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

1. Front Matter

   Pages i-viii

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2. Introduction

   • Elena Guardo, Adam Van Tuyl

   Pages 1-6

3. The biprojective space \(\mathbb{P}^{1} \times \mathbb{P}^{1}\)

   • Elena Guardo, Adam Van Tuyl

   Pages 7-17

4. Points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\)

   • Elena Guardo, Adam Van Tuyl

   Pages 19-39

5. Classification of ACM sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\)

   • Elena Guardo, Adam Van Tuyl

   Pages 41-52

6. Homological invariants

   • Elena Guardo, Adam Van Tuyl

   Pages 53-69

7. Fat points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\)

   • Elena Guardo, Adam Van Tuyl

   Pages 71-97

8. Double points and their resolution

   • Elena Guardo, Adam Van Tuyl

   Pages 99-115

9. Applications

   • Elena Guardo, Adam Van Tuyl

   Pages 117-125

10. Back Matter

    Pages 127-134

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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.

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

“The present monograph is nicely written, contains some interesting examples, and almost self-contained. It provides a nice introduction to the interpolation problem for products of projective spaces. In my view, this monograph is adequate for advanced undergraduate students. … I can fully recommend this monograph.” (Piotr Pokora, zbMATH 1346.13001, 2016)

• University of Catania, Dipartimento di Matematica e Informatica, University of Catania, Dipartimento di Matematica e Informatica, Catania, Italy

  Elena Guardo

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

  Adam Van Tuyl

Elena Guardo, PhD, is an Associate Professor in the Department of Mathematics and Computer Sciences at the University of Catania in Italy.
Adam Van Tuyl, PhD, is an Associate Professor in the Department of Mathematics and Statistics at McMaster University in Hamilton, Ontario, Canada.

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