Lectures on Curves, Surfaces and Projective Varieties---A Classical View of Algebraic Geometry

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 that could form material for specialized courses and which are suitable for the final two years of undergraduate study or beginning graduate level, as well as interesting topics for a senior thesis. The prerequisites for understanding the material in this book have been deliberately limited to knowledge of only the 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 geome- try 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 ideal use for this text could well be to provide a solid preliminary course to be mastered before approaching more advanced and abstract books.