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History of Algebraic Geometry History of Algebraic Geometry History of Algebraic Geometry An Outline of the History and Development of Algebraic Geometry JEAN DIEUDONNE Translated by Judith D. Sally Northwestern University ES CHAPMAN & HALL l($)P An International Thomson Publishing Company New York • Albany • Bonn • Boston • Cincinnati • Detroit • London • Madrid • Melbourne • Mexico City • Pacific Grove • Paris • San Francisco • Singapore • Tokyo • Toronto • Washington This edition published by Chapman & Hall A division of International Thompson Publishing Inc. I(J)P The ITP logo is a trademark under license Printed in the United States of America For more information, contact: Chapman & Hall Chapman & Hall One Penn Plaza 2-6 Boundary Row New York, NY 10119 London SE1 8HN International Thompson Publishing Europe International Thomson Editores Berkshire House 168-173 Campos Eliseos 385, Piso 7 High Holbom Col. Polanco London WC1V 7AA 11560 Mexico D.F. Mexico England International Thomson Publishing Gmbh Thomas Nelson Australia Konigwinterer Strasse 418 102 Dodds Street 53227 Bom South Merlboume, 3205 Germany Victoria, Australia International Thomson Publishing Asia Nelson Canada 221 Henderson Road 1120 Birchmount Road #05-10 Henderson Building Scarborough, Ontario Singapore 0315 Canada, M1K 5G4 International Thomson Publishing Japan Hirakawacho-cho Kyowa Building, 3F 2-2-1 Hirakawacho-cho Chiyoda-ku, Tokyo 102 Japan Originally published as Cours de geometrie algebrique I © 1974 Presses Universitaires de France © 1985 by Wadsworth, Inc. All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means - graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems - without the written permission of the publisher. Library of Congress Cataloging-in-Publication Data Dieudonne, Jean Alexander, 1906- History of algebraic geometry. Translation of: Cours de geometrie algebrique. P. cm. Bibliography: p. Includes index. ISBN 0-412-99371-6 1. Geometry, Algebraic—History. I. Title. QA564.D513 1985 516.3’5 84-17213 CIP CONTENTS F oreword vii I. I ntroduction ix T able of N otations xi II. T he F irst E poch—Prehistory (ca. 400 B.C.-1630 a .d .) 1 III. T he S econd E poch—Exploration (1630-1795) 4 IV . T he T hird E poch— The golden age of projective geometry (1795-1850) 8 V. T he F ourth E poch— Riemann and birational geometry (1850-1866) 18 VI. T he F ifth E poch— Development and chaos (1866-1920) 27 1. The Algebraic School 27 2. The Geometric School and Linear Series 35 3. The “ Transcendental” Theory of Algebraic Varieties 42 4. The Italian School and the Theory of Linear Systems 50 5. Rational Surfaces and Cremona Transformations 54 VII. T he S ixth E poch— New structures in algebraic geometry (1920-1950) 59 1. Kahler Manifolds and Return to Riemann 59 2. Abstract Algebraic Geometry and Commutative Algebra 64 3. Specializations, Generic Points, and Intersections 67 4. Fields of Definition and Notions of Algebraic Variety 73 5. Local Rings and Their Completions in Algebraic Geometry 11 6. £eta Functions 81 7. Equivalence of Divisors and Abelian Varieties 83 VIII. T he S eventh E poch— Sheaves and schemes (1950- ) 91 1. The Riemann-Roch Problem for Varieties of Dimension > 2 and the Cohomology of Sheaves 91 2. The Varieties of Sene 101 3. Schemes and Topologies 105 V VI CONTENTS IX. R ecent R esults and O pen Problems 114 L Problems on Curves 114 2. Problems on Surfaces 120 3. Varieties of Arbitrary Dimension 123 4. Singularities 128 5. Divisors and Cycles 131 6. Topology of Algebraic Varieties 140 7. Abelian Varieties 147 8. Diophantine Geometry 151 9. Additional Topics 162 Annotated Bibliography 163 Index of C ited N ames 179 Index of T erminology 183 Foreword Algebraic geometry is undoubtedly the area of mathematics where the devi­ ation is greatest between the intuitive ideas forming its starting point and the abstract and complex concepts at the foundation of modern research. The purpose of this work is to help the willing reader bridge the gap. In this volume, an attempt is made to show how geometers have been led progressively to broaden their ideas. The reader will discover here several funda­ mental ideas that are revived time after time in different guises, marking the incessant effort toward better understanding of geometric phenomena: transfor­ mations and correspondences, invariants, “ infinitely near” points, extensions of geometric objects by the “ adJunction” of new “points.” The reader will also see how the arsenal of algebraic geometry is enriched, little by little, with loans from analysis and topology, where powerful tools have been found, and, more recently, from commutative algebra and homological algebra. Right now, algebraic geo­ metry appears as one of the components of a trinity, the other two members being number theory and the theory of analytic spaces (the modern form of the “ theory of several complex variables” for which, according to Serre, the name “ analytic geometry” is appropriate1). The universal tendency of all the research in these three areas (in which, moreover, the theory of Lie groups, differential geometry, and functional analysis, particularly the theory of partial differential equations, get entangled more and more tightly) is toward a fusion of their basic concepts and of their fundamental methods, so much so that it becomes more and more artificial to dissociate them. Unfortunately, the frame of this work does not permit a better illumination of this gradual fusion, and the author apologizes for giving a de­ formed image of the truth in describing the influences exercised on algebraic geo­ metry by her neighbors and not the inverse influences that are Just as important; the reader must be content, in this regard, with several indications in the bibliography. To conform with the spirit of this collection, knowledge of mathematical notions beyond the level of the first cycle at the university is not assumed from the start; the new concepts, introduced in algebraic geometry up to 1940, which are above this level, have been described succinctly as the text progresses. These descriptions are supplemented by references in the annotated bibliography at the 1 There is no reason to keep the old appellation of “ analytic geometry” to designate the technique of cartesian coordinates, which, for the most part, has been absorbed on the one hand in the notion (“ set- theoretic” or “ categorical” ) of “ product” and on the other hand in linear and multilinear algebra which has supplanted it, to great advantage, in the curriculum of the first years at the university. VII VIII FOREWORD end of this volume to the most accessible works for readers who desire to fix precisely their information on these ideas and, eventually, to understand the corresponding proofs. I must mention here the help J. P. Serre has given me in drawing up this volume. His remarks, criticisms, and suggestions have been extremely helpful. It is rare, indeed, for the historian to have the cooperation of one of the principal actors in the events he relates, and I am deeply grateful to him. I Introduction To facilitate the exposition, the history of algebraic geometry has been divided into seven “ epochs” : ca. 400 B.C.-1630 a.d . Prehistory. 1630-1795 Exploration. 1795-1850 The golden age of projective geometry. 1850-1866 Riemann and birational geometry. 1866-1920 Development and chaos. 1920-1950 New structures in algebraic geometry. 1950- Sheaves and schemes. A last chapter describing some recent results and some of the principal problems that are unsolved at this moment has been added. O f course, this division has been introduced only for convenience and has no intrinsic value. Moreover, the reader will verify that the ideas and theories belonging to one epoch often have roots much farther down that extend almost invariably upwards. Also, it is clear that there can be no question of undertaking, within the limits of this work, a truly detailed and comprehensive history of algebraic geometry; such an enterprise, which is certainly desirable to see realized some day, would be a work of huge dimensions. Thus, the author is satisfied to mention the ideas and facts that have seemed most important to him, and from this, it very evidently follows that our “ historical overview” can have no pretense of absolute objectivity (if history can ever attain this ideal). The works cited in the bibliography will allow the reader to form a personal opinion on the points that seem doubtful or insufficiently supported by the facts. The internal references are given in the form (A, n), where A is the number of the chapter in roman numerals and n is the number of the section in arabic numerals. IX Table of Notations B(D) vector bundle associated to a divisor D: VIII, 4. C field of complex numbers. C'. C" “ intersection cycle” of two algebraic cycles on a variety: VI, 37 and VII, 28. (cr. c") “ intersection number” of two algebraic cycles of comple­ mentary dimensions: VI, 37 and VII, 28. ‘,(E),c(E) Chern class of in d e x to tal Chern class of a vector bundle E: V III, 5. c) Chern class of index j of a holomorphic manifold: V III, 5. X(D) Euler-Poincare characteristic of a divisor: VIII, 12. |D| complete linear system generated by a divisor: VI, 9 and V II, 43. D + D', - D operations on divisors: VI, 6 and VI, 43. D < D ' order relation on divisors: VI, 6 and VI, 43. deg(D) degree of a divisor on a curve: VI, 6. dz differential of an element K (K a finite extension of C(X)): VI, 12. dco exterior differential of a />-form: VII, 2. A(co), A divisor of a meromorphic differential form of maximal degree (canonical divisor): VI, 13 and VI, 44.
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