Algebraic Geometry and Arithmetic Curves
The main topics of this book are arithmetic surfaces and reductions of algebraic curves. But these topics are systematically treated only in chapters 8, 9, and 10. The aim of the first seven chapters is to educate the reader in the theory of schemes. The author devotes the whole first chapter to commutative algebra. Nevertheless, one chapter on algebra is not sufficient for the purposes of algebraic geometry and the author has to make digressions and introduce further algebraic notions and results several times. (It is interesting to mention that the author had the idea to present algebraic aspects in a simpler form; he succeeded in avoiding homological algebra.) Thanks to these algebraic parts, the book is rather self-contained.
Next the theory of schemes is studied. This presentation is excellent indeed. It is not a survey but a systematic and thorough theory of schemes covering many aspects of them. In spite of many necessary details, the text reads very well and moreover, we are not lost in this rich theory and it is possible to keep a certain global overview. The author includes a substantial number of examples, which is very helpful for the reader. Every section is followed by a long series of exercises. They are of two kinds. Some of them are designed to increase understanding of the text, the others extend and complete the theory. These seven chapters can be strongly recommended as a basis for a course on the theory of schemes. I agree with the author that even a good undergraduate student can learn a lot from this book.
The second part uses techniques developed in the first seven chapters. It begins with a description of blowing-ups. Then fibered surfaces over a Dedekind ring and desingularizations of surfaces are studied. The next topic is intersection theory on an arithmetic surface. The last chapter is devoted to the reduction theory of algebraic curves. Special attention is paid to elliptic curves. At the end, stable curves and stable reductions are treated; in particular, the Artin-Winter proof of the stable reduction theorem of Deligne-Mumford. Some concrete computations are also presented. These last chapters introduce the reader to contemporary research. He or she will also find hints for further reading.