Selected Title s i n Thi s Serie s 166 Kenj i Ueno , A n introductio n t o algebrai c geometry , 199 7 165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n Galois theory, 199 7 164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis , 199 7 163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko , Probability theory : Collectio n o f problems, 199 7 162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d method s i n linea r statistical models , 199 7 161 Michae l Blank , Discretenes s an d continuit y i n problems o f chaotic dynamics , 199 7 160 V . G . OsmolovskiT , Linea r an d nonlinea r perturbation s o f the operato r div , 199 7 159 S . Ya . Khavinson , Bes t approximatio n b y linea r superposition s (approximat e nomography), 199 7 158 Hidek i Omori , Infinite-dimensiona l Li e groups, 199 7 157 V . B . Kolmanovski T an d L . E . Shaikhet , Contro l o f systems wit h aftereffect , 199 6 156 V . N . Shevchenko , Qualitativ e topic s i n intege r linea r programming , 199 7 155 Yu . Safaro v an d D . Vassiliev , Th e asymptoti c distributio n o f eigenvalue s o f partia l differential operators , 199 7 154 V . V . Prasolo v an d A . B . Sossinsky , Knots , links , braids an d 3-manifolds . A n introduction t o the ne w invariants i n low-dimensiona l topology , 199 7 153 S . Kh . Aranson , G . R . Belitsky , an d E . V . Zhuzhoma , Introductio n t o th e qualitative theor y o f dynamical system s o n surfaces , 199 6 152 R . S . Ismagilov , Representation s o f infinite-dimensional groups , 199 6 151 S . Yu . Slavyanov , Asymptoti c solution s o f the one-dimensiona l Schrodinge r equation , 1996 150 B . Ya . Levin , Lecture s o n entir e functions , 199 6 149 Takash i Sakai , Riemannia n geometry , 199 6 148 Vladimi r I . Piterbarg , Asymptoti c method s i n the theor y o f Gaussia n processe s an d fields, 199 6 147 S . G . Gindiki n an d L . R . Volevich , Mixe d proble m fo r partia l differentia l equation s with quasihomogeneou s principa l part , 199 6 146 L . Ya . Adrianova , Introductio n t o linea r system s o f differential equations , 199 5 145 A . N . Andriano v an d V . G . Zhuravlev , Modula r form s an d Heck e operators, 199 5 144 O . V . Troshkin , Nontraditiona l method s i n mathematica l hydrodynamics , 199 5 143 V . A . Malyshe v an d R . A . Minlos , Linea r infinite-particl e operators , 199 5 142 N . V . Krylov , Introductio n t o the theor y o f diffusio n processes , 199 5 141 A . A . Davydov , Qualitativ e theor y o f control systems , 199 4 140 Aizi k I . Volpert , Vital y A . Volpert , an d Vladimi r A . Volpert , Travelin g wav e solutions o f paraboli c systems , 199 4 139 I . V . Skrypnik , Method s fo r analysi s o f nonlinear ellipti c boundary valu e problems , 199 4 138 Yu . P . Razmyslov , Identitie s o f algebras an d thei r representations , 199 4 137 F . I . Karpelevic h an d A . Ya . Kreinin , Heav y traffi c limit s fo r multiphas e queues , 199 4 136 Masayosh i Miyanishi , Algebrai c geometry , 199 4 135 Masar u Takeuchi , Moder n spherica l functions , 199 4 134 V . V . Prasolov , Problem s an d theorem s i n linea r algebra , 199 4 133 P . I . Naumki n an d I . A . Shishmarev , Nonlinea r nonloca l equation s i n the theory o f waves, 199 4 132 Hajim e Urakawa , Calculu s o f variations an d harmoni c maps , 199 3 131 V . V . Sharko , Function s o n manifolds : Algebrai c an d topologica l aspects , 199 3 130 V . V . Vershinin , Cobordism s an d spectra l sequences , 199 3 129 Mitsu o Morimoto , A n introductio n t o Sato' s hype r functions, 199 3 (Continued in the back of this publication) This page intentionally left blank An Introduction t o Algebraic Geometr y This page intentionally left blank 10.1090/mmono/166 Translations o f MATHEMATICAL MONOGRAPHS Volume 16 6 An Introduction t o Algebraic Geometr y Kenji Ueno Translated b y Katsumi Nomiz u ^^^,^\Q America n Mathematical Societ y Providence, Rhode Islan d <$°^5eS^ Editorial Boar d Shoshichi Kobayash i (Chair ) Masamichi Takesak i ix m m M A P I DAISU KIK A NYUMO N (An introductio n t o algebrai c geometry ) by Kenj i Uen o Copyright © 199 5 b y Kenj i Uen o Originally publishe d i n Japanes e b y Iwanam i Shoten , Publishers , Tokyo , 199 5 Translated fro m th e Japanes e b y Katsum i Nomiz u 2000 Mathematics Subject Classification. Primar y 14-01 ; Secondary 14B05 , 14C40 , 14G10 , 14H52 , 14H55 , 14K25 . ABSTRACT. Thi s boo k offer s a n invitatio n t o algebrai c geometr y to student s a t a n earl y stage an d introduces the m t o th e subjec t wit h a s fe w prerequisite s a s possible . A historica l an d intuitiv e treatment explain s the spiri t o f algebrai c geometr y wit h numerou s examples . Library o f Congres s Cataloging-in-Publicatio n Dat a Ueno, Kenji , 1945 - [Daisu kik a nyumon , English ] An introductio n t o algebrai c geometr y / Kenj i Uen o ; translated b y Katsumi Nomizu . p. cm . — (Translation s o f mathematical monograph s ; v. 166 ) Includes bibliographica l reference s an d index . ISBN 0-8218-0589- 4 (alk . paper ) 1. Geometry , Algebraic . I . Title . II . Serie s QA564.U3713 199 7 516.3'5—dc21 97-303 0 CIP AMS softcove r ISB N 978-0-8218-1144- 3 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provided th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisition s Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 199 7 by the America n Mathematica l Society . Al l right s reserved . Reprinted wit h correction s b y the America n Mathematica l Society , 2008 . The America n Mathematica l Societ y retains al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . 10 9 8 7 6 5 4 3 1 3 1 2 1 1 1 0 09 0 8 Contents Preface t o the Englis h Edition i x Translator's Not e x Preface x i Chapter 1 Invitation to Algebrai c Geometr y 1 §1.1 Th e birth o f geometry 1 (a) Euclidea n geometr y 1 (b) Th e theory o f conies o f Apollonius 2 §1.2 Coordinat e geometr y 2 (a) Th e birth o f coordinate geometry 2 (b) Euclidea n geometr y and affin e geometr y 4 §1.3 Projectiv e geometr y 9 (a) Th e birth o f projective geometr y 9 (b) Th e projective plan e 1 3 §1.4 Introductio n o f complex numbers 2 0 (a) Th e introduction o f complex numbers 2 0 (b) Comple x plane curve s 2 3 §1.5 Th e birth o f algebraic geometr y 3 0 (a) Plan e curve s and intersection s 3 0 (b) Dua l curves and Pliicker' s formul a 3 7 (c) Th e developmen t o f algebraic geometr y 4 3 Problems 4 6 Grothendieck's schem e theory 4 8 Chapter 2 Projective Spac e and Projectiv e Varietie s 4 9 §2.1 Projectiv e line s 4 9 (a) Th e Riemann spher e and projectiv e line s 4 9 (b) Projectiv e transformation s 5 3 (c) Functio n field s 5 6 §2.2 Th e projective plan e and plan e curve s 5 7 (a) Th e projective plan e 5 7 (b) Dualit y an d projectiv e transformation s 6 0 (c) Th e functio n field o f the projective plan e 6 4 (d) Plan e curve s 6 4 (e) Rationa l mapping s an d algebrai c morphisms 6 8 §2.3 Plan e curve s 7 2 (a) Tangent s and singula r point s 7 2 (b) Th e intersectio n theor y fo r plane curve s 8 5 viii CONTENT S (c) Functio n field s fo r plan e curve s 8 8 §2.4 Projectiv e varietie s 9 1 (a) Projectiv e spac e 9 1 (b) Projectiv e set s and varietie s 9 3 (c) Projectiv e set s and homogeneou s ideal s 9 7 (d) Dimensio n o f projective varietie s an d functio n field s 10 2 (e) Singularities , nonsingula r point s an d tangent hyperplane s 10 7 (f) Th e product o f projective space s 11 1 §2.5 Th e resolutio n o f singularities 11 6 (a) Blowing-u p o n the projectiv e plan e 11 7 (b) Resolutio n o f singularities o f plane curve s 12 0 (c) Resolutio n o f singularities fo r a surfac e 12 7 Problems 13 1 Chapter 3 Algebraic Curve s 13 5 §3.1 Th e Riemann-Roc h theore m 13 5 (a) Divisor s 13 5 (b) Differentia l form s an d the genu s o f algebraic curve s 14 2 (c) Th e Riemann-Roc h theore m 14 5 §3.2 Geometr y o f algebraic curves 14 7 (a) Th e Hurwitz formul a 14 7 (b) Imbeddin g int o the projectiv e spac e 15 1 §3.3 Ellipti c curve s 15 5 (a) Curve s o f genus 1 15 5 (b) Th e group structure o n a n ellipti c curv e 16 1 §3.4 Congruenc e zet a function s fo r algebrai c curve s 16 6 Problems 17 7 Chapter 4 The Analytic Theor y o f Algebraic Curve s 17 9 §4.1 Close d Rieman n surface s 17 9 §4.2 Perio d matrice s 18 9 §4.3 Jacobia n varietie s 19 7 Problems 20 6 Appendix Commutativ e Ring s an d Field s 20 7 §A.l Integer s an d congruenc e 20 7 §A.2 Th e polynomial rin g Q[x] 21 3 §A.3 Commutativ e ring s and field s 21 9 §A.4 Finit e field s 22 8 §A.5 Localizatio n an d loca l rings 23 3 References 23 9 Index 24 1 Index fo r Definitions , Theorems , etc.