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Algebraic geometry. A first course. (English) Zbl 0779.14001
Graduate Texts in Mathematics. 133. Berlin etc.: Springer-Verlag. xix, 328 p. (1992).
Algebraic geometry has undergone a tremendous development over the last century. In its first stage, during the 19th century, algebraic geometry mainly consisted in studying concrete varieties in projective space by purely geometric constructions. This “classical” approach reached its culmination in the work of the Italian school around the turn of the century, when it ultimately became apparent that the purely geometric language and techniques of the subject were exceedingly exploited and a deeper foundation was needed. This was done in the twenties of our century, basically by O. Zariski and, independently, by A. Weil who (under the influence of the German school of abstract algebraists) put the whole subject of algebraic geometry on a firm algebraic foundation. Finally, using the new framework of sheaf theory and cohomology, J.-P. Serre and (above all) A. Grothendieck invented an even more powerful, radical new conceptual and methodical footing for algebraic geometry. Grothendieck’s theory of schemes, developed in the 1960’s, is since then (and for now) an extremely rich and striking tool in algebraic geometry, arithmetical geometry, algebraic number theory, complex-analytic geometry, hermitean differential geometry and nowadays also mathematical physics. Modern algebraic geometry, in its scheme-theoretic foundation, has flourished enormously over the past thirty years, and much of the brilliant, advanced work in classical (Italian) algebraic geometry could be firmly re-established and carried further. Thus everybody who wants to study algebraic geometry today will find himself in a position of having to acquire a vast spectrum of concepts, methods and powerful tools. This certainly raises a didactical problem: what is the best way to study (or to teach) topics in algebraic geometry? In the meantime, many outstanding textbooks on algebraic geometry, written from different viewpoints, on different levels and for different purposes, are available. However, most of them introduce the modern approach from the beginning on and develop the whole subject to be treated in these terms. This is perfect for someone whose ultimate goal is to work in that field, or to apply its methods precisely.
If, on the other hand, someone just wants to know what algebraic geometry is about, what kind of objects (and their properties) are studied, what sort of results one can obtain, and whether it is worth to learn the sophisticated, vast framework of modern algebraic geometry systematically, then it might be better to start with the classical geometric part of the theory, in its modern setting, and to avoid the (perhaps discouraging) technical side for the beginning. – This latter aspect is the guiding issue for the present textbook. It represents the author’s very welcome attempt at giving an introduction to algebraic geometry from the more geometric and elementary algebraic point of view, stressing the still glorious classical aspects with their fascinating wealth of concrete examples, deep problems and beautiful intrinsic structure. The book grew out of his various courses on the subject given at Harvard and Brown University, during the past decade, and the circulating manuscript of it was popular and used long before this book appeared.
This textbook is unique, both in its (just explained) aim and its content, and it certainly fills a gap in the current literature in algebraic geometry. It is really a “very first” introduction to algebraic geometry, omitting any fancy algebraic, sheaf-theoretic, scheme-theoretic, or cohomological framework, but nevertheless it is not elementary altogether. The text leads the reader from the discussion and illustration of various kinds of projective varieties and their correspondences to the frontiers of current research in the field, provided by topics such as classifying spaces for certain types of algebraic varieties (moduli spaces), families of varieties and their parameter spaces (Chow varieties, Hilbert varieties), geometric invariant theory and algebraic groups. All this is done with an absolute minimum of technical machinery, but (instead) with an extremely skillful emphasis on the geometric ideas behind everything, on typical examples and encroaching links. Many examples are dealt with several times, in the light of each new conceptual development in the text, and the reader is invited to study them more thoroughly by working on the numerous exercises (and the hints to them). As for the prerequisites from commutative algebra, it will suffice to use one of the small textbooks simultaneously, for example the concise introduction by M. F. Atiyah and I. G. MacDonald [“Introduction to commutative algebra” (1969; Zbl 0175.036)] or, even better, the forthcoming book by D. Eisenbud “Commutative algebra: With a view towards algebraic geometry” (in press), which has just been written as the algebraic counterpart to the author’s present textbook on algebraic geometry. On the other hand, as for further reading, D. Eisenbud and the author have already published the book “Schemes: The language of modern algebraic geometry” [(1992; Zbl 0745.14002)], which provides a brief introduction to Grothendieck’s theory of algebraic schemes. However, any of the recent advanced books on algebraic geometry will be perfectly suited for continued studies, and the reader of the author’s present introductory text will certainly be well-equipped with a profound geometric background, concrete examples, motivation, and appreciation for the subject to be studied.
The content of the present book is of great ampleness, much too stratified to be discussed in detail here. Basically, the text is divided into two main parts. Part one, entitled “Examples of varieties and maps”, is devoted to introducing basic varieties (such as affine and projective varieties, Grassmannians, cones, determinantal varieties, secant varieties, quadrics, flag varieties, etc.) and maps (regular maps, projections, incidence correspondences, rational and birational maps, etc.), whereas part two, entitled “Attributes of varieties”, is concerned with the basic notions associated with varieties (such as degree, dimension, smoothness, tangent spaces, Hilbert functions, Gauss maps, parameter spaces, moduli, etc.) and their significance.
The author even explains assertions and theorems whose proofs could not be given, because they are much beyound the scope of the text, but whose statements are already enlightening and inspiring. The book contains a lot of material that cannot be found in other places in the textbook literature (so far), for example such topics like Fano varieties (of determinantal varieties), their tangent varieties, varieties of secant lines, varieties of incident planes, etc. Such varieties are objects of intense research, and it is very inspiring, even for the actively working algebraic geometer, to have them discussed in a modern, user-friendly textbook.
Altogether, the present work is a highly welcome enrichment of the textbook literature in algebraic geometry. It helps to make the existing, more advanced textbooks and the current research literature easier accessible, and it perfectly serves its purpose of providing a very first, albeit far-going introduction to the fascinating, beautifully intricate field of algebraic geometry. It is really a joy, both mathematically and aesthetically, to study this book, in particular as one can easily do it by jumping around in the text, without losing track of the essentials.

14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14E05 Rational and birational maps
14A10 Varieties and morphisms
14Nxx Projective and enumerative algebraic geometry