Graduate Texts in Mathematics 44 Editorial Board F W Gehring P R Halmos M anagillg Editor c C Moore Keith Kendig Elementary Algebraic Geometry Springer-Verlag New York Heidelberg Berlin Dr Keith Kendig Cleveland State University Department of Mathematics Cleveland, Ohio 44115 Editorial Board P R Halmos F W Gehring C C Moore Managing Editor University of California Department of Mathematics Santa Barbara, California 93 \06 University of Michigan Department of Mathematics Ann Arbor, Michigan 48104 University of California at Berkeley Department of Mathematics Berkeley, California 94720 AMS Subject Classification \3-0 I, 14-01 Library of Congress Cataloging in Publication Data Kendig, Keith, 1938Elementary algebraic geometry (Graduate texts in mathematics; 44) Bibliography: p Includes index Geometry, Algebraic Commutative algebra I Title II Series QA564.K46 516'.35 76 22598 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag ©1977 by Springer- Verlag, New York Inc Softcover reprint of the hardcover 1st edition 1977 ISBN-13: 978-1-4615-6901-5 DOl: 10.1007/978-1-4615-6899-5 e-ISBN-13: 978-1-4615-6899-5 Preface This book was written to make learning introductory algebraic geometry as easy as possible It is designed for the general first- and second-year graduate student, as well as for the nonspecialist; the only prerequisites are a one-year course in algebra and a little complex analysis There are many examples and pictures in the book One's sense of intuition is largely built up from exposure to concrete examples, and intuition in algebraic geometry is no exception I have also tried to avoid too much generalization If one understands the core of an idea in a concrete setting, later generalizations become much more meaningful There are exercises at the end of most sections so that the reader can test his understanding of the material Some are routine, others are more challenging Occasionally, easily established results used in the text have been made into exercises And from time to time, proofs of topics not covered in the text are sketched and the reader is asked to fill in the details Chapter I is of an introductory nature Some of the geometry of a few specific algebraic curves is worked out, using a tactical approach that might naturally be tried by one not familiar with the general methods introduced later in the book Further examples in this chapter suggest other basic properties of curves In Chapter II, we look at curves more rigorously and carefully Among other things, we determine the topology of every nonsingular plane curve in terms of the degree of its defining polynomial This was one of the earliest accomplishments in algebraic geometry, and it supplies the initiate with a straightforward and very satisfying result Chapter III lays the groundwork for generalizing some of the results of plane curves to varieties of arbitrary dimension It is essentially a chapter on commutative algebra, looked at through the eyeglasses of the geometer v Preface Algebraic ideas are supplied with geometric meaning, so that in a sense one obtains a "dictionary" between commutative algebra and algebraic geometry I have put this dictionary in the form of a diagram of lattices; this approach does seem to neatly tie together a good many results and easily suggests to the reader a number of possible analogues and extensions Chapter IV is devoted to a study of algebraic varieties in en and IPn(q and includes a geometric treatment of intersection multiplicity (which we use to prove Bezout's theorem in n dimensions) In Chapter V we look at varieties as underlying objects upon which we mathematics This includes evaluation of elements of the variety's function field (that is, a study of valuation rings), a translation of the fundamental theorem of arithmetic to a nonsingular curve-theoretic setting (the classical ideal theory), some function theory on curves (a generalization of certain basic facts about functions merom orphic on the Riemann sphere), and finally the Riemann-Roch theorem on a curve (which ties in function theory on a curve with the topology of the curve) After the reader has finished this book, he should have a foundation from which he can continue in any of several different directions-for example, to a further study of complex algebraic varieties, to complex analytic varieties, or to the scheme-theoretic treatments of algebraic geometry which have proved so fruitful It is a pleasure to acknowledge the help given to me by various students who have read portions of the book; I also want to thank Frank Lozier for critically reading the manuscript, and Basil Gordon for all his help in reading the galleys Thanks are also due to Mary Blanchard for her excellent job in typing the original draft, to Mike Ludwig who did the line drawings, and to Robert Janusz who did the shaded figures I especially wish to express my gratitude to my wife, Joan, who originally encouraged me to write this book and who was an invaluable aid in preparing the final manuscript Keith Kendig Cleveland, Ohio VI Contents Chapter I Examples of curves I I Introduction The topology of a few specific plane curves Intersecting curves Curves over iQI 19 25 Chapter II Plane curves I 10 Projective spaces Affine and projective varieties; examples Implicit mapping theorems Some local structure of plane curves Sphere coverings The dimension theorem for plane curves A Jacobian criterion for nonsingularity Curves in ifD2(C) are connected Algebraic curves are orientable The genus formula for nonsingular curves 28 28 34 46 54 66 75 80 86 93 97 Chapter III Commutative ring theory and algebraic geometry Introduction Some basic lattice-theoretic properties of varieties and ideals The Hilbert basis theorem Some basic decomposition theorems on ideals and varieties 103 103 106 117 121 VlI Contents 10 II The Nullstellensatz: Statement and consequences Proof of the Nullstellensatz Quotient rings and subvarieties Isomorphic coordinate rings and varieties Induced lattice properties of coordinate ring surjections; examples Induced lattice properties of coordinate ring injections Geometry of coordinate ring extensions 124 128 132 136 143 150 155 Chapter IV Varieties of arbitrary dimension I Introduction Dimension of arbitrary varieties The dimension theorem A Jacobian criterion for nonsingularity Connectedness and orientability Multiplicity Bezout's theorem 163 163 165 181 187 191 193 207 Chapter V Some elementary mathematics on curves I Introduction Valuation rings Local rings A ring-theoretic characterization of nonsingularity Ideal theory on a nonsingular curve Some elementary function theory on a nonsingular curve The Riemann-Roch theorem Bibliography Notation index Subject index viii 214 214 215 235 248 255 266 279 297 299 301 CHAPTER I Examples of curves Introduction The principal objects of study in algebraic geometry are algebraic varieties In this introductory chapter, which is more informal in nature than those that follow, we shall define algebraic varieties and give some examples; we then give the reader an intuitive look at a few properties of a special class of varieties, the "complex algebraic curves." These curves are simpler to study than more general algebraic varieties, and many of their simply-stated properties suggest possible generalizations Chapter II is essentially devoted to proving some of the properties of algebraic curves described in this chapter Definition 1.1 Let k be any field (l.l.l) The set {(Xl' , xn) IXi E k} is called affine n-space over k; we denote it by k n , or by kx, x n • Each n-tuple of k n is called a point (1.1.2) Let k[XI, ,Xn] = k[X] be the ring of polynomials in n indeterminants X b , X n , with coefficients in k Let p(X) E k[X]\k The set V(p) = ((x) E k n Ip(x) = O} is called a hypersurface of k n , or an affine hypersurface (1.1.3) If {Pa(X)} is any collection of polynomials in k[X], the set I V({Pa}) = {(X) E kn each pb) = O} is called an algebraic variety in k n , and affine algebraic variety, or, if the context is clear, just a variety If we wish to make explicit reference to the field k, we say affine variety over k, k-variety, etc.; k is called the ground field We also say V({P,}) is defined by {p,} I: Examples of curves (1.1.4) P is called the affine plane If P E k[Xl' Xz]\k, V(P) is called a plane affine curve (or plane curve, affine curve, curve, etc., if the meaning is clear from context) We will show later on, in Section I1I,3, that any variety can be defined by only finitely many polynomials P:X Here are some examples of varieties in [Rz 1.2 (1.2.1) Any variety V(aXZ + bXY + cYz + dX + eY + f) where a, , f E [R Hence all circles, ellipses, parabolas, and hyperbolas are affine algebraic varieties; so also are all lines (1.2.2) The "cusp" curve V(Yz - X3); see Figure (1.2.3) The "alpha" curve V(y2 - X2(X + 1)); see Figure EXAMPLE y y -i"II;: - X + ~ -X Figure Figure (1.2.4) The cubic V(y2 - X(XZ - 1»; see Figure This example shows that algebraic curves in [R2 need not be connected (1.2.5) If V(Pl) and V(P2) are varieties in [R2, then so is V(Pl) U V(Pz); it is just V(Pl Pz), as the reader can check directly from the definition Hence one has a way of manufacturing all sorts of new varieties For instance, (X Z + y2 _ I)(XZ + yZ - 4) = defines the union of two concentric circles (Figure 4) (1.2.6) The graph V(Y - p(X» in [Rz of any polynomial Y = p(X) E [R[X] is also an algebraic variety (1.2.7) If Pi' PZ E [R[X, y], then V(Pl, pz) represents the simultaneous solution set of two polynomial equations For instance, V(X, Y) = {(O,O)} S; [R2, while V(X Z + y2 - 1, X - Y) is the two-point set {(fi fi) (_ fi _fi)} '2' 2' 7: The Riemann-Roch theorem [Suggestions: Reduce the theorem to the case of a curve in [p>2(C) having at worst ordinary singularities (Exercise 6.12 and Theorem 6.19) Use the definition of genus in Exercise IV, 7.6 and the birational invariance of proved in Exercise IV, 7.8 Note that the statements of the lemmas in this section generalize verbatim to arbitrary irreducible curves in [P>"(C).] 295 Bibliography Ahlfors, L V Complex Analysis New York: McGraw-Hill, 1953 Bochner, S., and Martin, W T Several Complex Variables Princeton, N.J.: Princeton University Press, 1948 Borevich, Z.I., and Shafarevich, I R Number Theory New York: Academic Press, 1966 Cairns, S S Introductory Topology New York: The Ronald Press Company, 1961 Eisenbud, D., and Evans, E G., Jr Every algebraic set in n-space is the intersection of n hypersurfaces Inventiones Math., 19 (1973),107-112 Fulton, W A(qebraic Curves New York: W A Benjamin, Inc., 1969 Lang, S Introduction to Algebraic Geometry New York: Interscience Publishers, Inc., 1958 Markov, A A The problem ofhomeomorphy Proceedings of the International Congress of Mathematicians (1958) [Russian] Cambridge: The Cambridge University Press, 1960 Massey, W S Algebraic Topology: An Introduction New York: Harcourt, Brace & World, Inc., 1967 Narasimhan, R Analysis on Real and Complex Manifolds Amsterdam: North-Holland Publishing Company, 1968 Reid, C Hilbert New York: Springer-Verlag, 1970 Spivak, M Calculus on Manifolds New York: W A Benjamin, Inc., 1965 van der Waerden, B L Modern Algebra vols New York: Frederick Ungar Publishing Co., 1949 Vick, J W Homology Theory New York: Academic Press, 1973 Walker, R J Algebraic Curves New York: Dover Publications, Inc., 1962 Wells, R 0., Jr Differential Analysis on Complex Manifolds Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1973 Zariski, 0., and Samuel, P Commutative Algebra vols New York: Springer-Verlag, 1976 297 Notation index (A,M,n) Bp BpC(p) + cod V qX)* ( )' degD deg V deg (VI· V2 ) !?fl(f), !?fl xj(f) dfldg dim V dimp V div (f) div (F) div (w) D(V), DlP'oo·-I(V), Dj(V) ~, ~(a),,1· lEI ( )" fp g G*