Preface

Summary

The main purpose of the present treatise is to give an account of some of the topics in algebraic geometry which, while having occupied the minds of many mathematicians in previous generations, have fallen out of fashion in modern times. In the history of mathematics new ideas and techniques often make the work of previous generations of researchers obsolete; this applies especially to the foundations of the subject and the fundamental general theoretical facts used heavily in research. Even the greatest achievements of past generations, which can be found for example in the work of F. Severi on algebraic cycles or in O. Zariski's work in the theory of algebraic surfaces, have been greatly generalized and clarified so that they now remain only of historical interest. In contrast, the fact that a nonsingular cubic surface has 27 lines or that a plane quartic has 28 bitangents is something that cannot be improved upon and continues to fascinate modern geometers. One of the goals of this present work is to save from oblivion the work of many mathematicians who discovered these classic tenets and many other beautiful results.

In writing this book the greatest challenge the author has faced was distilling the material down to what should be covered. The number of concrete facts, examples of special varieties and beautiful geometric constructions that have accumulated during the classical period of development of algebraic geometry is enormous, and what the reader is going to find in the book is really only the tip of the iceberg; a work that is like a taste sampler of classical algebraic geometry.