Lectures on curves, surfaces and projective varieties. A classical view of algebraic geometry. Transl. from the Italian by Francis Sullivan.

*(English)*Zbl 1180.14001
EMS Textbooks in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-064-7/hbk). xv, 491 p. (2009).

The book is an introductory text in Algebraic Geometry, aiming to provide a support for an undergraduate course, at an advanced level. It is, by far, not an easy task to write down a self contained introduction to the methods and problems of Algebraic Geometry, for undergraduate students. The authors reach this target using the point of view of classical geometers of late nineteenth – early twentieth centuries. The only prerequisites, for reading the book, are rudiments of Algebra and Geometry (including properties of projective spaces), usually provided in standard basic courses. Links of complex projective geometry with the theory of several complex variables, or with non trivial commutative algebra, are mostly avoided.

Through the book, varieties are introduced as sets of zeroes of collections of polynomials, and maps are defined between embedded varieties. Derivations provide the notion of tangent spaces and tangent cones to a variety, which in turn provide the definition of dimension of an affine variety. The link between the notion of dimension and the theory of transcendental extensions of fields, is simply outlined. Rudiments of intersection theory (as Bézout theorem) follow from the study of the resultant of a set of polynomials. The fundamental notion of linear system is defined first on projective spaces and then on algebraic curves. The last three chapters deal with important examples of algebraic varieties: rational surfaces, Segre and Veronese varieties, Grassmannians.

Experience suggests that, among the main difficulties for a student facing a senior or a PhD thesis in Algebraic Geometry, a big deal concerns the language used in modern papers on the field, often full of references to sheaves, vector bundles and cohomology, subjects not treated in the book under review. Under this respect, after reading the book, the student will find that some gap must be filled, before he can handle the modern terminology. So, the book remains at a level which is more elementary than the level of other introductory texts (e.g. the ones of Hartshorne or Shafarevich). Nevertheless, the book provides a deep insight into the basic properties of many, very concrete instances of projective varieties (curves, rational surfaces, Grassmannians, ...) and, besides being a nice, self–contained exposition, furnishes the interested student with a strong background, for breaking easily into the understanding of many recent results in Algebraic Geometry.

Through the book, varieties are introduced as sets of zeroes of collections of polynomials, and maps are defined between embedded varieties. Derivations provide the notion of tangent spaces and tangent cones to a variety, which in turn provide the definition of dimension of an affine variety. The link between the notion of dimension and the theory of transcendental extensions of fields, is simply outlined. Rudiments of intersection theory (as Bézout theorem) follow from the study of the resultant of a set of polynomials. The fundamental notion of linear system is defined first on projective spaces and then on algebraic curves. The last three chapters deal with important examples of algebraic varieties: rational surfaces, Segre and Veronese varieties, Grassmannians.

Experience suggests that, among the main difficulties for a student facing a senior or a PhD thesis in Algebraic Geometry, a big deal concerns the language used in modern papers on the field, often full of references to sheaves, vector bundles and cohomology, subjects not treated in the book under review. Under this respect, after reading the book, the student will find that some gap must be filled, before he can handle the modern terminology. So, the book remains at a level which is more elementary than the level of other introductory texts (e.g. the ones of Hartshorne or Shafarevich). Nevertheless, the book provides a deep insight into the basic properties of many, very concrete instances of projective varieties (curves, rational surfaces, Grassmannians, ...) and, besides being a nice, self–contained exposition, furnishes the interested student with a strong background, for breaking easily into the understanding of many recent results in Algebraic Geometry.

Reviewer: Luca Chiantini (Siena)

##### MSC:

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14E05 | Rational and birational maps |

14H99 | Curves in algebraic geometry |

14E99 | Birational geometry |

14J26 | Rational and ruled surfaces |