Non-complete Algebraic Surfaces
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857 Masayoshi Miyahishi
Non-complete Algebraic Surfaces
Springer-Verlag Berlin Heidelberg New York 1981
Author
Masayoshi Miyanishi Department of Mathematics Faculty of Sciences, Osaka University Toyonaka, Osaka 560 Japan
AMS Subject Classifications (1980): 14020, 14J15 ISBN 3-540-10703-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10703-7 Springer-Verlag New York Heidelberg Berlin This work IS subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
These notes were prepared for my lectures delivered at Department of Mathematics, the University of Chicago, in the fall quarter, 1980.
Section 4 of Chapter I was lectured at Department
of Mathematics, McGill University, in September, 1980.
I would
like to express my sincere gratitude to both institutions for providing me with opportunities to give lectures and for warm hospitality during my stay.
Especially, I would like to thank all
members of Algebraic Geometry Seminar at the University of Chicago; attention and patience shown by them are greatly appreciated. I am indebted to S. Iitaka for inspiring ideas, e.g. logarithmic Kodaira dimension, to T. Fujita for the result on the Zariski decomposition of a pseudo-effective divisor and among others, to Y. Kawamata for almost all results in Chapters II and III; these chapters are nothing more than detailed explanations of the results of Kawamata's [14, 15]. I believe there is an abundant room in the theory of noncomplete algebraic surfaces waiting for further exploitations, and hope that these notes bring some lights to this purpose. January, 1981 M. Miyanishi
CONTENTS
Introduction .
VII
Notations and conventions .
XVI
Chapter I.
Non-complete algebraic surfaces with logarithmic Kodaira dimension -
00
•
Section 1.
Logarithmic Kodaira dimension
Section 2.
Algebraic surfaces containing cylinder like open sets
Section 3.
Fujita's theory of pseudo-effective divisors.
Section 4. Section 5.
4
subrings of a polynomial ring .
92
Normal affine surfaces containing cylinerlike 119
open sets Chapter II.
47
Examples of non-complete algebraic surfaces with logarithmic Kodaira dimension
Section 6.
36
Non-complete algebraic surfaces with logarithmic Kodaira dimension
Section 1.
130
or
The Zariski decomposition of a pseudo-effective divisor
Section 2.
0
130
O+K V
The structure theorem for relatively minimal non-complete algebraic surfaces with logarithmic Kodaira dimension
0
or
154
Section 3.
The proof of the structure theorem, I
157
Section 4.
The proof of the structure
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