Arithmetically Cohen-Macaulay sets of points in P¹ × P¹ [electronic resource] / Elena Guardo, Adam Van Tuyl.

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Arithmetically Cohen-Macaulay sets of points in P¹ × P¹ [electronic resource] / Elena Guardo, Adam Van Tuyl.

9783319241661 electronic book
3319241664 electronic book
9783319241647

Cham : Springer, [2015].

©2015

English

1 online resource.

QA251.3

512/.44

This brief presents a solution to the interpolation problem for arithmetically Cohen-Macaulay (ACM) sets of points in the multiprojective space P̂1 x P̂1. 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 P̂1 x P̂1, the bigraded Hilbert function, and ACM sets of points. The authors then show how, using a combinatorial description of ACM points in P̂1 x P̂1, 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 P̂1 x P̂1 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.

Includes bibliographical references and index.

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