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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.
List of contents
Introduction.- The Biprojective Space P^1 x P^1.- Points in P^1 x P^1.- Classification of ACM Sets of Points in P^1 x P^1.- Homological Invariants.- Fat Points in P^1 x P^1.- Double Points and Their Resolution.- Applications.- References.
About the author
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.
Summary
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.
Additional text
“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)
Report
"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)
