This text is intended to introduce graduate students to the methods and results of abstract algebraic geometry as practised today. No such exposition can succeed unless it enables the reader to make the drastic transition between the basic, intuitive questions about affine and projective varieties with which the subject begins, and the elaborate general methodology of schemes and cohomology employed currently to answer (or attempt to answer) these questions. The present text, notable for generality and depth, is also notable for its author's concern, throughout, to keep the important issues about varieties clearly in the foreground. Varieties. An opening chapter (57 pp.) introduces affine and projective varieties, as embedded point sets. Morphisms and regular functions lead to the function field; then smooth and singular points, dimension, basic facts about smooth curves, intersections in Pn follow. (Examples abound. In the first 6 pages, for instance, there are 11, with 6 more in the exercises.) The chapter ends with a survey entitled ``What is algebraic geometry?'', aimed at motivating the subsequent developments. Here we have a discussion of several unsolved (or partly solved) problems which have motivated current as well as past research. Classification questions, both discrete and continuous, receive strong emphasis, especially moduli problems, birational classification of surfaces, and classification of singularities. (This discussion makes good expository sense, because the reader has already looked at models of one-dimensional function fields, has encountered monoidal transformations, and has blown up some embedded multiple points.) Sufficient motivation is provided by arithmetic questions to justify working over arbitrary ground fields, while reducible loci and multiple components urge acceptance of the most general coordinate rings. (This leads, in the next chapter, directly to the definition of schemes.) Granting the value of posing the key motivating questions early, it is still obvious that no introductory text, alone, can do full justice to the work done, and being done, toward their solution. Nonetheless, there should be clear connecting links between the abstract methods to be presented and the specific questions they are intended to answer. Given the logical need to lay down general foundations first, the burden here falls largely (though not exclusively) upon the two concluding chapters, about curves and surfaces, where the generalities are systematically applied. Hence it may be best to examine these later chapters first. Curves (62 pp.). The Riemann-Roch theorem, Hurwitz's formula and Clifford's theorem are proved via sheaf cohomology. The canonical linear system is examined carefully, accompanied by a brief survey of moduli of curves, with examples. (Exercises include explicit deformations for curves of low genus, and Hurwitz's bound for the number of automorphisms when finite.) Elliptic curves are classified by j-invariant, using the Riemann-Roch theorem. The group structure is treated geometrically first, then using the bijection with Pic0. Over C, there is a good sketch of the connection with elliptic functions, including (with proof) a characterization of the curves with complex multiplication, in terms of the lattice structure. In characteristic p, the Hasse invariant is defined via Frobenius on H1(O), and the curves with Hasse=0 are classified. There follows a short aside on rationality questions. The chapter concludes with a rigorous classification of all smooth complete curves of degree≤7 in P3. Main theorems here are those of Halphen (a curve in Pn of genus g≥2 has a non-special very ample divisor of degree d if and only if d≥g+3) and Castelnuovo (a curve of degree d in P3, not contained in any plane, has d≥3, and, for the genus g, we have g≤14d2−d+1 (d even), g≤14(d2−1)−d+1 (d odd), with equality if and only if the curve lies on a quadric). Surfaces (67 pp.). In the first 10 pages we find basic facts about linear equivalence and intersections of curves on a smooth projective surface, and then the following results: the Riemann-Roch theorem in its classical form, the adjunction formula, the Hodge index theorem, and the Nakai-Moišezon criterion for ample curves. (As corollaries, the inequality of Castelnuovo-Severi and its application to the Riemann hypothesis for curves are given as exercises!) There follows a thorough study of ruled surfaces, viewed as projective bundles P(E), for E locally free of rank 2 on a smooth curve. Rational and elliptic ruled surfaces classified via normal forms for E, and criteria for ample and very ample curves are deduced. A section on monoidal transformations culminates in a proof of embedded resolution for curves on a smooth projective surface. Then the nonsingular cubic surfaces in P3 are treated via the system of plane cubics through 6 general points; symmetries of the 27 lines yield a criterion for ample and very ample curves. There follows another 10-page section of generalities, this time on birational maps: factorization, invariance of pa, Castelnuovo's criterion for contractible curves, existence of relatively minimal models. A survey of Enriques's classification, with references, concludes the exposition. (This chapter, far more than that on curves, requires the full power of schemes and cohomology: semicontinuity and base change, for example, unlock the ruled surfaces, while the theorem on formal functions verifies, in Castelnuovo's criterion, that the contracted surface is smooth.) The two middle chapters on schemes (140 pp.) and cohomology (91 pp.) give a very good introduction to the essential ideas. Care has been taken to avoid setting up excessively elaborate machinery, without (the reviewer feels) unduly weakening the important theorems. Duality for projective varieties is neatly streamlined; coherence of direct images, Zariski's theorem on formal functions, and the semicontinuity and base-change theorems are given for projective, rather than proper, morphisms. The resulting exposition, although less general than that of EGA and its satellites, is much shorter, while still sufficient for the author's purposes. The abstract development itself is interwoven with some applications, for example to proving several versions of Bertini's theorem. The text concludes with three appendices, sketching further developments (intersection theory and the Grothendieck Riemann-Roch theorem, algebraic varieties versus complex manifolds, the Weil conjectures) with references to the literature. A course along the lines of this text, according to the introduction, ran for five quarters at Berkeley. Knowledge of basic commutative and homological algebra (or a willingness to learn it) is assumed; some familiarity with complex analysis might also help. A further necessary commitment is that of working and (as needed) discussing a sizable portion of the book's 464 exercises. (These exercises include important theorems, additional examples, alternate treatments of some topics, as well as historical and technical asides. The style of the exposition seems to draw the reader into the problems, so the experience of reading this book may be more active than is usual at this level.) Granting these necessary commitments, the present text succeeds admirably, in the reviewer's opinion, in introducing its difficult subject at a level appropriate for preparing future workers in the field.
Reviewer: Speiser, Robert [form MathSciNet]