Ample Subvarieties of Algebraic Varieties Notes written in Collabora
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156 Robin Hartshorne
Ample Subvarieties of Algebraic Varieties Notes written in Collaboration with C. Musili
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Robin Hartshorne Department of Mathematics, University of California, Berkeley Berkeley, CA 94720, USA
1st Edition 1970 2nd Printing 1986 ISBN 3-540-05184-8Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-05184-8Springer-Verlag New York Heidelberg BerlinTokyo This work is subject to copyright. Alrrights 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 1970 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
These notes are an enlarged version of a three-month course of lectures I gave at the Tata Institute of Fundamental Research during the winter of 1969-70, while on sabbatical leave from Harvard. style is informal.
Their
I hope they will serve as an introduction to some
current research topics, for students who have had a one year course in modern algebraic geometry. They contain some known material, mostly available only in original research papers, and some new material published here for the first time.
In particular, I have added Ch. !
810, Ch. II 85,
Ch. III 886,7,8, and Ch. VI, which contain new results mentioned only briefly in the lectures.
For the benefit of future researchers there
are a number of exercises, open problems, and conjectures.
There is
an extensive bibliography at the end. I wish to thank the Tata Institute for inviting me to come to Bombay, and the auditors of the lectures for their questions and stimulating conversation.
My special thanks go to C. Musili who
recorded the lectures, wrote them up for inclusion in these notes, and assisted in all aspects of the preparation of the manuscript. Finally, I wish to thank my wife, whose support has been invaluable. Robin Hartshorne Bombay April, 1970
CONTENTS
Preface
III
Introduction
VII
Main implications of Chapters III,IV,V
XI
Notations, terminology and conventions
XII
Table of conjectures and open problems
XIV
Ample Divisors
1
Chapter I
...
§
0
Generalities on divisors
8
1
Cartier divisors
§
2
Linear systems
12
8
3
Ample divisors
18
§
4
Functorial properties
23
8 5
Nakai's criterion
29
§
6
Pseudo-ampleness
§
7
Seshadri's criterion
37
§
8
The ample cone
40
§
9
Chevalley's conjecture
45
Appendix: Curves on ruled surfaces, and examples of Mumford and Ramanujam
50
§ 10
5 7
...
34
v
Chapter I I
Affine open subsets
59
s
1
Serre's criterion for affineness
61
§
2
Sufficient conditions for the complement of a subvariety to be affine
64
§
3
Necessary conditions
66
§
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