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Masayoshi Miyanishi Recent Titles in This Series 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 theor y 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, Functions o n manifolds : Algebrai c an d topologica l aspects , 199 3 130 V . V . Vershinin, Cobordisms an d spectra l sequences , 199 3 129 Mitsu o Morimoto , A n introductio n t o Sato' s hyperfunctions , 199 3 128 V . P. Orevkov , Complexit y o f proof s an d thei r transformation s i n axiomati c theories , 199 3 127 F . L. Zak , Tangent s an d secant s o f algebrai c varieties , 199 3 126 M . L . Agranovskii , Invarian t functio n space s o n homogeneou s manifold s o f Li e group s an d applications, 199 3 125 Masayosh i Nagata , Theor y o f commutativ e 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 , Theory o f entir e an d meromorphi c functions : Deficien t an d asymptoti c value s and singula r directions , 199 3 121 I . B. Fesenko an d S. V . Vostokov, Loca l fields and thei r extensions : A constructive approach , 199 3 120 Takeyuk i Hid a an d Masuyuki Hitsuda , Gaussia n processes , 199 3 119 M . V . Karasev an d V . P. Maslov, Nonlinear 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 , Arithmetic o f probabilit y distributions , an d characterizatio n problem s o n abelia n groups, 199 3 115 Nikola i V . Ivanov, Subgroup s o f Teichmiille r modula r groups , 199 2 114 Seiz o Ito , Diffusio n equations , 199 2 113 Michai l Zhitomirskii , Typica l singularitie s o f differentia l 1-form s an d Pfaffia n equations , 199 2 112 S . A . Lomov , Introduction t o th e genera l theor y o f singula r perturbations , 199 2 111 Simo n Gindikin , Tube domain s an d th e Cauch y problem , 199 2 110 B . V . Shabat, Introductio n t o comple x analysi s Par t II . Function s o f severa l variables , 199 2 109 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 problem s i n rea l analysis, 199 2 106 G.-C . Wen , Conformal 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 globa l shap e fro m loca l interaction, 199 2 103 A . K . Tsikh , Multidimensional residue s an d thei r applications , 199 2 102 A . M. Il'in , Matching o f asymptoti c expansion s o f solution s o f boundary valu e problems, 199 2 101 Zhan g Zhi-fen , Din g Tong-ren , Huan g Wen-zao , an d Don g Zhen-xi , Qualitativ e theory o f differentia l 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 ellipti c type , 199 2 98 V . A. Vassiliev , Complement s o f discriminant s o f smoot h maps : Topolog y an d applications , 199 2 (revised edition , 1994 ) 97 Itir o Tamura , Topology o f foliations : A n introduction . 199 2 96 A . I . Markushevich , Introductio n t o th e classica l theor y o f Abelian functions , 199 2 (Continued in the back of this publication) This page intentionally left blank Algebrai c Geometr y This page intentionally left blank 10.1090/mmono/136 TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME 136 Masayoshi Miyanishi Algebrai c Geometr y Translated by Masayoshi Miyanishi && DAISU KIKAGAK U (Algebraic Geometry ) by Masayosh i Miyanish i Copyright © 199 0 b y Shokab o Publishin g Co. , Ltd . Originally publishe d i n Japanese b y Shokabo Publishin g Co. , Ltd., Toky o in 1990 . Translated fro m th e Japanes e b y Masayosh i Miyanish i 2000 Mathematics Subject Classification. Primar y 14-01 ; Secondary 13-01 , 14A10 , 14A15 . ABSTRACT. Thi s boo k cover s algebrai c geometr y fro m th e beginning s t o a n introductio n o f algebrai c surfaces, viz., t o th e gat e fro m whic h th e classificatio n o f algebrai c surface s starts . Th e boo k ha s thre e parts. Th e first par t provide s th e necessar y basi c result s fro m commutativ e algebra s an d th e theor y o f sheaves and its cohomologies. Th e second part i s on schemes and algebrai c varieties. Th e third part i s on algebraic curve s an d surfaces , placin g emphasi s o n th e us e o f linea r system s an d th e associate d rationa l mappings. Library o f Congres s Cataloging-in-Publicatio n Dat a Miyanishi, Masayoshi , 1940 - [Daisu kikagaku . English ] Algebraic geometry/Masayosh i Miyanishi ; translate d b y Masayosh i Miyanishi . p. cm . — (Translation s o f mathematical monographs , ISS N 0065-9282 ; v . 136 ) Includes bibliographica l reference s an d index . ISBN 0-8218-4615- 9 (acid-free ) 1. Geometry, Algebraic . I . Title . II . Series . QA564.M5713 199 4 94-201 8 516.3'5—dc20 CI P © Copyrigh t 199 4 b y th e America n Mathematica l Society . Al l right s reserved . Reprint wit h correction s 1998 . Translation authorize d b y th e Shokab o Publishin g Co. , Ltd . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . 0 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 . Information o n Copyin g an d Reprintin g ca n b e foun d i n th e bac k o f thi s volume . This volum e wa s typese t usin g AMS-T^X, the America n Mathematica l Society' s T^ X macr o system . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 0 6 0 5 0 4 0 3 0 2 0 1 Contents Preface t o th e Englis h Editio n i x Preface x i Part I. Preliminarie s 1 Chapter 1 . Theore m o f Liirot h 3 Chapter 2 . Theor y o f Sheave s an d Cohomologie s 2 5 Part II . Scheme s an d Algebraic Varietie s 5 9 Chapter 3 . Affin e Scheme s an d Algebrai c Varietie s 6 1 Chapter 4 . Scheme s an d Algebrai c Varietie s 8 1 Chapter 5 . Projectiv e Scheme s an d Projectiv e Algebrai c Varietie s 10 5 Chapter 6 . Nonsingula r Algebrai c Varietie s 13 3 Part III . Algebrai c Surface s 15 7 Chapter 7 . Algebrai c Curve s 15 9 Chapter 8 . Intersectio n Theor y o n Algebrai c Surface s 16 5 Chapter 9 . Pencil s o f Curve s 17 5 Chapter 10 . Th e Riemann-Roc h Theore m fo r Algebrai c Surface s 19 1 Chapter 11 . Minima l Algebrai c Surface s 20 1 Chapter 12 . Rule d Surface s an d Rationa l Surface s 21 1 Solutions t o Problem s 22 1 Part I 22 1 Part I I 22 6 Part II I 23 0 List o f Notation 23 7 Bibliography 24 1 Index 24 3 This page intentionally left blank Preface t o th e Englis h Editio n The interesting theory, where one first experiences the beauty in learning algebraic geometry, i s perhaps the theory o f algebraic curves and surfaces . Th e theory o f alge- braic surfaces i s particularly important whe n on e further studie s higher-dimensiona l algebraic varieties . In orde r t o approac h thi s theor y throug h moder n algebrai c methods , on e i s required t o hav e a basic knowledge o f local rings , sheaves , cohomologies, projectiv e varieties, linear systems of divisors, etc. Thi s basic knowledge may include all results given i n introductory textbook s devote d to thes e subjects , an d i t takes som e time t o get familia r wit h them .
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