An Introduction to the Theory of Algebraic Surfaces Notes by James C
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Oscar Zariski Harvard University, Cambridge, Mass.
An Introduction to the Theory of Algebraic Surfaces 1969
Notes by James Cohn, Harvard University, 1957 - 58
Springer-Verlag Berlin· Heidelberg· New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin· Heidelberg 1969 Library of Congress Catalog Card Number 68-59477 Printed in Germany. Title No. 3689
PREFACE
These are the lecture notes of a course which I gave at Harvard University in 1957-58. As the supply or the original mimeographed copies of these notes has been exhausted some years ago, and as there seemes to be some evidence of a continued demand, I readily accepted a proposal by the Springer Verlag to republish these old notes in the current series of "Lecture Notes in Mathematics. '.' I refrained from making any changes or revisions, for I feel that these notes can best serve their purpose if they are published in their exact original form. The purpose of these notes is to acquaint the reader with some basic facts of the theory of algebraic varieties, and to do that by self-contained, direct and I would almost say - ad hoc methods of Commutative Algebra, without overwhelming the reader with a mass of material which has a degree of generality out of all proportion to the immediate object at hand. I should also mention, incidentally, that the title of these lecture notes is somewhat misleading, for only three of the sixteen sections (namely, sections 7, 14, and 19) deal specifically with algebraic surfaces; the remaining thirteen sections deal with varieties of any dimension.
Oscar Zariski Harvard University December, 1968
TABLE OF CONTENTS
1. 2.
3.
4. 5. 6.
7. 8. 9. 10.
11. 12.
13.
14. 15. 16.
Homogeneous and non-homogeneous point coordinates •••• Coordinate rings of irreducible •••••••••••• Itorma1 varieties ••••••••••••••••••••••••••••••••••••• Divisoria1 cycles on a normal projective variety ••••. Divisors on an arbitrary variety V••••••••••••••••••• Intersection theory on algebraic surfaces •••••••••.••
§l. Let
k be our ground field; it need not be algebraically closed.
An will denote an n-dimensional affine space, and Sn an n-dimensional projective space. Def. 1.1:
These spaces have coordinates in a universal domain.
If P = (x J".'X ) A , then the local field, n
1
is
=
k(x)
n
•• "JXn)' and dim(p/k)
keF), of
p/k
= t.d.(k(x)/k).
This definition is ind6pendent of the choice of the affine coordinates in An Ik. Def. 1.2:
= (y o , ... ,yn )E.S n ,
If P
field generated over and dim (PJk) of
then the local field,
k(P),
y/:o
k by the ratios
i
= t,d.(k(P)ftc). keF)
is the
is independent of the choice of homogeneous
coordinates in S Ik. n No'tie:
If we fix Yi"; 0, then k(P) = k(yjyi , · .. , yr!Yi)' To see thi3, let y. =: y'0 and assume y1 J; 0. Then for arr:! j,
= (Yj/yo) /(:lII'10 ) ,
hence
Y.jY E key Iy s •• ", l u 1 0
Let
H 1
y Iy ).
be a hyparp'l ane in S,
n-
n
n
0
given by
faY i=O
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