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Arithmetically Cohen-Macaulay Sets of Points in P^1 x P^1

  • Elena Guardo
  • Adam Van Tuyl

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Elena Guardo, Adam Van Tuyl
    Pages 1-6
  3. Elena Guardo, Adam Van Tuyl
    Pages 19-39
  4. Elena Guardo, Adam Van Tuyl
    Pages 53-69
  5. Elena Guardo, Adam Van Tuyl
    Pages 71-97
  6. Elena Guardo, Adam Van Tuyl
    Pages 99-115
  7. Elena Guardo, Adam Van Tuyl
    Pages 117-125
  8. Back Matter
    Pages 127-134

About this book

Introduction

This brief presents a solution to the interpolation problem for arithmetically Cohen-Macaulay (ACM) sets of points in the multiprojective space P^1 x P^1.  It collects the various current threads in the literature on this topic with the aim of providing a self-contained, unified introduction while also advancing some new ideas.  The relevant constructions related to multiprojective spaces are reviewed first, followed by the basic properties of points in P^1 x P^1, the bigraded Hilbert function, and ACM sets of points.  The authors then show how, using a combinatorial description of ACM points in P^1 x P^1, the bigraded Hilbert function can be computed and, as a result, solve the interpolation problem.  In subsequent chapters, they consider fat points and double points in P^1 x P^1 and demonstrate how to use their results to answer questions and problems of interest in commutative algebra.  Throughout the book, chapters end with a brief historical overview, citations of related results, and, where relevant, open questions that may inspire future research.  Graduate students and researchers working in algebraic geometry and commutative algebra will find this book to be a valuable contribution to the literature.

Keywords

Arithmetically Cohen-Macaulay Sets of Points Cohen-Macaulay Ring Fat Points Hilbert Functions Minimal free graded resolutions Mulitprojective Space

Authors and affiliations

  • Elena Guardo
    • 1
  • Adam Van Tuyl
    • 2
  1. 1.University of Catania, Dipartimento di Matematica e InformaticaUniversity of Catania, Dipartimento di Matematica e InformaticaCataniaItaly
  2. 2.Dept of Mathematics and Statistics, McMaster UniversityDept of Mathematics and Statistics, McMaster UniversityHamiltonCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-24166-1
  • Copyright Information The Authors 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-24164-7
  • Online ISBN 978-3-319-24166-1
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • Buy this book on publisher's site