http://dx.doi.org/10.1090/stml/020 Elementary Algebraic Geometry This page intentionally left blank STUDENT MATHEMATICAL LIBRARY Volume 20 Elementary Algebraic Geometry Klaus Hulek Translated by Helena Verrill AMERICAN MATHEMATICAL SOCIETY Editorial Board David Bressoud, Chair Carl Pomerance Robert Devaney Hung-Hsi Wu Daniel L. Goroff Originally published in the German language by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, D-65189 Wiesbaden, Germany, as "Klaus Hulek, Element are Algebraische Geometrie, I. Auflage (1st edition)", © by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 2000. Translated from the German by Helena Verrill 2000 Mathematics Subject Classification. Primary 14-01. Library of Congress Cataloging-in-Publication Data Hulek, Klaus. [Elementare algebraische Geometrie. English] Elementary algebraic geometry / Klaus Hulek ; translated by Helena Verrill. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 20) Includes bibliographical references and index. ISBN 0-8218-2952-1 (pbk. : acid-free paper) 1. Geometry, Algebraic. I. Title. II. Series. QA565 .H85 2003 516.3'5—dc21 2002038457 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294, USA. Requests can also be made by e-mail to reprint-permission@ams. org. © 2003 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 15 14 13 12 11 10 Contents Preface vii Chapter 0. Introduction 1 Exercises 15 Chapter 1. Affine Varieties 17 §1.1. Hilbert's Nullstellensatz 17 §1.2. Polynomial Functions and Maps 37 §1.3. Rational Functions and Maps 49 Exercises 58 Chapter 2. Projective Varieties 63 §2.1. Projective Space 63 §2.2. Projective Varieties 66 §2.3. Rational Functions and Morphisms 75 Exercises 94 Chapter 3. Smooth Points and Dimension 97 §3.1. Smooth and Singular Points 97 §3.2. Algebraic Characterization of the Dimension of a Variety 103 Exercises 112 VI Contents Chapter 4. Plane Cubic Curves 115 §4.1. Plane Curves 115 §4.2. Intersection Multiplicity 118 §4.3. Classification of Smooth Cubics 126 §4.4. The Group Structure of an Elliptic Curve 136 Exercises 139 Chapter 5. Cubic Surfaces 143 §5.1. The Existence of Lines on a Cubic 144 §5.2. The Configuration of the 27 Lines 151 §5.3. Rationality of Cubics 161 Exercises 163 Chapter 6. Introduction to the Theory of Curves 167 §6.1. Divisors on Curves 167 §6.2. The Degree of a Principal Divisor 172 §6.3. Bezout's Theorem 182 §6.4. Linear Systems on Curves 184 §6.5. Projective Embeddings of Curves 193 Exercises 204 Bibliography 207 §A. Books on Commutative Algebra 207 §B. Books on Algebraic Geometry 207 §C. Other Literature 208 Index 211 Preface The German title of this book is "Elementare algebraische Geome- trie". This indicates that the aim of this book is an introduction to algebraic geometry using only a minimum of required prerequisites. This book originated from a one semster course aimed at 3rd or 4th year students at the University of Hannover who had attended the standard courses on algebra and complex analysis. Beyond this very little is required. It is one of my main aims to illustrate the relationship between the general theory on the one hand and various applications and examples on the other hand. I have decided not to introduce any sheaf theory or cohomology theory at all. Whereas these topics are essential for anybody who wants to do research in the subject, I believe that one can give an introduction into algebraic geometry without using these concepts. I hope, however, that this book will help the reader to progress to more advanced textbooks such as Hartshorne's. Clearly this book is influenced by previous textbooks. As the reader will notice, this is most notably the case with M. Reid's book Undergraduate Algebraic Geometry. In particular, chapter V about the 27 lines on a cubic surface is very closely based on his treatmant. I have been in contact with M. Reid at various stages while writing my book and I am grateful for his encouragement to publish this book. vn Vlll Preface My particular thanks go to H. Verrill, who translated this book into English. She has done a very good job, and her impact goes far beyond that of a normal translator. She has made very useful suggestions for corrections and improvements throughout the entire text and has thus helped to improve the presentation considerably. We are grateful to S. Endrass for allowing us to incorporate a picture of the Clebsch diagonal cubic which was produced with the help of his software package Surf. Klaus Hulek Bibliography A. Books on Commutative Algebra [AM] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley 1969. [E] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Graduate Text in Mathematics 150, Springer-Verlag 1995. [Kul] E. Kunz, Introduction to Commutative Algebra and Algebraic Ge• ometry. Birkhauser Boston 1985. [La] S. Lang, Algebra. Revised third edition. Graduate Texts in Math• ematics, 211. Springer-Verlag 2002. [Ma] H. Matsumura, Commutative Algebra. W. A. Benjamin 1970. [Rl] M. Reid, Undergraduate Commutative Algebra. LMS Student Texts 29, Cambridge University Press 1995. [ZS] O. Zariski and P. Samuel, Commutative Algebra I, II, reprint, Springer-Verlag 1979. B. Books on Algebraic Geometry [BK] E. Brieskorn and H. Knorrer, Plane Algebraic Curves. Birkhauser Verlag 1986. 207 208 Bibliography [C] H. Clemens, A Scrapbook of Complex Curve Theory. Plenum 1980. [EH] D. Eisenbud and J. Harris, The Geometry of Schemes. Springer- Verlag 2000. [Fil] G. Fischer, Plane Algebraic Curves, Student Mathematical Li• brary, 15. American Mathematical Society 2001. [Fu] W. Fulton, Algebraic Curves. An Introduction to Algebraic Ge• ometry, reprint, Addison-Wesley 1989. [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry. John Wiley and Sons 1978. [GD] A. Grothendieck and J. Dieudonne, Elements de Geometric Algebrique I. Springer-Verlag 1971. [H] J. Harris, Algebraic Geometry: A First Course. Springer-Verlag 1992. [Ha] R. Hartshorne, Algebraic Geometry. Springer-Verlag 1977. [Ke] G. Kempf, Algebraic Varieties. LMS Lecture Notes Series 172, Cambridge University Press 1993. [Ki] F. Kirwan, Complex Algebraic Curves. LMS Student Texts 23, Cambridge University Press 1992. [Mu] D. Mumford, Algebraic Geometry I: Complex Projective Vari• eties. Springer-Verlag 1996. [R2] M. Reid, Undergraduate Algebraic Geometry. LMS Student Texts 12, Cambridge University Press 1988. [S] I. R. Shafarevich, Basic Algebraic Geometry 1, 2, second edition, Springer-Verlag 1994. C. Other Literature [B] T. S. Blyth, Categories. Longman, Harlow; John Wiley and Sons, New York, 1986. [En] S. Endrass, Surf, A program for drawing surfaces. See http://www.mathematik.uni-marburg.de/~tbauer/clebsch.htm [Fi2] G. Fischer (Ed.), Mathematische Modelle. (Dual German-English text.) Friedr. Vieweg and Sohn, Braunschweig, 1986. [FB] E. Freitag, R. Busam, Funktionentheorie. Zweite erweiterte Au- flage, Springer-Verlag 1995. [He] A. Henderson, The 27 Lines upon a Cubic Surface. Cambridge University Press 1911. Bibliography 209 [Hu] D. Husemoller, Elliptic Curves. Springer-Verlag 1987. [L] G. W. Leibniz, Mathematische Schriften. Herausgegeben von C. I. Gerhardt, Letter to I'Hospital 28 April 1693, volume II, p.239, Vorwort volume VII, p 5. Georg Olms Verlag 1971. [Si] J. H. Silveman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer-Verlag 1986. [Wj H. Weyl, Die Idee der Riemannschen Fldche. Herausgegeben von R. Remmert, B. G. Teubner Verlagsgesellschaft 1997. This page intentionally left blank Index abstract affine variety, 58 cubic surface, 144 admissible transformation, 133 cuspidal cubic, 5 affine cone, 72, 112 affine hypersurface, 26 degree of a divisor, 167 affine part of projective space, 65 degree of a map between curves, 172 affine space, 2 degree of a plane curve, 116 affine subspace, 3 degree of an embedded curve, 186 affine variety, 22 determinant trick, 27, 28 algebraic group, 14 determinantal variety, 12 algebraic set, 2, 3 differential of a function at a point, ascending chain condition, 20 193 dimension, 101, 103 base point, 186 dimension of projective space, 63 base point free linear system, 186 Diophantine problem, 14 Bezout's theorem, 120, 182 discrete valuation, 170 birational, 83 discrete valuation ring, 170 birational equivalence, 83 discriminant, 129 birational immersion, 202 divisor, 167 blow-up, 90, 113, 164 divisor class group, 171 divisor denned by a function, 170 divisor group, 167 canonical divisor of a curve, 192 domain of definition, 49, 77, 81 canonical embedding,