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© 2018

Singular Algebraic Curves

With an Appendix by Oleg Viro

Benefits

  • Systematically treats the global geometry of equisingular families of algebraic curves on algebraic surfaces

  • Accumulates the material spread over numerous, recent and classical journal publications, and elaborates it into a unified theory which allows one to approach all main problems in the subject and to answer several classical questions in this area

  • Provides a guide to a variety of methods, results and applications of singular algebraic curves and their families

  • Offers a detailed presentation of the background stuff (including the global deformation theory and the original Viro patchworking construction) which leads the reader to the main ideas of the theory

Book

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

Table of contents

  1. Front Matter
    Pages i-xx
  2. Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin
    Pages 1-110
  3. Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin
    Pages 111-267
  4. Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin
    Pages 269-332
  5. Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin
    Pages 333-487
  6. Back Matter
    Pages 489-553

About this book

Introduction

Singular algebraic curves have been in the focus of study in algebraic geometry from the very beginning, and till now remain a subject of an active research related to many modern developments in algebraic geometry, symplectic geometry, and tropical geometry. The monograph suggests a unified approach to the geometry of singular algebraic curves on algebraic surfaces and their families, which applies to arbitrary singularities, allows one to treat all main questions concerning the geometry of  equisingular families of curves, and, finally, leads to  results which can be viewed as the best possible in a reasonable sense. Various methods of the cohomology vanishing theory as well as the patchworking construction with its modifications will be of a special interest for experts in algebraic geometry and singularity theory. The introductory chapters on zero-dimensional schemes and global deformation theory can well serve as a material for special courses and seminars for graduate and post-graduate students.Geometry in general plays a leading role in modern mathematics, and algebraic geometry is the most advanced area of research in geometry. In turn, algebraic curves for more than one century have been  the central subject of algebraic geometry both in fundamental theoretic questions and in applications to other fields of mathematics and mathematical physics.  Particularly, the local and global study of  singular algebraic curves involves a variety of methods and deep ideas from geometry, analysis, algebra, combinatorics and suggests a number of hard classical and newly appeared problems which inspire further development in this research area.

Keywords

14-02; 14Hxx; 14H10; 14H20; 14F17 14B07; 14M25; 14J60; 14P25;14Q05 Algebraic Curves Equisingular Families of Curves Zero-Dimensional Schemes Cohomology Vanishing Theory

Authors and affiliations

  1. 1.Fachbereich MathematikTU KaiserslauternKaiserslauternGermany
  2. 2.Fachbereich MathematikTU KaiserslauternKaiserslauternGermany
  3. 3.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Bibliographic information

  • Book Title Singular Algebraic Curves
  • Book Subtitle With an Appendix by Oleg Viro
  • Authors Gert-Martin Greuel
    Christoph Lossen
    Eugenii Shustin
  • Series Title Springer Monographs in Mathematics
  • Series Abbreviated Title Springer Monographs in Mathematics
  • DOI https://doi.org/10.1007/978-3-030-03350-7
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-030-03349-1
  • eBook ISBN 978-3-030-03350-7
  • Series ISSN 1439-7382
  • Series E-ISSN 2196-9922
  • Edition Number 1
  • Number of Pages XX, 553
  • Number of Illustrations 74 b/w illustrations, 0 illustrations in colour
  • Topics Algebraic Geometry
  • Buy this book on publisher's site
Industry Sectors
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Reviews

“The book is written in a very clear style, many topics are illustrated by various nice pictures, visual diagrams, etc. … this book is comprehensible, interesting and useful for graduate and post-graduate students; it is also very valuable for advanced researchers, lecturers, and practicians working in singularity theory, algebraic geometry, topology, combinatorics, tropical geometry and in other fields of mathematics and its applications.” (Aleksandr G. Aleksandrov, zbMATH 1411.14001, 2019)