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Ideals of Powers and Powers of Ideals

Intersecting Algebra, Geometry, and Combinatorics

  • Book
  • © 2020

Overview

Authors:
  1. Enrico Carlini

    1. Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

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  2. Huy Tài Hà

    1. Department of Mathematics, Tulane University, New Orleans, USA

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  3. Brian Harbourne

    1. Department of Mathematics, University of Nebraska, Lincoln, USA

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  4. Adam Van Tuyl

    1. Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

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  • First book to contain a summary of known results on the associated primes of edge ideals
  • First book to contain a summary of known results on the regularity of powers of monomial ideals
  • Contains open problems for graduate students
  • With a chapter for early career researchers on "how to do mathematics research"
  • Provides an up-to-date and comprehensive list of references of papers in the area
  • Written by authors who have made numerous contributions to these areas
  • First book to introduce the ideal containment problem (e.g. Waldschmidt contstant resurgence symbolic defect)

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

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Table of contents (20 chapters)

  1. Front Matter

    Pages i-xix
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  2. Associated Primes of Powers of Ideals

    1. Front Matter

      Pages 1-1
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    2. Associated Primes of Powers of Ideals

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 3-11

    3. Associated Primes of Powers of Squarefree Monomial Ideals

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 13-32

    4. Final Comments and Further Reading

      • 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
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    2. Regularity of Powers of Ideals and the Combinatorial Framework

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 37-49

    3. Problems, Questions, and Inductive Techniques

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 51-58

    4. Examples of the Inductive Techniques

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 59-66

    5. Final Comments and Further Reading

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 67-67

  4. The Containment Problem

    1. Front Matter

      Pages 69-69
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    2. The Containment Problem: Background

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 71-75

    3. The Containment Problem

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 77-80

    4. The Waldschmidt Constant of Squarefree Monomial Ideals

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 81-88

    5. Symbolic Defect

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 89-98

    6. Final Comments and Further Reading

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 99-99

  5. Unexpected Hypersurfaces

    1. Front Matter

      Pages 101-101
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    2. Unexpected Hypersurfaces

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 103-110

    3. Final Comments and Further Reading

      • Enrico Carlini, Huy Tài Hà, Brian Harbourne, Adam Van Tuyl
      Pages 111-111

  6. Waring Problems

    1. Front Matter

      Pages 113-113
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Keywords

About this book

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.

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)

Authors and Affiliations

  • Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Enrico Carlini

  • Department of Mathematics, Tulane University, New Orleans, USA

    Huy Tài Hà

  • Department of Mathematics, University of Nebraska, Lincoln, USA

    Brian Harbourne

  • Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

    Adam Van Tuyl

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

  • DOI: https://doi.org/10.1007/978-3-030-45247-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

  • Softcover ISBN: 978-3-030-45246-9Published: 22 May 2020

  • eBook ISBN: 978-3-030-45247-6Published: 21 May 2020

  • Series ISSN: 1862-9113

  • Series E-ISSN: 1862-9121

  • Edition Number: 1

  • Number of Pages: XIX, 161

  • Number of Illustrations: 21 b/w illustrations

  • Topics: Algebraic Geometry, Commutative Rings and Algebras, Number Theory

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