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Algebraic varieties. (English) Zbl 0780.14001

London Mathematical Society Lecture Note Series. 172. Cambridge: Cambridge University Press. x, 163 p. (1993).
The present book grew out of the author’s courses on algebraic geometry, which he taught during the recent years at Johns Hopkins University. Accordingly, it provides an introduction to the subject, and may be regarded as another textbook on algebraic geometry. Compared to the numerous already existing textbooks in this area of mathematics, which – on their part – differ among themselves by the distinct approaches and viewpoints they offer, the author’s attempt at introducing algebraic geometry is of quite another kind, and that in many regards.
First of all, the book provides an introduction to algebraic geometry in its modern language and framework, from the very beginning. It does not start, as most textbooks, do, with affine algebraic sets, the Zariski topology, and the classical aspects of algebraic varieties, but it introduces algebraic varieties as particular ringed spaces (i.e., as topological spaces with distinguished “regular” functions). Then their existence is proved, by developing and using the standard commutative algebra of finitely generated algebras over an algebraically closed field, and the fundamental concepts in the theory of algebraic varieties are introduced and investigated from that natural viewpoint of particular spaces with functions. This approach makes the early introduction of sheaves and their cohomology a very conceivable thing, and that is the second (and most significant) peculiar feature of the text: it discusses algebraic varieties, their algebraic functions and morphisms thoroughly (and utmost efficiently) from the sheaf-theoretic/cohomological point of view. The author has spent a lot of effort at realizing this methodically substantial and ambitious program. The result is amazing and impressive: on about 145 pages, which is surprisingly little for a textbook, he succeeds in providing a huge amount of the most important results on algebraic varieties, and that in a likewise systematic, methodically austere, mathematically original (sometimes ingeniously easy), lucid and comprehensive manner. The conciseness of the text, which is only possible due to the original, efficient approach elaborated by the author, and which is not in the least achieved at the expenses of the substantial content of the material, constitutes the third characteristic of the book under review. The reader acquires a wealth of knowledge on algebraic varieties on just a few pages, exactly by following the powerful sheaf- theoretic and cohomological treatment of the subject, and that means: with a minimum of expenditure the author achieves a maximum of mathematical depth and teaching effect. Of course, the reader is required to work quite hard in order to comprehend the tight exposition, in particular the sometimes rather short proofs, and to solve the many advanced problems left as exercises, but on principle he gets all the tools and methods he needs from the text. In this regard, the book is fairly self-contained, and a basic knowledge in algebra and topology is all of the prerequisites that are assumed.
Although the author does not discuss scheme theory, the reader gets implicitly prepared for it, so that he can easily continue his studies in algebraic geometry by further reading of more advanced textbooks dealing with the general theory of algebraic schemes. As for that, he will find the necessary sheaf- and cohomology-theoretical framework at his disposal. With regard to concrete examples, there are a few of them in the text, which admittedly focuses on the theoretical aspects, yet the reader is recommended to consult also other textbooks discussing classical examples, perhaps J. Harris’ recent book “Algebraic geometry: a first course”, Grad. Texts Math. 133 (1992).
The contents of the book under review are arranged as follows. Chapter 1 gives an introduction to algebraic varieties via ringed space. Their existence is proved by investigating \(\text{Spec}(A)\) for finitely generated \(k\)-algebras \(A\). This includes Hilbert’s Nullstellensatz and its consequences, furthermore affine and projective spaces, as well as determinantal varieties in the guise of first examples. Chapter 2 provides some more commutative algebra, just along the discussion of the irreducible components of an affine variety, the concepts of affine and finite morphisms, and the notion of dimension for algebraic varieties. Chapter 3 deals with the basic constructions for varieties, such as products, graphs of morphisms, affine cones of projective varieties, blowing-up, projective embeddings of complete varieties (Chow’s lemma), and a first discussion of elliptic curves as algebraic groups.
Chapters 4 and 5 concern general sheaf theory and the various kinds of sheaves occuring in algebraic geometry (quasi-coherent sheaves, coherent sheaves, invertible sheaves, and the basic operations on them). This is applied in chapter 6 to the Zariski cotangent space, tangent cones, algebraic differentials, the concept of smoothness for varieties and morphisms, and to the construction of affine morphisms and normalizations. This chapter concludes with a short, beautiful proof of Bertini’s theorem.
Chapter 7 is devoted to algebraic curves, their sheaves and morphisms. This discussion is continued in chapter 8, where the cohomology theory for sheaves on curves is introduced. This chapter culminates in the proof of the Riemann-Roch theorem for curves, its applications, and some residue theory. Here the proof of the Riemann-Roch theorem is new and particularly elegant, short and beautiful.
Chapter 9 then deals with general Čech cohomology of sheaves on affine and projective varieties, the Künneth formula, direct images of flat sheaves, base change, and the dimension behavior in families of cohomology groups. The concluding chapter 10 discusses various applications of the cohomological theory of algebraic varieties: projective embeddings by linear systems, the cohomological characterization of affine varieties, the Plücker formula and Bezout’s theorem for plane curves, the classification of line bundles of degree one over elliptic curves, Grothendieck’s theorem on the structure of vector bundles over \(\mathbb{P}^ 1\), the regularity in codimension one for rational maps, the classification of one-dimensional algebraic groups, correspondences between algebraic curves, and the Riemann-Roch theorem for algebraic surfaces.
At the end of the book, there are three appendices on localization in commutative algebra, direct limits in categories, and eigenspaces of endomorphisms of vector spaces, respectively. Also, there is a very carefully compiled glossary of notation, which makes it easier to work with the book.
Apart from the peculiar features of the author’s text already mentioned above, it should be emphasized that he has found many new and elegant proofs of old results, which fascinate by their brevity and enlightening conceptual compactness. By his book, the author has convincingly demonstrated how significant and powerful the cohomological method in algebraic geometry is, and that it can be efficiently conveyed even in a textbook for beginners in algebraic geometry. – His book is a real enrichment of the existing literature in algebraic geometry, and that for both newcomers and experts. Students, teachers, researchers and experts from other areas in mathematics will find this booklet highly valuable and inspiring, regardless of the perhaps surprising fact that some sections of the chapters occupy one page only, or even less. But just that might have the effect of being encouraging and challenging.
Altogether, this is a highly interesting, sophisticated and unique textbook on algebraic geometry, recommendable to anyone who wants to study the subject from an advanced point of view, but with as few prerequisites as possible, and who does not need extensive elementary motivation for that.

MSC:

14A10 Varieties and morphisms
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14E99 Birational geometry
14C20 Divisors, linear systems, invertible sheaves
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
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