Bücher von Joe Harris

This book is based on one-semester courses given at Harvard in 1984, at Brown in 1985, and at Harvard in 1988. It is intended to be, as the title suggests, a first introduction to the subject. Even so, a few words are in order about the purposes of the book. Algebraic geometry has developed tremendously over the last century. During the 19th century, the subject was practiced on a relatively concrete, down-to-earth level; the main objects of study were projective varieties, and the techniques for the most part were grounded in geometric constructions. This approach flourished during the middle of the century and reached its culmination in the work of the Italian school around the end of the 19th and the beginning of the 20th centuries. Ultimately, the subject was pushed beyond the limits of its foundations: by the end of its period the Italian school had progressed to the point where the language and techniques of the subject could no longer serve to express or carry out the ideas of its best practitioners.

Grothendieck¿s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.

Aims Theaimofthisbookistoprovideaguidetoarichandfascinatings- ject: algebraic curves, and how they vary in families. The revolution that the ?eld of algebraic geometry has undergone with the introd- tion of schemes, together with new ideas, techniques and viewpoints introduced by Mumford and others, have made it possible for us to understandthebehaviorofcurvesinwaysthatsimplywerenotpos- ble a half-century ago. This in turn has led, over the last few decades, to a burst of activity in the area, resolving long-standing problems and generating new and unforeseen results and questions. We hope to acquaint you both with these results and with the ideas that have made them possible. The book isn't intended to be a de?nitive reference: the subject is developing too rapidly for that to be a feasible goal, even if we had the expertise necessary for the task. Our preference has been to - cus on examples and applications rather than on foundations. When discussing techniques we've chosen to sacri?ce proofs of some, even basic,results-particularlywherewecanprovideagoodreference- inordertoshowhowthemethodsareusedtostudymoduliofcurves. Likewise, we often prove results in special cases which we feel bring out the important ideas with a minimum of technical complication.

Grothendieck's beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.