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A royal road to algebraic geometry. (English) Zbl 1237.14001

Berlin: Springer (ISBN 978-3-642-19224-1/hbk; 978-3-642-19225-8/ebook). xiii, 364 p. (2012).
The book under review offers a first introduction to algebraic geometry of a special kind, which the author calls “a royal road to algebraic geometry”. This title is inspired by the famous anecdote about Ptolemy II King of Egypt, who once asked Euclid if there really existed no simpler way to study geometry than to read all volumes of his “Elements”.
Euclid is said to have answered that “there is no royal road to geometry”, meaning that there is no simple, or even effortless, way to learn all of geometry profoundly. On the other hand, Herodotus of Halicarnassus described the so-called “Persian Royal Road” in the fifth century BC, which was a famous trade-route at that time, full of fine rest-houses and other facilities along the whole way of about 2880 kilometers. Also, he depicted how a pleasant journey along this road could be conducted in roughly ninety days, thereby seeing and enjoying a large number of wonderful things.
In this sense, the author of the present book draws up an outline of a road to modern algebraic geometry, which is enjoyable, enlightening, manageable, and topical at the same time.
More precisely, the author introduces a number of concepts, methods, and important theorems in algebraic geometry, thereby focussing on basic ideas and their interplay. Thus, if the reader manages to study four or five pages of the book each day, stopping at some of the many resting places, he should complete the journey in ninety days in a comfortable manner, just like the ancient “Persian Royal Road” as once described by Herodotus.
As for the contents, the text is divided into two major parts, each of which consists of several chapters and their sections.
Part I is titled “Curves” and introduces the basic concepts of algebraic geometry in the classical context of affine and projective curves. This part comprises the first five chapters treating the following topics:
Chapter 1 deals with affine and projective spaces, algebraic subsets therein, the Theorem of Desargues, and the notion of duality for the projective plane. Chapter 2 studies curves in the affine and in the projective plane, including conic sections, singularities, tangency, and multiplicities. Chapter 3 does some higher geometry in the projective plane, with the elementary study of projective plane curves being the main theme. Chapter 4 is devoted to the algebraic aspects of plane curves, including coordinate rings, local rings, valuation rings, algebraic derivatives and the Jacobian, elementary elimination theory, intersection multiplicities, Bézout’s Theorem, linear systems of plane curves, inflection points and Hessians, the twisted cubic curve, and further examples. Chapter 5 discusses the most important classical results on higher-dimensional projective varieties over an algebraically closed ground field, with focus on general smooth projective curves, Hilbert polynomials and projective invariants, and an explanation (without proof) of the Riemann-Roch Theorem for non-singular curves.
Part II comes with the title “Introduction to Grothendieck’s Theory of Schemes”. Containing the remaining seventeen chapters, this part turns to modern abstract algebraic geometry and its frameworks of both category theory and homological algebra.
Chapter 6 gives a brief introduction to categories and functors, ending up with adjoint functors, representable functors, universal properties, and Yoneda’s Lemma. Chapter 7 discusses various constructions of products and coproducts in categories, whereas Chapter 8 explains abelian categories, presheaves, sheaves, Grothendieck topologies, sheafifications, and the category of abelian sheaves on a topological space. Chapter 9 introduces the concept of the affine spectrum of a commutative ring.
Thereafter, in Chapter 10, the category of schemes is described, while Chapter 11 contains such important concepts as sheaves of modules and algebras on schemes, quasi-coherent and coherent sheaves, reduced and irreducible schemes, function fields, separated schemes and morphisms, irreducible components of noetherian schemes, and other fundamental properties of morphisms of schemes.
Chapter 12 turns to locally free modules and vector bundles on a scheme, which is followed by a list of further properties of morphisms in Chapter 13. Projective schemes and bundles form the main topic of Chapter 14, and further properties of morphisms of schemes are surveyed in the subsequent Chapter 15. The reader meets here affine and projective morphisms, valuational criteria for separatedness and properness, very ample sheaves, the Segre embedding, and other topics. Chapter 16 introduces conormal sheaves, sheaves of Kähler differentials, projective bundles, the Segre embedding for projective bundles, base extensions of projective morphisms, and regular schemes. Chapter 17 is devoted to the cohomology of schemes, including both Grothendieck’s cohomology via derived functors and Čech cohomology. Some aspects of intersection theory are touched upon in Chapter 18, with focus on divisors, Chow homology and Chow cohomology, Chow’s Moving Lemma, and the Chow ring of a smooth projective variety. Chapter 19 explains various concepts of characteristic classes in algebraic geometry, together with some related important theorems. This includes Chern classes, Chern characters, Todd classes, homological Segre classes, and the Grothendieck group. A brief survey of the Serre Duality Theorem concludes this chapter. The main theme of Chapter 20 is the Riemann-Roch Theorem for smooth projective varieties. The author starts with Hirzebruch’s variant, deduces from it the classical Riemann-Roch theorems for curves and surfaces, states then the general Grothendieck-Riemann-Roch Theorem, and concludes Hirzebruch’s Riemann-Roch Theorem from the latter. Chapter 21 gives some basic constructions in the category of projective varieties: the blowing-up of a closed subscheme and of subbundles, Grassmann bundles, the parameter variety for joining lines, the secant variety, and the join. The final Chapter 22 returns to the concept of duality in projective algebraic geometry. In this context, the dual variety, the conormal scheme of an embedded projective variety reflexivity and biduality, duality of hyperplane sections and projections, the Hefez-Kleiman Theorem (1984), and a related theorem of A. H. Wallace (1956) are discussed.
As one can see, quite a variety of topics from modern algebraic geometry is presented to the reader, often in a sketchy or survey-like manner, but always with hints for further reading. The sections of the single chapters are mostly short, and the presentation is nowhere lengthy or tedious. Although the proofs of difficult theorems are often omitted, their basic ideas are always well explained. In this vein, the book must be seen as a charming invitation to algebraic geometry, along some sort of “royal road” (of pleasure and diversity).

MSC:

14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14Axx Foundations of algebraic geometry
14A15 Schemes and morphisms
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C40 Riemann-Roch theorems
14H50 Plane and space curves
18A05 Definitions and generalizations in theory of categories
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