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© 2020

Ideals of Powers and Powers of Ideals

Intersecting Algebra, Geometry, and Combinatorics

Book

Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 27)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Associated Primes of Powers of Ideals

    1. Front Matter
      Pages 1-1
    2. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 3-11
    3. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 13-32
    4. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 33-34
  3. Regularity of Powers of Ideals

    1. Front Matter
      Pages 35-35
    2. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 37-49
    3. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 51-58
    4. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 59-66
    5. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 67-67
  4. The Containment Problem

    1. Front Matter
      Pages 69-69
    2. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 71-75
    3. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 77-80
    4. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 81-88
    5. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 89-98
    6. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 99-99
  5. Unexpected Hypersurfaces

    1. Front Matter
      Pages 101-101
    2. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 103-110
    3. Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 111-111
  6. Waring Problems

    1. Front Matter
      Pages 113-113

About this book

Introduction

This book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning  our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms.  Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.

Keywords

Associated Primes Castelnuovo-Mumford Regularity Containment Problem Edge and Cover Ideals of Finite Simple Graphs Monomial Ideals Powers of Ideals Symbolic Defect Symbolic Powers of Ideals Unexpected Curves and Hypersurfaces Waldschmidt Constant Waring Problems

Authors and affiliations

  1. 1.Department of Mathematical SciencesPolitecnico di TorinoTorinoItaly
  2. 2.Department of MathematicsTulane UniversityNew OrleansUSA
  3. 3.Department of MathematicsUniversity of NebraskaLincolnUSA
  4. 4.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Bibliographic information

  • Book Title Ideals of Powers and Powers of Ideals
  • Book Subtitle Intersecting Algebra, Geometry, and Combinatorics
  • Authors Enrico Carlini
    Huy Tài Hà
    Brian Harbourne
    Adam Van Tuyl
  • Series Title Lecture Notes of the Unione Matematica Italiana
  • Series Abbreviated Title Unione Mat.Italiana LN
  • DOI https://doi.org/10.1007/978-3-030-45247-6
  • Copyright Information The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-030-45246-9
  • eBook ISBN 978-3-030-45247-6
  • Series ISSN 1862-9113
  • Series E-ISSN 1862-9121
  • Edition Number 1
  • Number of Pages XIX, 161
  • Number of Illustrations 21 b/w illustrations, 0 illustrations in colour
  • Topics Algebraic Geometry
    Commutative Rings and Algebras
    Number Theory
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking

Reviews

“This is a very interesting monograph providing a fast introduction to different fields of research devoted to modern aspects and develompents of commutative algebra, algebraic geometry, combinatorics, etc.” (Piotr Pokora, zbMATH 1445.13001, 2020)