LECTURES ON CURVES, SURFACES AND PROJECTIVE VARIETIES. A CLASSICAL VIEW OF ALGEBRAIC GEOMETRY

1 Prerequisites
2 Algebraic Sets, Morphisms, and Rational Maps
3 Geometric Properties of Algebraic Varieties
4 Rudiments of Elimination Theory
5 Hypersurfaces in Projective Space
6 Linear Systems
7 Algebraic Curves
8 Linear Series on Algebraic Curves
9 Cremona Transformations
10 Rational Surfaces
11 Segre Varieties
12 Grassmann Varieties
13 Supplementary Exercises

This book offers a wide-ranging introduction to algebraic geometry along classical lines. It consists of lectures on topics in classical algebraic geometry, including the basic properties of projective algebraic varieties, linear systems of hypersurfaces, algebraic curves (with special emphasis on rational curves), linear series on algebraic curves, Cremona transformations, rational surfaces, and notable examples of special varieties like the Segre, Grassmann, and Veronese varieties. An integral part and special feature of the presentation is the inclusion of many exercises, not easy to find in the literature and almost all with complete solutions.
The text is aimed at students of the last two years of an undergraduate program in mathematics. It contains some rather advanced topics suitable for specialized courses on the advanced undergraduate or beginning graduate level, as well as interesting topics for a senior thesis. The prerequisites have been deliberately limited to basic elements of projective geometry and abstract algebra. Thus, for example, some knowledge of the geometry of subspaces and properties of fields is assumed.
The book will be welcomed by teachers and students of algebraic geometry who are seeking a clear and panoramic path leading from the basic facts about linear subspaces, conics and quadrics to a systematic discussion of classical algebraic varieties and the tools needed to study them. The text provides a solid foundation for approaching more advanced and abstract literature.

Authors
• Mauro C. Beltrametti (University of Genova, Italy)
• Ettore Carletti (University of Genova, Italy)
• Dionisio Gallarati (University of Genova, Italy)
• Giacomo Monti Bragadin (University of Genova, Italy)