Cox Rings

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. '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 WarsawIntroduction; 1. Basic concepts; 2. Toric varieties and Gale duality; 3. Cox rings and combinatorics; 4. Selected topics; 5. Surfaces; 6. Arithmetic applications.