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 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 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 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 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. 24 5

Errata 247 Preface t o the Englis h Editio n

Today plays an important role in several branches o f science and technology. Th e present book is written fo r non-specialists to explain the main ideas o f algebrai c geometry . Fortunately , i n Japa n th e origina l Japanes e editio n has bee n widel y accepte d a s a n introductor y boo k fo r algebrai c geometry . I hop e the present Englis h edition wil l serve the sam e role. My specia l thank s ar e du e to Professo r Nomizu , wh o no t onl y translate d th e Japanese editio n int o Englis h bu t als o suggested improvement s o f severa l parts o f the tex t s o tha t th e presen t Englis h editio n i s mor e readabl e tha n th e origina l Japanese edition . I als o expres s m y sincer e thank s t o Mr . I . Sasak i o f Iwanam i Shoten, Publisher s an d m y colleagu e Dr . Y . Shimizu , wh o foun d mistake s an d misprints i n the origina l Japanes e edition . December 199 6

Kenji Uen o Translator's Not e

Professor Ala n Landman's suggestion s were highly helpful i n getting started i n the translation o f the book .

x Preface

This boo k offer s a n invitatio n t o algebrai c geometry . Algebrai c geometry , a s the geometry o f figures defined b y a number o f equations, came into existence when coordinate geometry was introduced by Descartes and Fermat. I n the 18th and 19t h centuries, mathematicians working on problems in coordinate geometry with the aid of geometric intuition encountered various paradoxes, which necessitated a rigorous development o f the theory. I n the firs t hal f o f this century, Zariski , who wanted t o put th e algebrai c geometr y o f the Italia n Schoo l o n a firm ground, an d Weil , wh o wanted application s to number theory , develope d a rigorous foundation o f moder n algebraic geometry . Th e theory ha s mad e remarkabl e progress . I n particular , th e theory o f schemes o f Grothendieck pushe d th e algebrai c treatment o f geometry t o its limit, thus makin g a great contribution . It was thought fo r a long time that algebrai c geometry was a purely mathemat - ical theory withou t an y applications . Today , however , it s relationship s t o variou s branches o f natural science s an d engineerin g hav e bee n revealed . Weil' s algebrai c geometry ove r finite fields has been applied b y Goppa to coding theory. I n theoret - ical physics, clos e relationships betwee n solito n theory an d the theory o f algebrai c curves a s wel l a s between strin g theor y an d th e modul i theory o f algebraic curve s have been found. Furthermore , elliptic curves now play an important rol e in testing for prime numbers. The y were also essential in Wiles' recent proof o f Fermat's Las t Theorem. Al l thi s seem s t o indicat e tha t moder n mathematics , havin g achieve d a hig h leve l o f sophisticatio n fo r theoretica l needs , ha s no w reache d th e leve l o f maturity to mak e varied application s possible . Now it i s ofte n sai d that studyin g algebrai c geometr y i s rather difficult . Thi s may b e partl y du e t o th e fac t tha t eve n i n a n introductor y boo k w e ru n int o ideals, sheave s an d cohomology , makin g i t difficul t t o understan d wha t algebrai c geometry i s all about. O f course, to develo p algebraic geometry properly, there ar e many prerequisites. Th e need fo r a variety o f preparations comes , on the one hand, from th e wa y algebrai c geometr y wa s develope d — needing rigorou s construction s of the theory to avoi d paradoxe s arisin g fro m naiv e intuition — and, o n the othe r hand, fro m th e fac t tha t algebrai c geometer s develope d the theory b y avidl y usin g whatever tool s they neede d fro m man y branche s o f mathematics. In thi s boo k w e hav e trie d t o develo p algebrai c geometr y wit h minima l pre - requisites. I n Chapte r 1 , w e giv e a n introductio n t o algebrai c geometr y fro m a historical poin t o f view. Man y o f the concept s that appea r ar e refine d i n Chapte r 2. Thi s i s becaus e w e wan t th e reader s t o acquir e intuitiv e understandin g fro m the beginnin g s o that the y kno w wh y certai n argument s ar e necessar y an d under - stand th e significanc e o f the theory . W e g o through Chapter s 3 and 4 i n a hurr y compared t o Chapte r 2 . W e try t o explai n b y example s the meanin g o f theorem s

xi xii PREFAC E and to deriv e further result s rather than t o prove theorems. W e hope that reader s will taste variou s aspect s o f algebrai c geometr y an d will be better prepare d whe n they ge t t o wor k with mor e advance d book s o n algebrai c geometry . Furthermore , in the Appendix , w e explain elementar y fact s i n the theor y o f commutativ e ring s and field s that ar e needed i n algebraic geometry. W e give a lot o f detail so that th e Appendix ca n b e usefu l a s a n introductio n an d reader s will ge t use d t o abstrac t thinking. Throughou t th e boo k w e ofte n quot e previou s theorems , lemmas , an d examples. T o help readers i n locating them w e provide an index fo r thes e items . A particular featur e o f this boo k i s that w e dare to repea t explanations . Th e reader wil l notic e that on e an d th e sam e objec t come s u p i n differen t form s an d from differen t viewpoints . W e als o giv e a s man y concret e example s a s possible . In reading a boo k i n mathematics i t i s necessary to fin d concret e examples , verif y abstract results , an d mak e them you r own . Man y book s avoi d repetition a s muc h as possibl e an d expec t th e reader s t o rea d betwee n th e line s o n thei r own . Th e author's experienc e i n teaching a t colleges , however, indicate s that th e number o f students wh o d o no t understan d th e importanc e o f readin g betwee n th e line s o n their ow n i s increasing. Thi s phenomenon i s probably du e to the overemphasi s o n success in entrance examinations a t the expense o f "th e importance o f thinking fo r oneself." In this book w e have tried to examine many examples from various viewpoints. While doin g so we often nee d concrete computations. W e recommend that reader s have paper an d penci l ready an d verif y th e result s b y their ow n computations. A s was said i n ancient Greece , "ther e i s no royal road to geometry." Th e royal road i n mathematics i s to compute on your own until you convince yourself. Whe n you fee l that thi s book gives too much detail and repeats the same thing too often, yo u will have graduated fro m it . Tha t i s the time when yo u shoul d begi n to wor k with a n authoritative systemati c introductio n t o algebrai c geometry (se e the references) . This book is based o n a draft fo r the Iwanami Lecture Serie s in Applied Mathe- matics. I wrote what I wanted to write about algebrai c geometry without worryin g about space and without assumin g much background. Fo r the Lecture Series, it had to be cut down to about on e half. Mr . Hisa o Miyauchi o f Iwanami Shoten, Publish - ers, propose d tha t th e origina l draf t b e published a s a monograph. W e thank Mr . Miyauchi, Ms . A . Hamakado, an d U . Yoshida o f the Editorial Department . Muc h valuable advice on misprints, errors, and displays was received from the productio n staff, whos e help i s greatly appreciated . December 199 4

Kenji Uen o This page intentionally left blank References

[1] M . Reid , Undergraduate Algebrai c Geometry , Londo n Math . Soc . Studen t Texts 12 , Cambridge Universit y Press , 1988 . This i s a good introduction fo r those who have just finished readin g ou r book. [2] K . Iwasawa , Algebrai c Functions , Trans . Math . Monographs , vol . 118 , American Mathematical Society, Providence, 199 3 (translation o f the revised Japanese edition , Iwanami Shoten , 1973) . This celebrate d book deal s with the algebrai c theory o f algebraic func - tions base d o n th e theor y o f s an d th e analyti c theor y o f close d Riemann surfaces . I t i s recommended fo r al l students o f mathematics, re - gardless o f their specia l interests . [3] C.L . Siegel, Topics in Complex Function Theory, I, II, III, Wiley-Interscience, 1969, 1971 , 1973. This authoritativ e treatis e gives a luci d treatment , fro m ellipti c func - tions to the theory o f Jacobian varieties . [4] H . Matsumura, Commutativ e Algebra , 2n d ed . Benjamin , 1980 . This i s a very readable book o n the theory o f commutative rings . [5] D . Mumford , Th e Re d Boo k o f Varietie s an d Schemes , Lectur e Note s i n Math. No . 1358 , Springer, 1988 . A well-known boo k givin g an introduction t o the theory o f schemes. [6] A . Grothendiec k an d J.A . Dieudonne , Element s d e Geometri e Algebrique , Publications Mathematique s d e l'lnstitu t de s Haute s Etude s Scientifiques , vol. 4 , 8, 11, 17, 20, 24, 28, 32, 1960-67. Revise d edition o f EGA I, Springer , 1971. If yo u hav e th e prerequisite s o n commutativ e ring s an d homologica l algebra, yo u migh t wan t t o rea d thi s voluminou s unfinishe d work . I t i s easier than som e other introductor y books . [7] I.R . Schafarevitch , Basi c Algebraic Geometr y I , II, Springer , 1994 . [8] J.P . Serre , Faisceau x algebrique s coherents , Ann . o f Math. 61(1955) , 197 - 278 (reprinted i n his Complete Works , Vol . 1 , pp. 301-339) . This i s a pioneering wor k on algebraic geometr y that use s theory . You may lear n shea f cohomolog y theory fro m thi s book a s a supplement t o your reading o f [5j . [9] J.P . Serre, Geometrie algebrique et geometrie analytique, Ann. Inst . Fourie r 6(1956), 1-4 2 (reprinte d i n his Complete Works , Vol . 1 , pp. 402-443) . This paper i s generally cite d a s "GAGA" . [10] J.W.S . Cassels , Lecture s o n Ellipti c Curves , Londo n Math . Soc . Studen t Texts 24 , Cambridge Universit y Press , 1991.

239 240 REFERENCES

This boo k and the nex t ar e unique introductions t o ellipli c curves . [11] J . H . Silverma n an d J . Tate , Rationa l Point s o n Elliptic Curves , Springer , 1992. Recommended fo r thos e wh o wan t t o stud y ellipti c curves , includin g computer-aided experimenta l aspec t o f the theory o f elliptic curves . [12] J . H . Silverman , Th e Arithmetic o f Elliptic Curves , Springer, 1986 . [13] D.A . Co x and D. O'Shea, Ideals , Varieties, and Algorithms, Springer , 1992 ; 2nd ed. , 1996 . A good book dealing with ideal theory in polynomial rings and algebrai c varieties. Yo u will profit b y using softwar e suc h a s Maple o r Mathematica . [14] C . Moreno, Algebraic Curves over Finite Fields, Cambridge University Press, 1991. This book and the next dea l with application s o f algebraic geometr y t o coding theory . [15] J.H . va n Lin t an d G . va n de r Geer , Introductio n t o Codin g Theor y an d Algebraic Geometry , Birkhauser, 1989 . [16] D.M . Bressoud , Factorizatio n an d Primalit y Testing , Springer , 1989 . The las t chapte r deal s with application s o f s to primalit y testing, which i s also included i n [10] . [17] D . Mumford, Algebrai c Geometry I: Complex Projective Varieties, Springer , 1976. This contains important result s in algebraic geometry ove r the comple x numbers. [18] P . Griffith s an d J. Harris, Principles o f Algebraic Geometry, J. Wiley , 1978 . We recommend Chapter 2 of the book fo r complex curves and Rieman n surfaces. [19] C.H . Clemens , A Scra p Boo k o f Comple x Curv e Theory , Plenu m Press , 1980. The book contains interesting informatio n o n algebraic curve s and Rie- mann surfaces . [20] W . Fulton , Algebrai c Curves , Benjamin, 1969 . The boo k contains a n algebrai c treatment o f algebraic curves . Index

Abelian functio n theory , 4 3 Abel's theorem , 19 9 of type (2,3) , 8 0 affine plane , 58 , 7 3 of type (2,5) , 12 6

complex, 5 8 of type ( Piq), 12 6 affine plan e curve , 6 6 affine transformation , 8 defining equation , 65 , 9 5 affine transformatio n group , 8 degree algebraic curve , 13 5 of a differentia l form , 14 2 algebraic geometry , 6 4 of a divisor , 136 , 18 4 algebraic morphism , 29 , 70, 14 7 of a plan e curve , 23 , 6 5 algebraically close d field, 22 8 of a regular mapping , 14 9 almost everywhere , 72 , 10 3 of a surface , 9 3 analytic set , 18 0 discriminant, 15 8 Atiyah-Singer inde x theorem, 18 5 divisor 135 , 18 2 canonical, 142 , 18 4 base point, 36 , 17 7 positive, 14 1 Bezout's theorem , 3 4 principal, 137 , 18 3 birational equivalence , 10 6 , 3 8 , 4 4 dual projectiv e plane , 38 , 6 1 birational mapping , 7 2 principle , 4 3 birational transformation, 6 4 bitangent, 4 3 elliptic curv e 140 , 16 0 blow-up, 11 5 defining equatio n of , 16 0 blowing-up, 117 , 119 , 12 7 , 18 2 branch o f a curve , 8 1 elliptic functio n field, 9 1 Euclidean transformation , 6 canonical curve , 15 5 Euler's identity , 4 6 canonical divisor , 142 , 18 5 exceptional curve , 11 5 canonical map , 15 4 exceptional surface , 12 8 characteristic, 22 8 Chow's theorem , 18 0 Fermat, 4 8 class o f a curve , 3 8 field, 21 3 commutative field, 213 , 219, 22 0 finite, 22 9 commutative ring , 210 , 22 0 of definition , 16 0 complete linea r system , 17 7 of quotients, 23 4 complex manifold , 85 , 17 9 finite extension , 10 5 complex projectiv e space , 9 1 formal Lauren t series , 23 7 complex toru s formal powe r series , 149 , 23 6 one-dimensional, 182 , 18 7 formula o f Hurwitz, 149 , 151 , 17 7 ^-dimensional, 19 7 Prey curve , 48 , 17 8 congruence modul o a number, 20 7 function field, 56 , 64, 88, 10 4 congruence transformation , 6 fundamental theore m o f algebra, 22 8 congruence zet a function , 16 7 fundamental period , 18 2 coordinate neighborhood , 18 0 covering surface, 4 4 GAGA o f Serre , 57 , 18 2 Cremona transformation , 7 1 generalized Hurwit z theorem , 15 1 curve define d ove r a subfield , 16 0 generators o f an ideal , 9 8

241 242 INDEX

genus, 145 , 151, 154, 155 , 159 , 18 2 on a , 181- 2 group structure , 16 2 meromorphic function , 5 7 Mordell-Weil theorem, 5 7 Hironaka's theorem , 12 7 multiple poin t (se e singular point ) Hilbert's basi s theorem, 9 9 of order n , 3 2 Hilbert's zer o point theorem , 88 , 99, 22 8 multiplicatively close d set , 23 3 holomorphic differentia l form , 18 1 multiplicity, 31 , 78 homogeneous component , 9 8 homogeneous coordinates , 13 , 56, 9 2 n-fold point , 3 3 homogeneous ideal , 9 7 nonsingular plan e curve , 7 4 homogeneous polynomial , 18 , 5 6 nonsingular point , 7 4 group , 19 1 of a plane curve , 7 4 , 22 3 of a variety , 10 7 Hurwitz theore m (formula ) normal rationa l curve , 15 2 (see formula o f Hurwitz ) normalized basis , 19 6 , 140 , 15 4 normalized perio d matrix , 19 5 hyperelliptic functio n field , 9 1 hyperplane, 9 2 ordinary cusp , 33, 7 9 at infinity , 9 2 ordinary doubl e point , 33 , 7 9 hypersurface, 9 3 orthogonal group , 5 of degre e (d , e), 11 4 orthogonal matrix , 5 of degre e d , 9 3 pencil, 3 6 ideal, 98 , 214, 22 3 period matrix , 192 , 19 5 generated by , 22 3 plane curve , 23, 65 maximal, 22 6 irreducible, 2 3 prime, 101 , 217, 22 6 of degree d, 6 5 identity element , 22 0 reducible 23 , 65 image, 22 5 Pliicker's formula , 4 2 imbedding, 15 1 point o f indeterminancy, 7 0 inflection point , 43 , 16 1 polar curve , 4 1 inhomogeneous coordinates , 13 , 9 2 pole, 56 , 13 6 injection, 6 9 of a differentia l form , 14 2 integral domain , 22 6 polynomial ring , 98 , 22 2 intersection multiplicity , 34 , 8 5 prime field, 22 9 intersection number , 19 0 prime ideal , 101 , 217, 22 6 , 8 5 primitive element , 23 2 inverse, 21 2 projection, 10 , 10 2 projective genera l linea r group , 5 5 irreducible plan e curve , 2 3 projective geometry , 6 4 irreducible polynomial , 23 , 21 3 , 2 5 isolated singularity , 12 9 isomorphic curves , 16 1 complex, 2 7 isomorphism, 160 , 22 4 real, 5 1 projective plane , 14 , 5 8 jf-invariant, 158 , 16 0 complex, 22 , 5 8 , 19 8 projective set , 9 5 inP|m xP|n , 111 , 114 fc-rational point , 16 6 irreducible, 9 7 kernel, 22 3 reducible, 9 7 projective space , 9 1 Legendre's canonica l form , 16 1 projective submanifold , 11 5 line, 6 0 projective transformation , 15 , 24, 54 , 6 2 at infinitiy , 14 , 5 8 , 9 7 linear equivalence , 141 , 183 linear frationa l transformation , 9 , 5 3 quadratic transformation , 7 1 local intersectio n number , 34 , 19 0 quadric, 6 7 local parameter, 84 , 10 9 local ring , 23 3 radical, 9 9 ramification index , 14 8 meromorphic differentia l form , 18 1 ramification point , 14 8 INDEX 24 3

rank o f an ellipti c curve , 16 5 rational differentia l form , 14 2 rational function , 5 7 in a projectiv e variety , 10 4 of n variables , 10 4 on a plane curve , 8 8 on P|1(C), 5 6 on P|2(C), 6 4 rational mapping , 39 , 56 , 70 . 7 2 reduced ideal , 10 0 regular differentia l form , 14 2 regular function , 13 6 regular mapping , 70 , 14 7 (see algebrai c morphism ) residue class , 208 , 215, 22 4 residue ring , 217 , 22 4 resolution o f singularities, 116 , 120 , 12 7 for a plane curve , 120 , 12 7 for a surface , 12 7 Riemann, 44 , 4 5 Riemann conjecture , 16 8 Riemann constant , 20 2 Riemann's inequality , 14 5 Riemann's relation , 19 2 Riemann's singularit y theorem , 20 4 Riemann-Roch theorem , 145 , 18 5 , 23 , 49 Riemann surface , 44 , 85, 180, 18 1 scheme, 4 8 Siegel's uppe r half-space , 19 7 singular point , 32 , 7 4 Spec Z , 4 8 strict transform , 12 2 surjective injection , 2 3 symplectic basis , 19 0 symplectic group , 19 6 tangent cone , 32 , 7 9 tangent hyperplane , 11 0 tangent wit h multiplicity , 3 1 theta divisor , 20 1 theta function , 20 0 torsion subgroup , 16 6 total transform , 12 2 transcendental degree , 10 5 , 9 6

Weierstrass canonica l form , 15 7 Weierstrass p-function , 18 7 Weil's conjecture , 16 8

Zariski topology , 10 0 zero divisor, 212 , 217, 22 6 zero element, 209 , 21 9 zero o f a differentia l form , 14 2 zero o f a rational function , 56 , 13 6 zeta function , 16 7 This page intentionally left blank Index fo r Definitions , Theorems , etc . (The page numbers ar e give n in parentheses )

Definitions 1.1(14) 2.1(54); 2.2(62); 2.3(74) ; 2.4(97); 2.5(98); 2.6(107 ) 3.1(141); 3.2(145) ; 3.3(154) ; 3.4(160) ; 3.5(167 ) A.l(208); A.2(214); A.3(216); A.4(219); A.5(223); A.6(223); A.7(226 ) Theorems 1.1(34); 1.2(42); 1.3(43) 2.1(62); 2.2(89); 2.3(99); 2.4(99); 2.5(100) ; 2.6(105) ; 2.7(114); 2.8(127 ) 3.1(145); 3.2(149) ; 3.3(151)3.4(152) ; 3.5(154) ; 3.6(159) ; 3.7(160) ; 3.8(165) ; 3.9(166); 3.10(168) ; 3.11(168) ; 3.12(172 ) 4.1(180); 4.2(182); 4.3(185); 4.4(186); 4.5(192); 4.6(198); 4.7(199); 4.8(202); 4.9(203); 4.10(204 ) A.l(209); A.2(212); A.3(217); A.4(226); A.5(228); A.6(231); A.7(232 ) Propositions 3.1(151); 3.2(152) ; 3.3(156) ; 3.4(160) ; 3.5(161) ; 3.6(162) ; 3.7(163) ; 3.8(165) ; 3.9(171) Lemmas 2.1(54); 2.2(55); 2.3(57) ; 2.4(61); 2.5(61) ; 2.6(62) ; 2.7(63); 2.8(64); 2.9(64) ; 2.10(74); 2.11(98); 2.12(100) ; 2.13(101) ; 2.14(108); 2.15(119 ) 3.1(137); 3.2(137) ; 3.3(141) ; 3.4(142) ; 3.5(145) ; 3.6(146) ; 3.7(147) ; 3.8(152) ; 3.9(154); 3.10(169 ) 4.1(201) Al(208); A.2(208); A.3(211); A.4(213); A.5(214); A.6(223); A.7(223); A.8(228); A.9(232) Corollaries 2.1(61) 3.1(141); 3.2(141) ; 3.3(145) ; 3.4(146) ; 3.5(166) ; 3.6(169) ; 3.7(169) ; 3.8(171 ) 4.1(196); 4.2(202); 4.3(203) ; 4.4(203); 4.5(204 ) A.l(212); A.2(214); A.3(218); A.4(233 ) Examples 1.1(32); 1.2(33); 1.3(39) ; 1.4(39) ; 1.5(41) 2.1(67); 2.2(70 ) 2.3(71) ; 2.4(75); 2.5(76) ; 2.6(79); 2.7(80 ) 2.8(82) ; 2.9(83) ; 2.10(83); 2.11(86) ; 2.12(87) ; 2.13(89); 2.14(90) ; 2.15(90) ; 2.16(91) ; 2.17(94) ; 2.18(95); 2.19(96); 2.20(101); 2.21(103); 2.22(104); 2.23(105); 2.24(108); 2.25(109); 2.26(109); 2.27(110) ; 2.28(115) ; 2.29(122) ; 2.30(123) ; 2.31(124) ; 2.32(126 ) 3.1(136); 3.2(137) ; 3.3(143) ; 3.4(143) ; 3.5(150) ; 3.6(150) ; 3.7(153) ; 3.8(155) ; 3.9(164); 3.10(165) ; 3.11(165) ; 3.12(165) ; 3.13(167) ; 3.14(173) ; 3.15(174) ;

245 246 INDEX FO R DEFINITIONS , THEOREMS , ETC .

3.16(175); 3.17(175) ; 3.18(175) 4.1(181); 4.2(182); 4.3(187) A.l(209); A.2(210); A.3(216); A.4(216); A.5(217); A.6(218); A.7(218); A.8(220); A.9(221); A.10(222); A.ll(224); A.12(225); A.13(226) ; A.14(227); A.15(230); A.16(230); A.17(231); A.18(234); A.19(235); A.20(236 ) Errata—An Introduction to Algebraic Geometry by Kenj i Uen o

Line — 2 means 2 nd lin e u p fro m th e botto m o f the page .

line-2: q(X,Y) shoul d b e q(x',y f) line -8: (0 , /? : -a) shoul d b e ( 0 : /? : -a)

line 18 : (a 0, ai: a 2) G E7 o shoul d b e (a 0 : ai: a 2) G I70

line 20 : (1 : x0, y 0) shoul d b e (1 : x0: yo) 2 2 line —4 : |§ - shoul d b e | f line 4 : a i ^ 0 shoul d b e a o ^ 0 line —14 : ^ f shoul d b e un line -3: G C shoul d b e G C line 5 : ^=^ shoul d b e ^ ^

m A V 2) line 14 : A ~ VAf )j?( a) shoul d b e ^ A^ F(a) line —11 : correction simila r t o abov e bottom line : correctio n simila r to abov e line -6: = ( n + 2)( n + 3)/ 2 shoul d b e = ( n + 2)( n + l)/ 2 line —7 : /(xo,^i,X2) shoul d b e F{XQ,XI,X2) line 14 : i n P2(C) corerspond s shoul d b e i n P2(C)* correspond s line 8 : (P 2(C)2)* shoul d b e P 2(C)* line 4 |J{(0: 1) } shoul d b e |J{(1 : 0) } line 3 o f Figure 2.2 : you r wil l shoul d b e yo u wil l line 4 : U\ shoul d b e UQ line 4 : in UQ shoul d b e Ui line 9 : x lx^ shoul d b e XQX{ line 3 : a s shoul d b e a t 2 line —9 : a 2 shoul d b e a

line —3 : (6 0;6i) shoul d b e (b 0: &i )

bottom line : (xo : £i; x2) shoul d b e (xo : x±: x 2) line 11 : (1 : 0: 1 ) shoul d b e (1:0:0 )

lines 7 , 8 , 9 : (a 0: a±: a 2) shoul d b e (ai,a 2,a2) d x d l line 10 : a ~ shoul d b e a Q~

2 line —3 : x26 shoul d b e | f

247 248 ERRATA— AN INTRODUCTION TO ALGEBRAIC GEOMETRY B Y KENJ I UEN O p. 79, line 11 : point with multiplicit y shoul d b e poin t a t (0,0 ) wit h multiplicity p. 81, lin e 2 : y— shoul d b e y+ (fo r on e o f the factors , preferabl y th e secon d one—see p. 83 ) p. 83, bottom line : 3/2s 4 shoul d b e (3/2)s 4 p. 84, line 2 after Figur e 2.10 : P(C) 2 shoul d b e P 2(C) p. 84, line 1 2 after Figure : djyj shoul d b e a^yi p. 88, annotation to Figur e 2.12(b) : E: y 2 = y 3 shoul d b e E: y 2 = x 3 p. 90, line —6 : g2 — shoul d b e g\ — p. 90, line -2: t o C shoul d b e t o C p. 91, lin e 5 : denominato r o f LHS c{x) — d(x, y) shoul d b e c(x) + d(x)y p. 93, line —5 : after : x n insert : o f degree m p. 97, line —6 : gj shoul d b e Gj p. 98, line 1 3 (equation): Hj shoul d b e Fj p. 99, line -6: ( G ± H) m^n* shoul d be (G ± H) m^+rn* p. 99, line -4: (II ) shoul d b e (II ) p. 100, line -17: V(a x) shoul d b e V(a x) p. 100, line -4: F G shoul d b e G G p. 102, line -7: Pni shoul d b e P n_1 p. 102, line -6: Pni shoul d b e P 71"1 p. 104, line —3 : z 2 = z\z 2 shoul d b e z 2 = z\ p. 105, line -18: C(x ) shoul d b e C(X) p. 105, line — 5: z± shoul d b e Zi (to include the case s fo r i = 1,2 , 3 ) p. 105, line —5 : XI/XQ shoul d b e Xi/xo p. 107, line 9 : F shoul d b e Fj p. 107, line 1 2 after: o f the insert : d-dimensiona l (sinc e the Definitio n a s it stands refer s to ran k n — d, but ther e ha s bee n n o prior mentio n o f d) p. 108, line 9 (two corrections i n this): —#o~ 2#i^(f1-^2-) shoul d b e Z\ ^XQ XO X x 0 29zidz\ \U_ xo, ' xoXn ) p. 108, line —2 : a^r shoul d b e a n p. 109, line 12 : a^ shoul d b e a n p. 115, line 12 : ) ( shoul d b e ), ( p. 115, line 13 : i s surjection shoul d b e i s a surjectio n p. 115, line -16: P 2 shoul d b e P 1 p. 115, line -11: (at, fit) shoul d b e (at : fit) p. 116, line 2 after Figure : Gr shoul d b e (D -1 p. 116, line 1 1 after Figure : (a , /3) G P2 shoul d b e (a , /?) G P1 p. 117, line 16 : U\ U U2 shoul d b e U 0 U Ux p. 117, line 17 : eU shoul d b e G U0 p. 117, line 21: onto G UQ shoul d b e G Uo p. 118, line 15 : conside r UQ shoul d b e conside r UQ p. 118, line 16 : conside r U\ shoul d b e conside r U\ ERRATA—AN INTRODUCTION TO ALGEBRAIC GEOMETRY B Y KENJ I UEN O 24 9

line 20: o f U shoul d b e o f U line 21: } . shoul d b e } , line 25: {(0,0) } shoul d b e {(1,0,0) } line 1 : ( M - {p} shoul d b e ( M - {p}) line — 4: hi shoul d b e U$ line 4: = E shoul d b e = z n£ line —11 : i > 0 shoul d b e j > 0 annotation to Figur e 2.18(b) : . E shoul d b e C line 2 : /2 shoul d b e t 2 line —12 : a the origi n shoul d b e a t the origi n line — 3: y should b e y -f ^0 line —10 : highter shoulshould be highe r line —4 : l/x k shoul d b e l/x l line 2: ( n - •2K" -3n2„2 x\ shoul d b e ( n — 2)Xn-3^,Q X 2; line 13 : x = u/v should b e x = v/u line —11 : (1 : 0: oij) shoul d b e (1:^:0) line 15 : dg shoul d b e fdg line 9 : regula r regula r shoul d b e regula r

line—8: f x(x,y)+ shoul d b e f x(x,y)dx-\- i 3: v = - shoul d b e v = - y a ? line 2: (<#,<# ) shoul d b e (dg: oj ) line 11: y — y ~ a shoul d b e y — t — a should b e h n- line 1 1 (tw o corrections): /i n-q line 8 : V shoul d b e C line —11 : a^x^x 2 shoul d b e azx\x\ line —3 : qj shoul d b e gj line 14 : curv e shoul d b e curve s line —13 : 0. shoul d b e O. line 7 : Th e denominato r o f RHS shoul d rea d ( 1 — u)(l — qu) This page intentionally left blank Selected Title s i n This Serie s (Continued from the front of this publication)

128 V . P . Orevkov , Complexit y o f proofs an d thei r transformation s i n axiomatic theories , 1993 127 F . L . Zak , Tangent s an d secant s o f algebraic varieties , 199 3 126 M . L . AgranovskiY , Invarian t functio n space s o n homogeneou s manifold s o f Li e group s and applications , 199 3 125 Masayosh i Nagata , Theor y o f commutative fields , 199 3 124 Masahis a Adachi , Embedding s an d immersions , 199 3 123 M . A . Akivi s an d B . A . Rosenfeld , Eli e Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hou , Theor y o f entire an d meromorphi c functions : deficien t an d asymptotic value s an d singula r directions , 199 3 121 LB . Fesenk o an d S . V . Vostokov , Loca l field s an d thei r extensions : A constructiv e approach, 199 3 120 Takeyuk i Hid a an d Masuyuk i Hitsuda , Gaussia n processes , 199 3 119 M . V . Karase v an d V . P . Maslov , Nonlinea r Poisso n brackets . Geometr y an d quantization, 199 3 118 Kenkich i Iwasawa , Algebrai c functions , 199 3 117 Bori s Zilber , Uncountabl y categorica l theories, 199 3 116 G . M . Fel'dman , Arithmeti c o f probability distributions , an d characterizatio n problem s on abelia n groups , 199 3 115 Nikola i V . Ivanov , Subgroup s o f Teichmuller modula r groups , 199 2 114 Seiz o Ito , Diffusio n equations , 199 2 113 Michai l Zhitomi r skit, Typica l singularitie s o f differential 1-form s and Pfaffia n equations , 1992 112 S . A . Lomov , Introductio n t o the genera l theory o f singular perturbations , 199 2 111 Simo n Gindikin , Tub e domain s an d th e Cauch y problem , 199 2 110 B . V . Shabat , Introductio n t o comple x analysi s Part II . Functions o f several variables , 1992 109 Isa o Miyadera , Nonlinea r semigroups , 199 2 108 Take o Yokonuma , Tenso r space s an d exterio r algebra , 199 2 107 B . M . Makarov , M . G . Goluzina , A . A . Lodkin , an d A . N . Podkorytov , Selecte d problems i n rea l analysis , 199 2 106 G.-C . Wen , Conforma l mapping s an d boundar y valu e problems , 199 2 105 D . R . Yafaev , Mathematica l scatterin g theory : Genera l theory, 199 2 104 R . L . Dobrushin , R . Kotecky , an d S . Shlosman , Wulf f construction : A global shap e from loca l interaction, 199 2 103 A . K . Tsikh , Multidimensiona l residue s and thei r applications , 199 2 102 A . M . Il'in , Matchin g o f asymptotic expansion s o f solutions o f boundary valu e problems , 1992 101 Zhan g Zhi-fen , Din g Tong-ren , Huan g Wen-zao , an d Don g Zhen-xi , Qualitativ e theory o f differential equations , 199 2 100 V . L . Popov , Groups , generators , syzygies , an d orbit s i n invarian t theory , 199 2 99 Nori o Shimakura , Partia l differentia l operator s o f elliptic type , 199 2 98 V . A . Vassiliev , Complement s o f discriminants o f smooth maps : Topolog y an d applications, 199 2 (revise d edition , 1994 ) 97 Itir o Tamura , Topolog y o f foliations : A n introduction , 199 2 96 A . I . Markushevich , Introductio n to th e classica l theory o f Abelia n functions , 199 2 95 Guangchan g Dong , Nonlinea r partia l differentia l equation s o f second order , 199 1 94 Yu . S . Il'yashenko , Finitenes s theorem s fo r limi t cycles , 199 1 93 A . T . Fomenk o an d A . A . Tuzhilin , Element s o f the geometr y an d topolog y o f minimal surface s i n three-dimensional space , 199 1 92 E . M . Nikishi n an d V . N . Sorokin , Rationa l approximation(See the AMsS an catalod orthogonalityg fo r earlie r, 199titles1 )