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Introduction to algebraic geometry. (Corso introduttivo alla geometria algebrica.) (Italian) Zbl 0964.14001
Appunti dei Corsi Tenuti da Docenti della Scuola. Pisa: Scuola Normale Superiore, 247 p. (1998).
These lecture notes contain the material of an introductory course in algebraic geometry taught by the author, during the 1997/98 academic year, at Scuola Normale Superiore in Pisa, Italy. This course was designed for third-year students in mathematics, who already had a profound knowledge in basic algebra and geometry, and its main goal was to present a panorama of those methods and results of algebraic geometry that illustrate the development of the subject from its classical origin to its contemporary state of art. – This is certainly a very ambitious and rewarding goal, which makes this text into a rather unique one, within the current literature, and it ought to be stated already at the beginning that the author has produced, in this regard, a true masterpiece. These notes really throw a bridge from classical Italian algebraic geometry to modern algebraic geometry, and that in a way that possibly only an Italian expert in algebraic geometry could master.
Chapter I provides some preliminary material from commutative algebra (rings, ideals, polynomial rings, and modules), together with a brief introduction to projective spaces. – Chapter II is devoted to the fundamentals of classical invariant theory, especially to the invariants and covariants of binary forms. – Then, in chapter III, classical elimination theory (resultants and discriminants) is discussed, along with Hilbert’s Nullstellensatz in its affine and projective form. – Plane curves are the subject of chapter IV, where the focus is on the elementary theory, including tangent lines, multiplicities, intersections of plane curves, and linear systems. – Selected topics in the theory of projective plane curves are presented in chapter V, where a wealth of fascinating classical results is touched upon. Among those are things like the Poncelet polygons, Salmon’s theorem on the configuration of flexes on a plane cubic, the group law on a plane cubic, Max Noether’s \((AF+BG)\)-theorem, and the degree-genus formula for plane curves. Many more classical geometric results are presented and required to be proved in the complementary list of exercises. – Chapter VI deals with the algebra of power series. The reader gets here acquainted with Hensel’s lemma, the fundamental theorem of algebra, the theorem of Newton-Puiseux, and the Weierstrass preparation theorem. – This is then used, in chapter VII, to study singularities of projective plane curves from the analytical point of view. – It is only in chapter VIII, and rather unusual for a modern text on algebraic geometry, that the author turns to abstract topics. The subject of this chapter is the Zariski topology of affine and projective varieties, accompanied by the discussion of hypersurfaces, fiber dimensions of morphisms, the geometric version of Krull’s Hauptidealsatz, and the classical Plücker-Clebsch principle. – Chapter IX is devoted to the general theory of algebraic varieties, without any sheaf theory or cohomology being used, but including a detailed description of morphisms, regular and singular points, the Zariski tangent space, the structure sheaf of rational functions, function fields, and birational morphisms. – Selected topics in the theory of projective varieties are discussed in chapter X. The author describes incidence varieties, presents the theorem on the 27 lines on the smooth cubic surface in projective 3-space, and concludes this chapter with the Bertini-Sard theorem. The final chapter XI explains the Riemann-Roch theorem for smooth projective curves. This chapter is a little longer than most of the others, as the author has to do a great deal of the basic theory of algebraic curves. The topics included here are, among others: the degree of a morphism between curves, divisors and linear systems on curves, separable field extensions and rational differentials, the abstract residue theorem, ramification, the Riemann-Hurwitz formula, Clifford’s theorem, and the two parts of the Riemann-Roch theorem.
There are about 450 exercises added to the text, including the sets of “complementary exercises” at the end of the respective chapters. Most of these exercises are by far not easy, and the degrees of difficulty vary remarkably. However, many of these exercises provide the willing (and ambitious) reader with more fascinating results in classical geometry, as they can only be found in the very ancient literature. Other exercises are well-known deep theorems in (classical and modern) algebraic geometry, such as the Noether normalization lemma, the theorem of Lüroth, the Gieseker lemma, and others, and the tackling of those certainly requires further reading and study.
Alltogether, the author has made a lot of effort at honoring the great achievements of classical algebraic geometry, and to place them in the framework of modern algebraic geometry. This is very gratifying, rewarding and admirable, and it is fair to say that the author has rendered a great service to the mathematical community by having published his lecture notes. It remains to wish that these notes will get translated and published in English, in the not too far future, so that the bigger part of the mathematical community can profit from this oustanding text, too.

MSC:
14Axx Foundations of algebraic geometry
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14Nxx Projective and enumerative algebraic geometry
14B05 Singularities in algebraic geometry
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