Classical Algebraic Geometry: A Modern View

Classical Algebraic Geometry: a modern view IGOR V. DOLGACHEV Preface 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. Often in the history of mathematics new ideas and techniques make the work of previous generations of researchers obsolete, especially this applies to the foundations of the subject and the fundamental general theoretical facts used heavily in research. Even the greatest achievements of the past generations which can be found for example in the work of F. Severi on algebraic cycles or in the work of O. Zariski’s 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 then 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 a tip of the iceberg; a work that is sort of a taste sampler of classical algebraic geometry. It avoids most of the material found in other modern books on the subject, such as, for example, [10] where one can find many of the classical results on algebraic curves. Instead, it tries to assemble or, in other words, to create a compendium of material that either cannot be found, is too dispersed to be found easily, or is simply not treated adequately by contemporary research papers. On the other hand, while most of the material treated in the book exists in classical treatises in algebraic geometry, their somewhat archaic terminology iv Preface and what is by now completely forgotten background knowledge makes these books useful to but a handful of experts in the classical literature. Lastly, one must admit that the personal taste of the author also has much sway in the choice of material. The reader should be warned that the book is by no means an introduction to algebraic geometry. Although some of the exposition can be followed with only a minimum background in algebraic geometry, for example, based on Shafarevich’s book [577], it often relies on current cohomological techniques, such as those found in Hartshorne’s book [311]. The idea was to reconstruct a result by using modern techniques but not necessarily its original proof. For one, the ingenious geometric constructions in those proofs were often beyond the authors abilities to follow them completely. Understandably, the price of this was often to replace a beautiful geometric argument with a dull cohomological one. For those looking for a less demanding sample of some of the topics covered in the book the recent beautiful book [39] may be of great use. No attempt has been made to give a complete bibliography. To give an idea of such an enormous task one could mention that the report on the status of topics in algebraic geometry submitted to the National Research Council in Washington in 1928 [582] contains more than 500 items of bibliography by 130 different authors only in the subject of planar Cremona transformations (covered in one of the chapters of the present book.) Another example is the bibliography on cubic surfaces compiled by J. E. Hill [326] in 1896 which alone con