##
**Algebraic geometry and arithmetic curves.**
*(English)*
Zbl 0996.14005

Oxford Graduate Texts in Mathematics. 6. Oxford: Oxford University Press. xv, 576 p. (2002).

This textbook originated from notes distributed to the participants of a course on arithmetic algebraic surfaces for graduate students. The aim of that course was to describe the foundations of the geometry of arithmetic surfaces as presented in the works of I. R. Shafarevich [“Lectures on minimal models and birational transformations of two-dimensional schemes”, Tata Institute of Fundamental Research. 37 (1966; Zbl 0164.51704)] and S. Lichtenbaum [Am. J. Math. 90, 380-405 (1968; Zbl 0194.22101)], and also the theory of stable reduction of algebraic curves as developed in the celebrated paper by P. Deligne and M. Mumford on the irreducibility of the moduli spaces of algebraic curves [Publ. Math., Inst. Hautes Étud. Sci. 36, 75-100 (1969; Zbl 0181.48803)].

Alas, in writing up those lecture notes, the author had to face the amazing fact that, in spite of the importance of recent developments in this field, there does not exist any suitable book in the literature that treats these subjects in a comprehensive and systematic manner, and at a level that is accessible to students or other non-specialists in the field.

With a view to this difficulty in teaching advanced arithmetic algebraic geometry, the author decided to let his notes grow into such a still missing, systematic and comprehensive textbook. Thus the aim of the book under review is to gather together all the relevant concepts, methods and results, now classical and absolutely indispensable in arithmetic geometry, in order to make them more easily accessible to a larger audience.

The outcome is a thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting, that is from A. Grothendieck’s modern and most general point of view, followed by a second part on the arithmetic theory of algebraic curves and surfaces.

As to the first, purely algebro-geometric part of the book, it seems fair to say that this is, after A. Grothendieck’s voluminous treatise “Éléments de géométrie algébrique. I–IV” (EGA I–IV), the most comprehensive and detailed elaboration of the theory of algebraic schemes available in (text-)book form, whereas the second, merely arithmetic part provides the very first systematic and coherent introduction to the advanced theory of arithmetic curves and surfaces at all. Moreover, the entire text is arranged in such exhaustive a way that the book is essentially self-contained, keeping the prerequisites at a minimum, and perfectly suitable for seasoned graduate students. Another feature of this highly valuable book on algebraic and arithmetic geometry is provided by the vast amount of illustrating, theoretically important examples as well as by the approximately six hundred included exercises. Now, as to the concrete contents of the book, the text is divided into ten chapters, where the first seven chapters discuss the basic theory of algebraic schemes and their morphisms (à la Grothendieck), while the remaining three chapters are devoted to the arithmetic theory of algebraic curves and surfaces.

Chapter 1 presents some relevant material from higher commutative algebra that is frequently used throughout the entire text. This includes, amongst other topics, a detailed discussion of the concept of flatness and formal completion of a ring.

Chapter 2 gives an introduction to schemes, their general properties, and their dimension theory, whilst chapter 3 turns to the study of morphisms of schemes and their behavior under base change. As for applications of the general theory of schemes, the special case of algebraic varieties over a field is discussed in a separate section of the third chapter. Chapter 4 is devoted to some important local properties of schemes such as normality, smoothness, flat and smooth morphisms, étale morphisms, and Zariski’s Main Theorem. Coherent sheaves and their Čech cohomology is the topic of chapter 5, with a special emphasis on projective schemes. This includes those fundamental results like the behavior of higher direct images of sheaves under flat base change, the connectedness principle (Zariski), and the cohomology of the fibers of a projective morphism. Chapter 6 treats the differential calculus on schemes via Kähler differentials, sheaves of relative differentials, canonical sheaves on smooth schemes, and the according duality theory (à la Grothendieck). In this context, local complete intersections and regular immersions are also discussed. Chapter 7 deals with divisors (Cartier divisors and Weil divisors), including Van der Waerden’s purity theorem, and turns then to the specific theory of algebraic curves over a field. This encompasses the basic standard material: the Riemann-Roch theorem, the Hurwitz formula, hyperelliptic curves, the classification of curves of small genus, singular curves, and Picard varieties of curves.

After this basic course on algebraic geometry, provided by chapters 1-7, the author takes up his original goal of the course, which was to give an introduction to the advanced theory of arithmetic curves and surfaces. To this end, he discusses, in chapter 8, the fundamental framework for the birational geometry of algebraic surfaces. This includes the technique of blowing-up, the notion of excellent schemes, catenary schemes, and the study of fibered surfaces. The latter topic is crucial for the heart piece of the book, namely the study of relative curves over a Dedekind scheme. Chapter 9 is devoted to regular fibered surfaces, i.e., to regular fibered, connected noetherian schemes of dimension two. The author develops the relevant intersection theory on regular surfaces, starting from the local definition, studies then the relation between morphisms of fibered surfaces and intersection numbers, discusses Castelnuovo’s criterion and minimal surfaces in the sequel, and ends this chapter with investigating minimal regular fibered surfaces and canonical models for minimal arithmetic surfaces, including Artin’s contractibility criterion for regular fibered surfaces and Weierstrass models of arithmetic elliptic curves.

The concluding chapter 10 is entitled “Reduction of algebraic curves”. After discussing general properties that essentially follow from the study of arithmetic surfaces conducted before, the author describes the different types of reduction of arithmetic curves in detail (models and reductions, reduction of elliptic curves, stable curves, stable reduction, and stable models). The end of this final chapter is devoted to the stable reduction theorem of Deligne-Mumford and its beautiful proof by H. Artin and G. Winters [Topology 10, 373-383 (1971; Zbl 0221.14018)].

The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research.

With arithmetic geometry in mind, the author has kept the outset of the text as general as possible. In particular, it is almost never supposed that the ground field is algebraically closed, nor of characteristic zero, nor even perfect. The advantage of this approach, which to this extent cannot be found somewhere else in the textbook literature, is that the reader acquires the right (general) reflexes from the beginning on. As for the study of algebraic varieties, there are many other excellent (specific) textbooks that can be consulted. As stated before, this book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive, in addition to all the other mentioned advantages of the book.

Alas, in writing up those lecture notes, the author had to face the amazing fact that, in spite of the importance of recent developments in this field, there does not exist any suitable book in the literature that treats these subjects in a comprehensive and systematic manner, and at a level that is accessible to students or other non-specialists in the field.

With a view to this difficulty in teaching advanced arithmetic algebraic geometry, the author decided to let his notes grow into such a still missing, systematic and comprehensive textbook. Thus the aim of the book under review is to gather together all the relevant concepts, methods and results, now classical and absolutely indispensable in arithmetic geometry, in order to make them more easily accessible to a larger audience.

The outcome is a thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting, that is from A. Grothendieck’s modern and most general point of view, followed by a second part on the arithmetic theory of algebraic curves and surfaces.

As to the first, purely algebro-geometric part of the book, it seems fair to say that this is, after A. Grothendieck’s voluminous treatise “Éléments de géométrie algébrique. I–IV” (EGA I–IV), the most comprehensive and detailed elaboration of the theory of algebraic schemes available in (text-)book form, whereas the second, merely arithmetic part provides the very first systematic and coherent introduction to the advanced theory of arithmetic curves and surfaces at all. Moreover, the entire text is arranged in such exhaustive a way that the book is essentially self-contained, keeping the prerequisites at a minimum, and perfectly suitable for seasoned graduate students. Another feature of this highly valuable book on algebraic and arithmetic geometry is provided by the vast amount of illustrating, theoretically important examples as well as by the approximately six hundred included exercises. Now, as to the concrete contents of the book, the text is divided into ten chapters, where the first seven chapters discuss the basic theory of algebraic schemes and their morphisms (à la Grothendieck), while the remaining three chapters are devoted to the arithmetic theory of algebraic curves and surfaces.

Chapter 1 presents some relevant material from higher commutative algebra that is frequently used throughout the entire text. This includes, amongst other topics, a detailed discussion of the concept of flatness and formal completion of a ring.

Chapter 2 gives an introduction to schemes, their general properties, and their dimension theory, whilst chapter 3 turns to the study of morphisms of schemes and their behavior under base change. As for applications of the general theory of schemes, the special case of algebraic varieties over a field is discussed in a separate section of the third chapter. Chapter 4 is devoted to some important local properties of schemes such as normality, smoothness, flat and smooth morphisms, étale morphisms, and Zariski’s Main Theorem. Coherent sheaves and their Čech cohomology is the topic of chapter 5, with a special emphasis on projective schemes. This includes those fundamental results like the behavior of higher direct images of sheaves under flat base change, the connectedness principle (Zariski), and the cohomology of the fibers of a projective morphism. Chapter 6 treats the differential calculus on schemes via Kähler differentials, sheaves of relative differentials, canonical sheaves on smooth schemes, and the according duality theory (à la Grothendieck). In this context, local complete intersections and regular immersions are also discussed. Chapter 7 deals with divisors (Cartier divisors and Weil divisors), including Van der Waerden’s purity theorem, and turns then to the specific theory of algebraic curves over a field. This encompasses the basic standard material: the Riemann-Roch theorem, the Hurwitz formula, hyperelliptic curves, the classification of curves of small genus, singular curves, and Picard varieties of curves.

After this basic course on algebraic geometry, provided by chapters 1-7, the author takes up his original goal of the course, which was to give an introduction to the advanced theory of arithmetic curves and surfaces. To this end, he discusses, in chapter 8, the fundamental framework for the birational geometry of algebraic surfaces. This includes the technique of blowing-up, the notion of excellent schemes, catenary schemes, and the study of fibered surfaces. The latter topic is crucial for the heart piece of the book, namely the study of relative curves over a Dedekind scheme. Chapter 9 is devoted to regular fibered surfaces, i.e., to regular fibered, connected noetherian schemes of dimension two. The author develops the relevant intersection theory on regular surfaces, starting from the local definition, studies then the relation between morphisms of fibered surfaces and intersection numbers, discusses Castelnuovo’s criterion and minimal surfaces in the sequel, and ends this chapter with investigating minimal regular fibered surfaces and canonical models for minimal arithmetic surfaces, including Artin’s contractibility criterion for regular fibered surfaces and Weierstrass models of arithmetic elliptic curves.

The concluding chapter 10 is entitled “Reduction of algebraic curves”. After discussing general properties that essentially follow from the study of arithmetic surfaces conducted before, the author describes the different types of reduction of arithmetic curves in detail (models and reductions, reduction of elliptic curves, stable curves, stable reduction, and stable models). The end of this final chapter is devoted to the stable reduction theorem of Deligne-Mumford and its beautiful proof by H. Artin and G. Winters [Topology 10, 373-383 (1971; Zbl 0221.14018)].

The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research.

With arithmetic geometry in mind, the author has kept the outset of the text as general as possible. In particular, it is almost never supposed that the ground field is algebraically closed, nor of characteristic zero, nor even perfect. The advantage of this approach, which to this extent cannot be found somewhere else in the textbook literature, is that the reader acquires the right (general) reflexes from the beginning on. As for the study of algebraic varieties, there are many other excellent (specific) textbooks that can be consulted. As stated before, this book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive, in addition to all the other mentioned advantages of the book.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14Exx | Birational geometry |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14A15 | Schemes and morphisms |