Description
Cox rings are significant global invariants of algebraic varieties, naturally generalizing homogeneous coordinate rings of projective spaces. This book provides a largely self-contained introduction to Cox rings, with a particular focus on concrete aspects of the theory. Besides the rigorous presentation of the basic concepts, other central topics include the case of finitely generated Cox rings and its relation to toric geometry; various classes of varieties with group actions; the surface case; and applications in arithmetic problems, in particular Manin's conjecture. The introductory chapters require only basic knowledge in algebraic geometry. The more advanced chapters also touch on algebraic groups, surface theory, and arithmetic geometry. Each chapter ends with exercises and problems. These comprise mini-tutorials and examples complementing the text, guided exercises for topics not discussed in the text, and, finally, several open problems of varying difficulty.
This book provides a largely self-contained introduction to Cox rings and their applications in algebraic and arithmetic geometry.
About the Author
Ivan Arzhantsev received his doctoral degree in 1998 from Lomonosov Moscow State University and is a professor in its department of higher algebra. His research areas are algebraic geometry, algebraic groups and invariant theory. Ulrich Derenthal received his doctoral degree in 2006 from Universitat Goettingen. He is a professor of mathematics at Ludwig-Maximilians-Universitat Munchen. His research interests include arithmetic geometry and number theory. Jurgen Hausen received his doctoral degree in 1995 from Universitat Konstanz. He is a professor of mathematics at Eberhard-Karls-Universitat Tubingen. His field of research is algebraic geometry, in particular algebraic transformation groups, torus actions, geometric invariant theory and combinatorial methods. Antonio Laface received his doctoral degree in 2000 from Universita degli Studi di Milano. He is an associate professor of mathematics at Universidad de Concepcion. His field of research is algebraic geometry, more precisely linear systems and algebraic surfaces and their Cox rings.
Reviews
'An excellent introduction to the subject, featuring a wide selection of topics, careful exposition, and many examples and exercises.' David Cox, University of Massachusetts, Amherst
'This book is a detailed account of virtually every aspect of the general theory of the Cox ring of an algebraic variety. After a thorough introduction it takes the reader on an impressive tour through toric geometry, geometric invariant theory, Mori dream spaces, and universal torsors, culminating with applications to the Manin conjecture on rational points. The many worked examples and exercises make it not just a comprehensive reference, but also an excellent introduction for graduate students.' Alexei Skorobogatov, Imperial College London
'This book provides the first comprehensive treatment of Cox rings. Firstly, its broad and complete exposition of the fundamentals of the general theory will be appreciated by both those who want to learn the subject and specialists seeking an ultimate reference on many subtle aspects of the theory. Secondly, it introduces readers to the most important applications that have developed in the past decade and will define the direction of research in the years to come.' Jaroslaw Wisniewski, Institute of Mathematics, University of Warsaw
'Cox rings are very important in modern algebraic and arithmetic geometry. This book, providing a comprehensive introduction to the theory and applications of Cox rings from the basics up to, and including, very complicated technical points and particular problems, aims at a wide readership of more or less everyone working in the areas where Cox rings are used ... This book is very useful for everyone working with Cox rings, and especially useful for postgraduate students learning the subject.' Alexandr V. Pukhlikov, Mathematical Reviews
Book Information
ISBN 9781107024625
Author Ivan Arzhantsev
Format Hardback
Page Count 472
Imprint Cambridge University Press
Publisher Cambridge University Press
Weight(grams) 850g
Dimensions(mm) 235mm * 158mm * 34mm
This book provides a largely self-contained introduction to Cox rings and their applications in algebraic and arithmetic geometry.
About the Author
Ivan Arzhantsev received his doctoral degree in 1998 from Lomonosov Moscow State University and is a professor in its department of higher algebra. His research areas are algebraic geometry, algebraic groups and invariant theory. Ulrich Derenthal received his doctoral degree in 2006 from Universitat Goettingen. He is a professor of mathematics at Ludwig-Maximilians-Universitat Munchen. His research interests include arithmetic geometry and number theory. Jurgen Hausen received his doctoral degree in 1995 from Universitat Konstanz. He is a professor of mathematics at Eberhard-Karls-Universitat Tubingen. His field of research is algebraic geometry, in particular algebraic transformation groups, torus actions, geometric invariant theory and combinatorial methods. Antonio Laface received his doctoral degree in 2000 from Universita degli Studi di Milano. He is an associate professor of mathematics at Universidad de Concepcion. His field of research is algebraic geometry, more precisely linear systems and algebraic surfaces and their Cox rings.
Reviews
'An excellent introduction to the subject, featuring a wide selection of topics, careful exposition, and many examples and exercises.' David Cox, University of Massachusetts, Amherst
'This book is a detailed account of virtually every aspect of the general theory of the Cox ring of an algebraic variety. After a thorough introduction it takes the reader on an impressive tour through toric geometry, geometric invariant theory, Mori dream spaces, and universal torsors, culminating with applications to the Manin conjecture on rational points. The many worked examples and exercises make it not just a comprehensive reference, but also an excellent introduction for graduate students.' Alexei Skorobogatov, Imperial College London
'This book provides the first comprehensive treatment of Cox rings. Firstly, its broad and complete exposition of the fundamentals of the general theory will be appreciated by both those who want to learn the subject and specialists seeking an ultimate reference on many subtle aspects of the theory. Secondly, it introduces readers to the most important applications that have developed in the past decade and will define the direction of research in the years to come.' Jaroslaw Wisniewski, Institute of Mathematics, University of Warsaw
'Cox rings are very important in modern algebraic and arithmetic geometry. This book, providing a comprehensive introduction to the theory and applications of Cox rings from the basics up to, and including, very complicated technical points and particular problems, aims at a wide readership of more or less everyone working in the areas where Cox rings are used ... This book is very useful for everyone working with Cox rings, and especially useful for postgraduate students learning the subject.' Alexandr V. Pukhlikov, Mathematical Reviews
Book Information
ISBN 9781107024625
Author Ivan Arzhantsev
Format Hardback
Page Count 472
Imprint Cambridge University Press
Publisher Cambridge University Press
Weight(grams) 850g
Dimensions(mm) 235mm * 158mm * 34mm
