Cox Rings

This book provides a largely self-contained introduction to Cox rings and their applications in algebraic and arithmetic geometry.

Ivan Arzhantsev (Author), Ulrich Derenthal (Author), Jürgen Hausen (Author), Antonio Laface (Author)

9781107024625, Cambridge University Press

Hardback, published 29 August 2014

472 pages, 19 tables 120 exercises
23.5 x 15.8 x 3.4 cm, 0.85 kg

'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

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.

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

Subject Areas: Algebraic geometry [PBMW], Geometry [PBM], Algebra [PBF], Mathematics [PB]