Algebraic geometry. An introduction.
(Géométrie algébrique. Une introduction.)

*(French)*Zbl 0842.14001
Paris: InterÉditions. Paris: CNRS Éditions. x, 301 p. (1995).

What is a good concept for a basic course in algebraic geometry? One might find several completely different positions for such an introduction, between a rather concrete point of view with a bunch of examples, e.g. curves, surfaces etc., and particular questions, and the more sophisticated point of view for the foundation, like schemes etc. on the other side. For a course in one or two terms one should interest the students in concrete results on the subject and should give them the right guide for further studies in order to learn more abstract techniques, sometimes two tendencies contracting each other for lack of time.

From the reviewer’s point of view the introduction under review is a masterpiece for an elegant combination of both of these requirements. It is an original introduction to affine and projective algebraic sets, motivated by Bezout’s theorem and rational curves, with a view towards sheaves and varieties. Central themes are the cohomology of sheaves and the Riemann-Roch theorem. Besides of these one can find also an introduction to dimension theory, tangent spaces and rational morphisms. There is also an introduction to liaison, for the first time in a textbook.

The textbook is inspired by several other ones. But by his choice, organization and presentation of the material the author finds his own originality. The book grew out of courses given by the author at the University of Paris (Sud) several times. One might feel the author’s intention to encourage his students for the field of algebraic geometry. Each of the ten chapters is completed by a series of exercises. Furthermore, over fifty pages the author summarizes problems and examples to his courses, including the solutions of the examples. The problems might be used for exercises parallel to the course. A large part of the needed commutative algebra is developed parallel in the text and also in some exercises. In the chapter about linkage there is a proof of the Apery-Gaeta-Peskine-Szpiro result that any arithmetical Cohen-Macaulay (ACM) curve in \(\mathbb{P}^3\) is in the linkage class of a complete intersection. There is also a discussion about genus, degree, and the cohomology in the non ACM case.

An active student will benefit from the study of this introductory text. For a wider distribution of this original access the reviewer wishes the book a translation into English.

From the reviewer’s point of view the introduction under review is a masterpiece for an elegant combination of both of these requirements. It is an original introduction to affine and projective algebraic sets, motivated by Bezout’s theorem and rational curves, with a view towards sheaves and varieties. Central themes are the cohomology of sheaves and the Riemann-Roch theorem. Besides of these one can find also an introduction to dimension theory, tangent spaces and rational morphisms. There is also an introduction to liaison, for the first time in a textbook.

The textbook is inspired by several other ones. But by his choice, organization and presentation of the material the author finds his own originality. The book grew out of courses given by the author at the University of Paris (Sud) several times. One might feel the author’s intention to encourage his students for the field of algebraic geometry. Each of the ten chapters is completed by a series of exercises. Furthermore, over fifty pages the author summarizes problems and examples to his courses, including the solutions of the examples. The problems might be used for exercises parallel to the course. A large part of the needed commutative algebra is developed parallel in the text and also in some exercises. In the chapter about linkage there is a proof of the Apery-Gaeta-Peskine-Szpiro result that any arithmetical Cohen-Macaulay (ACM) curve in \(\mathbb{P}^3\) is in the linkage class of a complete intersection. There is also a discussion about genus, degree, and the cohomology in the non ACM case.

An active student will benefit from the study of this introductory text. For a wider distribution of this original access the reviewer wishes the book a translation into English.

Reviewer: P.Schenzel (Halle)