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Arithmetically Cohen-Macaulay sets of points in P^1 x P^1 / by Elena Guardo, Adam Van Tuyl.
Springer Nature - Springer Mathematics and Statistics eBooks 2015 English International Available online
View online- Format:
- Book
- Author/Creator:
- Guardo, Elena, Author.
- Van Tuyl, Adam, Author.
- Series:
- SpringerBriefs in Mathematics, 2191-8198
- Language:
- English
- Subjects (All):
- Commutative algebra.
- Commutative rings.
- Geometry, Algebraic.
- Geometry, Projective.
- Commutative Rings and Algebras.
- Algebraic Geometry.
- Projective Geometry.
- Local Subjects:
- Commutative Rings and Algebras.
- Algebraic Geometry.
- Projective Geometry.
- Physical Description:
- 1 online resource (136 p.)
- Edition:
- 1st ed. 2015.
- Place of Publication:
- Cham : Springer International Publishing : Imprint: Springer, 2015.
- Language Note:
- English
- 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.
- 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.
- Notes:
- Description based upon print version of record.
- Includes bibliographical references and index.
- ISBN:
- 3-319-24166-4
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