In hindsight one wonders why no one had noticed that the way a projective variety is embedded in different projective spaces can be collected in a natural graded sheaf, whose ring of global homogeneous sections generalize the homogeneous coordinate ring of projective space with its standard grading. David Cox (*The homogeneous coordinate ring of a toric variety. *J. Alg. Geom. 4, 17–50, 1995) introduced this ring, and computed it, for the case of toric varieties.

As usually happens when a fundamental new idea is introduced, there were several instances where this concept and its properties provided a natural framework to formulate or interpret some advances in related areas. One such instance was previous work in the 1970s by Colliot-Thélène and Sansuc who introduced universal torsors in arithmetic geometry to study rational points on algebraic varieties, where the explicit representation of these torsors in terms of Cox rings is now at the center of a research program, for example around a conjecture of Manin on an asymptotic bound for the set of rational points on some projective varieties. Another such instance was initiated in the work of Hu and Keel in the 1990s on the unexpected relation between geometric invariant theory and Mori’s minimal model program afforded by the finite generation of the corresponding Cox ring. These two examples illustrate, but not exhaust, the interest of Cox rings in two mainstream branches of mathematics. This well-written monograph is a timely publication and may perhaps help to introduce this topic to a broader audience. I must qualify the term *monograph*, since this book could as well serve as a textbook at the graduate level for its detailed presentation, carefully chosen examples, and lists of exercises at the end of each of its chapters.

The book starts by assembling some general facts on algebras graded by commutative monoids, algebraic (quasi) tori actions, and their quotients. Cox rings are then introduced, as global sections of a sheaf built from representatives of divisor classes of the given normal variety with only constant invertible functions and finitely generated divisor class group. Then, some algebraic properties of the Cox ring are proven. The first chapter ends with some properties of the relative spectrum of the Cox sheaf, its char*acteristic space*. The main result is a presentation of the underlying variety as a quotient of its characteristic space by the action of a quasitorus. For factorial varieties this is the universal torsor of Colliot-Thélène and Sansuc.

Chapter two treats in detail the original case of toric varieties. Toric varieties are fundamental in the theory, since if any finitely generated Cox ring is a quotient of the Cox ring of some toric variety. This fact allows some of the combinatorial concrete descriptions of a toric variety and its Cox ring to be translated into combinatorial descriptions of some invariants of the given variety. Chapter three is devoted to this approach via geometric invariant theory.

Chapter four is a collection of various topics on Cox rings, for example on the Cox ring of a variety embedded into another variety in terms of the Cox ring of the ambient variety. Another such topic is the computation of the Cox ring of a variety obtained by some birational morphism from another variety. Several other important topics are also covered, such as quotient presentations or general criteria for finite generation of the Cox ring.

Chapter five is devoted to the special case of Cox rings of surfaces. The results here are more concrete and detailed. Examples are abundant, including results on Cox rings of del Pezzo, K3 and some Hirzebruch surfaces.

The last chapter focuses on the arithmetic side of the theory, starting with the link between Cox rings and universal torsors. The main underlying thread is the existence of rational points, over a given number field, of some projective varieties defined over that field. The main considered aspects include the Brauer-Manin obstruction to the Hasse principle, the distribution of rational points on varieties over number fields, specifically Manin’s conjecture on an asymptotic bound in terms of height functions and geometric invariants of Fano varieties. The authors survey several important examples, giving many details of this approach via universal torsors.

Although the book is mainly addressed to graduate students or researchers working in related areas, it could be used with profit as an introduction to these important areas by anyone with a basic background on algebraic geometry. The exposition is detailed, illustrated with carefully chosen examples, and enriched with exercises in each of its chapters.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is [email protected].