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.

• The first book to give a rigorous, complete introduction to Cox rings • Combines geometric and arithmetic aspects • The basic theory is developed in a self-contained way and is accompanied by many examples

### Contents

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

### 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.' Jarosław Wiśniewski, 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