##
**Algebraic surfaces. Transl. from the Romanian by V. Maşek.**
*(English)*
Zbl 0965.14001

Universitext. New York, NY: Springer. x, 258 p. (2001).

The book under review is the translation into English of the Romanian original (Bucuresti 1981; Zbl 0583.14011). Back then, twenty years ago, there were scarcely any tolerably modern textbooks on the contemporary theory of algebraic surfaces, although especially this part of algebraic geometry had undergone a tremendous development during the second half of the 20-th century. Of course, there were the then recent books of I. R. Shafarevich and his students [I. R. Shafarevich, B. G. Averbukh, Yu. R. Vajnbert, A. B. Zhizhchenko, Yu. I. Manin, B. G. Mojshezon, G. N. Tjurina and A. N. Tjurin, “Algebraic surfaces”, Proc. Steklov Inst. Math. 75 (1967); translation from Trudy Mat. Inst. Steklova 75 (1965; Zbl 0154.21001)], P. Griffiths and J. Harris [“Principles of algebraic geometry” (New York 1978; Zbl 0408.14001)] and A. Beauville [“Surfaces algébriques complexes”, Astérisque 54 (1978; Zbl 0394.14014)] which covered large parts of the theory of complex algebraic surfaces, but did not equally reflect the then recent achievements in the study of algebraic surfaces over an algebraically closed field of arbitrary characteristic, in particular the Bombieri-Mumford approach to a general Enriques classification theory for algebraic surfaces.

The author’s book, a treatise that was awarded an academic prize in Romania back then, was probably the first textbook in the literature to fill this gap at the introductory level. Unfortunately, it has not been translated from Romanian into English, during the following twenty years, and has therefore only been accessible for a rather limited mathematical audience. As this excellent textbook on general algebraic surfaces has not suffered loss from its actuality or significance, over the past two decades, it is very gratifying to see that it finally has conquered its deserved place within the international literature in algebraic geometry, thank to the now existing English translation.

The main goal of this book is (and was) to introduce the reader to basic algebraic surface theory by means of a completely algebraic approach, which makes it possible to treat the Enriques classification of surfaces in arbitrary characteristic just as well and within a unified framework.

In contrast to the Romanian original, the English translation comes with two additional chapters as well as with sets of exercises provided at the end of each chapter. The main body of the book is its second part, which deals with the unifying Bombieri-Mumford approach to the classification of algebraic surfaces in arbitrary characteristic. The first part, consisting of the first six chapters, provides numerous basic facts and methods from the theory of algebraic surfaces, mainly in their algebraic setting and with a view towards their significance in classification theory, but also by including any other aspects that are interesting in themselves. More precisely, the text is structured as follows:

Chapter 1: Cohomological intersection theory and the Nakai-Moishezon criterion of ampleness; Chapter 2: The Hodge index theorem and the structure of the intersection matrix of a fiber;

Chapter 3: Criteria of contractibility and rational singularities;

Chapter 4: Properties of rational singularities;

Chapter 5: Noether’s formulae, the Picard scheme, the Albanese variety, and plurigenera;

Chapter 6: Existence of minimal models;

Chapter 7: Morphisms from a surface to a curve, elliptic and quasielliptic fibrations;

Chapter 8: Canonical dimension of an elliptic or quasielliptic fibration;

Chapter 9: The classification theorem according to canonical dimension;

Chapter 10: Surfaces with canonical dimension zero;

Chapter 11: Ruled surfaces and the Noether-Tsen criterion;

Chapter 12: Minimal models of ruled surfaces;

Chapter 13: Characterization of ruled and rational surfaces;

Chapter 14: Zariski decomposition and applications;

Chapter 15: Appendix (hints for further reading).

The concluding chapter 14 has been added to the original text. The author presents here Zariski’s theory of finite generation of the graded algebra associated to a divisor on a surface, together with some more recent developments related to this theory. Another updating is provided by the appendix (chapter 15), which however is very short and restricted to hints for further reading. These concern those themes in algebraic surface theory which have undergone substantial progress in the last twenty years and, on the other hand, the mentioning of those monographs on algebraic surfaces which have appeared in the meantime. In view of the very algebraic approach described in the book under review, it should be mentioned that the very recent monograph “Open algebraic surfaces” by M. Miyanishi [Providence (2001; Zbl 0964.14001)], is partly related, and may be seen as another possibility for further reading too.

Altogether, this finally translated monograph is a textbook on algebraic surfaces at the advanced level. The reader is required to have a profound knowledge of basic algebraic geometry, including sheaf cohomology and some singularity theory, but taken that for granted, the text is fairly self-contained. The exposition is very clear, thorough, rigorous, elegant, and positively creative. This makes the book into a still very valuable source for graduate students in algebraic geometry, teachers in this field, and even for researchers in algebraic geometry and related areas in mathematics. Without any doubt, this is one of the very best books on algebraic surfaces, now as before.

The author’s book, a treatise that was awarded an academic prize in Romania back then, was probably the first textbook in the literature to fill this gap at the introductory level. Unfortunately, it has not been translated from Romanian into English, during the following twenty years, and has therefore only been accessible for a rather limited mathematical audience. As this excellent textbook on general algebraic surfaces has not suffered loss from its actuality or significance, over the past two decades, it is very gratifying to see that it finally has conquered its deserved place within the international literature in algebraic geometry, thank to the now existing English translation.

The main goal of this book is (and was) to introduce the reader to basic algebraic surface theory by means of a completely algebraic approach, which makes it possible to treat the Enriques classification of surfaces in arbitrary characteristic just as well and within a unified framework.

In contrast to the Romanian original, the English translation comes with two additional chapters as well as with sets of exercises provided at the end of each chapter. The main body of the book is its second part, which deals with the unifying Bombieri-Mumford approach to the classification of algebraic surfaces in arbitrary characteristic. The first part, consisting of the first six chapters, provides numerous basic facts and methods from the theory of algebraic surfaces, mainly in their algebraic setting and with a view towards their significance in classification theory, but also by including any other aspects that are interesting in themselves. More precisely, the text is structured as follows:

Chapter 1: Cohomological intersection theory and the Nakai-Moishezon criterion of ampleness; Chapter 2: The Hodge index theorem and the structure of the intersection matrix of a fiber;

Chapter 3: Criteria of contractibility and rational singularities;

Chapter 4: Properties of rational singularities;

Chapter 5: Noether’s formulae, the Picard scheme, the Albanese variety, and plurigenera;

Chapter 6: Existence of minimal models;

Chapter 7: Morphisms from a surface to a curve, elliptic and quasielliptic fibrations;

Chapter 8: Canonical dimension of an elliptic or quasielliptic fibration;

Chapter 9: The classification theorem according to canonical dimension;

Chapter 10: Surfaces with canonical dimension zero;

Chapter 11: Ruled surfaces and the Noether-Tsen criterion;

Chapter 12: Minimal models of ruled surfaces;

Chapter 13: Characterization of ruled and rational surfaces;

Chapter 14: Zariski decomposition and applications;

Chapter 15: Appendix (hints for further reading).

The concluding chapter 14 has been added to the original text. The author presents here Zariski’s theory of finite generation of the graded algebra associated to a divisor on a surface, together with some more recent developments related to this theory. Another updating is provided by the appendix (chapter 15), which however is very short and restricted to hints for further reading. These concern those themes in algebraic surface theory which have undergone substantial progress in the last twenty years and, on the other hand, the mentioning of those monographs on algebraic surfaces which have appeared in the meantime. In view of the very algebraic approach described in the book under review, it should be mentioned that the very recent monograph “Open algebraic surfaces” by M. Miyanishi [Providence (2001; Zbl 0964.14001)], is partly related, and may be seen as another possibility for further reading too.

Altogether, this finally translated monograph is a textbook on algebraic surfaces at the advanced level. The reader is required to have a profound knowledge of basic algebraic geometry, including sheaf cohomology and some singularity theory, but taken that for granted, the text is fairly self-contained. The exposition is very clear, thorough, rigorous, elegant, and positively creative. This makes the book into a still very valuable source for graduate students in algebraic geometry, teachers in this field, and even for researchers in algebraic geometry and related areas in mathematics. Without any doubt, this is one of the very best books on algebraic surfaces, now as before.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

14Jxx | Surfaces and higher-dimensional varieties |

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

14J17 | Singularities of surfaces or higher-dimensional varieties |

14D06 | Fibrations, degenerations in algebraic geometry |

14J10 | Families, moduli, classification: algebraic theory |