-81
  5. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 83-122
  6. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 123-170
  7. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 171-191
  8. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 193-207
  9. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 209-249
  10. Hubert Flenner, Liam O’Carroll, Wolfgang Vogel
    Pages 251-275
  11. Back Matter
    Pages 277-307

About this book

Introduction

Dedicated to the memory of Wolfgang Classical Intersection Theory (see for example Wei! [Wei]) treats the case of proper intersections, where geometrical objects (usually subvarieties of a non­ singular variety) intersect with the expected dimension. In 1984, two books appeared which surveyed and developed work by the individual authors, co­ workers and others on a refined version of Intersection Theory, treating the case of possibly improper intersections, where the intersection could have ex­ cess dimension. The first, by W. Fulton [Full] (recently revised in updated form), used a geometrical theory of deformation to the normal cone, more specifically, deformation to the normal bundle followed by moving the zero section to make the intersection proper; this theory was due to the author together with R. MacPherson and worked generally for intersections on algeb­ raic manifolds. It represents nowadays the standard approach to Intersection Theory. The second, by W. Vogel [Vogl], employed an algebraic approach to inter­ sections; although restricted to intersections in projective space it produced an intersection cycle by a simple and natural algorithm, thus leading to a Bezout theorem for improper intersections. It was developed together with J. Stiickrad and involved a refined version of the classical technique ofreduc­ tion to the diagonal: here one starts with the join variety and intersects with successive hyperplanes in general position, laying aside components which fall into the diagonal and intersecting the residual scheme with the next hyperplane; since all the hyperplanes intersect in the diagonal, the process terminates.

Keywords

Algebra Bezout's theorem Cohomology Intersection theory connectedness theorems join varieties residual intersections

Authors and affiliations

  • Hubert Flenner
    • 1
  • Liam O’Carroll
    • 2
  • Wolfgang Vogel
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Department of Mathematics and StatisticsUniversity of EdinburghEdinburghUnited Kingdom

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-03817-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 1999
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08562-8
  • Online ISBN 978-3-662-03817-8
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site