Geometric theory of algebraic space curves (Lecture notes in mathematics ; 423)

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

423 S. S. Abhyankar A. M. Sathaye

Geometric Theory of Algebraic Space Curves

Springer-Verlag Berlin.Heidelberg New York 1974

Prof. Dr. Shreeram Shankar Abhyankar Purdue University Division of Mathematical Sciences West Lafayette, IN 4?907/USA Prof. Dr. Avinash Madhav Sathaye University of Kentucky Department of Mathematics Lexington, KY 40506/USA

Library of Congress Cataloging in Publication Data

Abhy~Lkar~ Shreeram Shankar. Geometric theory of algebraic space curves. (Lecture notes in mathematics ; 423) Includes bibliographical references and indexes. i. Curves, Algebraic. 2. Algebraic varieties. I. Sathaye~ Avinash Madhav, 1948joint author. Iio Title. III. Series: Lecture notes in mathematics (Berlin) ; 423. Q~3.L28 no. 423 [QA567] 510'.8s [516'.35] 74-20717

A M S Subject Classifications (1970): 14-01, 14 H 99, 14 M 10

ISBN 3-540-06969-0 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06969-0 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

PREFACE

The o r i g i n a l Montreal

Notes

(36.9),

namely

m a i n part of this b o o k w a s

[ 3 3that

The m a i n

"All i r r e d u c i b l e

degree

at most

ground

field are c o m p l e t e

completely

five and genus

proved

and circulated, self-contained of the proof,

it had

nonsingular

at most one over

in 1971.

published.

We

obsolete

space

intended

or rather,

the p r e p a r a t o r y

material,

finally

started

1973,

a completely

in the process,

and s o m e w h a t

in June

closed

the T h e o r e m A w a s

to give

and

clearer

of

of the proof h a v e b e e n w r i t t e n

of the Theorem,

to b e c o m e

curves

an a l g e b r a i c a l l y

treatment

October

by p r o v i n g

1973, that

however,

the size

enlarged;

sharper.

and w e

Murthy

while

the

The p r e s e n t

finally

at m o s t one o v e r an a l g e b r a i c a l l y

plete

intersections.

Conjecture" k

his p r o o f

[123-

decided

that

is a l g e b r a i c a l l y

closed

modules

closed)."

over

detailed

proofs

itself,

by u s i n g

a concrete

three

elements

given

in the M o n t r e a l

curves

the w e l l k n o w n

can sometimes

are

"Serre's

free in

be more u s e f u l

description

of a basis

nonsingular

[ 3 3 as one of the main

of

field are com-

k[X,Y,Z~

for the ideal of an i r r e d u c i b l e Notes

ground

space

H o w e v e r he also i l l u s t r a t e d

concrete

the t h e o r e m

our m a i n T h e o r e m

nonsingular

In fact he proved

that all p r o j e c t i v e

that,

rendered

"All i r r e d u c i b l e

genus

than

to the T h e o r e m

As such,

Two v e r s i o n s

to the

to be a book.

During

(when

the p r o o f

intersections."

but none

proof c o n t i n u e d version was

part was

just a s e q u e l

steps

space curve in his

proof. Another genus

and d i f f e r e n t i a l s

function developed 1973.

important

feature

of the book

of a s e p a r a b l y

field over an a r b i t r a r y by A b h y a n k a r

One v i r t u e

any a r t i f i c i a l

during

the

of the p r e s e n t

devices

is a new t r e a t m e n t

of the

generated

one-dimensional

field.

The t r e a t m e n t was

ground Purdue

Seminar

treatment

such as r e p a r t i t i o n s

is,

in S u m m e r

that it does

or d e r i v a t i o n s

of

and Fall not need of the

IV

g r o u n d field.

(See C h a p t e r III for the t r e a t m e n t and §40 for the com-

parison with other treatments.) R e t u r n i n g to the proof of the c o m p l e t e note that it has b a s i c a l l y two parts: p r o j e c t i o n of the space curve w i t h (26.12))

i n t e r s e c t i o n theorem, we

one part is to o b t a i n a "nice"

a "nice" a d j o i n t

(our T h e o r e m

and the second part is to c o n s t r u c t a basis of two elements

for the ideal of the curve

(our T h e o r e m

(36.7)).

In the present v e r s i o n of the proof,

the second part that is

needed was already d e v e l o p e d in [ 3 ], but w e include a p r o o f for the sake of completeness. namely,

In an older version,

a g e n e r a l i z a t i o n was needed;

the t r e a t m e n t of the s o - c a l l e d chains of e u c l i d e a n domains,

and

is p r e s e n t e d in §39, m a i n l y b e c a u s e it is of interest in itself. The t r e a t m e n t of the first part about p r o j e c t i o n s different

from its c o u n t e r p a r t in ~ 3 3-

is e s s e n t i a l l y

in [ 3 I, w e w r i t e down expli-

cit e q u a t i o n s of the curve and carry out the p r o o f by c o m p l i c a t e d calculations;

here a s i m i l a r m e t h o d was

too cumbersome.

W h a t we p r e s e n t h e r e is a g e o m e t r i c a r g u m e n t in w h i c h

w e never e v e n need a c o o r d i n a t e system. to c o n v i n c e anybody,

self-contained

However,

that this is geometry,

a v o i d e d the use of g e o m e t r i c terms, ours,

first tried and proved to be

it m i g h t be d i f f i c u l t

for we h a v e d e l i b e r a t e l y

so that the proof may stay rigor-

and still r e a s o n a b l y short.

Thus we h a v e taken the useful g e o m e t r i c concepts,

translated

t h e m into precise a l g e b r a i c terms and a l m o s t never gone b a c k to the g e o m e t r i c terms.

For a g e o m e t r i c m i n d e d reader, however, we h a v e pro-

v i d e d a d i c t i o n a r y in §43 so that he may be able to read the underlying g e o m e t r i c 'geometry'

argument very easily.

The name of this b o o k owes its

to this arrangement.

The p r o o f of e x i s t e n c e of the "nice p r o j e c t i o n " may be a p p r o p r i a tely d e s c r i b e d as repeated a p p l i c a t i o n s of "Bezout's Theorem." need b a s i c a l l y the special

(but m o s t w e l l known)

"Bezout's Little Theorem" in the present book,

case,

namely,

We

termed as the case of the

intersection

of a h y p e r s u r f a c e

§ (23.9).

The general

presented

in §38;

case

mainly

because

Bezout's

is, however,

study

to e m b e d

jective

terminology

Chapter

II,

is treated

in

is

p r o o f of this

To avoid clashes only

get a p r o j e c t i v e

counterpart

of the

(§26)

theorem.

and

This made

in a p r o j e c t i v e

concentrate

space,

in affine

on p r o j e c t i v e

then return

theorem

it

and

and procurves

in

about exist-

to affine

curves

in

IV.

Bezout's

theorem

multiplicity;

also needs

for p r o j e c t i v e

We d e v e l o p

the

for affine

curves

theory

in C h a p t e r

is to provide

a precise

as w e l l

for p r o j e c t i v e

The only use of C h a p t e r cerned

curve

first

of nice p r o j e c t i o n s

Chapter

accessible

a projective

completion. we

case

literature.

the g i v e n affine

its p r o j e c t i v e

This

of two h y p e r s u r f a c e s

a readily

in the

Theorem

a curve.

of i n t e r s e c t i o n

is not a v a i l a b l e

necessary

ence

case

with

III,

I §5,6;

theory of i n t e r s e c t i o n

as affine curves

curves,

in C h a p t e r

in our case. II §23,24

and put it t o g e t h e r

so far as c o m p l e t e

a p r o o f of the w e l l k n o w n

in c h a p t e r

intersections genus

and

formula

are

IV.

con-

for plane

curves. We h a v e their

taken

coordinate

coordinate sections

rings

appear

they d e s c r i b e Here

to be almost

spaces

abstract, in case

in d u p l i c a t e

results

for these

is a general

summary

of various

I gives

the theory

essentially

some general

Chapter

cepts

if they are

of a m b i e n t

at a point,

contains

sent

rings,

that v a r i e t i e s

the same

Chapter curves

the v i e w p o i n t

II gives

abstract

for p r o j e c t i v e

curves.

they

in the

are embedded.

(§5,6;

§23,24

two types

by

etc.)

Several because

of varieties.

chapters. multiplicity

or local

curves.

of two It also

terminology. a treatment

irreducible

of i n t e r s e c t i o n

and by ideals

of i n t e r s e c t i o n for affine

are r e p r e s e n t e d

of

projective

multiplicity,

"homogeneous varieties. projection,

domains"

which

It d e v e l o p s tangential

repre-

the con-

spaces

etc.

VI

C h a p t e r III gives a t r e a t m e n t of d i f f e r e n t i a l s generated

function fields.

in s e p a r a b l y

It also has various genus

a b s t r a c t and e m b e d d e d p r o j e c t i v e curves.

formulas

for

This c h a p t e r is almost

s e l f - c o n t a i n e d except for some use of C h a p t e r I and some a l t e r n a t i v e proofs u s i n g C h a p t e r II. C h a p t e r IV studies affine irreducible curves w i t h an equivalence class of affine c o o r d i n a t e systems preassigned; algebraically,

translated

such curves have c o o r d i n a t e rings w h i c h are

ered domains"

"filt-

We also study the concepts of taking a p r o j e c t i v e

c o m p l e t i o n and taking an affine piece; tion and d e h o m o g e n i z a t i o n . the main T h e o r e m

algebraically,

homogeniza-

Then we go on to finish the proof of

(36.9).

C h a p t e r V is a supplement.

It deals w i t h g e n e r a l i z a t i o n s

some concepts of the first four chapters

of

and has several statements

w h o s e p r o o f are only s k e t c h e d or referred to other sources. An e l e m e n t a r y k n o w l e d g e of general algebra is assumed to be a v a i l a b l e to the reader Lemma'

(for example,

results

, "Krull's I n t e r s e c t i o n Theorem',

There is only one

"official exercise"

like

'Nakayama's

'~ eif i = n formula'

etc.).

(in §15), but several pro-

perties stated in C h a p t e r I, II and IV may very w e l l be treated as exercises w i t h v a r y i n g degree of difficulty. The contents

are intended to give b r i e f d e s c r i p t i o n s

in

g e o m e t r i c w o r d s of w h a t is b e i n g treated in the r e l e v a n t sections. A list of i n t e r d e p e n d e n c e s of sections

follows the contents.

S h r e e r a m S. A b h y a n k a r A v i n a s h Sathaye

CONTENTS CHAPTER

I.

LOCAL

GEOMETRY

OR LENGTH

§l.

General

§2.

Principal

§3.

T o t a l q u o t i e n t r i n g and c o n d u c t o r . (3.1). L o c a l i z a t i o n of the c o n d u c t o r .

§4.

N o r m a l model. (4.1). Divisor (4.2). Divisor (4.3) . The " ~

§5.

Length affine curve. (5.1 (5.2 (5.3 (5.4)

.

(5.5 (5.6 (5.7

(5.8) (5.9

(5.10 (5.1l) (5.12).

.

terminology. ideals

and p r i m e

ideals.

of a f u n c t i o n . of zeros of a f u n c t i o n . e.f. = n " formula. ll

in a o n e - d i m e n s i o n a l n o e t h e r i a n domain, or i n t e r s e c t i o n m u l t i p l i c i t y on an i r r e d u c i b l e V a l u e s of local i n t e r s e c t i o n m u l t i p l i c i t y . V a r i o u s cases. L o c a l e x p a n s i o n of i n t e r s e c t i o n m u l t i p l i c i t y over a divisor. Global intersection multiplicity. L o c a l e x p a n s i o n of length. I n t e r s e c t i o n m u l t i p l i c i t y e q u a l s l e n g t h in the i n t e g r a l c l o s u r e (R*) . Local intersection multiplicity equals a l e n g t h (in R) for a p r i n c i p a l ideal. G l o b a l i z a t i o n of (5.6). S p e c i a l case of (5.6) - the n o r m a l case. S p e c i a l case of (5.7) - the n o r m a l case. D e f i n i t i o n . M u l t i p l i c i t y of a local d o m a i n of d i m e n s i o n one. D e f i n i t i o n and p r o p e r t i e s . C o n d u c t o r , its length; and a d j o i n t s . L e m m a on o v e r a d j o i n t s .

§6.

L e n g t h in a o n e - d i m e n s i o n a l noetherian homorphic image, or a f f i n e i n t e r s e c t i o n m u l t i p l i c i t y on an e m b e d d e d i r r e d u c i b l e curve. (6.1). V a l u e s of local i n t e r s e c t i o n m u l t i p l i c i t y . V a r i o u s cases. (6.2). L o c a l e x p a n s i o n of i n t e r s e c t i o n m u l t i p l i c i t y in the p r e i m a g e . (6.3). Global intersection multiplicity. (6.4). Case of a l g e b r a i c a l l y c l o s e d g r o u n d field (6.5). Case w h e n a c u r v e is t h o u g h t to be e m b e d d e d in itself. (6.6) to (6.9). R e s t a t e m e n t s of (5.6) to (5.9) for the case of a h o m o m o r p h i c image.

20

§7.

A c o m m u t i n g lemma for length.-(7.1). F o r two e m b e d d e d i r r e d u c i b l e curves, at a c o m m o n s i m p l e point, the i n t e r s e c t i o n m u l t i p l i c i t y of e i t h e r one w i t h the o t h e r is the same. (7.2). G l o b a l i z a t i o n of (7.1) o v e r a d i v i s o r .

27

VIII

(7.3)

C o m p l e t e g l o b a l i z a t i o n of

(7.1).

§8.

L e n g t h in a t w o - d i m e n s i o n a l regular local d o m a i n . . . . . . (8.1). I n t e r s e c t i o n m u l t i p l i c i t y of curves e m b e d d e d in a regular surface. Local case. (8.2). For two curves e m b e d d e d in a regular surface, the i n t e r s e c t i o n m u l t i p l i c i t y of either one w i t h the other is the same. Local case. (8.3). A d d i t i v i t y of i n t e r s e c t i o n m u l t i p l i c i t y of curves e m b e d d e d in a regular surface. Local case.

28

§9.

M u l t i p l i c i t y in a regular local domain (9.1). M u l t i p l i c i t y of an irreducible curve (embedded in a regular surface) at a point is the order of its d e f i n i n g equation. (9.2). T e c h n i c a l lemma for (9.1).

30

§i0. Double points of algebraic curves i0.i). Theorem. D e s c r i p t i o n of a double point of a curve. 10.2). Lemma. D e s c r i p t i o n of h i g h nodes. 10.3). Lemma. D e s c r i p t i o n of h i g h cusps. 10.4). Lemma. D e s c r i p t i o n of n o n r a t i o n a l h i g h cusps.

33

C H A P T E R II.

88

PROJECTIVE G E O M E T R Y OR H O M O G E N E O U S DOMAINS . . . . . .

§ii. F u n c t i o n fields and p r o j e c t i v e models.

86

§12. H o m o g e n e o u s h o m o m o r p h i s m .

68

§13. H o m o g e n e o u s ideals and h y p e r s u r f a c e s

68

(projective varieties)

§14. H o m o g e n e o u s subdomains, flats (linear varieties), projections, b i r a t i o n a l projections, and cones. (14.1). D i m e n s i o n and e m b e d d i n g d i m e n s i o n of a h o m o g e n e o u s subdomain. (14.2) and (14.3). D i m e n s i o n and e m b e d d i n g d i m e n s i o n of a (homogeneous) h o m o m o r p h i c image.

70

§15. Zeroset and h o m o g e n e o u s l o c a l i z a t i o n . _ (15.1), (15.2) and (15.3). E x t e n s i o n to (homogeneous) localization. (15.4) and (15.5) A l t e r n a t i v e (affinized) d e s c r i p t i o n of the (homogeneous) localization. (15.6). C o r r e s p o n d e n c e b e t w e e n h o m o g e n e o u s prime ideals and h o m o g e n e o u s localization. (15.7), (15.8) and (15.9). R e s t a t e m e n t of (15.1), (15.2) and (15.3) for e m b e d d e d varieties. (15.10). Lemma. Number of conditions imposed on a linear s y s t e m of h y p e r s u r f a c e s .

75

§16. H o m o g e n e o u s c o o r d i n a t e systems.

85

IX

§ 17.

§18.

§ 19.

§2o.

§21.

§22.

Polynomial rings as h o m o g e n e o u s domains . . . . . . (iZl) to (17.5). E q u i v a l e n t d e s c r i p t i o n s and p r o p e r t i e s of h o m o geneous domains w h i c h are p o l y n o m i a l rings over a field. O r d e r on an e m b e d d e d ( i r r e d u c i b l e ) curve and i n t e g r a l projections. (18.1) and (18.2). O r d e r of a h y p e r s u r f a c e at a v a l u a t i o n of an e m b e d d e d curve. (18.3) and (18.4). O r d e r of an ideal at a v a l u a t i o n of an e m b e d d e d curve. (18.5). Zerosets of ideals. (18.6) to (18.10). O r d e r at a v a l u a t i o n of an e m b e d d e d curve behaves like a valualtion. (18.11). P r o j e c t i o n lemma. P r o j e c t i o n of v a l u a t i o n and order, from a v e c t o r space. (18.12). Projection lemma. P r o j e c t i o n of v a l u a t i o n and order, from a flat (linear variety). (18.13). Corollary-definition. C o n d i t i o n for a ~ - i n t e g r a l p r o j e c t i o n (where ~ is a hyperplane). (18.13.1) . S p e c i a l case of ( 1 8 . 1 3 ) - p r o j e c t i o n from a c e n t e r not m e e t i n g the curve. O r d e r on an a b s t r a c t (irreducible) curve and integral p r o j e c t i o n s . (19.1) to (19.12). V e r s i o n s of (18.1) to (18.12) w h e n a curve is t h o u g h t of as e m b e d d e d in itself. (19.13) and (19.13.1). V e r s i o n s of (18.13) and (18.13.1) for an a b s t r a c t curve. (19.14). Remark. "Integral"ness of p r o j e c t i o n commutes w i t h h o m o m o r p h i c image. V a l u e d v e c t o r spaces. (20.1) to (20.13). S t r u c t u r e and p r o p e r t i e s v e c t o r space.

86

88

93

95

of a v a l u e d

O s c u l a t i n g flats and integral p r o j e c t i o n s of an e m b e d d e d (irreducible) curve. (21.1) D e f i n i t i o n and s t r u c t u r e of o s c u l a t i n g flats. (21.2), (21.3) and (21.4). A p p l i c a t i o n of §20 to the p r o p e r t i e s of o s c u l a t i n g flats. (21.5). Properties of o s c u l a t i n g flats in special cases. (21.6). C o n d i t i o n for integral p r o j e c t i o n in terms of o s c u l a t i n g flats at the center of projections. O s c u l a t i n g flats and integral p r o j e c t i o n s of an a b s t r a c t (irreducible) curve. (22.1) to (22.6). R e s t a t e m e n t s of (21.1) to (21.6) w h e n a curve is thought of as e m b e d d e d in itself (22.7). Remark. O s c u l a t i n g flats commute w i t h h o m o m o r p h i c image.

109

118

X

§23.

I n t e r s e c t i o n m u l t i p l i c i t y w i t h an e m b e d d e d (irreducible projective) curve. (23.1), (23.2) and (23.3). Properties of i n t e r s e c t i o n m u l t i p l i c i t y w i t h an embedded curve. (23.4). Case of a l g e b r a i c a l l y closed ground field. (23.5). A d d i t i v i t y of i n t e r s e c t i o n m u l t i p l i c i t y . (23.6). I n t e r s e c t i o n m u l t i p l i c i t y equals length for a p r i n c i p a l ideal. (23.7). All points of an e m b e d d e d line are simple. (23.8) and (23.9). Bezout's Little Theorem. Definition. D e g r e e of an elabedded (irreducible) curve. Remark. Affine i n t e r p r e t a t i o n of degree. (23. i0). Lemma. If there are enough rational points (23. ii). then e m b e d d i n g d i m e n s i o n of an embedded curve is less than or equal to its degree. Lemma. If the degree of an embedded curve (23. 12). is one, then its e m b e d d i n g d i m e n s i o n is one. Remark. H y p e r p l a n e s h a v e degree one. (23. 13). P r o j e c t i o n formula. P r o j e c t i o n of v a r i e t i e s (23. 14). from flats. Special p r o j e c t i o n formula. Degree of (23. 15). the projection. Remark. Case of an a l g e b r a i c a l l y closed (23. 16). ground field. Lemma. B i r a t i o n a l i t y of the p r o j e c t i o n from (23. 17). the generic point on a line. D e f i n i t i o n and p r o p e r t i e s of tangents to (23. 18). an e m b e d d e d curve. C o m m u t i n g len~as. V e r s i o n s of (23. 19) and (23.20). (7.1) and (8.2) for p r o j e c t i v e curves.

§24.

I n t e r s e c t i o n m u l t i p l i c i t y w i t h an a b s t r a c t (irreducible) curve. (24.1) to (24.17). V e r s i o n s of (23.1) to ( 2 3 . 1 7 ) for an a b s t r a c t curve. (24.18). D e f i n i t i o n and p r o p e r t i e s of tangents to an a b s t r a c t curve. (24.19). Remark. Relations b e t w e e n an e m b e d d e d curve [A,C] and an a b s t r a c t curve A/C. (24.20). Lemma on overadjoints. Projective v e r s i o n of (5.12). (24.21). Lemma on u n d e r a d j o i n t s . E x i s t e n c e of c e r t a i n type of p r o j e c t i v e u n d e r a d j o i n t s w h i c h are true adjoints in an affine piece.

§25.

T a n g e n t cones and q u a s i h y p e r p l a n e s . (25.1). Definition. L e a d i n g form of a h y p e r s u r f a c e . (25.2) and (25.3). Le~ma-definition. Definition and p r o p e r t i e s of tangent-cones. ~-quasihyperplane. (25.4) . Definition. C h a r a c t e r i z a t i o n of ~ - q u a s i h y p e r p l a n e s . (25.5) . Lemma. A h y p e r p l a n e (different from ~) (25.6) . Lemma. is a ~ - q u a s i h y p e r p l a n e . Lemma. Quadric ~ - q u a s i p l a n e s . (25.7) . Lemma. V e r s i o n of (9.1) for p r o j e c t i v e curves. (25.8) . Lemma. D e g r e e of an e m b e d d e d plane curve is (25.9). the d e g r e e of its d e f i n i n g equation.

121

138

148

XI

(25. i0) . (25.11) . (25.12) . (25.13) . (25.14) .

§26.

Lemma. Characterization o f t a n g e n t lines of plane projective curves. Definition Intersection multiplicity of two hypersurfaces. A d d i t i v i t y of the i n t e r s e c t i o n m u l t i p l i c i t y of two hypersurfaces. Lemma. V e r s i o n o f (8.2) f o r p r o j e c t i v e curves. Bezout's Theorem. Intersection of two plane projective curves.

2-equimultiple plane projections of projective space quintics. (For n o t a t i o n see b e g i n n i n g of §26.) (26.1). Lemma. Most lines through a d-fold point o f a n i r r e d u c i b l e c u r v e are d - s e c a n t s . A l s o , if d < d e g r e e of the curve, t h e n t h e r e are (d l l ) - c h o r d s t h r o u g h the p o i n t in e v e r y p l a n e t h r o u g h the p o i n t . Lemma. A n i r r e d u c i b l e c u r v e of d e g r e e m 2 26.2). h a s 2 - c h o r d s in e v e r y p l a n e . 26.3) Lemma. S u f f i c i e n t c o n d i t i o n for e x i s t a n c e of 4-chords. 26.4) Lemma. S u f f i c i e n t c o n d i t i o n for e x i s t a n c e of 2-secants. 26.5) Lemma. P r o j e c t i o n f r o m p o i n t s o n an (n-l)-secant. 26.6) Lemma. Projection from points on 2-secants o f an i r r e d u c i b l e q u a r t i c . 26.7) Lemma. P r o j e c t i o n f r o m p o i n t s on c e r t a i n 3-secants. 26.8) Cone Lemma. 26.9). Plane Lemma. (26.10) . Q u a d r i c L e m m a . (26.11). Proposition. Detailed description of proj e c t i o n s of c u r v e s of d e g r e e at m o s t 5. (26.12) . T h e o r e m . C o n d e n s e d v e r s i o n o f (26.11) for r e f e r e n c e .

CHAPTER

III.

BIRATIONAL

GEOMETRY

155

180

OR GENUS_

§27.

Different. (27.1) . D e f i n i t i o n s . (27.2) . D e d e k i n d ' s f o r m u l a . "{y = ~ dx " (27.3) . L e m m a . Condition for an u n r a m i f i e d e x t e n s i o n . (27.4) . T e c h n i c a l lemma. (27.5) . L e m m a . Another characterization of an unramified extension. Description of an unramified extension. (27.6). F i n i t e n e s s o f the s e t o f (27.7) . L e m m a . ramified primes. Separably generated (27.8) . D e f i n i t i o n . function fields. (27.9) . L e m m a . Conditions for separably generated function fields.

180

§28.

Differentials. (28.1). ~formulatiQn of ( 2 7 . 2 ) in t e r m s ~ivlsors. i n t e g r a l case.

189

of

XII

(28.2).

C o n v e r s i o n f o r m u l a for d i f f e r e n t . I n t e g r a l case. (28.3) and (28.4). E x c h a n g e lemmas. (28.5). G e n e r a l c a s e of (28.2). (28.6). G e n e r a l c a s e of (28.1). (28.7). Definition. ordv(~,x) ; (~,x) (28.8)

to

(28.11). (28.12) . (28.13) . (28.14) . (28.15) . (28.16) . (28.17) . (28.18) . (28.19). (28.20) . (28.21). (28.22) .

i n t e n d e d to be r e p l a c e d by ~dx. (28.10). Lemma. (~,x) behaves so far as ord V is c o n c e r n e d .

~dx,

T e c h n i c a l d e f i n i t i o n of genus. Theorem. A r a t i o n a l curve h a s genus zero Definition. Usual differentials and their properties. U s u a l d e f i n i t i o n of genus. Genus f o r m u l a s . Remark. A l t e r n a t i v e p r o o f to a genus formula. Remark. Definition. Uniformizing parameter and c o o r d i n a t e . Lemma. P r o p e r t i e s of u n i f o r m i z i n g coordinates. Example. Valuations with unseparable r e s i d u e fields. Remark. Lemma. (28.19) r e f o r m u l a t e d u s i n g differentials.

§29.

Genus of an a b s t r a c t curve. (29.1) and (29.2). Genus f o r m u l a s for p l a n e p r o j e c t i v e curves. R a t i o n a l i t y of a c u r v e of genus zero. (29.3). Remark. (29.4). D i r e c t p r o o f of r a t i o n a l i t y of a conic. (29.5). D i r e c t p r o o f of r a t i o n a l i t y of a line. (29.6). D i r e c t c o m p u t a t i o n of the genus of a cubic. (29.7). Theorem. A p p l i c a t i o n of (24.21) to p l a n e p r o j e c t i v e c u r v e s of genus ~ 1 and d e g r e e 4 or 5.

220

§30.

G e n u s of an e m b e d d e d curve. (30.1) and (30.2). Genus f o r m u l a s . of a c u r v e of genus zero. (30.3). Theorem. Combined version and (26.12) for r e f e r e n c e .

231

CHAPTER

IV.

AFFINE

GEOMETRY

domains.

OR FILTERED

Various

Rationality of

(29.7)

DOMAINS-

definitions.

234 234

§31.

Filtered

§32.

Homogenization or t a k i n g p r o j e c t i v e c o m p l e t i o n . (32.1). Definition. Degree (32.3) to (32.8). P r o p e r t i e s of h o m o g e n i z a t i o n .

236

§33.

Dehomogenization or t a k i n g an a f f i n e p i e c e (33.1). Definition. Dehomogenization (33.2) to (33.4). P r o p e r t i e s of d e h o m o g e n i z a t i o n

238

XIII

§34.

Relation b e t w e e n h o m o g e n i z a t i o n and d e h o m o g e n i z a t i o n . . . .

241

§35.

P r o j e c t i o n of a filtered domain. (35.1). Definition. Projection. (35.2). R e l a t i o n w i t h h o m o g e n i z a t i o n and dehomogenization. (35.3). Lemma. C o n d i t i o n s for integral projections. (35.4). Definition. Degree, genus. (35.5). Theorem. Affine v e r s i o n of (30.3) and (26.12) .

244

§36.

C o m p l e t e intersections. (36.1) . Definition. C o m p l e t e intersections, essentially hyperplanar. (36.2) . Lemma. E s s e n t i a l l y planar space curve is a c o m p l e t e intersection. (36.3) . C o r o l l a r y to (36.2). (36.4) . Lemma. A case of complete intersection. (36.5) . Theorem. A s u f f i c i e n t c o n d i t i o n for c o m p l e t e intersection. (36.6) . E l e m e n t a r y t r a n s f o r m a t i o n s . (36.7) . Theorem. Another sufficient condition for complete intersection. (36.8) . C o r o l l a r y to (36.7). (36.9) . Main theorem of complete intersection.

248

C H A P T E R V. §37.

257

APPENDIX.

Double points of a l g e b r o i d curves. t r e a t m e n t of most of §i0.

§38.

Bezout's

§39.

Chains of e u c l i d e a n curves. v e r s i o n of (36.7).,

§40. §41.

theorem.

An a l t e r n a t i v e 257 268

The general case

Treatments of d i f f e r e n t i a l s A short survey.

A generalized 274

in d i m e n s i o n one.

A g e n e r a l i z a t i o n of D e d e k i n d ' s c o n d u c t o r and d i f f e r e n t . _

280 formula about 281

§42.

The general adjoint condition.

285

§43.

G e o m e t r i c language. Geometric motivations behind the various notations.

287

§44.

Index to notations.

295

§45.

Index to topics.

298

I n t e r d e p e n d e n c e o f sections. In the following,

§b e §a I ..... §a r

d i r e c t l y referred to in §b. other p r e r e q u i s i t e s previous

sections

§l, §2, §3, §4,

for §b

Except

means

§al ..... §r

for such references,

are the notations

and d e f i n i t i o n s

and they may be located from the index.

§5,

basic.

are

the only from

XIV

§6 ~ §5.

§7 4- §6. §8, §9,

independent.

§10 ~ §s. §ii, §18 ~

§12,

§13,

§14,

§15,

§16,

§17, basic.

§15.

§19 ~ §18. §20 4- §18,§19. §21 ~ §18,§20. §22 ~ §21. §23 ~ §4,§5,§15,§17,§18. §24 4-- §5,§10,§15,§23. §25 ~ §8,§9,§15,§23. §26 ~ §18,§21,§23,§25. §27, basic. §28 ~ §27

(§3,§5,§23

and §25 optional use).

§29 ~ §4,§15,§17,§24,§25,§28. §30 ~ §25,§26,§29. §31,§32,

basic.

§33 4- §32. §34 ~ §15,§25,§32,§33. §35 ~- §19,§26,§30,§33,§34. §36 4- §10,§26,§30. §37 ~- §i0. §38 ~ §23,§25. §39 ~ §36. §40,

independent.

§41,

related to §28.

§42 ~ §29,§30 §43,§44,§45,

(also related index.

to §i0).

CHAPTER

I:

LOCAL

gEOMETRY

OR LENGTH

§i. We

shall

assume

(1.7),(1.8),(1.9) tions

and

I.

elementary

ful r e s u l t s

about

General

terminology.

the

terminology

This

includes

properties

local

introduced

general

algebraic

of m o d e l s

rings.

We

shall

in [i:

and

(1.1), (1.6),

terms,

defini-

some w e l l k n o w n

also

u s e the

and u s e -

following

nota-

tion. By card w e d e n o t e For E = a

subsets

module

positive generated

over

and A

we d e n o t e

For of

by

the

a module

n, b y

set

also put

Ix I + x 2 + . . . + X n :

E

x i ~ Ji}.

jn

[E

: A~ = sup{n:

a ring

note

that, if

dimension

of

there

exists

For

: A]

is e i t h e r

is a field, t h e n

E

over

a ring

A

n > 1

(A) = the

subgroup

a subset

J

of

of

In c a s e

is an

J

A

A

subgroup

A, b y

[E:A~

we denote

a sequence

of

ideal

the

length

of

E

E 0 c E 1 c E2c...cE n with

a nonnegative [E

: A3

integer

is s i m p l y

A.

Principal

ideals

and p r i m e

ideals.

we define:

set o f all

nonzero

principal

ideals

in

or

w.

Also

the v e c t o r - s p a c e -

w

F

is a

~ E2~.-.~En~

A

§2.

x i ~ Ji }.

i.e.,

E0 ~ E1 [E

by

the a d d i t i v e

x i e J}.

of A - s u b m o d u l e s

then

(for i n s t a n c e

integer,

additive

For

we denote

[XlX2...Xn:

over

as an A - m o d u l e ,

that

the

E

jO = A.

E

Note

A, w h e r e

we denote

[XlX2...Xn:

integer

group

is a p o s i t i v e

of a r i n g

JiJ2...Jn

set

n

set

by

generated A, w e

the

integer, the

of an a d d i t i v e

where

J i , J 2 .... 'Jn

a positive

in

a ring),

subsets

by

number.

J i , J 2 ..... Jn

Jl + J 2 + ' ' ' + J n For

cardinal

A,

1.2

~(A)

= the

~i(A)

set of all p r i m e

= [P c ~(A):

dim A/P

ideals

in

A

= i]

for any ~i(A,x)

= ~(A,x)

~(A,I)

I c p]

1

for any

= ~(A,I)

~([A, I3)

x ¢ A

,

N ~i(A)

= {p ~ ~(A):

~i(A,I)

,

N ~i(A)

I c A

,

= ~(A,I) for any

I ¢ A

or

I c A

,

~i ([A, I]) = ~i (A, I) and ~([A,I],J)

= ~(A,I)

~i([A,I~,J)

We

note

~(A)

that

= ~(A,0)

Z(A, IA) I c A rad

= ~

n 9(A,J)

([A,I~,J)

~

(A,I) = ~(A) ~ ~

I e A

or

I c A

i

and

J e A

or

J c A

any

,

I c A,

I =

for any

then

= 9([A,0],I) or

n ~i(A)

1

~

= 9([A,I],0)

and

for any

= ~([A,I])

ideals

I

and

= D(A,I) J

P I c

radA{0}

~(A,I)

=

I = A

~(A,I)

= ~(A,J)

~

9(A,I)

O 9(A,J)

= ~(A,I

n J)

~(A,I)

n ~(A,J)

= ~(A,I

+ J)

radAI

= radAJ =

A, IJ)

and

§3.

For

a nonnull

Total

quotient

ring

R

we

rlng

define

and c o n d u c t o r .

in

for any A

we have:

.

1.3

~(R)

= the

total

quotient

ring

of

R.

and we define ~(R) ~(R),and of

R

we

in

~(R)

For

= the c o n d u c t o r note ~(R), = the

we

largest

{X

~

R:

xR

=

{X

e

R

: xR

=

{x

c ~(R):

upon

in the

letting

ideal C

integral

R

to be

closure

the

of

R

integral

in

closure

c

C

in

R

which

remains

an ideal

in

R*

R} R}

xR

ideal

~([A,C])

upon

R

then have

=

a nonunit

and,

that,

of

c R}.

in a r i n g

A

we

define

= ~(A/C)

letting

f: A ~ A / C

to be

the c a n o n i c a l

epimorphism,

we

define

~([A,C~) We

take

= f-I(~(A/C)) note

of the

fact

that

the c o n d u c t o r

localizes

properly,

i.e.:

(3.1) i__nn ~(R) respect

If

R

is a d o m a i n

is a finite to some

model,

For a domain the =

For

~(R)

and

multiplicative

§4.

I(R,k)

such

R

and

set

of all

~ (V) =

local

ring.

= I (R, R)

R

we

S

Normal

in

define

and

integral

R, t h e n

closure r i n q of

~(R)S

=

of

R

R

with

(S).

model. k

of

subrings R)

the

is the q u o t i e n t

system

a subring

that

a domain

that

V V

R of

we ~(R)

define with

k c V

is a o n e - d i m e n s i o n a l

such

regular

1.4

and for any

Q e R

~(R,Q)

or

Q c R

we d e f i n e

= {V e ~(R) : O r d v Q

For a ring

A

and any

> 0}.

C e 9(A)

we define

([A,C3) = ~(A/C). F o r a ring f: A ~ A / C

A, C e ~(A),

and

to b e the c a n o n i c a l

ord[A,C~,vQ

Q e A

or

epimorphism,

= Ordvf(Q)

for any

Q c A, u p o n

letting

we d e f i n e

V ~ ~([A,C])

and w e d e f i n e

~([A,C~,Q) We observe

We note

= ~(f(A),

that,

if

field e x t e n s i o n )

(K,k)

K

c o m p l e t e m o d e l of

is a f u n c t i o n

m o d e l of K

k

over

sor o f a f u n c t i o n

i, t h e n

I__ff K

for e v e r y

is zero",

.

over k

(i.e. a f i n i t e l y t r d e g k K = i, t h e n

k

and:

I (K,k)

all w h o s e m e m b e r s

fact that "the d e g r e e

field o v e r a field

k

{ i I if if

x ~ 0 x = 0.

that b y c o n v e n t i o n

real n u m b e r

of the d i v i -

with trdegkK =

we have,

(ordVX) [V/M(V) : k~ =

positive

is the

are normal.

V e ~ (K,k)

We remark

gener-

i.e.:

is a f u n c t i o n x e K

field with

K

W e a l s o take n o t e of the w e l l - k n o w n

(4.1)

> 0~

f(Q))

o v e r a field

is a p r o j e c t i v e

unique

ord[A,C?,vQ

that t h e n

~([A,C~,Q)

ated

= {V ¢ ~ ( [ A , C ] ) :

t i m e s ~ = ~ times p o s i t i v e = ~ times ~--- c o

real n u m b e r

1.5

Further, upon

for a n y

family

a (U)us U

where

is an i n t e g e r

a(u)

or

letting

U'

= [u ¢ U:

a(u)

~ 0},

=

a(u)

< 0}.

u

= {u c U:

a(u)

= ~]

,

ww

u

(u ~ u:

by convention

we have

0

if a(u)

u~U'

a (u)

U'

=

if U' , U =

is a n o n e m p t y

finite

set and

if

U'

is a n o n e m p t y

finite

s e t and

if

U'

is a n i n f i n i t e

=

u~U

finite

We follows fact

recall from

that

(4.1)

(4.3)

about

(4.2)

If

extensions

K

U

~

is a

set.

follows

" ~ eif i = n "

set and

U

immediately

formula,

i.e.,

of Dedekind

is a f u n c t i o n

field

from

(4.2) w h i c h

in t u r n

f r o m the w e l l - k n o w n

domains.

over

a field

k

with

w

trdegkK

= I, then,

closure

of

k

upon

i__nn I
i.

(10.1.2)

(I) c a r d

over

xI e R

emdim

it is e a s y

Now we

and

(10.1.4)).

and

Since

(i0.1.4)

emdim

= 1

from

(10.1.13)

we have

of

(5.5)

(10.1.12)

and h e n c e

either

and

(I0.i.i0).

(i0.i.i),

member

rational

and

and

(10.1.2)

From

(using

(5.4)

= Jd ) ; and a l s o

(10.1.7),

by

and

(i0.I.ii)

~(R)

implied

of this,

for letting

1.37

w e get

x0 ~ R

clearly

have

with

ordvx 0 = 1 = ordwx 0

ordVX

finally,

clearly

there

z, b y N a k a y a m a ' s

-- 1 = o r d w x

exists

lemma,

z ~

we h a v e

; also

~ l(R,x)

(R

R

for a n y

x ~ R

we

= 2 ;

~ M(V))\M(W)

and

for any

such

= R[z3. W

Since have

l(R)

that

R n

~(R)

= 2, =

IV],

(M(V)\M(V) 2) =

clearly

if

(2) h o l d s ,

V

then:

is r e s i d u a l l y

~, R N

upon

rational

(M(V) 2\M(V) 3) ~

clearly

there

lemma,

Since

and

l(R)

that

9(R)

for a n y

exists

= 2, =

R

if

2

z ~ M(V)\M(V)

we have

(3) h o l d s

then:

[V/M(V)

: R/M(R)]

we

clearly

have

there

exists

g: V - V/M(V)

by Nakayama's Now, deduction, and

as w e l l shall

for the

z ¢ V

is the

lemma,

(10.1.14)

(10.1.8)

(10.4)

for a n y

x ¢ R

we

and

such

upon

such

z, b y

V = R

we

(M(V)kM(V2))

= ~,

= 2 ;

V/M(V)

epimorphism;

-- g ( R ) ( g ( z ) ) for any

such

z,

R* = R[z3.

can b e d e d u c e d as p r o o f s

of

be p r e s e n t e d (I),

letting

= 2, R N

that

canonical

we have

cases

for a n y

= R[z~.

{V],

x ~ R

clearly

where

R,

-- 2 ;

x ~ M ( V ) \ M ( V ) 2 ~ l(R,x)

also,

V = R , we

have

Nakayama's

have

over

~, and

x e M ( V ) 2 \ M ( V ) 3 ~ k (R,x)

also,

letting

(2)

from

(10.1.5)

(10.1.4), in the and

(10.1.9).

(10.1.5),

Lemmas

(3)

and

This

(10.1.6),

(10.2),

(10.3)

(10.1.7) and

respectively. w

REMARK. R the

is like

Geometrically

the

singularity

local is

ring what

speaking:

In c a s e

of a s i n g u l a r i t y may be called

(i),

R

represents

of an a l g e b r a i c

a "high

node";

curve

the h i g h

(or,

when

node

is

w

an " o r d i n a r y called

node"

a "high

if

cusp";

d =

I.

the h i g h

In case cusp 37

(2),

R

represents

is an " o r d i n a r y

cusp"

what if

may be

d = i.

1.38

In case

(3),

R

represents

what

may be

called

a "nonrational

high

cusp" °

(10.2) V

and

are

W

LEMMA be

the d i s t i n c t

residually

such

that

n_ote the

ON H I G H

rational

ordVX

NODES.

Assume

members

of

over

R.

= 1 = ordwx,

set o f a l l p a i r s

that

~(R).

Assume

that

a_nd fix a n y

(p,q)

card~(R)

= 2, and

Assume

that

there

exists

such

x ¢ R.

of nQnneqative

V

let

and

w

x ¢ R

Let

inteqers,

G

d__ee-

let

W

G

and

--- { (p,q)

for e v e r y n o n n e g a t i v e

G

For

everv

I c R

inteqer

=

n

~ G: p = q]

n

[ (p,q)

le__~t

~ G: p a n ~ q}.

le__~t

G(I)

=

{ (ordvr,

or~r):

0 ~ r e I}

; w

(Note t h a t G(I)

then

G(I)

c G,

is a s u b s e m i g r o u p

every

(p,q)

e G(I)

and

of

and

if

G,

I

is an ideal

i.e.,

p',q')

(p + p ' , q

in

+ q')

R

or

e G(I)

R

, then when-

e G(I).).

Let w

P = R

(Note t h a t

then:

ideals

R

in

.

P

and

Further

R

For

every

w

N M(V)

Q

are

and

Q = R

exactly

N M(W).

all the

distinct

maximal

we h a v e

Q

P =

R

n

Q

=

R

Q

(PQ)

-- M(R).

(re,n) ¢ G, w e h a v e w

S(V) m N M(W) n = pm Q Qn = pmQn =

{r ¢ R : o r ~ r

~ m

and o r d w R

and G ( p m Q n) =

[ (p,q)

38

~ G: p ~ m

and

q ~ n}

;

a n~

1.39

and,

in p a r t i c u l a r ,

for

every

nonnegative

M(V) n n M(W) n = pn Q Q n = p n Q n

integer

= {r c R * :

n

we

(ordVr, ordwr)

have

e G n U {~,~]]

and G(pnQn)

= Gn .)

Let d =

Then

we

have

the

(10.2.1)

: R].

following:

For

a = min(ordVY,

[R/~(R)

any

y ~ R

orgy),

we

with

have

ordVY

paQa

c

~ orgy,

(x,y)R

upon

; whence

letting

in p a r t i c u l a r ,

G a c G((x,y)R). (10.2.2)

We

have

ordv~(R ) = or~(R) ~(R)

= pdQd

G(R)

= G

and

= d = a positive k~(R)

integer

= 2d

and W

For

every

unique

integer

ideal

moreover

we

Ji

have

every

with

i__nn R

and

emdim

0 ~ i ~ d

with

~(R)

R = 2.

we have

c Ji

such

that that

there [R/J i

exists

a

: R] = i ;

have

Ji = ~(R)

For

i

U Gd

+ xiR

y c R

(x,y)R

and

with

= M(R).

k ( R , J i)

orgy For

c ~(R) ~R

any

= 2i

~ orgy

= ~(R)

we

> 2d

~ k(R,e)

39

0 < i ~ d

and

~ ~ R

~ k(R,~)

for

min(ordVY,or~y)

have

,

= 2d

.

,

= d, w__ee

1.40

.a_nd

l(R,~)

(10.2.3) dimensional

Let

f: A - R

regular such

< 2d = I (R,~)

local

anv

elements

that

Then

Ker

f c

((~,~)A) 2

PROOF

OF

(10.2.1).

be

ring.

R

an epimorphism Let

= R[z],

Let

=- 0 ( 2 ) .

any

z ~ R

f(~)R

, ~ c A

=

y £ R

where

~(R),

A

is a t w o -

and

and

~ e A

f(~)z

be

given.

First

there

exists

r e

we

b__ee

=

f(8).

claim

that:

if

a = ordVY

such

that

Namely, 8 e R\M(R)

< ordWY

or~r

since such

= a

W

that

= b,

then

and

ordwr

(x,y) R

> b.

is r e s i d u a l l y

rational

ordW(Y

- 8x b)

> b;

(I) w e

get:

over

R,

it s u f f i c e s

there

exists

to t a k e

r = y - 8x b. By

I

induction

if

on

a = orgy

m,

from

< ordWY

and

m

is a n y

nonnegative

integer,

then

(2) there

Next

(3) I

if


q - p + a + xq

claim

such

that

ordvr

= a

and

ordwr

> m.

that:

a = ordVY

there

(x,y)R

and

(x,y)R

can

(p,q) such

find

r e

; it s u f f i c e s

in c a s e

p < q.

that:

40

e G

that

with ordVS

(x,y)R to t a k e

such

a ~ p = p

that

s = xp

~ q,

and

then

ordws

ordvr in c a s e

= q.

= a p = q,

1.41

if

a = ordVY

such

that

Namely, e R\M(R)

y,

= y - 6,x a.

such

(3)

I if

and

ordvY

then

a = orgy'

since

6'

By

< orgy,

V

there

ordV(Y

(4) w e

get

~ orgy,

y'

¢

(x,y)R

< ordvY°

is r e s i d u a l l y

that

exists

rational

over

R,

there

- 6'x a) > a ; it s u f f i c e s

exists

to take

that:

then

G a c G((x,y)R)

,

(5) where

a = min(ordvY

Now we

claim

if o r d V Y (6)

that:

~ orgy

min(or~y,or~y), (x,y) R Namely,

elements

I, and

(since

residually or~(D

now

there

in

R

that

or~

,

62

e R\M(R) ~'

(since

that

61

to take W

such

R

we

and n o w

~'

over

find

R) w e and

such

V

can

find

- 62t2)

such

W

~'

= ~ - 60t 0 rational

ordv( ~ - 61t I) > e

and

in case

R) w e now

is

that

is r e s i d u a l l y

over

> e,

find

(since

to take

that

rational

41

~ - ~'

= e = ordvD,

can

then

p e R

(since

= ~ - 62t 2.

(5) w e

ordv~

= ~ - 61t I ; finally,

or~(~

that

O r d v t 0 = e = o r d v t 0, o r d v t I =

it s u f f i c e s

~ R\M(R)

such

by

2 ; in c a s e

can

is r e s i d u a l l y that

e R

m a =

> min(ordV~,or~).

rational

= e < or~,

find

~'

ordv( ~ - 60t 0) > e

over

ordv~

can

exists

min(ordv~,Or~)

e = min(or~,or~), such

- 60t 0 - pt I) > e,

R) w e

with

min(or~',ordwD')

rational

it s u f f i c e s

take

then

is r e s i d u a l l y

such

pt I ; in case over

0 ~ ~ c R

ordvt 2 > e = or~t

v

60 c R\M(R)

and

letting

t0,tlt 2

e < or~t first

and

upon

, orgy).

can

ordv~

, and > e =

find

it s u f f i c e s

to

1.42

By induction if

l

(7)

on

i, from

ordVY ~ ordwY ., i

any element

in

NOW

~' c R*

, ordw~')

l

ordVY ~ ordwY

paQa c

integer,

min(ordv~,Ordw~) such that

and

~

is

~ min(ordVY,ordwY)

~ - 4' ¢ (x,y)R

and

> i.

(7) can clearly be reformulated

if

(8)

is any nonnegative

R , with

then there exists

min(ordv~'

(6) we get:

and

(x,y)R + piQi

thus:

a = min(ordVY,or%y),

for every nonnegative

then

integer

i.

Next we claim that:

f there exists

a positive

integer

u

such that

(9) pUjQuj Namely, positive get

c M(R) j since

integer

~(R) u

puQU c M(R);

integer

for every positive is a nonzero

such that

it follows

integer

ideal

puQU c ~(R):

that

pUjQuj

in

j. R , we can find a

since

c M(R) j

~(R) c R, we then for every positive

j.

Finally we claim that:

(lO)

I~ f

ordVY / o r % y

aQa c

Namely, positive

and

a = min(ordVY,OrdwY)

then

(x,y)R. by

integer

(8) and

(9) we get

paQa c

j, and by the Krull [(x,y)R + M(R) j] =

(x,y)R + M(R) j

intersection

for every

theorem we have

(x,y)R.

j=l This completes PROOF OF w

such that

the proof of

(10.2.2). for some

Let ~ ~ R

Q

(10.2.1). be the set of all nonnegative

we have

OrdW~ = w.

42

ordv~ ~ ordw~

and

integers

min(ordv~,

1.43

w

Since

~(R)

Q ~ ~.

c R

Upon

and

is a n o n z e r o

~R)

a = min{w:

that

in

R , we

see t h a t

letting

(1) we get

ideal

a

(2)

w

is a p o s i t i v e

ordVY

h e n c e f o r t ~ ........fix . any

~ orgy

such

e Q}

integer,

and

and

for s o m e

rain (Ordvy,

y 8 R

ordWY)

= a

we have

;

y ¢ R.

By a s s u m p t i o n

x e R

(3) and

so

G*\{0,0]

we have

with

ordvx

--- G ( [ x , x 2,x 3,...])

G a c G((x,y)R);

(4)

G~(R))

In v i e w

of

(10.2.1)

and

paQa

clearly

c G((x,y)R)

consequently,

= G((x,y) R) =

(5)

and hence

= 1 = ordwx

c

in p a r t i c u l a r

; by

in v i e w of

(G \[ (0,0)]

U Ga

(10.2.1)

(i) we g e t

and

G(R)

and that

= G

U G a-

(2) w e h a v e

(x,y)R

paQa

c

~(R)

; also,

for any

r e R*\paQ a

have min(ordvr,ordwr)

~- a

w

and we

can

find

s ~ rR

such

that

m i n ( o r d V r , o r d w r) = m i n ( o r d V s , o r d W S )

and then

in v i e w

thus we have

proved

(6)

By

(7)

of

(2)

(4) w e

< max(ordVs,ordw

see t h a t

s ~ R

that

~(R)

= paQa

.

(6), w e get ordv~(R ) = ordw~R) 43

= a

and h e n c e

s)

r ~

~(R)

;

we

i. 44

and,

since

V

and

W

are

(8)

residually

k~(R)

Since (6) w e

V

and

W

are

rational

over

R, w e

also

get

rational

over

R, b y

(4) and

= 2a.

residually

see t h a t

(9)

for any

~ e R

w e have:

~ e

~(R)

(i0)

for a n y

~ ¢ R

w e have:

c~R

for any

~ e R

w e have:

I(R,~)

~ k (R,~)

a 2a

w

=

~(R)

~ ~ (R,~)

= 2a

and

(ii)

We

claim

(R A

rational then

{ ¢ R

given

over

in v i e w

of

reverse

can

we

inclusion

claim

[(R N

Namely,

e R n find

= ordw@

have

= i < a

we have

(piQl).

(piQi),

p ~ R

(since

such

ordw(g

v

that

-p~)

is r e s i d u a l l y

Ordv(~

> i

-Og)

> i

and h e n c e

; thus

(pi+iQi+l))

+ ~R D

is o f c o u r s e

induction

(R n ( p a Q a ) )

Next

(14)

~

(4) w e m u s t

By decreasing

(13)

ordv{

+ {R = R N

(pi+iQi+l)

(R n

the

any

R) w e

- pg e R n

and

with

(pi+iQi+l))

Namely,

~ 0(2).

that

for any (12) I

~ 2a = k(R,~)

on

(piAi. U ) ,

obvious.

i, in v i e w o f

+ xlR = R N

.pi_i, ~ ~ )

(3) and

for

(12) w e get:

0 ~ i ~ a.

that:

(piQi))/(R

N

(pi+IQi+l))

for e v e r y

44

: R3 =

1

for

0 ~ i < a.

and

1.45

~

by

(4) w e m u s t h a v e

(R n

o r d v ~ -- i = o r d w ~

(R n this shows

(pi+iQi+l))+

(pi+iQi+l))

and h e n c e b y

~R = R N

(piQl)

(12) w e h a v e

;

that [ (R N

(i4 i) by

(piQi))\(R n

(piQi))/(R n

(pi+iQi+l))

: R~ s 1 ;

(3) xi e

(R n

(piQ1))\(R N

(pi+iQi+l))

and h e n c e (142 )

R N

now by

(141 ) and

(piQi)~

R Q

view of

(pi+iQi+l)

; hence by applying

: R~ = 1

(13) w i t h

i = I, in

(x,y)R = M(R)

(15) we h a v e

e m d i m R < 2; n o w

R

is not r e g u l a r b e c a u s e

we m u s t h a v e

(16)

emdim R = 2 . Since

R N

(17)

(6),

(3),

(pOQ0) = R, u p o n

J. = 1

(13) and

(18) By

;

(5) w e get

therefore

by

(PIQI))/(R Q

(PQ) = M(R)

(15)

By

(pi+iQi+l)

(142 ) we get [ (R n

Now

R N

setting

~(R)

,

(14) w e get

[R/J i : R~ = i (6) and

+ xiR

(17) we h a v e 45

for

0 < i s a .

~(R) ~ i;

1.46

(19)

X ( R , J i) =

We

claim

given

2i

for

0 ~ i ~ a.

that:

any

ideal

J

in

R

with

~(R)

c J,

upon

letting

i = ordvJ, (20) we have

Namely, 0 < i < a,

that

i

in v i e w

is a n

(4)

and

J c R n

i = a

that

then

since

~(R)

J = J a ; so h e n c e f o r t h

ordv~

=

i

and

then

=

(203 )

c J)

(2)

and

in v i e w

6x I + ~ y

(6) w e h a v e

y

(204 )

e

xi c

since

(R) c J, b y that

is

an

J = Ji"

integer

with

;

by

(3) , (6), (17)

that

(6),

~(R)

(13),

hence

of

i < a; w e

(2),

with

~(R),

and can

(201 ) w e

see

take

(3)

8 e R\M(R)

and hence

+ ~R

(17),

and

by

(15) w e

and

(pOQ0)

= R, b y

by

the

definition

of

(6),

(203 ) w e

(201),

(13)

and

we have

a = d.

46

get

;

: R] = a

d

get

~ e R

(202 ) a n d

(204 ) w e

1

R N

[R/~(R)

(21)

i

that

and

J = J..

Since

and

0 ~ i ~ a,

~ e J

with

clude

see

(piQi)

assume

(202 )

by

(6) w e

with

and

(201 )

if

integer

(14) w e

get

con-

1.47

Now

in v i e w

(Ii),

and

PROOF

of

(21),

(15) to

OF

the p r o o f

of

(10.2.2)

R

= R[z],

is c o m p l e t e

by

(4),

(20).

(10.2.3).

Since

and

so,

upon

relabelling

z ~

P.

Now,

since

V

V

and

W

we must

suitably,

is r e s i d u a l l y

(I)

rational

have

we may over

z ~

that

R, w e c a n

find

that

(2)

z + f(8)

¢ P.

w

NOW

R

= R[z

+ f(8)],

and h e n c e

(3)

z + f(8)

Since

f(~)R*

=

~(R),

(4)

by

.

we get

that

-- d = o r d w f ( ~ )

~ =

clearly

~ +

66

have

(6)

By a s s u m p t i o n

(~,9)A =

f(~)z

=

consequently,

(7)

By a s s u m p t i o n

by

(2),

=

and h e n c e

f(~)(z

(3) a n d

ordwf(~)

OrdVx

(~,~)A

f(8),

f(N)

(8)

have

letting

(5)

we

~ Q

(10.2.2)

ordvf(~)

Upon

we must

+ f(6))

(5) w e h a v e

;

(4) w e g e t t h a t

= d < Ordvf(~)

= 1 = OrdwX,

{ e A

by

with

and h e n c e

f(~)

47

-- x

.

upon

P n Q,

suppose

8 ~ A\M(A)

such

(6) to

fixing

any

i. 48

we h a v e

(9)

Ordvf(~)

NOW,

in v i e w

of

(7) and

(I0)

By

= 1 = ordwf( 0,

in v i e w of

(7) w e

also h a v e

(4) w e g e t

~ ~ ~A

~ e M(A)

; consequently,

; in v i e w

in v i e w

of

of

(4)

(ii) we

can

write

(12)

and

~ =

b

6'~ b + D~

is a p o s i t i v e

where

integer;

(13)

By

now,

b = d

(12)

and

(13) w e

see t h a t

6' c A\M(A)

and

by

(9) a n d

(4),

(7),

p e A

(12) w e g e t

.

({d,n)A =

(e,9)A,

and h e n c e

by

(6) w e

get (~d,~)A =

(14)

Let then NOW,

y ¢

any

(({d,9)A)2,

in v i e w of

(15) such

y e A

with and

(11), w e

f(y)

(~,~)A.

= 0

in v i e w

b e given.

of

(14)

We

this will

shall

show that

complete

the p r o o f .

can w r i t e

y = r + s~ + t~ 2

with

r,s,t

in

A

that I

either

r = 0

(16) or

r = 60 ~a

with

60 e A\M(A)

48

and n o n n e g a t i v e

integer

a

1.49

l

either

(17) [ o r

By

(9),

s = 0

s = 81 gb

(16) and

(18)

f(~)

(19) N o w by

= or~f(r)

= 0, in v i e w o f

ordvf(r) (15),

(16),

and

integer

b.

that

and

ordvf(S)

(7),

= oral(r) (17)

and n o n n e g a t i v e

8 1 e A\M(A)

(17) w e d e d u c e

ordvf(r)

Since

with

(15) and

a 2d (19)

and

= or~f(s)

(18) w e c o n c l u d e

ordvf(S)

it f o l l o w s

that

that

= or~f(s)

z d .

y c ((~d,9)A)2.

w

(10.3) {V] R N

LEMMA ON HIGH CUSPS.

and that

V

is r e s i d u a l l y

for e v e r y

n c G

I c V

[2m

Assume R.

that

(M(V)2\M(V)3)-

Let

let

: m ~ G]

{2m+l

: m ~ G

,

with

m ~ n]

let G(I)

= {ordvr

: 0 ~ r e I]

Let d = [R/~ (R) Then we have

: R]

the f o l l o w i n g ,

(10.3.1).

d

is a p o s i t i v e

ordv~(R)

integer,

= 2d = X~(R)

and

49

and

that

Assume

let

Gn For every

over

x c R N

integers,

G

V = R .

rational

(M(V)\M(V) 2) = ~, and fix any

b the s e t of all n o n n e g a t i v e

and

Let

~(R)

= M(V)

2d

~(R)

G

=

1.50

G(R)

For

every

unique

inteqer

ideal

moreover

i

Ji

= G

with

i__nn R

U Gd

and

0 ~ i ~ d

with

~(R)

emdim

we have

~ Ji

R =

2.

that

such

that

2i

for

there

exists

a

[ R / J i : R~ =

i ;

we have

Ji = ~(R)

_FOr e v e r y F__oor an%
e ; n o w it

W

suffices

to take

0' = 9 - 6 t.

By i n d u c t i o n on (6)

M(V) 2a c

i, from

(5) we get:

(x,y)R + M(V) i

for e v e r y n o n n e g a t i v e

integer

i.

N e x t w e c l a i m that:

(7) I

there

exists

a positive

M(V) uj c M(R) j Namely, positive t h e n get positive By

since

integer

(6) and

is a n o n z e r o

such t h a t

M(V) u c M(R) integer

u

for e v e r y p o s i t i v e

~(R) u

integer

j. (7) w e get

51

integer

ideal

M(V) u ~

; it f o l l o w s

such t h a t

that

~(R)

in

j V, w e can find a

; since

~(R) c R, we

M(V) uj c M(R) j

for e v e r y

1.52

M(V) 2a c and by

the

(x,y)R + M(R) j

Krull

for e v e r y

intersection

[(x,y)R

theorem

+ M(R)J~

=

positive

integer

j

we have

(x,y)R

;

therefore

j=l (8)

M(V) 2a c

and h e n c e we

can

s ~

in p a r t i c u l a r

find

R

s ¢rR

and h e n c e

r ~ ~(R)

; thus

(i0)

ordv~(R)

(i) w e

~(R)

M(V) 2a c ~(R) ordVS

~(R)

Since

.

with

(9)

and

(x,y)R

=

=

2a - 1

we h a v e

= M(v)2a

see that,

and

X~(R)

= 2a

(12)

~ c {(R)

(13)

~R

and

any

then by

proved

*

r ~ R \M(V)

2a

(4) w e h a v e

that

. V

for any

(ll)

given

and hence

= 2a

IV]

; also

is r e s i d u a l l y

rational

~ ¢ R, we h a v e

the

over

R, b y

(4)

following.

.

~ l(R,~)

~ 2a

w

= {(R)

~ k(R,~)

-- 2a

.

and (14)

I (R,~)

We

claim

{

2a

=

l(R,~)

~

0(2)

.

that

for a n y

{ e R

with

ordw{

we have

(R N M(V) 2i+2)

= 2i

where

i

is an

integer

< a ,

(15) + ~R = R N M(V) 2i

w

Namely, rational

given

over

any

{

R) we c a n

e R N M(V) 2i, find

p e R

(since

such

that

V

is r e s i d u a l l y

o r d v ( { -p~)

> 2i

w

then *

in v i e w of

(4) w e m u s t

- p{ ¢ R N M(V)

2i+2

have

the

reverse

By d e c r e a s i n g

~ 2i + 2, and h e n c e

; thus

(R N M(V) 2i+2) and

o r d v ( { -p{)

inclusion

+ {R D R Q M(V) 2i

is of c o u r s e

induction

on

i,

52

,

obvious.

in v i e w

of

(3) and

(15) w e get

and

1.53

(16)

(R N M(V) 2a)

+ x i R = R O M(V) 2i

for

0 ~ i < a .

for

0 < i < a .

N e x t we c l a i m that (17)

[(R N M(v) 2 i ) / ( R

Namely,

(R n M ( v ) 2 i ) \ ( R

(4) we m u s t h a v e

o r d v ~ = 2i

(R N M(V) 2i+2)

this

shows

N M(V) 2i+2)

and h e n c e b y

+ ~R

=

(15) we h a v e

R Q M(V) 2i

that [(R N M(v) 2 i ) / ( R

(171 ) by

: R]

for e v e r y ~

by

N M(V) 2i+2)

N M(V) 2i+2)

: R~ ~ 1 ;

(3) we h a v e x

i

C

(R N M(V)

2i\

(R N M

(V) 2i+2)

and h e n c e (17 2 )

(R Q M(v) 2 i ) \ ( R

now by

(171 ) and

~ ~ ;

(172 ) we get [(R N M ( v ) 2 i ) / ( R

Now by assumption M(R)

N M(V) 2i+2)

; consequently,

R P

N M(V) 2i+2)

: R~ = 1 .

(M(V)\M(V) 2) = ~, and h e n c e

by applying

(16) w i t h

R e M(V) 2 =

i = I, in v i e w of

get (18) By

(x,y)R = M(R).

(18 we get

R N (19)

emdim

R < 2 ; now

(M(V)\M(V) 2) = ~ ; t h e r e f o r e emdim

R

is not r e g u l a r

we m u s t h a v e

R = 2

53

because

(8) we

i. 54

Since

R n M(V) 0 = R,

J

(20)

by

(9),

(16)

and

(21)

By

(3),

(9) a n d

(22)

claim

given

f

i =

(23) I

J

we have

that

i

in v i e w and

0 < i < a .

in

R

with

~R)

c J,

upon

letting

is an

of

(4)

integer

and

with

(9) w e

0 ~ i ~ a,

see

that

(2),

(3)

i

and

J = Ji

is an

integer

clearly ;

~cJ

and

=

2i

and

~ =

6x I + ~y

then

(9) w e h a v e

(234 )

clude

.

take

(233 )

since

for

J c R N M(V) 2i

ordv~

(2)

0 < i ~ a

for

(i/2)or~J,

(232 )

by

i

2i

ideal

(231 )

with

that

that:

0 ~ i ~ a,

can

,

(20) w e h a v e

any

Namely, with

letting

+ xlR

see

: R~ =

(R,J i) = We

we

--- ~(R)

l

(17) w e

[R/J i

upon

xi c

~(R) that

c J, b y

(9),

in v i e w

y ~

~(R)

of

with

8 e R\M(R)

~(R),

and

+ ~R

(16),

hence

and

and

by

(18)

~ ~ R

(233 ) w e

we

get

;

get

;

(20),

J = J.

l

54

(231),

(232 ) a n d

(234 ) w e

con-

1.5~

Since

R N M(V) 0 = R, b y

(9),

[R/~(R)

and h e n c e

b y the d e f i n i t i o n

(24)

Now

in v i e w

(14),

and

(18)

PROOF

OF

rational 8 e A

of

(24), to

such

(17) we get

d

we h a v e

of

(10.3.1)

.

the p r o o f

is c o m p l e t e

R, and

By assumption R n

R

= R[z],

V

(M(V)\M(V) 2 = ~, t h e r e f o r e

that

(I)

o r d V (z+f (8)) =

1

and

(2)

Since

f(6)R

either

8 e A\M(A)

or

=

by

w e get

~(R),

(3)

~ =

clearly

that

.

~ + 66

have

(5)

(~,~)A =

By a s s u m p t i o n

f(e)z

=

f(~)

consequently (6)

= 2d

6 = 0 .

letting

(4)

we

(i0.3.1)

ordvf(C~)

Upon

by

by

(4),

(9) to

(23).

(10.3.2).

over

and

: R] = a

of

a = d

(16)

(i) and

f(~),

=

(~,8)A

and h e n c e

f(~)(z

+ f(6))

(3) w e g e t

ordvf(D)

by

that

--- 2d + 1

55

(4) we h a v e

;

is r e s i d u a l l y we

can

find

1.56

By a s s u m p t i o n

ordvx

(7)

=

2

{ e A

and hence,

upon

with

= x

f({)

fixing

any

,

we have

(s)

ordvf(~)

NOW,

in v i e w

of

(6) and

(9)

(7), b y

(f(~),

By

(10.3.1)

clude

we h a v e

(10.3.1)

we g e t

that

f ( ~ ) ) R = M(R).

emdim

R =

2, and h e n c e

in v i e w

of

(9) w e

con-

that

(i0)

(~,~)A = M(A)

Since and

= 2

d > 0,

in v i e w

(6) w e a l s o h a v e

~ ~

of

.

(3) w e g e t

~A

~ ~ M(A)

; consequently,

; in v i e w

in v i e w

of

of

(3)

(i0) w e

can

write

=

(II)

and

p

6' {P + p~

is a p o s i t i v e

integer;

(12)

By

where

nOw,

p = d

(ii)

and

(12)we

6' e A \ M ( A )

and

by

(8) and

(3),

(6),

p e A

(ii) w e g e t

°

see that

({d,~)A =

(~,~)A,

and h e n c e

by

(5)

w e get

(13)

(@d,~)A =

Let a n y then proof.

(14)

y ~

y ¢ A

((~d,z)A)2,

Now,

in v i e w

with

f(y)

and of

(~,$)A.

= 0

in v i e w

(i0),

y = r + sD + t~ 2

we

of

be given. (13)

We

this will

can w r i t e

with

56

r,s,t

in

A

shall

show that

complete

the

i. 57

such

that

[

either

r = 0

or

60 ~a

(15) r =

with

6 0 e A\M(A)

and

nonnegative

integer

a

with

81 ¢ A\M(A)

and

nonnegative

integer

b

and

( I either

s = 0

(16) or

Since

s = 61 ~b

f(y)

= 0, b y

(6),

(8),

r ~ {2dA

(17)

Now by

(14)

and

(17)

(14), and

it f o l l o w s

(15) s

and

¢ {dA

that

y ~

(16) w e d e d u c e

that

.

((@d,9)A)2 @

(10.4) that

~(R)

=

LEMMA

ON N O N R A T I O N A L

IV}.

Assume

R n M(V)\M(V) 2 ~ g: V ~ V/M(V)

~

and

b e the

that

T h e n we h a v e

the

(10.4.1)

CUSPS.

[V/M(V)

fix an v

canonical

d =

HIGH

x

Let

: R/M(R)]

¢ R n

= 2.

(M(V)\M(V) 2) .

epimorphism.

[R/~(R):

V = R .

Assume

Assume

that

Let

Let

R]

followinq

We h a v e

or%{(R)

= d = a positive

~(R) = M ( v ) d

and

integer

k ~(R) = 2d

and emdim For

everv

unique

inteqer

ideal

J

in 1

moreover

i

with R

R = 2 .

0 ~ i ~ d

with

~(R)

we h a v e c J

such

that that

- -

there

exists

[R/J.

: R~ 1

we have Ji ~

~(R)

+ x±R

and

k(R, Ji)

57

= 2i

for

0 ~ i ~ d

o

a =

i

;

1.58

For any

0 ~ ~ ¢ R

we have:

d ~ ordv@

For every

y ¢ R

(x,y)R = M(R).

I

there exists

[

ordv@'

~ith

= ordv@

ordVY = d

For an~

@' ¢ R

such that

and

and

g(9'/@)

~ g(R).

g(y/x d) ~ g(R), we have

~ 6 R

we have

-- {(R)

~ X(R,~)

= ~(R)

~ X (R,~) -- 2d

w

aR

~ 2d

w

~R and

X(R,~) (10.4.2). dimensional

Let

~ 2d = I (R,~) -= 0(2).

f: A - R

reqular l o c a l

be a n eDimorphis m where

rinq.

Let

z ¢ R

W

an y elements Then

such that

Ker f c

((~,~)A) 2

PROOF OF

(10.4.1).

R

, ~ C R

A and

is a two~ ~ R

be

*

= R[z],

f(~)R

= {(R),

and

f(~)z = f(~).

We claim that:

there exists a positive

integer

u

such that

(I) I M ( V ) uj c M(R) j Namely, positive get

since

integer

M(V) u c M(R);

integer

for every positive

{(R) u

is a nonzero

such that

integer

ideal in

M(V) u c ~(R);

it follows that

j.

V, we can find a

since

M(V) uj c M(R) j

~(R)

c R, we then

for every p o s i t i v e

j.

Recall that b y assumption (2)

x c R Let

some

Q

~ ~ R

g(V) ~ g(R),

with

ordVX = I.

be the set of all nonnegative with

or%~

and hence

= w

we have

in v i e w of

integers

w

such that for

g (~/x w) ~ g (R).

By assumption

(i) we see that 58

~ ~ ~.

1.59

Now upon

letting

(3)

a = min[w:

we get that

a

w ¢ Q}

is a p o s i t i v e

i n t e g e r and t h e r e

exists

y ¢ R

such

that

(4)

ordvY = a

henceforth

fix a n y

such

We c l a i m that

(5)

a

OrdV8

I OrdVS'

if

ordve = ordvxb cOnsequently if and

e

and

g(@'/e)

hence by

and e i t h e r

;

such that

and

g(e'/8)

then by

to t a k e in

then either

(3) w e m u s t h a v e

we h a v e

0' e R

g(yxb-a/8)

are e l e m e n t s

~ g(R),

@ c R

= OrdV8

ordv8 = b m a

it s u f f i c e s 8'

0 ~

exists

~ Namely,

g ( y / x a) ~ g(R)

y ~ R.

for any

I there

and

(4) w e see t h a t ~ g(R)

6' = y x b - a R

~ g(R).

or or

such that g(~'/xh)~g(R)

ordVYX

b-a

g(xb/8) ~ g(R) 8' = x b.

or~8' or

= ;

Conversely,

= ordv@ = b ~ g ( 8 / x b) ~

g(R)

b ~ a.

Next we claim that

given any

0 ~ ~ e V

such that

q - ~' e

Namely, elements

p

since

and n o w it s u f f i c e s By induction

M(V) a c

(7) By

(i) and

(x,y)R

[V/M(V) p*

and

w i t h ord V = e ~ a, t h e r e

on

in

R

to t a k e i, from

and

: R/M(R)]

OrdV~'

> ordv~

= 2, in v i e w of

such t h a t 4' = ~ - px

exists

~' ¢ V

.

(4) we c a n

find

o r d v (qx -e - p - p*yx -a ) > 0 e

* e-a - p yx

(6) w e get:

(x,y)R + M(V) i

for e v e r y n o n n e g a t i v e

(7) w e g e t

59

integer

i.

;

i. 60

M(V) a c

and by

the

Krull

(x,y)R + M(R) j

intersection

for

theorem

every

positive

integer

j

we have

¢0

n j=l

[x,y)R

+ M(R) j] -- ( x , y ) R

;

therefore

(8)

and

M(V) a c

hence

ordvr

= p < a,

s ~ V that

in p a r t i c u l a r

with either

upon

OrdvS r ~

R

M(V) a c ~(R)

letting = p

and

or

s ~

; also,

s = r y / x a, b y g(s/r) R

~(R)

(9)

(x,y)R

~

; thus

given

any

(2) a n d

g(R),

and

we have

(4) w e

then

proved

r ¢ V

by

get

with that

(5) w e

see

that

= M(V) a

and hence

(io)

ordv~(R)

~(I{) =

Since see

{V}

and

[V/M(V)

= a

.

: R/M(R)]

=

2,

in v i e w

¢

~(R)

of

(i0) w e

that

(n)

X~(R)

= 2a

(12)

for any

~ ¢ R

we have:

(13)

for

any

~ ¢ R

we have:

~R

for any

~ ¢ R

we have:

X(R,~)

~ k (R,~)

m 2a

@

=

~(R)

~ k(R,~)

= 2a

and

(14) We

claim

that

60

&

2a = k ( R , ~ )

=- 0(2)

1.61

/ for any

~ ¢ R

with

o r d v ~ = i < a, we h a v e

(15) (R N M(V) i+l)

Namely,

given

+ ~R = R N M(V) i

any

~ * ~ R N M(V) i

we must h a v e

*

then o b v i o u s l y ,

and

if

ordv~

(if

ordv~*

>

= i, then b y

(5))

g(~ /~)

e g(R)

W

h e n c e we can find

p e R

such that

(R N M(V) i+l)

and the r e v e r s e

inclusion

By d e c r e a s i n g (16)

~

- p~ ¢ R N M ( v ) i + I

; thus

+ ~R ~ R D M(V) i ,

is of course

induction

on

obvious.

i, in v i e w of

(R q M(V) a) + x i R = R Q M(V) i

for

(2) and

(15) we get

0 ~ i < a .

Next we claim that (17)

[ (R D M(v) i/(R D M(V) i+l)

Namely,

by

for

0 ~ i < a .

(R N M(v) i ) \ ( R N M(V) i+l)

(15) we h a v e (R N M(V) i+l)

which

+ {R = R D M(V) i

shows that

(171 ) by

: R~ = 1

for e v e r y ~

[ (R n M ( v ) i ) / ( R

N M(V) i+l)

: R] ~ 1 ;

(2) we h a v e xi ~

(R n M(v) i ) \ ( R D M(V) I+I)

and h e n c e

(172 )

i,

W

(R N M(v) i ) \ ( R Q M(V) i+l) @ @ ;

61

and

1.62

now by

(171 ) a n d

(172 ) w e

get

[ (R n M ( v ) i ) / ( R

Now view

of

R N M(V) (8) w e

= M(R)

and hence

by

; R] = 1 .

applying

i =

(16) w i t h

I,

in

get

(18)

BY

D M ( V i+l)

(x,y)R = M(R)

(18) w e

g(V)

get

~ g(R)

emdim

R < 2 ; now

; therefore

we must

(19)

emdim

Since

R N M(V)

0

= R,

J

(20)

R

is n o t

regular

because

have

R =

upon

2 .

letting

=

~(R)

see

that

+ xlR

,

1

by

(9),

(16)

and

(17) w e

(21)

for

[ R / J i : R] = i

Since

[V/M(V)

: R/M(R)]

(22)

=

2 , by

k ( R , J i) =

We

claim

given

2i

0 < i < a

(2),

for

(9) a n d

0 < i < a

.

(20) w e h a v e

.

that:

any

ideal

i = or~

J

we have

that

J

in

R

with

~(R)

c J,

upon

letting

,

(23)

Namely, 0 < i < a,

in v i e w and

i

is a n

of

(9) w e

i = a

see

with

that

i

0 ~ i ~ a,

is a n

and

integer

J = Ji

"

with

clearly

J c R N M(V) i ;

(231 )

if

integer

then

(since

~(R)

c J) b y

62

(2),

(9),

(20)

and

(231 ) w e

see

i. 63

that

J = J a ; so h e n c e f o r t h

(232)

with

ordv~

= i

and

then

~ =

(4) and

(9) w e h a v e

~(R)

c J, b y

that

J = J

Since

(9),

i

(16),

by

in v i e w

0

have of

and

(18)

PROOF

OF

g(z) ~ (10.4.1),

(1)

(24), to

(9),

by

and

(233)

~ e R

;

w e get

(231),

(232 ) and

(234 )

we

con-

(16)

and

(17) w e g e t

of

d

we h a v e

of

(10.4.1)

.

is c o m p l e t e

by

(5),

(9) to

(23).

Since

g(R).

Since

we get

that

ordvf(~)

(2)

6 ¢ R\M(R)

(18) we get

: R] = a

the p r o o f

(10.4.2).

By assumption

V = R[z]

f(~)R

--- d = o r d v f ( ~ )

Ordvx

e A

= 1

with

=

and

~(R)

and hence,

f(~)

or~f(~)

= x

= 1 .

63

and

g(V)

and

f(~)z

g(f(~)/f(~))

we h a v e

(3)

take

+ ~R ;

(20),

= R, b y

the d e f i n i t i o n

of

(4) and

and h e n c e

¢ ~(R)

a = d

(14),

can

l

R n M(V)

(24)

Now

i < a ; we

(2),

with

[R/~(R)

and h e n c e

of

y ~ ~(R),

x

since

in v i e w

6x I + ~y

(234 )

clude

that

~ e J

(233 )

by

assume

upon

,

~ g(R), =

f(~),

~ g(R).

fixing

any

we m u s t in v i e w

i. 64

In v i e w

(i), t h e r e

g(f(9)/f([d))

that A

of

such

is a p e r m u t a t i o n

~ g(R).

Note

that

now

of

(9,{) ~

(~,~)

such

are e l e m e n t s

and

in

that

(4)

o r d v f (~3) = d

(5)

g(f(~])/f(~d))

(6)

ordvf(~)

(7)

g(f({)/f(~))

(~ g(R)

= d

~ g(R)

and

(8)

(~,~])A =

In v i e w

of

(2),

(4) and

(9)

By

(5), b y

(f(~),

(10.4.1)

(c~,8)A

.

(10.4.1)

we get

f ( ~ ) ) R = M(R)

we also have

emdim

R = 2, and h e n c e

by

(9) we

conclude

that

(i0)

(~,~)A = M(A)

Since (6) and

d > 0,

in v i e w

(7) w e a l s o h a v e

of

(6) w e get

~ ~

~(A)

~ ~ M(A)

; consequently,

; in v i e w in v i e w

of

of

(4),

(i0) w e

can w r i t e

(Ii)

and

~ =

p

6'~ p + p~

is a p o s i t i v e

where

integer;

6'

now by

e A\M(A)

(3),

(4),

and

p e A

(6),

(7) and

(ii) w e

get

(12) By

(ii)

p = d and

(12) w e

see t h a t

. (~d,~)A =

get

64

(~,~)A,

and h e n c e

by

(8) w e

1.65

(13)

(~d,~)A = Let any

then

y e A

with

y ¢ ((~d,~)A)2,

Now, in view of (14)

(~,~)A .

f(y) = 0

be given.

and in view of

We shall show that

(13) this will complete the proof.

(i0), we can write

y = r + s~ + t~ 2

with

r, s, t

in

A

such that

(15) I either i or

r = 0

r = 60 ~a

with

60 ¢ A\M(A)

and nonnegative

integer

a

with

61 ¢ A\M(A)

and nonnegative

integer

b .

and I either

s = 0

(16) I

or

s = 81gb

Since

f(y) = 0, by

(4),

r e ~2dA

(17) NOW by

(3),

(14) and

(5), and

(14),

(15) and

(16) we deduce that

s e ~dA .

(17) it follows that

65

y c ((~d,~)A)2

CHAPTER

IIo

PROJECTIVE

By a h o m o g e n e o u s family

[Hn(A)]0mn 0. In the r e s t o f C h a p t e r

§ll.

Function

II,

let

A

be a homogeneous

fields and projective

domain.

models.

We define H(A)

=

U

H n (A) .

0gn 0, w e m a y

flat w h e r e

We

also

£(A, Q I , Q 2 ..... Qs )

e = E m d i m [ A , Q i , Q 2 .... ,Qs])

QI,Q2,...,Qs reference

call

to

note

; from A

these

is c l e a r

phrases from

i__n A

" in A

the

flat

spanned

(or the e-

by

" m a y be d r o p p e d

when

the

the context.

that H I(A)

= ~r-l(A)

F o r any

where

[0] fi N ~ ~

r = Emdim (A), w e

A.

define

A N = A N N H I (A)

Again tion

we

observe

that

of the s e t of all

homogeneous

subdomains

N - AN

gives

nonzero

members

of

A,

and

a

(inclusion

preserving)

of

onto

~

the

inverse

{0}

~ N e ~

(A)

bijection

the

set of all

is g i v e n

B - HI(B)A. In the r e s t

of §14,

let any

73

(A)

biject-

be qiven.

by

2.9

For

any

J c A

we d e f i n e jN,A = A N N J

and w e

call

jN,A

words,

projection

corresponding to

A

from

member

is c l e a r

and w e

the ~ r o j e c t i o n

simply

from

call

of

N

is the

N

n HI(A)

the

context,

J

from

same

thing

of

~(A).

N

of

J

A N = A N'A = the p r o j e c t i o n

A.

In o t h e r

as p r o j e c t i o n Again,

we m a y w r i t e

it the p r o j e c t i o n

in

jN

from

of

when

the

instead N.

A

from the

We

from

reference

of

note

N

jN,A that,

(in A)

,

and Emdim A N = Emdim We o b s e r v e

inverse

particular c N] ~

j ~ jN

* [J ¢ ~i(A)

tion of and the

that

: J c N]

bijection

~ , N

onto

gives

HI(AN),

and

A - Emdim[A,N]

gives

a

(inclusion

onto

~ i*- e - i (A N )

preserving) where

is g i v e n b y

K ~ KA.

a

preserving)

(inclusion

the

inverse

- 1 .

We

bijection

bijec-

e = Emdim[A,N~,

note

that

then,

[~ c HI(A)

is a g a i n

in

:

given by

@A.

We

define H

(A,N)

=

{~ ¢ H

(A)

: ~ =

and w

*

Hn(A,N) and we n o t e Hn(A,N) We

also

that

onto note

H

= H

w

(A,N)

~ , ~N (AN),

and

*

to m e a n ideal

in

gives

(inclusion

inverse

W

(A,A (A)) = H

shall

say t h a t

that

~ ( A N) = ~(A).

C

the

a

preserving)

bijection

bijection

is g i v e n

by

of

~ ~ ~A.

that

H We

n Hn(A)

in

A, we

W

(A)

and

the p r o j e c t i o n

shall

Given

from

N

= H n(A) in

any n o n m a x i m a l

say t h a t

74

*

H n ( A , A (A))

A

is b i r a t i o n a l

homogeneous

the p r o j e c t i o n

of

C

prime from

N

in

2.10

A

is b i r a t i o n a l

to m e a n t h a t

N ~ C

and

Again,

f r o m t h e s e two p h r a s e s w e m a y d r o p

to

is c l e a r

A

LR([A,C~):R([AN,cN)~ " in A " w h e n

= i.

the r e f e r e n c e

f r o m the c o n t e x t .

§15.

Zeroset

and h o m o g e n e o u s

localization.

We define ~(A)

= the s e t of all n o n m a x i m a l

homogeneous

prime

ideals

in

A.

W e also d e f i n e

~I(A)

= {P ¢ n(A)

: E m d i m A / P = D i m A/P]

hi(A)

= {P ~ n(A)

: D i m A / P = i}

I(A)

and w e n o t e

= ~I(A)

N n i (A)

that

~ i (A) c ~ i + l (A) We observe A ~ A/P, With

every

1

hi(A)

and

t h a t for any

*

c ~i(A)

P c ~(A),

[R([A, P3)

i.

v i a the c a n o n i c a l

H 0 ( A / P ) - v e c t o r - s p a c e becomes a

this u n d e r s t a n d i n g

for all

epimorphism

H0(A)-vector space.

we have

: H0(A)~

= [R([A,P])

: H 0 (A/P)

= {P ~ •(A)

: [R([A,P~)

: H 0(A)]

and 9 0(A)

< ~}

We define neg~A, P3 = [R([A, P3)

: H 0(A) ] for all

and w e n o t e t h a t t h e n

Deg[A, P3 = D e g ( A / P )

75

for all

P c Do(A)

P e ~0(A)

2.11

We also note that 1 n0(A ) = [p ¢ n0(A)

: Deg[A,P]

= i]

and in particular: H0(A)

algebraically

closed = n0(A) = ~ ( A )

We define n(A,x)

= [P ¢ n(A)

~l(A,x)

: x ¢ P]

= ~(A,x)

n ZI(A)

~i (A,x) = •(A,x)

Q ~i(A)

for any

x ¢ H(A)

~}(A,x) : ~(A,x) n ~(A) 1 n(A,I)

: [P c ~(A)

: I n H(A) c P]

n I(A,I) = [%(A,I) e ~I(A) for any

I cA

n i (A, I) = n (A, I) N n i (A) nI(A,I)

= n(A,I)

n nl(A~l

n(A,\I)

= ~(A)\n(A,I)

nl(A,\I)

= n(A,\I)

N nl(n)

ni(A,\I)

= ~3(A,\I)

N ~i(A)

~(A,\I)

= ~(a,\i) n e~(a)

n([A,I])

= •(A,I)

nl([A,I])

= •I(A,I)

ni([A,I])

= ni(A,I)

1 ni([A'I])

I(A,I ) = [%i

I

76

for any

I c H(A)

or

I cA

for any

I ¢ H(A)

or

I cA

2.12

n([A,I~,J)

= n(A,I)

N ~(A.,J)

nl([A,I~,J)

= n([A,I~,J)

n nl(A)

for a n y

I ¢ H(A)

or

ni([A,I~,J)

= n([A,IT,J)

N hi(A)

and a n y

J ~ H(A)

or J c A

nl([A,I],J) i

= ~([A, IT,J)

n nl(A) i

n([A,I~,\J)

= ~3(A,I)\~3(A,J)

~I(A,I],\J)

= ~I([A,I~,\J)

I c A

and

Q ~I(A)

~i([A,I~,\J)

= ni([A,I~,\J)

n ~i(A)

~ i1( [ A , I ] , \ J )

= n Ii ( [ A , J ~ , \ J )

N ~I(A)

We note that then ~(A) •(A,xA)

= •([A,0],x)

= ~(A,0)

= ~([A,x],O)

= n([A,0],I)

n(A,I)

= ~ ~ HI(A)

~(A,I)

is a finite

set ~ n(A,I)

~ n0(A,I) (radAI)

=

J

= e ~ ~0(A,I)

c n(A,J)

~ n0(A,~)

and

n(A,I)

(radAI) n(A,I)

I

= n(A)

~ n0(A,I)

and a n y

J ~ H(A)

or J c A

= Z(A,x) ,

= ~([A,I])

= n(A,I)

I c A ,

n(A,i)

= n(A,J)

or

,

= n([A,I],0)

ideals

I e H(A)

l

x e H(A)

for a n y and for a n y h o m o g e n e o u s

for a n y

= Z([A,x])

for a n y n(A, (I N H ( A ) ) A )

I

D0(A)

in ~

A Z =

w e have: [0]

c radAI

,

,

= ~0(A,I)

= n0(A,J) P

(HI(A)A)=

(radAJ)

Q

(HI(A)A)

•

(radAJ)

Q

(HI(A)A)

,

c n0(A,J ) n

(HI(A)A)

77

~

I c A

2.13

n(A,I)

U n(A,J)

= ~(A,I

N J) = n(A, IJ)

n(A,I)

n n(A,J)

= n(A,I

+ J)

,

and

We observe sing)

injection

~(A). A/C

that,

in p a r t i c u l a r of

~(A)

Finally we note

bijection

G i v e n any

t h a t for any

~([A,C~)

P e ~(A)

= {x/z

~(A,I,P)

=

; ~(A,P)

~(A,P)/M(~(A,P))

for e v e r y

n

=

I c A

subsets

letting

gives

a

of

f: A

(inclusion

~(A/C).

for any

x ~ Hn(A)

for any

I c A ,

,

(A)

is a local d o m a i n w i t h q u o t i e n t

= dim A - Dim A/P

isomorphic with

R([A,P~)

we have

~ ~ ~(A,x,P)

(x/z)~(A,P)

we have

for e v e r y

~(A,I,P)

= ~(A, (I N H(A))A,

M(~(A,P))

field

H 0(A) ; d i m ~ ( A , P )

= ~(A,P,P)

and

78

;

P)

;

;

;

c ~(A,P)

z ¢ Hn(A)\P

c ~(A,P)

{ ~(A,I,P)~(A,P)

(15.3)

~(A,P)

x c H

~(A,x,P)~(A,P)

(inclusion rever-

= ~(A,A, P) ,

is n a t u r a l l y

I1511 (15.2)

onto

: z c Hn(A)\P ~

is a s p o t o v e r

for e v e r y

upon

p - f(P)

U ~(A,x,P) XcINH(A)

a n d w e note t h a t then:

an

we define

(A,P)

(A)

C e ~(A),

epimorphism,

of

~(A,x,P)

gives

into the set of all n o n e m p t y

to be the c a n o n i c a l

preserving)

P - ~(A,P)

and

and

;

2.14

given

any

I' =

z ¢ HI(A)\P

U {xz - n 0~n 0 = x

e P.

For

ord([A,C~,x,V)

(18.3)

For

ord([A,C~,I,V)

any

x

e Hn(A)

we

have

= ordvf(x)/f(z)

any

I c A

we

for

all

z

have

= ord([A,C~,

(I D H ( A ) ) A , V )

--- o r d ( [ A , C ~ ,

(I n H ( A ) ) A

= min[ord([A,C~,x,V): =

£ Hn(A) \ P

a nonnegative

+ C,V) x

integer

~ I n H(A)} or

and ord([A,C~,I,V)

= ~ ~

I n H(A)

c C

,

ord([A,C~,I,V)

> 0 ~

I Q H(A)

c P

.

88

~

,

.

then

2.24

In v i e w o f

(18.1)

and

(18.3) we get

(18.4)

ord([A,C],J,V)

(18.5)

For any

@([A,C],Q)

= IV'

In v i e w o f (18.6),

(18.7),

m ord([A,C],I,V)

Q e H(A)

(18.8)

(18.2) and

ord([A,C],a,V)

(18.7)

For any

ord ([A,C],x+y,V)

where

H n (A)

If

of

0]

.

or~

w e a l s o get

for all

and

0 # a e H0(A).

0 ~ a e H0(A)

we have

= ord([A,C],x,V)

and

y

in

Hn(A)

we h a v e

m min(ord([A,C],x,V),ord([A,C],y,V))

equality holds

(18.9)

J c I c A.

we have

and p r o p e r t i e s

x ~ H(A)

x

(18.5):

for all

: ord([A,C],Q,V')>

= 0

ord ([A,C],ax,V)

For a n y

Q c A

and

(18.9):

(18.6)

(18.8)

or

~ ,~([A,C])

(18.1),

(18.4)

in case

[V/M(V)

ord([A,C],x,V)

: H0(A) ] =I,

,

/ ord([A,C],y,V)

t h e n g i v e n any

x

and

,

y

in

s u c h that ord ([A,C],x,V)

there exists

a unique

a ¢ H 0(A)

ord([A,C],x-

moreover:

m ord ([A,C],y,V)

ay,V)

a = 0 ~ ord([A,C],x,V)

# ~ ,

such that > ord ([A,C],y,V)

>ord

;

([A, C],y,V)

W e s h a l l n o w prove:

(18. I0) L E M M A . ord([A,C],xy,V)

For any

x e Hm(A)\C

= ord([A,C],x,V)

89

and

y ~ Hn(A)\C

+ ord([A,C],y,V)

we have

2.25

and ord ([A,C],xn,v)

PROOF.

- ord([A,C],ym,v)

We can take

= Ordvf(xn)/f(ym)

z ~ H 1 (A)\P

and t h e n we h a v e

ord ([A, C],xy,V) = o r d v f (xy)/f (zre+n) = [ordvf(x)/f(zm ) + -- o r d ( [ A , C ] , x , V )

[or~f(y)/f(z

by

(18.2)

by

(18.2),

by

(18.2)

n)

+ ord([A,C],y,V)

and we also h a v e ([A,C],xn,v)

- ord([A,C],ym,v)

= [Ordvf(xn)/f(zmn)]

- [Ordvf(ym)/f(zmn)]

= Ordvf(xn)/f(ym ) N o w w e shall prove: (18.11) and

D = C N.

PROJECTION Assume

(Note that,

Let

h:

that

if

D ~ %(B)

LEMMA.

Given

[0] ~ N e ~(A),

let

B = AN

D e ~3I(B).

H 0(A)

is a l g e b r a i c a l l y

closed,

then

~ IN : H0(A) ] - [C A N : H0(A) ] ~ 2.)

R([B,D])

~ R(f(B))

b e the c a n o n i c a l

isomorphism,

and let

W = h - l ( v D R(f(B))). (Note that now: h(W)

= v ~ R(f(B)), Then

for all

ord([A,e],x,V) and

for all

ord([A,C],N,V) and

x ¢ N\C

= N, W ~ ~([B,D]),

is a p o s i t i v e

integer.)

we h a v e

- ord ([A,C],N,V)

J c 7](A)

ord([A,e],J,V)

ordvM(h(W))

~ ~, HI(B)

= [ordvM(h(W)) ]lord ([B,D],x,W) ] ,

wit____hh J c N

- ord([A,C],N,V)

and

J ~ C

we h a v e

-- [ordvM(h(W)) ] [ o r d ( [ B , D ] , J , W )

90

2.26

PROOF. Q = ~

Let

g: B ~ B / D

([B,D],W).

b e the c a n o n i c a l

epimorphism.

Let

We can fix

z ¢ N\Q

and then,

in v i e w of

ordWg(x)/g(z)

Upon multiplying

and

(18.2),

= ord([B,D~,x,W)

the a b o v e

I for all

(i)

(18.1)

ordvf(x)/f(z)

=

ordvM(h(W))

(18.10)

(2)

in v i e w of

(3)

ord([A,C~,x,V)

z ¢ N, b y

(5)

for all

(18.1)

(18.3)

and

(2) a n d

and

ord([A,C],x,V)

[

=

(18.3)

and

we have

J ~ C

and

ord ([A,C],J,V)

that

for all

x c N.

(3) w e get

= o r d ([A,C~,z,V) that

- ord([A,C],N,V)

[ordvM(h(W))][ord([B,D],x,W)

In v i e w of

,

(i), w e c o n c l u d e

(4) it f o l l o w s

/

x ~ N\C

~ ord([A,C~,z,V)

ord ([A,C~,N,V) (i),

integer.

- ord([A,C~,z,V)

and hence,

Now by

w e see t h a t

[ordvM(h(W)) ] [ o r d ( [ B , D ] , x , W )

= ordvf(X)/f(z)

(4)

integer.

we h a v e

ord([A,CT,x,V)

Since

we h a v e

we have:

= a nonnegative By

x ¢ N\D

= a nonnegative

equation by

x ¢ N\D

for all

(5) w e see t h a t

]

for all

for all

x ¢ N\C

J c ~(A)

- ord([A,C],N,V) =

[ordvM(h(W)) ][ord([B,D],J,W) ] .

91

.

with

JcN

2.27

In v i e w of

(18.12) and

D = C N.

(18.3) b y

PROJECTION Assume

(Note that,

h:

LEMMA.

that

if

D e ~(B)

Let

(18.11) w e get:

Given

is a l g e b r a i c a l l y

~ Emdim[A,e~N]

-

*

(A), let

B

=

AN

D e ~I(B)-

H0(A)

R([B,D])

{0] ~ N e ~

R(f(B))

closed,

- Emdim[A,N]

then:

~ 2.)

b e the c a n o n i c a l

isomorphism,

and let

W = h - l ( v N R(f(B))).

(Note t h a t now: V n R(f(B)), Then

and

and

for all

ord([A,C],N,V)

ordvM(h(W) ) x e

is a p o s i t i v e

(N D H I ( A ) ) \ C ,

ord([A,C],x,V)

- ord([A,C],N,V)

for all

(A)

J ¢ ~

~ ~, W e ~ ( [ B , D ] , h(W))

with

=

integer.)

w_e have,

-- [ o r d v M ( h ( W ) ) ] [ o r d ( [ B , D ] , x , W ) ) ] ,

J c N

and

J ~ C, u p o n

lettinq

K = jR, we h a v e

ord

([A,C],J,V)

-

ord([A,C],N,V)

= [OrdvM(h(W)) ][ord([B,D],K,W) ] . (18.13) be given with t h a t the

COROLLARY-DEFINITION. ~ c N.

following

D ¢ ~(B)

I

Let

B = AN

two c o n d i t i o n s

and,

upon

to b e the c a n o n i c a l

(*)

h - l ( v ' n 9(f(B)))

D ¢ ~I(B)

Let and

~ e HI(A) D = C N.

and

By

N e ~

(A)

(18.12) we

see

are e q u i v a l e n t .

letting

h:

9([B,D])

isomorphism,

w e have:

e ~([B,D],17 N)

for all

- R(f(B))

V'

~ ,~([A,C],~).

and:

(**) ord([A,C],n,V')

> ord([A,C],N,V')

92

for all

V'

e ~([A,C],~)

2.28

We

shall

integral

to m e a n

satisfied. to

A

tion

say that

that one

From

is c l e a r

the p r o j e c t i o n

this

(and h e n c e

C

both)

phrase we may drop

f r o m the c o n t e x t .

(**), w h e r e a s

of

in C h a p t e r

N

o f the

" in A

In t h i s

IV w e

from

is n -

two conditions

" when

Chapter

shall use

i__nn A

we

the

is

reference

shall use

condition

(*).

condi-

We

note

that obviously the p r o j e c t i o n (18.13.1)

D ¢ nl(B)

and n0([A,C3,N)

of C from N

= ~ = is m - i n t e g r a l .

§19.

(19.1)

Order

to

R

P = ~

be

celarly

(19.i) by

assertion

and

integral

get

by

projections.

{0];

in addition B

identity

map

(18.i)

and

f

and

by

as

to

by

G i v e n V c ~(R),

R = i.

~B,D~

(19.10)

where, A

or

for

map

R - R.

(18.11)

and

(18.12)

replacements by

S;

D

we by

I c A.

1 ~ i ~ 10,

as w e l l

identity

from

above

I ¢ H(A)

replacing:

the

(19.12)

to the

for any

(19.1)

from

as w e l l R(S)

Dim

We define

assertions

(19.11)

replace:

domain with

= ord([R,{0}~,I,V)

is o b t a i n e d

R; C

where

a homogeneous

(R,V).

ord(R,I,V)

We

curve

(19.12).

Let let

on an abstract

as We

[A, C3 also get

respectively,

let

S = RN

and

{0};

and

by

h

the

- R(S).

w

(19.13) be

given with

COROLLARY-DEFINITION. ~ c N.

ing two conditions

(*) (**)

Dim

then

Let

Let By

S = R N.

*

~ c HI(R) (19.12)

we

and

N e ~

see t h a t

the

(R) follow-

are e q u i v a l e n t .

S = 1

and:

V'

N R(S)

Dim S = 1

and:

ord(R,~,V')

93

e ~ ( S , ~ N)

for all V'

> ord(R,N,V)

c O(R,~).

for a l l V'

¢ ~(R,n).

2.29

W e s h a l l say t h a t the p r o j e c t i o n to m e a n that o n e satisfied. to

R

tion

(and h e n c e both)

from

from the context.

(**), w h e r e a s

in C h a p t e r

in

R

of the c o n d i t i o n s

F r o m this p h r a s e we m a y o m i t

is c l e a r

N

i__ss~ - i n t e q r a l (*) and

(**)

is

,r in R '~ w h e n the r e f e r e n c e

In this C h a p t e r w e s h a l l use c o n d i -

IV w e s h a l l u s e c o n d i t i o n

(*).

We note that obviously

I

the p r o j e c t i o n

(19.13.1)

D i m S = 1 and

~0(R,N)

= ~

from

N

in R

is n - i n t e g r a l

and the p r o j e c t i o n (19.13.2)

from

N

in

R

is ~ - i n t e g r a l

I the p r o j e c t i o n

(19.14) canonical

REMARK.

of

G i v e n any

{0] from N in R

C ~ ~I(A),

let

is n - i n t e g r a l .

f: A - A / C

b e the

epimorphism. w

We n o t e that,

then,

for any

V

c ~([A,C~),

ord([A,C~,x,V

) = o r d (f (A) , f (x) ,V )

ord([A,C~,I,V

) -- o r d ( f ( A ) , f ( ( I

we c l e a r l y h a v e

for a n y

x c H(A)

and N H(A))A),V

)

*

We also n o t e that, n c N

and

for any

for any W

~ c HI(A)

and

N ~ ~

(A), w i t h

~ ~ C, w e c l e a r l y h a v e that:

the projection

of

the p r o j e c t i o n

C from

from

N

f(N)

in in

g4

A f(A)

is f ( ~ ) - i n t e g r a l is

f(~)-integral.

I c A.

2.30

§20.

By a v a l u e d a field,

A

Valued

vector

space we m e a n

is a k - v e c t o r - s p a c e

v: is a m a p p i n g , and

Z

where

is the

vector

Q(A)

a triple

with

[A : k~

A U fi(A)

is the

spaces.

(k,A,v)

where:

k

is

< ~ , and

~ Z

set of all k - v e c t o r - s u b s p a c e s

set o f all n o n n e g a t i v e

integers

together

of

with

A

, such

that: (1)

v(0)

=

(2)

v(x+y)

> m i n (v (x) ,v (y) )

(3)

v(x+y)

= min(v(x),

v(x)

~

;

~ v(y)

v(y))

v(zx)

= v(x)

for a l l

(5)

given

any

and

there

exists

such (6)

that v(L)

(*) [V/M(V)

(**)

(H 0(R),

(18.1),

: H0(A)]

H I(R),

in

> v(y)

(18.3),

and x

y

and

in y

]% ;

in

]% w i t h

0 ~ a ¢ k v(x)

;

> v(y)

~ ~ ,

; and for e v e r y

(18.7),

and

= i, we h a v e

and

I% w i t h

: x ¢ L}

any

that

(18.8)

V

L ¢ Q(A) and

(18.9)

¢ ~([A,C~)

(H0(A) , HI(A),

we

see that:

with ord([A,C3,.,V)

space.

(19.1),

(19.3),

F_or any h o m o q e n e o u s with

for all

x ¢ A

y

C ¢ ~I(A)

vector

In v i e w o f

V ¢ ~(R)

v(x-ay)

For any

is a v a l u e d

x

= min{v(x)

In v i e w o f

x

;

(4)

a ¢ k

for all

[V/M(V)

(19.7), domain

(19.8) R

with

: H 0(A) ] = i, we h a v e

ord(R,.,V))

is a v a l u e d

95

vector

and

(19.9)

Dim R = 1 that space.

we

see that:

and

any

2.31

In

§21

and

groundwork lemmas like

about

to

topic of

for

this

shall

defining

define

osculating

osculating vector

in m i n d

flats

flats,

space.

situations

osculating

To

(*)

much

shall

fix

and

more

we

flats.

the

(**).

thoroughly

To

now

idea, We

prove the

shall

than

prepare

several

reader

deal

needed

in t h e

defined For

every

=

[]i(L) Z(L)

rest

of

§20,

=

let

in

(k,A,v)

be

a valued

vector

let:

~ ( A ) : I c L}

[Ie

Q(L):

[I

=

[v(x): =

Y' (L) = =

Ix

x e L}

~ L:

[I~

,

: k]

Iv(I) : I ~ 0 (L)]

Y(L,j)

Y(L)

L ¢ Q (A)

[Ie =

Z' (L)

=

i}

,

,

v(x)

~ j}

,

q (L) : I = Y ( L , j )

[Ie

Q(L):

,

for

I = Y(L,v(I))]

some

j e 7}

,

,

w

(L)

=

{X e L:

v(x)

> v(L)}

(L)

=

[x e L:

V(X)

= ~]

, and

w

A

*

We

note

that,

in v i e w

w

(i),

w

(2)

and

: k]

.

(4),

we

then

have

w

A

(L)

e O (L).

Let w

p(n)

we

also

may

with

above.

Q(L)

T

the

the

book.

So, as

we

a valued

keep

of

§23

observe

that

=

[A

in v i e w C N L

A

in t h e

of

(18.1) in

case

and of

(19.1) (*)

(L)= [0]

Now,

(L)

rest

of

§20,

in c a s e

let

of

(**)

L ~ n d (A)

p = p (L) .

96

be

given

and

let

space

the rest

2.32

We

shall

first

state Lemmas

(20.1)

to

(20.13)

and t h e n p r o v e

t h e m o n e b y one. (20.1)

LEMMA.

Z(L) = Z' (L)

(20.2)

LEMMA.

c a r d Z'(L)

(20.3)

LEMMA.

Y(L)

(20.4)

LEMMA.

For any

over

j e Z(L),

then

{ d - p + i.

= Y' (L).

Y(L,j)

j ~ Z

we h a v e

~ Y(L)

(20.5)

LEMMA.

c a r d Y(L)

(20.6)

LEMMA.

I ~ Y(L)

(20.7)

LEMMA.

I__~f d > p

and

Y(L,j)

v(Y(L,j))

¢ ~(L);

if m o r e -

= j.

= c a r d Z(L). = Y(I)

c Y(L)

then

~

and

(L) c T

~

(I) = A (L).

(L) ~

Y(L)

N Qd_I(L)

W

and

v(T

that

(L)) > v(L);

v(I)

> v(L),

(20.8) Ld = L

then

LEMMA.

such t h a t

Moreover

we have

There

have

v(L and

i)

> v(

~i+I )

L'~I = Li

for

for

, i) > v ( L

whenever the said

characterized

sequenc e

and

sequence

v(L

Alternatively,

unique

a unique

sequence

v ( L i) > v ( L i + I) p ~ i ~ d, L i = T

%,

~

0d_l(L)

such

Lp c L p + i C . . . c for

p { i < d.

(Li+ I)

with

L ,e c L , e + i c . . . c L ,i+ I)

for

p ~ e ~ d,

,d = L

fo___~r ~ ~ i < d.

such that

Moreover

we

p { ~ { ~ ~ i ~ d. sequence

by saying

the

c Lp,+ic...c_L~ = L

for

of

= ~.

w e c a n say that g i v e n a n y

a unique and

is a n y m e m b e r

(L).

(L), an___~d V(Lp)

L , i = L~, i

completely

i_~f I

exists

L i e Y(L)

More generally

L , i ~ Qi(L)

I = T

Li ~ Qi(L)

p { i < d, Lp = ~

there exists

moreover,

%

c Lp+iC...eL

following: such that

P' ~ i ~ d, v( % ', , = -.

p ~ i ~ d.

97

-- L

can be a

there exists L:l ~ Qi (L)

Moreover

we h a v e

a and..... p' = p

2.33

(20.9)

We have

card

Y(L)

= card

Y' (L) = c a r d

card

Y(L)

n Qi(L)

Z' (L) = c a r d

Z(L)

= d - p + 1

and

Moreover,

with

~ = V(Lp)

is t h e

the

= card

Y' (L)

notation

N Qi(L)

of

(20.8)

=

1

for

we have

p

< i < d.

that:

> V(Lp+l)>...>V(Ld)

unique

descending

Y(L)

= Y' (L) =

Y(L)

N Qi(L)

= Y' (L)

LEMMA.

Let

labelling

{ L p , L p + 1 ..... Ld]

of

Z(L)

,

,

and

(20. i0 ) Given the Ji

J c ~ b (L)

unique c ~i(J)

and

In v i e w

of

and

L D J, w e

r(b)

~ d

of

Lp c Lp+l

n --- p (J)

let

sequence

(obtained

v(Ji) the

Q Ni(L)

fo r

set-theoretic

integers

get such

p

~ i < d.

be

L

(20.8)

as

in

(20.8)

to

J)

such

Ib e

that

n < i < b.

inclusions

a unique

for

J n c J n + l c" " °c Jb -- J

let

applying

> v ( J i + I)

clearly

[Li}

c. " . c L d =

and

by

=

Lp c Lp+iC...

sequence

p = r(~)

Ld = L

< r(~+l) ord(S,Ki+l,W)

HI(S)

integer.)

J0 c Jl c . . . c Jb = J

> ord(R, Ji+l,V)

be the u n i q u e by

~ ~ , H0(S)

is a p o s i t i v e

(obtained b y a p p l y i n g

c . . . c Kb = J ing

ord(R,N,V)

from

Also

Ji ¢ ~i (J) let

K0 c Ki

(20.8) b y r e p l a c -

K i ¢ Qi(J) Then we have

and

2.37

K. = J. 1

ord(S,Ki,W)

and

and

(20.10)

gives

L = A, i.e., P R O O F OF

P R O O F OF

(20.1).

By

for

for o b t a i n i n g

from the

0 < i ~ b

.

the s e q u e n c e s

sequences

is p a r t i c u l a r l y

(6)

and b y

(20.2).

V(Xl),

given

any e l e m e n t s

we have

(i),

Z' (L) c Z(L).

that

nonzero,

,

(Li)0~i~ d

significant when

L = HI(R).)

x k ¢ Q(L),

consequently

recipes

(ord(R,Ji,V)~i~b

(ord(R, L i , V ) ) 0 ~ i ~ d ; t h i s

we have

0 ~ i ~ b

= [ o r d ( R , J i , V ) - ord(R,N,V)]/orcIvM(W)

(Observe t h a t (Ji)0~i~b

for

1

(4)

Z(L) and

Therefore

(6)

in

w e get

v(xk)

X l , X 2 ..... x m

are all d i s t i n c t

al,a2, .... a m

For any

x ¢ L,

--- v(x)

Z(L) = Z' (L).

Given any elements

v ( x 2) ...... v ( x m)

c Z' (L).

k

in

L

and n o n i n f i n i t e ,

at l e a s t one o f w h i c h

such and is

we have for some

v ( a l x I + a 2 x 2 + . . . + a m X m) = v ( x i)

i

by

(3)

and

(4)

and h e n c e alx I + a 2 x 2 + . . . + a m X m It f o l l o w s

A (L)

that c a r d Z' (L)\[~} =

[L/A

(L)

= k]

and hence c a r d Z' (L) ~ [L/A*(L)

Clearly

[L/A*(L)

: k] + 1 .

: k~ = d - p, and h e n c e

P R O O F OF

(20.3).

Obvious

P R O O F OF

(20.4).

By

(I*),

in v i e w o f

(2*) a n d

102

c a r d Z' (L) ~ d - p + 1 .

(6*)

(4*) w e

see t h a t

;

2,38

W

Y(L,j)

~ O(L),

and n o w b y

Y(L,j)

e Y(L)

and

PROOF Y(L)

OF

into

Therefore

and by

OF

(20.6).

in v i e w o f

In v i e w

(i)

and

we

(20.3),

(6)

if

j ¢ Z(L),

then

gives

see t h a t

an

the

injective

said m a p

m a p of

is s u r j e c t i v e .

Z(L).

Obviously:

Therefore, of

I - v(I)

(20.4)

= card

see that,

= j.

Clearly

card Y(L)

PROOF

we

v(Y(L,j))

(20.5).

Z(L),

(6)

I ¢ Y' (L) = Y' (1) c y' (L)

w e get

we also

I ¢ Y(L)

see t h a t

= Y(I)

I ~ Y(L)

.

c Y(L)

= A

.

(I) = A

(L)

W

PROOF

OF

(20.7)°

Assume

that

d > p.

Then

in v i e w of

(6)

we have (i)

v (L)
v(L)

exists

y ¢ L

such

that

v(y)

(2) and

(5)

-- v(L)

= min[v(x)

: x e L}

*

In v i e w o f

(i),

W

we

see t h a t

y ~ L\T

(L)

and

the k -

W

vector-space

L/T

(L)

is

generated

by the

image of

y ; consequently

W

[L/T

(L)

: k] = 1

(3) Let

and h e n c e

T

(L) ¢ ~ d _ l ( L )

any

(4) be given

I ¢ ~d_l(L) such t h a t

(5)

then by

v(I)

(5) and

(6*) w e g e t

> v(L)

T*(L)

103

;

c I, and h e n c e

by

(4) and

(3) w e

2.39

get

I = T

(L)

PROOF

OF

paragraphs of

(i)

(20.8).

follow

and

(6)

The

assertions

from the assertio~ we

clearly

have

in the in the

L ~ Y(L)

second

and the third

first paragraph. N Qd(L)

In v i e w

and:

w

v(L)

Therefore

the assertions

d - p = 0.

Now,

first paragraph

PROOF

OF

= m ~ L =

in t h e

in v i e w

of

(20.9).

L e t the

(L)

.

first paragraph

(20.6)

in the g e n e r a l

A

and

case

(20.7),

the

follow by

notation

are o b v i o u s

assertions

induction

b e as in

in c a s e

(20.8).

on

in t h e

d - p

By

.

(20.8)

we have

{ L p , L p + 1 ..... Ld} c Y(L)

consequently

by

(20.1),

:

~, (~) :

Y(L)

(i)

(20.2),

(20.3)

£~,L+I

.....

and

(20.5)

we

conclude

that

T,d}

(20.8)

= c a r d Y' (L) = c a r d

Z' (L) = c a r d

Z(L)

= d - p + 1 .

we have

L i ¢ Y(L) and

c a r d [ L p , L p + 1 ..... Ld} = d - p + 1 ;

and

c a r d Y(L)

By

and

N Qi(L)

for

p ~ i ~ d

clearly

c a r d Y(L)

n ~i(L)

= c a r d Y(L)

p < i gd Consequently,

by

c a r d Y(L)

(i) w e

conclude

N ~i(L)

that

= card

Y' (L) N Q i ( L )

= 1

for

p

~ i ~ d

and Y(L)

N Qi(L)

= y' (L) N ~ i ( L )

104

=

{Li}

for

p ~ i ~ d

.

2.40

In v i e w o f

(i), b y

I

~ = V(Lp)

(20.8) we also

> V ( L p + l ) > . . . > v ( L d)

l is the u n i q u e PROOF

OF

see that

descending

(20.10).

labelling

By a p p l y i n g

(20.9)

of to

Z(L) L

and

J

we get that:

(I) {~ = V(Lp)> L(Lp+l)>...>v(L d) is the u n i q u e

descending

labelling

of

Z(L)

,

of

Z(J)

,

= v ( J n) > V ( J n + l ) > . . . > v ( J b) (2) is the u n i q u e

(3)

descending

labelling

Y(L)

Q ~i(L)

= {Li}

for

p ~ i ~ d ,

Y(J)

D Qi(J)

= [Ji}

for

n ~ i ~ b .

and

(4)

J c Q (L), we c l e a r l y h a v e

Since and

(2), t h e r e

d

(5) Since

(6) Since

of i n t e g e r s

Upon

a unique

J c L, b y

(3),

Ji = J n Lt(i) Ji c Qi (J) by

~ Z(L)

sequence

; consequently,

p = t(n)

by

(I)

< t(n+l) v(x)

and Fi =

in v i e w

of

(i),

e L

(3)

and

L s c L t (b)

(s)

we

n

< i v ( J i + l )}

by

(20.1)

we

for

U F1

have

n

Z' (J)

=

~ i < b

Z(J),

and

and

hence

in v i e w

of

(2)

;

we

also

get

(9)

By

J N

(7),

(8)

and

J N

(io) I

and

(9)

F. = 1

it

~

for

follows

n

< i ~ b

.

that

L t(b)

=

J N Ls

for

J N Lt(i)

=

J N Ls ~

and

~ s < t(i+l)

t (b)

~ s < d

,

and

hence

we

Consequently,

t(i)

must

have

upon

~ =

letting

J D

n

for

Lt(i+l)

n

~ i < b

,

and

t (i)

r(b+l)

= d

= r (i)

+ i, b y

for

n

(5),

(6)

~ i g b and

(i0)

. we

get

Ji =

J D Ls

for

r(i)

< s < r(i+l)1 for

v ( J i ) = v ( L r(i) )

The follows

special from

PROOF Then,

OF

in v i e w

the

claim

bracketed

(20.11). of

for

(20.9),

n

~ i gb

.

I

the

case

when

d - p = b

- n +

1

now

remark.

Let we

Lp c know

Lp+iC...~L that

106

d

=

L

be

as

in

(20.8).

2.42

(1)

v (Lp) = ~

and

V(Lp)

> V ( L p + l ) > . . . > ( L d)

(2) is the u n i q u e

Since

0 < n < p, w e can take

(2) w e can take

(3) By

descending

x.

c L

l

J

(2) and

(3)we

(4)

¢ nn(~

of

Z (L)

In view of

(L)).

(20.1)

and

with

v(x i) = V(Lr(i)) (i),

labelling

for

n < i < b

get

> V ( X n + l ) > . . . > v ( x b)

Let W

J = J Then in v i e w of *

J

+ Xn+l k + X n + 2 k + . . . + X b k

(4), b y

(l),

(3)

and

(4)

. we see that

J c ~b(L),

W

= A (J), n = p(J),

and

!

(5) Let ing

Z (J) = [ ~ , V ( X n + l ) , V ( X n + 2) ..... V(Xb)] Jn c Jn+l ~'" .c J n (20.8)

n < i < b,

to

J, such

= J

be the u n i q u e

that

V(Jn)

I

and

(obtained

v(J i) ~ v(Ji+l)

and

(6) By a p p l y i n g

Ji e ~i(J)

sequence

(20.1)

and

(20.9)

= to

J

we know

that

v(J n) > V ( J n + l ) > . . . > v ( J b) is the u n i q u e

and hence,

in view of

descending (4) and

labelling

of

(5), we m u s t h a v e

107

Z' (J)

by applyfor

2.43

(7) By

v(J i) = v(xi) (I),

(3),

(6) and

for

(7) we get

(8)

v(J i) = V(Lr(i))

In v i e w of p = r(n) the pair

(2) and

(8), by

< r(n+l) ord([A,C],Ji+l,V)

for

n ~ i < b ,

> ord([B,D~,Ji+l,W)

for

n ~ i < b

(I) we get that

ord([B,D],Ji,W)

and hence by the uniqueness part of

(20.8),

applied

to

B,D,W,J,

we

conclude that K.1 = J 1 therefore,

in view of

for

n ~ i < b ;

(i), it now follows that

ord ([B,D],Ki,W)

I for

= [ord(~A,C],Ji,V) PROOF OF

(20.13).

- ord([A,C],N,V) ]/ordvM(h(W)) By

(19.11)we

108

have

n < i ~ b.

2.44

(1)

ord (S,Ji,W) for

= [0rd(R,Ji,V)

0 < i ~b

- ord (R,N,V) ]/ordvM(W)

.

Since ord(R, Ji,V) by

> ord(R, Ji+l,V)

for

0 ~ i < b

> ord (S,Ji+I,W)

for

0 ~ i < b

(20.8),

applied

,

(i) w e g e t t h a t

ord(S,Ji,W)

and h e n c e b y the u n i q u e n e s s conclude

therefore,

in v i e w o f

for

C ~

(21.1)

1

for

= [ord(R, Ji,V)

S,W,J,

we

;

that

- ord(R,N,V)]/ordvM(W)

.

flats and i n t e g r ~

projections

of an e m b e d d e d

curve.

DI(A) LEMMA-DEFINITION.

: H0(A) ] = i.

Emdim[A,C,L],

0 ~ i ~ b

(I), it n o w f o l l o w s

0 < i ~b

Osculating

Let

= J

1

ord(S,Ki,W)

[V/M(V)

to

that K

§21.

p a r t of

F o r any

Let

V ~ O([A,C])

L e @d(A),

it is e a s y to get the

upon

be such that

letting

following by applying

p = (18.3)

and

(20.8). The______ree x i s t s L i c ~i(A) Moreover

an d

sequence

L = Ld D Ld+I~...~L p

ord([A,C],Li_l,V ) < ord([A,C],Li,V)

for

such t h a t d < i ~ p.

we have

L i ~ HI(A) for

a unique

= {x ~ Li_ 1 ~ HI(A)

: ord([A,C],x,V)

d < i ~ p ,

109

> ord([A,C],Li_l,V~

2.45

L p = A (A,C,L) , and

ord ([A, C] ,Lp, V) = ~

M o r e q e n e r a l l y w e can say that q i v e n there exists

a unique

L , i e ~i(A) d < i ~ ~.

and

Moreover we have

unique

sequence

,i_l,V)

L , i = LS, i

by sayinq

d ~ ~ ~ p , such that for

d ~ i ~ ~ ~ 8 ~ p.

L = L d D Ld+ID...DLp

L = L'd ~ Ld+l'D...~L'p,

,V) = ~.

whenever

the f o l l o w i n q :

ord([A,C],L~_I,V ) < ord([A,C3,LI,V) ord([A,C],L~,

with

< ord([A,C~,L~,i,V)

the s a i d s e q u e n c e

characterized

~

L = L ,d ~ L ~ , d + I D . . . ~ L ~ , ~

ord([A,C],L

Alternatively, completely

sequence

any

There exists

such that

for

L!z e ~i(A)

d < i ~ p'

Moreover we have

p' = p

c a n be a and

, and

and

L~z = L.l

for

d ~ i ~ b. We d e f i n e

Ti([A, C3,L,V) = Li

Ti([A,C],L,V) and w e n o t e t h a t then: negative

integer

for

1

for

d ~ i ~ p

= ord([A,C],Li,V ) Tp([A,C],L,V)

= ~, ~ i ( [ A , C ] , L , V )

d ~ i < p, and in v i e w of

is a non-

(18.3) w e h a v e

w

~d([A,C],L,V)

> 0 ~ L c ~

([A,C],V)

We also define

Ti([A,C],V ) = Ti([A,C],A(A),V) for

-i < i < E m d i m [ A , C ]

Ti([A,C3,V ) = Ti([A,C~,A(A),V )

W e note t h a t in v i e w of

(18.3) w e h a v e w

T_I([A,C'],V)

= 0

and

110

To([A,C],V)

= ~ ([A,C'],V)

2.46

Ti([A, C3,L,V) at

V

relative

f l a t of when

in

=.

L.

the r e f e r e n c e

to

A

By

(20.1)

(18.3) and

Let

A

and

(21.4)

Z(L)

m a y be c a l l e d

to

(20.12) w e

A

the o s c u l a t i n g

i-

" in A

immqediately g e t L e m m a s

L c ~d(A),

= [ord([A,C~,x,V):

Y'(L)

= [Ie

~

be s u c h t h a t

th@ set of all n o n n e g a t i v e

= {ord([A,C~,I,V):

upon

[V/M(V)

integers

" ,

(21.2),

: H0(A)~

together with

letting

I e ~

(A)

with

I c L}

x e L n HI(A)}

(A): I n HI(A)

= [x ~ L N Hl(A): j

for some = {I e M

C

f r o m the c o n t e x t .

V ¢ 3([A, C3)

Z'(L)

Y(L)

in

From these phrases we may drop is c l e a r

Let

denote

T h e n for any

i - f l a t of

as s t a t e d b e l o w .

LEMMA. Z

the o s c u l a t i n g

Ti([A, C3,V) V.

(21.2) = I.

to

at

(21.3)

C

m a y be c a l l e d

(A) : I N HI(A)

= [x e L n HI(A):

ord([A,C~,x,V)

~ j ]

e Z~ ord([A, C3,x,V)

> ord([A,e~,I,V)}} p = Emdim[A, C, L~

we have

the f o l l o w i n g :

(i)

Y(L)

(2)

c a r d Y(L)

(3)

Y(L)

Q 91i(A) = Y' (L) n ~

= c a r d Y' (L) n ~ i ( A )

= Y' (L) = [ T i ( [ A , C ~ , L , V ) :

Z(L)

c a r d Y(L)

for d ~ i ~ p.

I

is the u n i q u e

= 1

for

d ~ i < p.

d ~ i ~ p}.

= Z' (L).

= c a r d Y'(L)

Td([A, C3,L,V) (6)

(A) = { T i ( [ A , C ~ , L , V ) } w

Q ~i(A)

(4) (5)

;

= c a r d Z' (L) = c a r d Z(L)

< Td+I([A, C 3 , L , V ) < . . . < T p ( [ A , C ~ , L , V )

ascending

labelling

111

of

Z(L).

= p - d + I. = =

2.47

J

J - [ord([A,C],I,V):

I ~ ~

w

(A)

with

I c J]

qives a surjective map of (7) [J e

(A) : J c L]

onto the set of all nonempty

subsets of

Z(L) (21.3) =

LEMMA.

Let

V e ~ ([A,C])

be such that

[V/M(V):

H0(A)]

i.

Given any Emdim[A,C,L]

L ~ ~d(A) and

and

J e ~b(A)

with

J c L, let

p =

n = Emdim[A,C,J].

In view of the relations L = T d ([A,C],L,V)

(1')

Ts([A,C],L,V)

we clearly qet a unique (2')

e~(A)

for

L D J eg~b(A)

,

d ~ s ~ p

and

sequence

d < r(b) < r(b+l)gb+p-n-d

~ d

and { d , d + l ..... p } \ { r ( b ) , r ( b + l ) ..... r ( v ) ~ Whence, of

in p a r t i c u l a r ,

p - d > n - b,

the

H

----

unique

upon

we must

v = n.

with

d ~ u < p

) c J + Tp([A,C~,L,V)

and Tu([A,CI,L,V)

We

also

note

letting

integer

Tu+I([A,C~,L,V

have

= [gl,g 2 ..... g b + p _ n _ d ~

~Z J + T p ( [ A , C ~ , L , V )

113

such

that

that

in c a s e

2.49

and u

= maxis

¢ [d,d+l ..... p-l}: =

we

clearly

u= It

u

A(A,J,Ts+I([A,C],L,V))

~ ,

have

= gl

follows

r(i)

~(A,J,Ts([A,C],L,V))

that,

" if

p - d = n - b + i, then:

= p - n + i - 1

for

b

b - 1 ~ u + n - p < n

~ i ~ u + n - p

,

and r(i)

We v = n

= p - n + i

claim and,

that,

upon

r(i-1)

=

in t h e

=

particular, above

T i([A,C],J,V)

for

u + n - p < i ~ n.]

with

the

r(b-l)

(2')

J + Tp([A,C],L,V)

as d e f i n e d

=

we have

| for

b

~ i ~ n

.

Tr(i) ( [ A , C ] , L , V ) if

p - d = n - b + 1 remark,

the_____nn, w i t h

u

as d e f i n e d

we h a v e

A(A,J, T p _ n + i _ I ( [ A , C ] , L , V ) )

] for

T i ([A,C],J,V)

above,

= d - I, w e h a v e

A(A,J,Ts([A,C],L,V))

bracketed

=

sequence

c

< s ~ r(i)

~i([A,C],J,V) ID

Tp_n+i([A,C],L,V)

Letting

Ti([A,C],J,V) for

and

Tp_n+i_ I([A,C],L,V)

b

< i ~ u + n - p

I i

and

Ti([A'C]'J'V)

=

~(A'J'Tp-n+i([A'C]'L'V))

} fQr u + n - p < i

T i([A,C],J,V)

=

~p-n+i([A'C]'L'V)

114

~n

.

,

2.50

(We n o t e

that,

in case

Ti([A,C~,J,V)

= Ti([A,C],L,V),

Conversely, d ~ r(b)

A(A,C,L)

given

any

=

[0}, w e h a v e

for

u < i g n.)

L e '~d(A)

< r(b+l) 2.

let

h:

every

Let

~ ([B,D])

V ~

~[A,C])

let

n f(B)).

=

that

[s

(21.3)

Te+I([A,C],N,V)

for

every

V ~

e [ 0 , i ..... p]:

and

=

Te+ I([A,C],N,V) N =

(21.4)

Ts([A,C],V)

(with

A(A,N,Tm(v)

,q([A,C]),

J = N

([A,C],V))

and

upon

letting

J N]

,

L =

A(A)),

¢

~ * ( [ B , D ] , V N)

and w

m(V)

D = C N.

: I c N]

a bijection

D ~ ZI(B),

the

observe

of

~

that

m(V)

in v i e w

q = Emdim[A,C,N],

Let

that

J

Assume

let

closed.

= 0 ~ Tm(V) ([A.C],V)

= .~ ( [ A , C ] , V )

116

= V (~ ~ ( [ A , C ] , N )

we

get:

2.52

We

also

F(J)

observe

=

that

for any

[V ¢ ~ ( [ A , C ] )

J ¢ Q,

upon

letting

: Te+I([A,C],N,V)

= J}

,

we have F(J)

=

IV e ~ ( [ A , C ] )

=

{V ¢ ~ ( [ A , C ] , J )

= a finite

and,

in v i e w

of

Finally p = Emdim

we

:

we

: ord([A,C],J,V)

> ord([A,C],N,V)]

get

V ¢ F(J) } = ~ ( [ B , D ]

observe

A = q =

~ ord([A,C],N,V}

set

(21.4),

{V N

: ord([A,C],J,V)

that,

if

, jN)

A(A,C)

( E m d i m B)

+ 1

and

integer

with

-i

for

=

,

[0}

any

and V

e

e = 0,

then

.~[A,C]),

upon

letting

u(v)

=/

the

unique

Tu(V)+I

in v i e w

u(V)

of

(21.3)

= maxis

([A,C],V)

and

c N

(21.4),

¢ [ - I , 0 ..... p - l ] :

and

with

=

1 ~ N = ~

T i ( [ B , D ] , V N)

< p

such

Tu(V) ([A,C],V)

J = N

and

L =

that

~ N

,

A(A),

A(A,N,Ts+I([A,C],V))}

get:

,

([A,C~,V)

= Ti+ I([A,C],V) N for -i

T i ( [ B , D ~ , V N)

we

A(A,N,Ts([A,C],V)) =

u(V)

< u(v)

=

Ti+I([A,C~,V)

and

117

- ~0([A,C],V)

~ i ~ q-i

,

2.53

u(v) # -1 - N ~ S ([A,C],V) T i([B,D],v N) = A(A,N,T i([A,C],v))N for

-i ~ i < u(V)

for

u(V)

Ti([B,D],V N) = Ti([A,C],V) =

and T i ( [ B , D ] , V N) = T i + I ( [ A , C ] , v ) N ~ i < q - 1 .

T i([B,D],V N) = Ti+ I([A,C],V)

(21.6) Let

REMARK.

~ ~ H 1 (A)

and

q = Emdim[A,C,N].

I

Assume that N ~ ~ e (A)

is a l g e b r a i c a l l y

be given with

~ c N.

closed.

Let

We note that then clearly:

the p r o j e c t i o n

(21.6.1)~

H 0 (A)

of

C

from

N

in

A

is w-integral

q-e ~ 2

and

~ c Te+I([A,C],N,V)

for all

V e ~([A,C],~)

q-e a 2

and

~ c Te+I([A,C],N,V)

for all

V ¢ ~([A,C],N).

!

~

We also note that, the p r o j e c t i o n

of

C from N in A

is n-integral (21.6.2)

if e = 0,then:

{ q ~ 2

and

~ ~ TI([A,C~,V)

for all

V c ~([A,C],N).

§22.

Osculating Let

R

(22.1)

flats and integral projections

be a homogeneous

domain with

LEMMA-DEF INITION.

Let

in an abstract

curve.

Dim R = i.

p = Emdim R.

Let

V e ~(R)

be

w

such that

[V/M(V)

: H0(R) ] = i.

For any

(20.8) we get the following:

118

L ~ ~d(R),

by

(19.3)

and

2.54

There

exists

L i ¢ ~i(A) Moreover

and

a unique

ord(R, Li_l,V)

=

Ix ¢ L i _ for

clearly

< ord(R,Li,V)

for

{0}

M__ore q e n e r a l l y

we

exists

1 N H I(A) : ord (R,x,V)

d < i Kp

Lp =

there

L = L d ~ Ld+I~...oL

such

P

that

d < i ~ p.

we have

L i n H I(R)

and

sequence

a unique

and can

> o r d (R,Li_I,V)

,

o r d ( R , Lp,V) say

that

sequence

= =.

qiven

L = L

any

,d ~ L

ff

with

d ~ ~

,d+l~...~Lff,ff

~ p

such

, that

w

L

,i ¢ ~i(R)

Moreover

and

ord(R,L

we have

L

, i = LS, i

alternatively, completely

,i_l,V)

the

characterized

said by

< ord(R,L

whenever sequence

sayinq

, i , V ) for

d ~ i ~ ~

d < i ~ ~

~ 8 ~ P-

L =L d D Ld+I~...DLp

the

followinq:

.

There

can be exists

a

w

unique

sequence

ord(R,L[_l,V) Moreover We

L = L'd D Ld+l~..' .DL'p < ord(R,L~,V)

we have

T i(R,L,V)

integer

= p

and

also

d < i g p'

L!

1

= L

for

1

and

LIt ¢ ~ i ( R ) ord(R,L'p,,V)

and = =.

d ~ i ~ p.

-- L i for

d ~ i < p

= ~

, mi(R,L,V)

= ord(R,Li,V)

note

that,

then

for

d ~ i < p,

Tp(R,L,V) and

in v i e w

Td(R,L,V) We

that

define T i(R,L,V)

We

p'

for

such

of

(19.3)

> 0 ~ L c

is a n o n n e g a t i v e

we have

~*(R,V)

define T i(R,v)

= T i (R, ~ (R) ,V) for

T i(R,v)

=

T i ( R , A (R),V)

119

-i

~ i ~ p

2.55

and we n o t e that,

in v i e w o f

T_I(R,V) Ti(R,L,V) relative a__tt V. R

to

T0(R,V)

Ti(R,V)

= ,O*(R,V)

the o s c u l a t i n q

may be called

From these phrases we may drop

(22.2)

to

to

(19.3),

[0]; B

identity map

(22.7)

(20.1)

(22.6) w h i c h

(21.6) b y l e t t i n g

R

the o s c u l a t i n q

a__tt V

i-flat

in

R

" in R " w h e n the r e f e r e n c e

S = RN

to

(20.11),

and

are r e s p e c t i v e l y

and r e p l a c i n g :

A

(20.13),

to

obtained as w e l l

from as

(21.2)

[A,C]

as

[B,D]

by

by

[0];

f

R - R; and

h

b y the

identity map

R(S)

~ 9(S).

REMARK.

Given

any

C c ~I(A),

b e the c a n o n i c a l

[V/M(V)

We n o t e that,

: H0(A)]

S; D

we g e t a s s e r -

as w e l l

f: A ~ A / C

such that

in

(22.6).

(22.2)

R; C b y

i-flat

from the context.

In v i e w of

let

and

may be called

L.

is c l e a r

tions

= 0

(19.3), we h a v e ,

let

epimorphism.

by

b y the

r = Emdim[A,C] Let

to

and

V e ~([A,C])

be

= i.

t h e n clearly:

Emdim

f(T i([A,C],V)) = T i(F(A),V)

f(A) = r

and

,

T i([A,C],V)

= f-l(T i ( f ( A ) , V ) )

Ti([A'C]'V)

= Ti (f (A) 'V) '

for

-i < i < r .

' I #

More generally, p = Emdim[A,C,L],

we n o t e that,

w e c l e a r l y have:

for any f(L)

L e ~ d (A), u p o n

e 9~d+r_p(f(A))

f(T i([A,C],L,v)) = T i + r _ p (f (A) , f (L) ,v)

= L n f-i (Ti+r_ p (f (A) , f (L) ,V) ) ,

T i([A,C],L,v)

= T i + r - p (f (A) ' f (L) 'V)

120

and

,

Ti([A,C],L,V)

,

letting

for

d ~ i ~ p .

2.56

§23.

Intersection

Given

C ¢ ~(A),

For any

multiplicity let

I ¢ H(A)

or

with

f: A ~ A / C I c A

an e m b e d d e d

curve.

b e the c a n o n i c a l

and any

Q ¢ H(A)

epimorphism.

or

Q c A

we define: H ([A,C], I,Q) w

o r d ( [ A , C ] , I , V ) [V/M(V) : R ( [ A , ~ V~

([A,C],V) ]) ] ,

([A,C]) ,Q)

([A,C], I,\Q) =

([A, c],v) ]) ]

~ ord ([A, el, I,V) [V/M (V) : ~ ([A,~ Ve,~ ([A, C] ,\Q) ([A,C],I,Q)

=

~ ord([A,C],I,V)[V/M(V): V ¢ ~ ([A,C],Q)

H0(A)]

•

and o r d ( [ A , C ] , I , V ) [V/M(V) : H0(A) ]

([A.C], I . \ Q ) =

vc,q ([A,c],hQ) For any

I e H(A)

or

I c A

we define: w

u([A,C],I) For any

= u([A,C],I,0) Q ~ H(A)

~([A,C],Q)

=

U~([A,C],\Q)

U ~* ([A'C]'Q)

Adj ([A,C],Q)

Q c A

U

([A,C],I)

----

= U

,

~ ~.~(~([A,C],P)) Pe~]0 ([A,C],kQ)

.

f(k) (~ ([A,C], P) ~ ~"(Z P6~]0 ([A,C3,Q)

where

~ I~f(k) (~([A,C],P) PCZ]0 ([A,C],\Q) = [~ e H

(A): ~ ( [ A , C ] , # , P )

for all Adj ([A,C], \Q) = {~ e H

(A): 9 ( [ A , C ] , ~ , P )

k = H0(A)

, c adj(~([A,C],P))

p ¢ n 0([A,C~,\Q)]

121

where

k = H 0 (A) ,

C adj(9([A,C],P))

P ¢ ~0([A,C],Q)]

for all

([A,C],I,0)

we define:

~ I{(~([A,C],P)) P e n 0 ([A,C],Q) =

U * ([A.C~.\Q)=

or

and

,

2.57

Tradj([A,C3,Q)

--- [~ e H

(A): ~ ( [ A , C ] , ~ , P )

for a l l

¢ Tradj (~([A,C3,P))

P e n0([A,C~,Q)},

and

w

Tradj([A,C~,\Q)

= {~ ¢ H

(A): ~ ( [ A , C 3 , P )

for all By

an a d j o i n t

¢ tradj(~([A,C~,P)

P ¢ n 0([A,C3,\Q) ] ,

of

C

in

A

a__tt Q, we m e a n a m e m b e r of

B y an a d j o i n t o f

C

in

A

outside

Adj ([A,C3,\Q).

Q, we m e a n a m e m b e r o f

B y a _true a d j o i n t o f

m e m b e r of

Tradj ([A,C],Q).

Q, w e m e a n

a m e m b e r of

c

in

A

a_~t Q, w e m e a n a

B y a true a d j o i n t of

Tradj ([A,C],\Q).

d r o p " in A ", w h e n the r e f e r e n c e

Adj ([A,C~,Q).

to

A

C

in

A

outside

From these phrases is c l e a r

we may

from the context.

Finally we define and

Ad9 ([~,C~) = Adj ([A,C~,0) B y an a d j o i n t o f

C

a true adj0int of

in C

and

U~([A,C],0)

in

and

T r a d j ([A,C~) = T r a d j ([A,C],0).

A, we m e a n a m e m b e r of

Adj ([A,C~),

A, w e m e a n a m e m b e r o f

from these phrases we may drop clear

~([A,C])--

and b y

Tradj ([A,C~).

" in A ", w h e n t h e r e f e r e n c e

to

Again, A

is

from the c o n t e x t . In v i e w of

(5.1), (5.6), (5,8), (5.10), (5.11), (17.4), (18.1), (18.2),

(18.3), (18.4), (18.5), (23.1)

Let

and

(18.10), w e c l e a r l y g e t

k = H0(A).

any

P e ~0([A,CT),

I' =

(I N H ( A ) ) A

upon in c a s e

([A,cl,I,P)

Then

letting

for any I' = IA

(23.1)to

I ¢ H(A) in case

or

(23.7): I c A

I ¢ H(A)

and

and

I c A, w e have:

= ~ (~([A,c3,p),

~([A,C3,T,P))

= ~ (~(A,P),~(A,C,P) ~,~(~,T,p))

positive

integer,

if , if

U

([A,C~,I,P)

I' c P I' c C;

= u([A,c3,I,P)Deg[A'P~ = if(k) ( ~ ( [ A , C ~ , P ) , ~ ( [ A , C ~ , I , P ) )

= ~k ([~ (A, p) ,~ (A,C,p) 7,~ (A, I,P) )

122

and and

I' ~ C ,

2.58

0

I

a positive

For

any

P ¢ ~0([A,C]),

1 < card

we

~([A,C],P)

also

integer,

if

I' 9 ~ P

if

I' c p

if

I' c C

([A,C],P)

U~([A,C],P)

< u([A,C],P)

.

=

I(~([A,C],P))

=

~,([N(A,P),~(A,C,P)

= u([A,C],P)Deg[A,P]

=

=

xk([R(A,P),~(A,C,P)

],~(A,P,P))

=

X~(~([A,C],P))

integer

,

= a positive

= X~([~(A,P),~(A,C,P)

= ~([A,C],P)Deg[A,P]

],~(A,P,P))

I f(k) ( ~ ( [ A , C ] , P ) )

= a nonnegative U ~*( [ A , C ] , P )

I' ~ c ,

and

have:

= a positive U

,

f(k) },~

=

integer,

I)

integer,

(~ ( [ A , C ] , P ) )

k([~(A,P),~(A,C,P) = a nonnegative

])

integer.

and U~([A,C],P)

= 0

U~([A,C],P)

= 0

~ ([A,C],P)

is n o r m a l

~ ([A,C],P)

is r e g u l a r

u([A,C],I,P)

= ord~([A,C~,p)~([A,C],I,P) for

(23.2) Q c A,

upon

For

any

every

I ¢ H(A)

I c H(A) or

I c A

letting

IA + C

in c a s e

I ~ H(A)

if = (I n H ( A ) ) A

+ C,in

I cA

case

123

,

or

I cA

and

any

.

Q

c H(A)

or

2.59

f

QA + C

and

, in c a s e

Q e H(A)

,

Q' = 1 (Q N H ( A ) ) A

+ C,

Q cA

in c a s e

,

t we have :

u([A,C],I,Q)

= ~([A,C],I',

=

U

(radA(I'+Q'))

~

Pen0([A,C~,Q)

integer

-- U ( [ A , C ] , I ' ,

(radA(I'+Q')

=

~ u Pe~0 ([A,C],Q)

= a nonnegative

u([A,C],I,Q)

= ~ ~ U and

U ([A,C],I,Q)

N

(HI(A)A))

or

([A,C],I,P) integer

([A,C],I,Q) n([A,Cg,Q)

= 0 ~ ~

(HI(A)A))

u ([~,c],i,P)

= a nonnegative

([A,C],!,Q)

~

~

([A,C],I,Q)

or

~

,

= ~ ~ ~([A,C],I)

= ~([A,C])

~ ~ I' -- C

HI(A)

and

= 0 ~ ZI~,[A,C],I)

N

~ radAQ'

n([A,C~,Q)

HI(A ) c radA(I'+Q') *

[i ([A,C],I)

= U ([A,C],I,Q)

W

~ ~

([A,C],I)

= ~

~3([A,C],I) Q' N

U ([A,C],!,\Q)

c ~([A,C],Q)

(HI(A)A)

= U ([A,C],I',\(radAQ')

N

([A,C],I,Q)

c radAI'

(H I ( A ) A ) )

L~ ([A,C], I,P)

P~0 ([A,C],\Q) a nonnegative

U

([A,C], I,\Q)

= U

integer

or

([A, C T , I ' , \ ( r a d A Q ' ) N

~

,

(HI(A)A))

W

=

~

~

([A,c],i,p)

P~ n0 ([A,C], \Q) = a nonnegative

integer

124

or

~

,

,

,

=

e

2.60

([A,C~, I,\Q) = ~ ~ ~

([A,C~,\Q)

= ~ ~ ~([A.C~,I) ~I'

and

= n([A,C~)

=C~Q'

n ( [ A , C ~ , \Q) ~

,

and

U([A,C],I\Q) = 0 ~U

([A,C~,I,\Q)

= 0 ~ ~([A,C~,I) 9'

Moreover, c I'

for any

J ~ H(A)

with

N

J ~ I'

n n([A.C],\Q)

(HI(A)A)

or

J c A

=

c radAI' with

J n H(A)

w e have:

([A,C],J,Q)

U ([A,C~,J,Q)

~ ~ ([A,C~,I,Q) w ([A,C~,J,Q)

= ~I([A,C~,I,Q)

and . ~ U

([A,C~,I,Q)

~ U*([A,C~,J,Q)

= U

,

([A,C],I,Q)

p ([A.c3.J. p)~ ([A.c3.p)*

,

I = ~([a,c~,I,p)~([a,c],P~, for all u([A,C],J,\Q)

~ u([A,C3,I,\Q)

P e •0([A,C],Q)

and

([A,C~,J,XQ)

~ U

([A,C~,I.\Q)

,

and

u ([A,c],J,XQ) = u ([A,CT,I,\Q)

U

[

([A,C~,J,\Q)

= U

([A,C3,I,\Q)

~ ([A,C~, J, P)~ ([A,C~, P) * *

I = ~ ([A,C~, I, P)~ ([A, C~, P)

for all

where

~([A,C],P)

is the i n t e g r a l

P e ~0([A,C3,P)

closure

of

~([A.C~,P)

, in

([A,c~). For any

Q ¢ H(A)

or

Q c A

also h a v e

125

(in v i e w o f

(15.4)

and

(15.5) we

2.61

U~([A,C],Q)

~

W

([A,C],Q)

~([A,C],Q)

=

~ Q)~([A,C],P) Pcn0(~A,C~,

=

~ Pen 0 ( [A, C~, Q ) ~

W

= a nonnegative

integer

,

[A,C], P) = a n o n n e g a t i v e

integer

,

(

= 0 ~ ~([A,C~,Q)

= 0 ~ ~([A, C3,P)

is r e g u l a r

p ~ n0([A,C],Q) U ([A,C],~,Q)

~ U~([A,C],Q)

and

U

(~A,C],~,Q)

for all Adj ([A,C],Q)

Tradj([A,C],Q)

=

B

~ ~

(~A,C],Q)

~ c Adj([A,C],Q)

~ Adj ([A,C],p) Pen 0 ( [A, C], Q)

=

for all

,

,

Tradj(LA, C~,P) Pen0(~A,C~,Q)

= {~ e Adj([A,C],Q)

: ~([A,C],~,Q)}

= [~ c Adj([A, Cj,Q)

: ~

~(EA,

C],\Q)

=

A,~C],\Q)~([A,C], P6n 0 ( [

~([A,

C3,\Q)

=

~ ~([A,C],P) P~e O([A,c3,\Q)

~([A,C],\Q)

= 0 ~ U~([A,C],\Q)

([A,C~,~,Q)

> U~([A,C~,\Q)

integer

,

= a nonnegative

integer

,

= 0 ~ ~([A,C],P)

and

U

for all

([A,C],~,\Q)

is r e g u l a r

for all

, ~ ~([A,Cq~,\Q)

• c Adj (EA, C],\Q)

Adj([A,C~,\Q)

=

N Adj ([A,C],P) PeSO ( [A, C], \Q)

Tradj ([A,C],\Q)

=

N Tradj ([A,C],P) Pc• 0 ([A, C], \Q)

and

126

([A,C],Q)}

P) = a n o n n e g a t i v e

P ~ n0([A,C],\Q) ([A,C],~,\Q)

= ~

, ,

2.62

=

{~ ~ Adj([A,C3,\Q):

u([A,C],~,\Q)

=

[~ ¢ Adj([A,C3,\Q):

U

(23.3)

For

any

([A,C~,~,\Q)

I ¢ H(A)

IA + C

= ~([A,C~,\Q))

or

I c A,

, in case

(I n H ( A ) ) A

+ C,

= N~([A,C3,\Q)}

upon

letting

I ¢ H(A)

I cA

in case

•

,

,

we have:

u ([A,c~,i) = u ([A,C~,I')

ord([A,C],I,V)[V/M(V):

=

R([A,~

([A,C~,V)])]

V¢~([A,C3)

=

~

u ([A,c3,I,P)

p~n 0 ([A,C~, I,P) = a nonnegative

u

([A,c~,z)

= u

([A,c],z')

integer

or

o r d ([A, C ~, I,V) [ V / M (V) : H 0 (A) ~

= vc,q ([A,c))

w

=

~ u P e n 0 ([A,c3)

= a nonnegative

([A,c3, I,P) integer

or

~ ,

w

([A,C~,I)

u([A,C~,I) and

for any

=

~ ~ U

([A,C3,I)

= = ~ •([A,C3,I)

= ~([A,C3)

= 0 ~ ~

([A,C~,I)

= 0 ~ n([A,C~,I)

=

J ~ H(A),

with

J ~ I'

or

J c A

# ~ HI(A) with

~ u([A,C~,I)

and

U

([A,C~,J)

and

127

> U

([A,C~,I)

,

c radAI'

J N H(A)

we have u([A,C~,J)

= I' = C

c I'

2.63

~([A,C],J)

= ~([A,C],I)

~ Z

([A,C],J)

= Z

([A,C],I)

~ ([A,C], J, P) ~ ([A, C], P)

f where

~([A,C],P)

=

~ ([A,C], I, P) ~ ([A, C], P) * for all

P ¢ [30([A,C])

is the i n t e g r a l

closure

of

"

~([A,C],P)

in

([A,c]). Finally ~([A,C])

(in v i e w of

=

~

(15.4)

~

and

(15.5) we also have:

([A,C~,P)

= a nonnegative

integer

U~([A,C],P)

= a nonnegative

integer

P ~ 0 ([A,c]) ~ u~ ([A'c])

=

~

pe~0 ([A,c])

,

W

U~([A,C])

= 0 ~ U~([A,C])

= 0 ~ .~([A,C],P) p c n0([A,c])

is r e g u l a r ,

w

L~([A,C],~)

~ ~([A,C])

and

U

w

([A,C],~)

~ U~([A,C])

¢ Adj ([A,C]) Adj ([A,C]

=

N Pen 0

for all

Adj ([A,C],P)

for all

, ,

([A,C])

and Tradj ([A,C~

=

N Tradj ([A,C], P) P e n 0 ([A,C])

= {~ ¢ A d j ( [ A , c ~ ) :

U([A,C],~)

= {~ ¢ A d j ( [ A , C ] ) :

U*([A,C],~)

= U~([A,c])] w

(23.4). ([A,C],I,Q)

is a l g e b r a i c a l l y

If

H 0(A)

= ~

[A,C~,I,Q)

closed,

-- U ~ ( [ A , C ] ) ] then:

for any

I ¢ H(A)

or

I cA

and any

Q ¢ H(A)

or

Q cA

W

([A,C],I,\Q) ([A,C],I)

= ~ ([A,C],I,\Q) = ~([A,C],I)

for any

128

I ¢ H(A)

or

I cA

,

;

2.64

= ~([A,C],Q) Q

for a n y ~([A,C],\Q)

= ~([A,C],\Q)

or

H(A)

Q cA

,

f

,

and W

k~{([A,C]) = k ~ ( [ A , C ] ) (23.5).

We have

U ([A,C],xy,Q)

= U ([A,C],x,Q)

+ U ([A,C],y,Q)

for any

,

U{([A,C],xy,Q)

= [I*([A,C],x,Q)

+ U

([A,C],y,Q)

,

in

U ([A,C],xy,\Q)

= U ([A,C],x,\Q)

+ u ([A,C],y,\Q)

,

and any

W

U

W

([A,C],xy,\Q)=

U

x

and

Y

H (A)\C

W

([A,C],x,\Q)+

Q 6 H(A)

t] ([A,C],y,\Q)

or Q c A

and %

U ([A,Cq,xy)

= U ([A,C],x)

.

u

+ u ([a,C],y)

w

([A,C],xy)=

(23.6) Either

U

,

1 )

([A,C],y)

,]

w

([A,C~,x)+

Let

U

x

and

y

(

P ¢ ~0([A,C]),

assume that

for any

and let

in

H(A)\C

I e H(A)

~([A,C],I,P)~([A,C],P)

or

.

I c A.

is p r i n c i p a l

(note t h a t

w

this is c e r t a i n l y U ([A,C],P)

= i.

U ([A,C],I,P)

so if

I ~ H(A)

or

I z H

(A)); o r a s s u m e

Then,

= [~([A,C],P)/~([A,C],I,P)~([A,Cj~,P) =

that:

[~(A,P)/(~{(A,I,P)~{(A,P)

+ ~(A,C,P))

: ~([A,C],P)] : ~{(A,P)]

and ([A,C],I,P)

(23.7).

= [[~([A,C],P)/~([A,C],I,P)~([A,Cq,P) = [~(A,P)/(~(A,I,P)~(A,P)

+ ~(A,C,P))

If

U ([A,C],P)

Emdim[A,C]

= i, t h e n

P ~ n0([A,C].

129

: H0(A) ] : H0(A)] = 1

for all

2.65

Next we claim that: (23.8)~

LEMMA.

For any

x ¢ Hm(A)\C

and

y ¢ Hn(A)\C

w__ee

have n[z PROOF. n[~

([A,C],x) ]

=

m[~

([A,C],y) ] .

Namely

([A,C],x) ] - m[u

([A,C],y) ~_

=

~ Ve3([A,C])

[n(ord([A,C~,x,V))

=

~ [or~f(xn)/f(ym)~[V/M(V) Ve~([A,C])

= 0

by

(23.9)

- m(ord([A,C~,y,V))~[V/M(V)

(4.1),

since

DEFINITION-BEZOUT'S

there exists a unique positive

: H0(A) ~

: H0(A)]

by (18.10)

0 ~ f(xn)/f(y m) ~ R([A,C~)

LITTLE THEOREM. integer,

In v i e w of

t 9 be denoted by

(23.8),

Deg[A,C~,

such that U

([A,C~,~)

=

(23.10)

REMARK°

characterize Q ¢ H(A)\C

(Deg[A,C]) (DegAS)

Q c A

with

Q N H(A) ~ C

([A,C~,~,\Q)/degA~:

= max{u

([A,C~,~,\Q)/n:

Deg[A,C~

if

H0(A)

= max{~

way.

with

~ ~ C .

we can clearly Let any

be given.

Then

W

= max{~

is algebraically

([A,C~,~,\Q)/n: for all

~ c H

~ ¢ Hn(A)

for all large enough Moreover,

(A)

formula,

also in the following

*

Deg[A,C~

~ ~ H

In view of the above

Deg[A,C] or

for all

130

with

with

A ~ ~ ~ C]

~ ~ C}

n. closed,

~ c Hn(A)

n > 0 .

(A)

then

with

~) ~ C]

2.66

(23.11)

LEMMA.

,Assume t h a t

card{P

e ~30([A,C]):

(Note that this is a l g e b r a i c a l l y and,if

H0(A)

assumption

closed;

Deg[A,P]

= i] m 1 + D e g [ A , C ]

is a u t o m a t i c a l l y

namely, ~ 0 ( [ A , C ] )

is a l g e b r a i c a l l y

closed,

satisfied

is a l w a y s

then

.

if

H0(A)

an i n f i n i t e

Deg[A,P]

= 1

set,

for all

P c n0([A,c]).) Then

Emdim[A,C]

PROOF.

Let

d = Deg[A,C],

(i)

s =

By assumption D~ (A)

~ Deg[A,C]

there

[H I(A)

and

let

: H 0 (A) ]

exist pairwise

distinct members

L0,LI,...,L d

of

such t h a t

(2)

[L i : H 0 ( A ) ]

= s - 1 , for

0 < i < d ,

and (3)

LiA C ~ 0 ( [ A , C ] ) ,

Suppose,

if p o s s i b l e ,

~ H I(A) LoA,

such that

LIA ..... LdA

Z~0([A,~ ])

that

for

L 0 D LID...NL d ~ C ; then there exists

• ~ C

and

are p a i r w i s e

and hence,

0 ~ i ~ d .

~ c LiA

distinct members

in v i e w of

(23.1)

•

U

for

and

0 ~ i < d. of

Now

£0([A,C])

N

(23.3), w e see that w

([A,C],~)

> d + i.

However,

since

~ £ HI(A)

and

~ ~ C, b y

w

w e get

U

([A,C],~)) = d, w h i c h

L 0 D L I N . . . N L d c C, and h e n c e (4)

By

(5)

is a c o n t r a d i c t i o n . in v i e w of

(3) w e h a v e

L 0 n L I N . . . N L d = C N H I(A) (i),

(2) and

(4) w e see t h a t

[C N HI(A)

: H0(A ) ] ~ s - d - i

131

Therefore

(23.9)

2.67

By

(i) a n d

(5) w e g e t

(23.12)

Emdim[A,C]

LEMMA.

PROOF.

First

Deg[A,C3

suppose

g d = Deg[A,C]

.

= 1 ~ Emdim[A,C]

that

Emdim[A,C]

= 1 .

= I.

Then there exist w

P ¢ D0([A,C]) c p

and

(23.3),

with

~ ~ C.

(23.1)

Deg[A,P] Clearly

and

= i, a n d t h e r e e x i s t s ~ + C = P

and h e n c e ,

~ ¢ HI(A) in v i e w o f

with (23.9),

(23.6) w e get

w

Deg[A,C]

= U

([A,C],~)--~

Conversely

suppose

([A,C~,~,P)=

that

[~(A,P)/~(A,~C,P)

: ~(A,P)]

=

[~(A,P)/M(~(A,P)):

~(A,P)]

=

1

Deg[A,C]

•

= i.

Let

e = 1 + Emdim[A,C]. w

Then

e ~ 2

that

@i i C

and clearly for

1 ~ i ~ e

(i)

~(A,C)

Since for

there exist

Deg[A,C~

n0([A,C],@i)

=

in

HI(A)

such

and

+ ~i + ~2 + ' ' ' + %

= I, in v i e w o f

1 < i ~ e, w e h a v e

~ I , ~ 2 ..... %

(23.1),

= A(A)

(23.3)

and

c a r d ~ 0 ( [ A , C ] , ~ i) = 1

{Pi], w e h a v e

Deg[A,Pi]

= I.

. (23.9), w e see that,

and, By

upon

letting

(i) w e g e t

n 0([A,c3,~ 1)n...n~ 0([A,c3,~ e) = and hence

card[P

e n0([A,C]):

Therefore by (23.13) Deg[A,C]

Deg[A,P]

= i] ~ c a r d { P i , P 2 ..... Pe]

(23.11) w e c o n c l u d e REM~ARKo

= i, t h e n

Emdim[A,C~

As a consequence

u([A,C],P)

As a r e f o r m u l a t i o n

that

of

= 1

of

for all

= 1 .

(23.12), w e

nl(A):

132

see that,

P ¢ D0([A,C~)

(23.12) w e have:

£)I(A) = [ E ¢

z 2 .

Deg[A,E]

= i]

;

if

2.68

this motivates

As a consequence

if

~i1 (A)

the n o t a t i o n of

(23.12) w e a l s o

E m d i m A = D i m A, t h e n

The a b o v e o b s e r v a t i o n s

(23.14) be given

PROJECTION

such t h a t

~I(A)

= {E ¢ nl(A):

may henceforth

FORMULA.

J c N

see that:

and

(Note that,

if

H0(A)

C N ¢ hi(AN)

~ Emdim[A,C,N]

= i]

be used tacitly.

Let a n y

J c C.

Deg[A,E]

N ¢ ~

Assume

is a l g e b r a i c a l l y - Emdim[A,N]

(A)

that

and

J ¢ ~

(A)

cN ¢ ~i (AN) "

closed,

then:

~ 2.)

T h e n we h a v e w

.

u ([A,c],J) - ~ ([A,C],N) = [~([A,c]): ~([AN, cN])] ~*([A~,cN~,J N) PROOF°

R (f (B))

Let

B = A N , D = C N, and

b e the c a n o n i c a l

K = jN.

isomorphism.

Let

For every

h: R([B,D])

-

W ¢ ~([B,D])

let G(W) = {V ¢ .~([A,C~): V n R(f(B))

For e v e r y

W c ~([B,D])

h: R([B,D])

~ R(f(B))

V ~ V/M(V),

V/M(V)

g(V) =

and

V ¢ G(W),

and the c a n o n i c a l

becomes

[V/M(V):

a

= h(W)]

v i a the c a n o n i c a l epimorphisms

(W/M(W))-vector-space

W/M(W)]

and

p(W)

=

isomorphism

W ~ W/M(W) and,

[W/M(W):

upon

(1)

[V/M(V):

H0(A) ] = g(V)p(W)

Now

([A,c],J) =

~

- ~

([A,c],~)

[ord([A,C],J,V)

- ord([A,C],N,V)][V/M(V):

VC~([A,C])

133

letting

H0(B) ] ,

we have

H0(A) ]

and

2.69

=

~ We ~([B,D])

=

~ [ord([A,C],J,V) V~G(W)

~

(i)

(18o 1 2 )

V e G (W)

=

[R([A,C]): R([B,D])]

=

[R([A,C]) : R([B,D]) ] ~ ([B,D],K)

=

[~ ([A,C])

~ [ord([B,D],K,W)]p(W) w~|[B,O])

by

(4.3)

~([Am,cN]) ]~* tt~AN, cN~,~). J

:

(23.15) b e given.

by

[ordvM(h(W)) ] [ o r d ( [ B , D ] , K , W ] g ( V ) p ( W )

~

W e ~ ([B,D])

by

- ord([A,C],N,V)]g(v)p(W)

SPECIAL

Assume

(Note that,

PROJECTION

that if

FORMULA.

{0] J N ~ ~

Let any

(A)

C N e ~I(AN).

H0(A)

is a l g e b r a i c a l l y

C N ~ ~ I ( A N) ~ E m d i m [ A , C , N ]

closed,

- Emdim[A,N]

then:

~ 2.)

T h e n we h a v e (23.15.1)

Deg[A,C]

- ~/ ([A,C],N) = [R ([A,C])

: R ([AN, cN]) ]Deg[AN, c N]

and Deg[A,C]

- ~

([A,C],N)

. ~ ([~,CN]) ]

= [~([A,C])

Deg[AN, c N] = 1 (23.15.2)

PROOF. taking

= 1

Emdim[A,C,N]

- Emdim[A,N]

In v i e w of

J -- xA

(23.15.1),

Emdim[AN,cN]

with

(23.12)

and

(23.16)

REMARK.

*

~*

(23.9),

x c

(23.15.1)

(N N H I(A))\C.

= 2 . follows Now

from

(23.15.2)

= {N ~

follows

by from

(14.3). Assume

that

H 0(A)

is a l g e b r a i c a l l y

Let

N

(23.14)

(A): E m d i m [ A , C , N ]

- Emdim[A,N]

134

= 2]

closed.

2~70

i.e., N

= [0 / N ~ ~

~'

=

(A) : E m d i m [ A N , cN] = i}

.

Let

Then clearly Deg[A,C~

{N

N' ~ ~

~

N

n0([A,c~)

:

and b y

n n0([A,Nl)

(23.15) w e

see t h a t

= m a x { [ R([A,C~) : R ( [ A N , c N ~ ) ~ : = [ R([A,C])

: R([AN, cN])]

= ~}

N e N*]

for all

N ¢ N'

W

(23.17)

LEMMA.

(Note that,

if

Let

[0} ~ N eg~ I(A).

H0(A)

Assume

is a l g e b r a i c a l l y

that

closed,

C N e ~ I ( A N) ~ E m d i m [ A , C , N ~

C N e nl(A).

then:

a 3.)

Let = {P e ~ 0 ( A ) :

N c P

and

[R([A, C3) : R ( [ A P , c P ~ ) ~ ~ i}

and the set o f a l l s u b f i e l d s o f

R(f(AN))

T h e n we h a v e

and w h i c h

R([A,C])

are d i f f e r e n t

which

from

contain

R([A,C])

the f o l l o w i n q :

(23.17.1)

card ~ ~ c a r d ~'

o

(23.17.2)

If

([A,C~,N)

(Although we

Deg[A,C]

- ~

s h a l l not u s e this r e m a r k

~ 3, ~he_D

card ~ ~ 1 .

in t h i s b o o k ,

we note that

w

by a well-known overfield

fact

of a field

from algebra K',

then:

(namely, K

if

K

is an a l g e b r a i c

has a primitive

element over W

K' ~

there

are o n l y a f i n i t e n u m b e r of s u b f i e l d s

K') we k n o w t h a t over

~(f(AN)).

~'

is f i n i t e ~

Hence by

(23.17.1) 135

R([A,C~)

of

K

containing

has a p r i m i t i v e

w e see t h a t

if

R([A,C3)

element has

a

2.71

primitive cular,

element

if

over

H0(A)

R([A,C~)

has

R(f(AN))

(i')

~

is finite.

zero c h a r a c t e r i s t i c ,

is s e p a r a b l e

PROOF°

then

over

R(f(AN)),

or, more

then

~

So,

in p a r t i -

generally

if

is finite.)

Clearly

R(f(AN))

c R(f(AP))

c R([A,C])

for all

W

P e ~0(A) and h e n c e

to prove

(23.17.1)

with

N c p

it s u f f i c e s

to show

that

w

(*)

card{P

In v i e w of

(i'),

e ~0(A):

(*)

R(f(AP))

c K} ~ 1

is e q u i v a l e n t

for e v e r y

K e Q'

to:

R(f(AN)) ( R ( f ( A P ) ) , R ( f ( A Q ) ) )

= R([A,C~)

,

(**) for e v e r y

P ~ Q

in

~(A)

with

N c P

and

N c Q .

w

TO prove given.

(**)

let any

P ~ Q

in

~0(A)

Then w e can take e l e m e n t s N + X A = p,

with

X,Y,Z

N + Y A = Q,

in and

N c P HI(A) Z £ N\C

and

N c Q

such that .

Now clearly R(f(AP))

= R(f(AN)) (f(X)/f(Z)) R(f(AQ))

and

= R(f(AN))(f(Y)/f(Z))

;

also c l e a r l y (N n H I(A))

+ X H 0(A)

+ Y H 0(A)

= H I(A)

and h e n c e R([A,C3)

= R(f(AN))(f(X)/f(Z),

f(Y)/f(Z))

consequently R(f(AN)) (R(f(AP)),

R(f(AQ))) 136

=

R ( [ A , C ~) ~

;

;

be

2.72

and this proves By

(**).

(23.15)

(R([A,C]):

w e have, R(f(AN))~

~ Deg[A,C~

- ~*

([A, C3,N)

and h e n c e Deg[A,C3

- ~*([A,C3,N)

[R(LA, C~q): R ( f ( A N ) ) 3

~ 3 = 1 or 2 or 3

= card ~' < 1 ; therefore

(23.17.2)

(23.18) (*)

follows

DEFINITION°

[V/M(V):

([A,C~,P)

(23.17.1).

Let

H0(A) 3 = 1

(Note that c o n d i t i o n U

from

= l, or: H 0 ( A )

P ~ Do([A, C3). for all

that

V c ~([A,C3,P)

(*) is a u t o m a t i c a l l y is a l g e b r a i c a l l y

*

Assume

satisfied

closed.

if, either:

Also

note

that:

W

(*) = P ¢ ~o(A),

Deg[A, P3 = l, and

U

([A,C3,P)

= ~([A,C],P).)

We define: W

*

*

TI([A, c I,P) = {L 6 ~l(A) : L c P We note

that,

and

~

*

(EA,C],L,P)

g ~

([A,C3,P)}.

then clearly, W

TI(EA,C~,P)

= {L ¢ ~ l ( A ) :

*

([A,C],L,P)

> ~

([A,C],P)]

(23.18.1) W

for some

= {L ¢ ~I(A) : L = TI([A,C~,V) v c ~([A, C3,P)} and h e n c e (23.18.2)

1 < card TI([A, C3,P)

and so in p a r t i c u l a r

137

< c a r d 0([A, C3,P)




e

be a homogeneous

P ¢ ~0(R,~),

s(P)

P

Assume

of

.

m

and

2.80

then there

exists ¢ Tradj (R,\~)

such that

~

is i r r e d u c i b l e

® --- ~i~2 PROOF.

with

®i

~ Hm(R) in the

and

sense

~2

that:

i__nn H

(R) = ~i = R

o__rr ~2 --- R.

Let k = H 0 (R)

In v i e w of (i)

(*), b y

(i0.i.i0)

[~(R,P)/~(~(R,P))

and b y

(10.1.13)

I

: k3 =

we h a v e

w e H

we h a v e

(I/2)u~(R,P)

for all

P ¢ n0(R)

w < u~(R,P)

for all

P ¢ n0(R)

that, w

(R)

with

U

that

(R,~,P)

(2) w

= U

(R,~)

-= 0(2)

Let (3)

~2 = [P e •0(R,rr): k~(R,P) ~ 2}

Then b y by

(*) we h a v e

= 0

for all

P e £0(R,I~)\Q,

and h e n c e

(**) we get

(4)

s(P) = 0 For each

ideal (5)

#~(R,P)

.

J(P)

P e. Q, in

in v i e w of

~(R,P)

[~(R,P)/J(P)

for all

with

P e ~0(R,~)\Q

(**), b y

~(~(R,P))

: k~ = [ ~ ( R , P ) / ~ ( ~ ( R , P ) )

(10.1.12) c J(P)

there

exists

an

such that

: k~ - s ( P ) D e g [ R , P ~

and (6)

xk(9~(R,P),J(P))

For each

P e D0(R)

= k~k(~ (R,P))

- 2s(P)Deg[R,P~

we get an ideal

t45

I(P)

in

~(R,P)

by setting

2.81

(7)

J(P)

if

P e

~(~(R,P))

if

P ~ ~ o

I(P)

By

(i),

(3),

(4),

(8)

(5

and

(7)we

then have

: k] =

[~(R,P)/I(P) Pen 0 (R)

( 1 / 2 ) U ~ (R)

s(P)Deg[R,P]

-

.

Pcn0(R,~) Upon

(9)

letting

E = {x ¢ Hm(R):

by

(15.10)

(io) By

we see that

[E:k] +

(***) ,

9~(R,x,P) E

(8) and

for all

P ¢ f]0(R)]

is a k - v e c t o r - s u b s p a c e

~ [9(R,P)/I(P) p c ~ 0 (R)

: k]

of

z [Hm(R)

Hm(R)

: k]

with

.

(I0) we get

(ii)

[E:k] Upon

¢ I(P)

> 0

letting *

(12) by

A = [@ e Hm(R):

(9) we o b v i o u s l y U

(R,~,P)

and h e n c e

in v i e w of

(i3)

x ¢ E]

(3),

@ ¢ A

(4),

for all (6) and

e ~ A

(7) we

and

P ¢ ~]0(R)

see that

we have:

~ ~(R,P)

- 2s(P)Deg[R,P]

for all

P ¢ 90(R,~)

for all

P ~ ~]0(R'~)

and u

By

(R,G,P)

for some

get

m xk(~(R,P),I(p))

for e v e r y U

@ = xR

(R,~,P)

>_ ki{(m,P)

(13) we get

146

2.82

*

(14)

w

U

(R,®)

~ u~(R)

-

2s(P)Deg[R,p]

for all

® c A .

P¢~0 (R, ~) By

(3),

(7),

(9) and

(15) By

(12) w e

A c Adj (R,\~)

(9),

(Ii) and

A ~ ~ •

In v i e w of

(12), b y B e z o u t ' s

(17

u

(****),

(13),

U

( 18

(R,e,P)

Little

(R,®) = m n

(14) and

for e v e r y

(17)

® g A

Theorem

for all

(24.9) w e a l s o h a v e

~ g A .

it f o l l o w s

that

w e have:

= ~(R,P)

- 2s(P)Deg[R,p]

for all

P g ~0(R, TT)

for all

P c ~0(R,\n).

and U

By

A Hm(R)

(12) w e h a v e

(16)

By

also get

(R,®,P)

(15) and

(18) w e see t h a t

(19)

A c Tradj (R,\n)

NOW 2.

= u~(R,P)

let

In v i e w of

® ¢ A

such that

(2) and

= ®i~2

~ = ~i®2

with

®i ~ H

(R) for i = i,

(18) w e see that, U

and hence by Bezout's

N Hm(R)

(R,® i) =- 0(2)

Little

with

Theorem

® g A

and

,

(24.9) w e g e t t h a t

®i e H

(R)

for

i = 1,2

(20) = n ( D e g R ® i) =- 0(2) Now,

(21)

if

m ~ 2

and

D e g R ® i = 0(2),

and

for

i = 1,2

n # 0(2),

.

then by

(20)

DegR~ 1 + DegR® 2 ~ 2

147

it f o l l o w s

that

2.83

It

follows

®2 = R.

This

that

together

§25. Assume

we

with

Emdim

and

cones

A = Dim

DEFINITION.

for

(16)

Tangent

that

(25.1

DegR@ i = 0

For

some (19)

and

and hence

finishes

the

@I = R

or

proof.

quasihyperplanes.

A = r

any

i

.

• ~ H

(A)

and

any

P ~ ~[A,~])

define

([A,~],P)

We

note

that

=

k' ([~ ( A , P ) , ~ (A,~,P) ] , M ( ~ (A,P)))

then:

~' ( [ A , ~ ] , P )

= a nonnegative

U' ( [ A , ~ ] , P )

-- 0 ~ ~ ~ P

U' ( [ A , ~ , P ] )

and

so,

if

• ¢ e(A),

=

integer

,

,

1 ~ ~(A,P)/~(A,~,P)

is a r e g u l a r

local

domain

,

in p a r t i c u l a r ~

then:

U' ( [ A , ~ ] , P )

=

1 ~ • c P

Let

4} C H

and

~([A,~],P)

is

regular.

(25.2)

LEMMA-DEFINITION.

given.

We

can

take

P + XrA

= }{I (A)A.

Now

d

be

the

we

unique

integer

~i = 0

We the

claim

choice

of

Namely,

that ~ let

can

n n-i ~ ~iXr i---0

= Let

~ c H (A)

for

ord([A,C],P0,V )

upon letting

L = TI([A,C],P0,V) in v i e w of

now obviously

L

and

(or say in v i e w of

U ([A,C],L) and h e n c e

,

(21.1), w e h a v e

L ¢ nl(A,~)

P0 e ~ 0 (A'L)

> U ([A,C],P 0) = d = b - 1 ,

is a b - c h o r d .

LEMMA.

If

PROOF.

Clearly

(or say b y B e z o u t ' s

P0 e ~ 0 (A'TT)

follows by

;

(23.18))

(26.2)

exists

P ¢ ~o([A,C],~)\{Po}

in v i e w of B e z o u t ' s

U ( [ A , C ] , n , P O) = U ([A,C],~) Consequently

some

n ~ 2, t h e n t h e r e

with

exists

Little

a 2-chord

Theorem

L.

(23.9))

there

U ([A,C],P 0) >_ i, and h e n c e o u r a s s e r t i o n

(26. i) .

(26.3)

Assume

(*)

that

n ~ 4

~

and

[u ([A,C],P)

- 1] ~ 2

p~n 0 ([A,C],~) Then

there exists

PROOF. our assertion

If

a 4-chord

L.

U ([A,C],P 0) > 3

follows

from

for some

(26.1).

157

If

P0 e n 0 ( [ A , C ] , v ) ,

U ([A,C],P)

< 3

then

for all

2.93

P c £o(EA, C],~) bers

Pl

then,

and

P2

in v i e w

of

now

it c l e a r l y

(26.4).

(*) Then

that

u([A,C],~,P)

there

exists

PROOF. 3-secant,

By

then

Then,

Let

PI,P2,P3

(*),

by Bezout's

~o(A,L')

be

= 2.

So n o w

If

also

must have dicated

there

Little

to take

Let

Theorem

N2

and Let

L

is not

assume

that

N 3 e ~o(A,L*).

b y the

exists

the d i s t i n c t

following

N3 L

that

is not a 3 - s e c a n t .

a 2-chord L = L'

members

of

(23.9)

we

be

get

i = 1,2

158

L'

get

figure

that

a

L'

In v i e w

of

card ~0([A,C],n)\ members

Then

Then

of

clearly

it s u f f i c e s

let

is n o t

c a r d ~ o ( [ A , C ] , L ') = 3.

the d i s t i n c t

then

If

~o([A,C~,L').

is a 3 - s e c a n t .

suggestive

L'

So n o w a s s u m e

= A(A, PI,N2).

For

mem-

P e ~o(A,~)

(*), w e

a 3-secant L

two d i s t i n c t

and

which

of

find

L = &(A, PI,P2).

for all

L

in v i e w

~o(EA,C],~)\£0(A,L'). 2-chord.

~ 1

it s u f f i c e s

is a 3 - s e c a n t .

n = 5

can

i = 1,2,

to take

a 2-chord

(26.2)

such

a 2 , for

suffices

Assume

(*), w e

~o([A,C],~)

U ([A,C],Pi)

and

of

L

is a

to take

in v i e w

Li ~ £ iI(A,~)

of

L = L

.

(*) we be

as in-

2.94

| l

L2 %% L*I ~

N

.....

n I

P1

fl ~'/

$

% % %

i.e.,

let

2-secant

L i = A(A,Ni,Pi). for

i = 1,2.

REMARK.

Note

Namely,

Q0(A,~)

and we are

looking

So

Then

in v i e w

it s u f f i c e s

that this

lemma

of

to

(*) w e

take

is j u s t

see t h a t

L = L1

or

an e l e m e n t a r y

Li

is

L = L2 .

combinatorial

fact:

n,

contain

exactly

distinct,

lines

the

five points

and

since

the ten

ten

to choose then

(n-l)-chord.

There

from.

it a p p e a r

is n o t d i v i s i b l e does

LEMMA. Then

not

Assume L

five d i s t i n c t

for a l i n e w h i c h

three points.

lines which

(26.5)

is a set o f

is a n

joins

are

2

If a l i n e exactly

contain

that

contains

Emdim[A,C7

(n-l)-secant,

159

times

there must

exactly

in t h e p l a n e

two o f t h e m a n d d o e s = i0,

three

by three,

points

three

not

not

necessarily

exactly in t h e exist

three of

ten

lines;

a line

in

points.

= 3, and

let

L

be

any

2.95

card n 0 ( [ A , L ] , C ) and every

N ¢ ~0([A,L],\C)

PROOF. that

is a best projecting

The assertions

L ~ ~iI(A,~)

and

(i)

about card

center.

follow from our assumption

C ~ ~I(A,\~).

Now let any

N ¢ n0([A,L],\C)

be given.

Then clearly

C N ~ ~I(AN),

that the p r o j e c t i o n of Emdim[A,C]

Since

N

for any ~

~ n

(18.13.1)

is w-integral.

U ([A,C],~)

we also know

Since

(23.9) we get

e HI(A)

we see that,

for any

J e ZI(A)

A(A,L,J)

e HI(A)

U ([A,C],L) because

from

Little Theorem

N ~ ~0(A,C),

(3)

C

and by

= 3, by Bezout's

(2)

with

N e ~0(A,J)

and

J ~ L, we have

and

+ U ([A,C],J) ~ u([A,C],a(A,L,J))

clearly

L c ~](A,A(A,L,J)),J By

< ~ = card Z 0 ( [ A , L ] , \ C )

(2) and

e ~(A,h(A,L,J)),

and

n0(A,L)

N n0(A,J)

= IN].

(3) we get that

I for every

J ¢ ~iI(A)

with

N ¢ 90(A,J)

and

J ~ L, we have

(4) U ([A,C],L)

+ U ([A,C],J)

~ n .

By assumption (5)

U ([A,C],L)

Clearly there exists J' = ~(A,P,N) U ([A,C],J')

we get

~ i.

~ n - i .

P e ~0([A,C],\L) J' ~ ~31 I(A)

C o n s e q u e n t l y by

160

and then upon letting

with (4) and

N ~ ZI0(A,J'),

J' ~ L, and

(5) we see that

2.96

(6)

L1 ( [ A , C 3 , L )

i.e.,

L

is an

(n-l)-secant.

jection Formulas C

from

N

(23.14)

= n - 1

In v i e w of

and

(4) a n d

(23.15) w e d e d u c e

(6), b y the P r o -

t h a t the p r o j e c t i o n

is b i r a t i o n a l , Deg[AN, cN~ = D e g [ A , C ~

--- n ,

and U ([AN,cN~,Q)

Also

= 1

for all

Q e [30([AN,cN~,\LN)

clearly n 0 ( [ A N , c N ~ , \ N) c n 0 ( [ A N , C N ~ ,\LN)

Therefore

N

(26.6)

is a b e s t p r o j e c t i n g

LEMMA.

Assume

that

.

center.

n = 4.

Let

L

be a 2 - s e c a n t

and

le____t

(*)

N 6 ~0([A,L~,\C)

be such t h a t (**)

the project.ion of

Then

N

is a b e t t e r

PROOF.

Now

Deg[A,C3

In v i e w of

(i),

from

projectinq

N

of

= 4, and (*) and

C

from

N

L ¢ £iI(A,~)

and h e n c e b y

Deg[AN,cN3

= 4

and

161

(18.13.1)

is rT-integral. with

we k n o w

By a s s u m p t i o n

U ([A,C~ L) -- 2.

(**), b y the P r o j e c t i o n

see t h a t (2)

is b i r a t i o n a l .

center.

N e £0([A,~3,\C)

that the p r o j e c t i o n

(i)

C

Formula

(23.14),

we

of

2.97

U ([AN, cN], LN) = 2

(3)

Given

any

and

L N e ~ 0 ( [ A N , c N ] , ~ N)

Q e 9 0 ( A N , \ L N),

upon

letting

D = &(A N ,L N,Q)

we c l e a r l y

have

(4)

D ~ H I ( A N)

and (5)

U ([AN,cN],Q)

In v i e w of

(2) and

(4), b y B e z o u t ' s

(6)

Little

Theorem

(23.9)

(3),

(5) and

(7)

(6), we get U ([AN,cN],Q)

Therefore (26.7)

N

is a b e t t e r

LEMMAo

N e ~30(A,L)

Assume

be

~ 2 .

projecting

that

n = 5.

center. Let

L

be a 3-secant,

such that

(*)

U ([A,C],N)

(**)

U ([A,C],~,N)

> 1 > 1

and (***) Then

the p r o j e c t i o n N

is a qood

PROOF. (i)

we get

U ([AN, cN], D) = 4 ;

now b y

let

+ U ([AN,cN~, LN)~ U ([AN, cN], D)

By

of

C

from

projectinq

N

center.

(***) we h a v e EmdimEA,C,N]

~ 2 .

Since

162

is b i r a t i o n a l .

and

2.98

(2)

N ~ •0(A,L)

and (3)

L e ~i1 (A,Tt) ,

we get

(4)

N ~ f]0(A,n)

By(*),

(**)

and

(4) we get that

(5)

u ([A,c],~)

=

i

and (6)

card

G([A,C],N)

= i

and

rf c T I ([A,C],V)

where In v i e w of C

from

N

(i),

(4) and

(6), by

is ~-integral.

(7)

(23.14)

and

(***), (23.15)

(8)

(2),

(3),

(21.6.2)

~([A,C],N) we

see that the p r o j e c t i o n

Now by a s s u m p t i o n

U ([A,C],L)

In v i e w of

IV] =

= 3 (4) and

(7), by the P r o j e c t i o n

we see that Deg[AN, c N] = 4

and (9)

N ([AN,cN],L N) = 2 Given

any

and

L N ~ I]0([AN, cN],TT N)

Q e fl0 (AN,\L N) , upon D = A ( A N,L N,Q)

we c l e a r l y (i0)

have n £ H I (AN)

and

163

letting

Formulas

of

2,99

(ii)

~ ([AN,cN],Q)

In v i e w o f

(7) and

(12)

+ U ([AN, cN], LN)

(9), b y B e z o u t ' s

~ ~ ([AN,cN], D)

Little

Theorem

(23.9) w e g e t

U ([AN,cN], D) - 4 ;

now, b y

(8),

(I0) and

(13)

(ii), w e get,

U ([AN,cN],Q)

Therefore

(26.8)

N

is a g o o d p r o j e c t i n g

C O N E LEMMA.

(1)

~ 2 .

Let

Emdim[A,C]

L

center.

b e a 2-chord.

Assume

that

n ~ 5,

= 3 ,

and

(*)

~ ([A,C],P)

Also assume (**)

4 = 2 times

that

The 2

C

from

only composite

and

~ = cN'A

we clearly have

(3)

is not b i r a t i o n a l .

positive

integer

factorization

letting

(2)

such t h a t

Projection

-< 5

of 4. Formula

and

D e g [ A N' ,C N' ~ = 2 Upon

N'

.

i S a b e s t project_iin__g c e n t e r .

(*), b y the S p e c i a l

e ~ ( A N')

P ~ ~([A,C])

~ ~(A,L)

is the o n l y p r o p e r

(i) a n d

C N'

of

N'

N e %([A,L],\C)\[N']

PROOF.

view of

for all

that there exists

the projection

Then every

= 1

~ ¢ n(A)

,

164

u ([A,C],N')

~ 1 .

is 4, and

Therefore, (23.15), w e

in see

2.100

(4)

~ ¢ H

(5)

C ¢ nl(A,#)

and,

in v i e w o f

(i), b y

(25.9)

(6)

¢

NOW by

and h e n c e

~: n 0 ( A , L )

by

the

and

Projection

T,~'

(7)

(2) and

,

also

see t h a t

H 2 (A)

U ([A,C~,L)

Formula

a 2 > 1 a U ([A,C3,N')

(23.14)

c ~0 ( A N ' ' c

N l

we

see t h a t

)

(7) w e g e t

(8) By

we

,

assumption

~,

By

(A,N')

L ¢ ~(A,~)

(3) and

(6) w e

see that

(9)

Q(A,~)

Now

= ~ .

let any

(i0)

b e given.

N H I(A)

N ~ n0([A,L~,\c)\[N'] Let us k e e p

in m i n d

the

165

following

suggestive

figure:

2. i01

S,T

: Points of

C

necessarily

on

L, not

distinct

may contain

N'

and

S f

(') and

projection

C

(4) and

of

/

.

N

NOW in view of

T

(18.3.1)

from

N

we know that

is n-integral.

J

C N e ~ ( A N)

Since

and the

N' ¢ ~ ( A , L ) ,

by

(I0) we see that for any

J 6 ~(A)

with

N ~ ~(A,J)

and

J @ L, we have

(ii) w

By

A(A,N',J)

e HI(A)

A(A,N',J)

E n(A,@)

(9) and

and if

J ¢ ~(A,~),

then

(ii) we get that

for any

J e ~(A)

with

N e ~(A,J)

and J / L, we have

(12) I J ~ In view of

[~(A,~) (6) and

(12), by Bezout's

166

Little Theorem

(23.9) we conclude

2,102

that I for any (13)

J ¢ £]II(A) with

and

J ~ L, we have

/ I U ([A,J],~)

By

N 6 ~]0(A,J)

(8) and

~ 2

I

(i0) we know that

N ¢ Z]0(A,#)

and hence by

(13) we get

that I for any

J e 2 I(A)

with

N e D 0 (A, J)

and

J ~ L, we have

and hence by

(5) and

N e ~]0(A,J)

and

(14) ([A,J],~,\~)

By

(i0) we know that

~ 1 .

N ~ ~0(A,C),

(14) we get

that for any

J e 211 (A)

with

J ~ L, we have

(15) u ([A,J],C)

(because

~ 1

U ([A, J],C) = ~ ([A,J],C,\N)

by

(10)

u ([A,J],~,\N)

by

(5)

1

by

(14).)

In view of (*), by the Commuting Lemma ([A,J],C) = U ([A,C],J) and hence by I

for any

(23.19) we know that 1 (A) J c ~31

(15) we get that

for any

J e ~(A)

with

N e [30 (A,J)

and

J ~ L, we have

(16) ([A,c],J)

~ 1

Now in view of (16), by the Projection Formula that the projection of

C

from

N

167

(23.14) we conclude

is birational,

2.103

D e g [ A N , C N] = D e g [ A , C ]

= n

and (17)

u([AN,cN],Q)

Since

L e ~(A,~),

-- 1

by

U ([AN,cN],Q)

Therefore

N

Z ~ ( A , J I)

that, and

Further

O e 90 ([AN,cN],\~ N)

Assume

that

are d i s t i n c t

fo___~r A ( A , L , J I , J 2) = 9, say, we h a v e

that,

~(A,L)

then,

there

Z 0 ( A , J 2) assume

for all

L, Ji,J2

such that,

Note

= 1

LEMMA.

(A)

assume

(17) we get

center.

PLANE

Further

Q e ~ 0 ( [ A N , c N ] , \ L N)

is a b e s t p r o j e c t i n g

(26.9) o__~f ~

for all

N ~ ( A , J I) O ~ ( A , J 2 ) is a u n i q u e

; and then

member,

members

~ e HI(A).

= ~.

say

N, c.ommon to

N~(A,L).

that

(*)

U ([A,C],N)

{ 1

an___dd 2

(**)

t +

t = ~ U

~ U ([A,C],J i) ~ n + 2, w h e r e i--i

([A,C],L,P),

the s u m m a t i o n

P e ~(A,C)\(~0(A,JI)

U n0(A,J2)).

beinq

extended

over

all

T h e n we h a v e C ¢ ~(A,%) PROOF.

In v i e w of

(*), by

U ([A,C],Ji,N) and h e n c e

(i)

upon r e l a b e l l i n g

Jl

and h e n c e

Emdim[A,C]

(23.18.3)

we see that

for

~ 1 and

U ([A,C~,JI,N)

168

J2

i = 1 or 2

suitably

~ i.

< 2 .

we may

suppose

that

2. 104

By the d e f i n i t i o n

(2) By

of

~([A,C~,JI,N) (i) and

N

we clearly have

+ u([A,C],JI,\J2 ) = u(~A,C],JI)

(2) w e g e t

(3)

1 + u ( ~ A , C ] , J I , \ J 2) ~ u ( ~ A , C ] , J I)

By the d e f i n i t i o n

(4)

of

~

L ~ ~(A,@)

we have

and

Ji ¢ ~(A,~)

for

i = 1,2

.

Now clearly

U (~A,C],~)

~ t + ~ ( [ A , C ] , J I , \ J 2) + ~ ( ~ A , C ] , J 2) n + 1

and h e n c e b y B e z o u t ' s

(26.10) n

=

QUADRIC

by by

Little

Theorem

LEMMA.

Let

(23.9) w e g e t

L

(4)

(**) and

(3)

C ¢ ~I(A,~).

be a 2 - s e c a n t .

Assume

that

5p

(*)

~(~A,C],P)

= 1

for all

P e ~0([A,C],\L)

and t h e r e does

not e x i s t any

~ c H 2 (A)

such that

C c ~(A,~)

.

(**) is a n - q u a s i p l a n e

and

Let = {N ~ ~0(LA, L],\C) :

N

is a b e t t e r

projection

center}.

Then card ~0(~A,L],\C)\~) PROOF°

(i)

In v i e w of

~ 2

(**), by

and

card n0(A,L)\~

(25.6) w e

Emdim[A,C]

169

= 3.

see that

< ~ = c a r d ~.

2.105

Given jection number and

any

of

C

and

N ~ ~0([A,L],\C), from

N

we

(18.13.1)

is ~ - i n t e g r a l ,

Emdim[A,C]

(23.15)

by

and

we k n o w that

(because

5 is a p r i m e

= 3) in v i e w of £he P r o j e c t i o n

see that the p r o j e c t i o n

C

of

the pro-

from

Formula

N

(23.14)

is b i r a t i o n a l ,

Deg[AN, c N] = 5 ,

U ([AN,cN], JN) = H ([A,C],J)

and h e n c e

for e v e r y

j e ~II(A)

with

N ~ 90(A,J)

in p a r t i c u l a r

U ([AN, cN], LN) = U ([A,C],L)

= 2

LN~ n O ([AN,cN],~ N)

and

w

Thus

it o n l y r e m a i n s

to p r o v e

that

card

w

= {N ~ n 0 ( [ A , L ] , \ C ) : J ¢ ~(A)

u([A,C],J)

with

f)

£

2

where

for some

~ 3

N C ~0(A,J)] w

Suppose, (2)

if possible,

three distinct

that

members

> 3.

card NI,N2,N 3

of

T h e n we

can take

20 ([A, L], \C)

for w h i c h w e can find 1 Ji ~ nl(A)

(3)

with

N i ~ ~0(A,Ji)

for

i = 1,2,3

such that

(4) Since (5)

u ( [ A , C ] , J i) ~ 3 is a 2-secant,

L

n0(A,L)

Now we can choose suggestive

in v l e w of

N n 0 ( A , J i) = [Ni] Pij

~ g0(A)

for

i = 1,2,3

(2),

(3) and

and

Ni~

as i n d i c a t e d

figure

170

.

(4) we get

~0(A,C) in the

for

i = 1,2,3.

following

2 . lOG

I

I

I I

I l

/i

S

J2

J3

N2

N3

~P21

11

P22

13

IP32

P23

!

t P23

!

I l

I S,T

: Points

of

I

C

on

L, not n e c e s s a r i l y

distinct.

n0(~A, L3,C) = { S , T ]

i.e.,

w e choose

nine d i s t i n c t

members

P

. , i,j = 1,2,3, 13

of

0 (A)

such that

(6)

Pij ~ ~ 0 ( A ' J i )

for

i,j = 1,2,3

.

N o w we have,

[H 2 (A) : H 0(A) ] = number

of d i s t i n c t

monomials

of d e g r e e

2 in 4 i n d e t e r m i n a t e s

= i0 , (i.e.,

geometrically

projective

(7)

3-space)

speaking,

there

are

and h e n c e we can find

¢ ¢ H 2 (A)

such t h a t

171

~gquadrie

surfaces

in

2.107

(8) By

Pij 6 ~0(A,~) (6),

(7) and

for

(8) we have

~([A,Ji3,{)

~ 3 > 2 = (Deg[A, Ji3) (DegAS),

and hence by Bezout's (9)

Little Theorem

Ji 6 ~ ( A , ~ )

Consequently

by

Ni (2),

6

(7) and

(23.9) we conclude

for

i = 1,2,3.

~0(A,~)

for

i = 1,2,3.

that

~ 3 > 2 = (Deg[A, L3) (DegAS)

and hence by Bezout's

Little

(II)

L c £~(A,~). L

i = 1,2,3,

(I0) we have

~([A,L~,~)

Since

for

(3) we get

(I0) Now by

i,j = 1,2,3.

is a 2-secant,

(12)

Theorem we conclude

that

we have ([A,C],L)

= 2.

Now let s = ~ u([A,C~,~,P),

(131 )

over all In view of

(5),

Emdim[A,C3

NOW

(15)

(ii),

if possible

i,j c {I,2,3}.

(132), we see that

(26.9)

= 2 .

~0(A, Ji) Q ~0(A, Jj) ~ ~

In view of

for some dis-

(*), (2),(3), (4), (5), (12), (13 I)

can be applied

~ i, a contradiction

~0(A,Ji)

is extended

3 U £0(A, Ji) i=l (12) and (131) we have

s ~ u([A, C3,L)

Now suppose,

(14)

the summation

P e ~0(A,C)\

(132 )

tinct

where

to

(i).

to

L,J i

and

i,j e {1,2,3].

(9) gives, u([A,C~,~,J i) > ~([A,C~,Ji),

NOW

172

Jj; and we g e t

Thus we have proved

N £0(A, Jj) = ~, for distinct

for

i = 1,2,3.

and

2,108

([A,c],~) > s +

3 ~ U ([A,C],~,J i) i=l

by

(131 ) and

(14)

by

(15)

3

> s +

~ U ([A,C],J i) i=l

> ii

by

> 5 times

=

(4)

and

2 by

(Deg[A,C](DegA~)

and h e n c e b y B e z o u t ' s

Little

(16)

Theorem

(7)

(23.9) we m u s t h a v e

C ~ nl(A,~)

In v i e w of

(**),

(7),

(ii)

and

(16), b y

(25.7)

we c o n c l u d e

w

(17)

~ ¢ H

In v i e w of

(A,P)

for some

(14), we can take

(18)

permutation

P e ~0(A,L)

a

(e(1),

e(2),

e(3))

of

(1,2,3)

such that

(19) Upon

P ~ E]0(A,Je(i))

(14),(18)

(21)

for

4 i = A ( A , p , j e(i) ) and

4i

for

i = 1,2,

i = 1,2

with

(19) we get

~ H I(A)

for

and (22)

i = 1,2.

letting

(20) by

(132 )

Je (3) (~ ~ (A'4142)

173

41#42

that

2. 109

By

(7),

(9),

(17),

(18),

(20)

and

(21) w e

see

that

(231 By

(9)

and

(18) w e

also h a v e

(24)

Je(3)

Now

(22),

card Q

(23)

and

¢ ~ (A, ~)

(24) y i e l d

a contradiction;

therefore

we m u s t h a v e

< 2 .

(26. ii)

PROPOSITION. W

(26.11.1) such

that

If

Emdim[A,C3

C £ ~(A,~),

and

~ 3

there

then:

exists

a ~-quasiplane

and

(26.11.2)

If

n ~ 2

(26.11.3)

If

n = 3 = Emdim[A,C},

every

2-chord

every

then

Emdim~A,C]

and

N ¢ ~o(EA, L},\C)

any

there

I_ff n = 4, EmdimEA, C~

3-secant

exists L

a 3-chord;

we have

2-secant

exists

L

i__~s

Assume

that

~([A,C~,P)

= l

projecting

center.

PO e ~ 0 ( [ A , C ] , ~ )

3-chord

is a 3 - s e c a n t ;

projecting

Emdim[A,C~

for all

174

that

= 3, and

is a b e s t

n = 4,

a 2-chord;

we have

that

N e ~ o ( E A , L3,\C)

(26.11.5)

~

£0([A,L~,\C)

for some

every

there

card ~ 0 ( [ A , L ) , C ) < ~ = c a r d £ o ( [ A , L ] , \ C )

and e v e r y

that

~ 3

then:

for any

is a b e s t

# ( E A , C 3 , P O) ~ 1

then:

such

~ 6 H I(A)

C e ~(A,~).

is a 2 - s e c a n t ;

(26.11.4)

exists

~ ~ H 2 (A)

card no([A,L~,C ) < ~ = card and

there

= 3

center.

and

P 6 £o([A,C},~)

and

for

2.110

Then there

exists

fQ l l o w ! n g .

If

a 2-chord,

L

and for any 2-chord

is not a 2 - s e c a n t

card n 0 ( [ A , L ~ , C )

L

N ~ ~ 0 ( [ A , L3,\C)

a 2-secant

and

-- 1

,

is a b e s t p r o j e c t i n g

for all

we h a v e the

is a 3-secan______~t,

< ~ = card n 0 ( [ A , L ~ , \ C )

and every

~ ([A,C~,P)

then

L

center.

P e ~0([A,C~,\77)

I__ff L

i_ss

,

then upon letting

= {N ~ n 0 ( [ ~ , L ~ , \ c ) :

is a b e t t e r p r o j e c t i n g

N

center}

we have

card n0([A,L~,\C)\ ~ (26.11.6)

If

~ 1

and

n = 5, E m d i m [ A , C ~ [u([A,C~,P)

then:

there

exists

any 4-secant

L

a 4-chord;

Assume

(*)

2 >

(**)

u([A,C],P)

that:

assuming

- i] ~ 2 ,

every 4-chord

and

is a 4 - s e c a n t ;

is a best projecting

and

for

= 1

for all

C e ~I(A,~).

that there

exists

center.

n = 5;

~ [U ([A,C],P) P ~ n 0 ([A, C~, ~)

and t h e r e d o e s not e x i s t a n y ~uasiplane

-- 3, and

< ~ = card D 0 ( [ A , L ] , \ C )

N ¢ ~0([A,L~,\C)

(26.11.7)

< = = card Q .

we have that

card D 0 ( [ A , L ] , C )

and e v e r v

card D 0 ( A , L ) \ D

- i] ~ 0 ;

P e D0([A,C],\~)

e H 2 (A) [Note t h a t

such that (*) +

PO ~ D o ( [ A ' C ~ ' W )

175

(**)

is a Wis e q u i v a l e n t

such that:

to

2.111

U ([A,C],P 0) = 2, and

Then

there

exists

~ ([A,C],P)

a 2-secant

~ ([A,C],P)

and

for a n y

=

such

= 1

2-secant

= 1

L

such

for all

L

upon

IN e ~ 0 ( [ A , L ] , \ C ) :

N

~0([A,L],\C)\Q

and

for all

P e %([A,C],\P

0)

that

P ¢ ~([A,C],\L)

,

lettinq

is a b e t t e r

projecting

center]

we h a v e

card

(26.11.8)

Assume

~ 2

that:

card

e0(A,L)hQ

< = = card Q

n = 5;

(*)

~ [u([A,C],P) me n0 ([A, C],~)

- i]

(**)

U ([A,C],P)

P e n0([A,C],\~)

= 1

u([A,C],~,N')

and

there

does

quasiplane the

not

and

exist

such

ugon

any

some

=

N'

;

e ~0(A,~)

~ e H2(A) [Note

0

such

that

;

;

that

(*) +

~

(**)

is a ~is e q u i v a l e n t

to

that

~ ([A,C],P)

any

for

C e ~I(A,~).

assumption

Then

for all

> 1

.

= 1

there

exists

2-chord

L

for all

P ~ Z0([A,C]).]

a 2-chord

we have

the

L

such

that

following.

If

N' L

e ~0(A,L),

and

is a 2 - s e c a n t

for then

letting

=

{~ ~ ~ 0 ( [ A , T ~ , \ C ) :

N

is a b e t t e r

projecting

center}

we have

card

I_~f L

~0([A,L],\C)\Q

is a 3 - s e c a n t

and

~ 2

and

the p r o j e c t i o n

176

card~(A,L)\n

of

C

from

< = = card f]

N'

o

is b i r a t i o n a l

2.112

then

N'

is a g o o d

projection

of

C

from

card

and

every

neither

N

is a b e s t

is a 3 - s e c a n t

and

the

then

projectinq

then.

L

center.

is a b e s t

n =

I_~f

L

i__ss

is a 4 - s e c a n t e

< ~ = card n0([A,L~,\C)

that:

~

L

< ~ = card([A,L],\C)

a 3-secant,

Assume

If

is n o t b i r a t i o n a l

~ ~0([A,L],\C)

(26.11.9)

(*)

nor

n0([A,L],C)

N

center.

~0([A,L~,C)

a 2-secant

every

N'

c ~30([A,L],\C)

card

and

projecting

,

projectinq

center.

5;

13

[~ ( [ A , C ] , P )

-

-- 0

;

1

for all

P ¢ n0([A,C],\~)

for

P ¢ ~0(A,n)

Pen 0 ([A,CL~)

(**)

U ([A,C],P)

=

(***)

u([A,CT,n,P)

< 1

all

;

;

e

and

there

does

quasiplane

and

(*) +

(**)

not

exist

and

for

is e q u i v a l e n t

then

upon

Q =

there

any

~ ~ H 2 (A)

C ¢ Ill(A,#).

u[A,C],P)

Then

any_

=

such

L

to t h e

1

exists

[Note

we have

the

that

assumption

for all

a 2-chord

such

that

(***)

=

~ (*)

is a ~; also

note

that

P e n0([A,Cl).3

L

such

following.

that

L

is n o t

I__ff L

a 3-secant,

is a 2 - s e c a n t

letting

{N c ~ 0 ( [ A , L ~ , \ C ) :

N

is a b e t t e r

projecting

center~

we have

card n0([A,L~,\C)\O

I_~f

L

is n o t

a 2-secant

< 2

then

and

L

that

card n0(A,L)\Q.

is a 4 - s e c a n t ~

177

< ~ = card

n

.

2. 113

card ~ o ( [ A , L ] , C ) < ~ = card no([A,L],\C) and every

N 6 £0([A,L],\C )

PROOF. of

(25.6),

(26.11.2) (26.5), from and

the second assertion

(26.11.4)

(26.2),

To prove

there exists

such that

(26.11.9)

follows

(26.12)

L

(26.8).

(26.7),

Assume

~([A,C],P)

= 1

and

There exists

3 ~ n { 5

p r o j e c t i o n of

and

follows

follows

from

U ([A,C],P)

(*) there exists

follows (26.8)

= 1

from

and

(26.1)

for all (26.10).

(26.5).

(26.5).

that

n ~ 5

and

~ e H2(A )

(26.3)

~0 (A'L)" and finally note

and

for all

(26.11.8)

Finally,

P 6 ~]0([A,C],\n)

three situations such that:

C

from

and there exists N

~

min(4,n)

prevails. is a ~ - q u a s i p l a n e

the projection of

for all

C

from

and there exists N

C

Q 6 £ 0 ( [ A N , c N ] , \ ~ N)

') = 2

for some

N ~ ~o(A,TT)

is birational,

i__ssTT-integral, u([AN,cN],Q)

~([AN,cN],Q0

= 1

such that:

the from ,

~ Deg[AN,c N] < n .

4 < n ~ 5

p r o j e c t i o n of

N c ~o(A,~)

is birationa!,

i__ssTT-inteqral, ~([AN, cN],Q)

(3)

N

(26.11.6)

and

C ¢ £1(A'~) . (2)

N

in view

(26.2)

(26.11.5)

(26.10)

Then at least one of the following (i)

(26.5).

PO~

(26.11.7)

(26.4),

THEOREM.

from

first note that by

such that

(26.10),

from

and,

U ([A,C],P O) = 2, then note that by

the rest of

(26.1),

follows

and

(**) we must now have

P c ~o([A,C],\L); from

and

(26.11.7),

a 2-secant

(*) and

(26.1)

is obvious

from the first assertion.

(26.11.3)

from

(26.6)

in (26.11.1)

follows

(23.11).

follows

PO 6 ~o([A,C],~)

follows

from

(26.5),

(26.5).

that by

is a best projecting center.

The first assertion

follows

,

~ 2

the projection of

for all

C

the from

Q ¢ ~]o(AN),

Q0 6 ~o([AN,cN],\~N),

178

such that:

z([AN,cN],QI)

= 2

2.114

for some

Q1 ~ ~

PROOF.

([AN'CN~'~N)'

Follows

from

and min(4,n)

(26.11).

179

~ Deg[AN, cN~

< n .

CHAPTER

III:

BIRATIONAL

In this ~(Y)

in

chapter

Y , with

derivative

of

X,Y,Z

~(Y);

for a n y

X,Y,Z,

corresponding

subscript;

X

and

Y, w i t h

partial

derivative

denotes

the p a r t i a l

shall

(27.1) field

L.

field

now

thus

for i n s t a n c e ,

~(X,Y)

derivative

of

(S,K)

the

K

of

of

and

we

R c ~

note

that

(S,K).

We

S

L.

~

that

(27.2)

we

O(S,K)

~ {0};

shall

closure

indicated

in-

b y the

X,

with

~(X,Y)

denotes

and

the

~y(X,Y)

respect

to

Y.

o f the d i f f e r e n t .

domain with

of

S

quotient

in a f i n i t e

algebraic

define

module

TraceK/L(~)

(S,K)

is

an

of

~ S

S

i__nn K

for a l l

R-submodule

8 ~ R]

of

K

with

define

= {a ~ K:

note

t h e Y-

of the

~x(X,Y)

to

be a n o r m a l

We

~3(S,I 0

(3) and

-

(I/2)u~(R)

Theorem

-

(24.9)

(1/2) (n-l) (n-2)

we m u s t

(1/2) (n-l) (n-2)

(9), b y

(29.1)

~ 0

we get

have

;

that

/

/g (i0)

= (1/2)(n-l)(n-2) = a nonnegative

Henceforth such

(Ii)

-

(1/2)~(R)

integer.

assume t h a t

g = 0

that

m m max(l,

n-2)

223

and

let

any

integer

m

be

given

3.45

Then by

(i0) w e h a v e w

(12)

u{(R)

and h e n c e b y

(i),

=

(8) and

(13)

(n-l) (n-2)

(ii) we get t h a t

e (m) m 1

and w

(14)

e(m)

If

[E(m,e(m)

~£

Hm(R)

and t h e n b y

+ I)

with

(6) and

+ N~(R)

: H0(R)]

~ = CR

=mn

.

> 0, t h e n we c o u l d t a k e

for some

~ ~ E(m,e(m+l))

(14) we w o u l d h a v e w

(R,~) in c o n t r a d i c t i o n Therefore

[E(m,

(5) and

(15) a n d

(17) Again,

+ i)

in v i e w of

and h e n c e

(24.9).

: H0(R)]

= 0 .

: H0(R)]

~ 1.

(16) w e g e t

[E(m,

(18)

(19)

e(m)

[E(m,e(m))

(7),

Theorem

(8) w e h a v e

(16) By

Little

we m u s t h a v e

(15) By

to B e z o u t ' s

~ 1 + mn

e (m))

(13),by

[E(m, in v i e w o f

E(m,

e(m)

: H 0(R) ] = 1 .

(5) and

- i)

(8) w e h a v e

: H0(R) ] ~ 2

(7) w e g e t

e(m)

-I)\E(m,

224

e(m))

/

3.46

In v i e w

of

(17) w e

can

(20)

and

0 ~ x e E(m,e(m))

in v i e w o f

(19) w e

(21)

can

take

0 ~ y ¢ E(m,e(m)

In v i e w we

take

of

(4),

conclude

(14),

(20)

ord(R,x,V)

l

(21),

ord(R,x,W)

= e(m)

+ ordv~(~(R, ~

W

= or~(~(R,O

(23)

(R,W)))

whenever

In v i e w

and,

by Bezout's

of

(22)

in v i e w

and

of

(23),

(22)

and

by

by

we

(19.10)

= I, and

consequently

(29.3) above,

conic. But

by

REMARK. and

so also

latter

that would

be

can

genesis a good

(24.9)

Also

Let

assume

some

we

see

now

x/y

e R(R)

that

~ 0

V @ W

e 3(R)

;

of

(29.2.2)

ancient

as w e l l

pad

for m u c h

as of

its

of b i r a t i o n a l

i d e a of p a r a m e t r i z a t i o n

be d e d u c e d So h e r e

proof

as a c o r o l l a r y

of a

of

is the e l e m e n t a r y

(29.2.2). version,

§25.

OF A C O N I C .

k = H0(R) that

that

R (R) = H 0 ( R ) ( x / y ) .

facetious! after

= 1 + ord(R,y,V)

~ ~(R).

or~(x/y)

launching

and

PARAMETRIZATION

0 ~ z e HI(R).

have

of c o u r s e

a bit

any m a t e r i a l

(29.4)

n = 2.

The

is the e l e m e n t a r y

The

not using

(4.2) w e m u s t

(R,V)))

see

whenever

geometry,

Theorem

< ord(R,y,W)

V ~ W

(19.10)

(23),

or~(x/y)

given

Little

that

(22)

and

and

- l) \E (m, e (m) )

and

R' = k

V e ~(R,~)

225

Let

~ = zR

with

[HI(R)z-lJ. is r e s i d u a l l y

any

Assume

that

rational

over

3.47

k,

and

let

euclidean

P = ~

(R,V).

domain.

Then

Moreover,

i__nn H 1 (R)

with

xk

+ yk

(*)

R'

= k[x/z]

R'

= k[x/z,z/x]

Emdim

there

R ~

exist

+ zk = HI(R)

and

zy =

x

2

2,

g =

0,

nonzero and

case

z2

in

R'

elements

(x,z)R

in

and

=

P

~o(R,~)

is x

such

=

[P]

an

~nd

y

that:

,

and

(**) NOTE. figures P

is

conic

xy

writing

down

the

two

cases

where,

at

~

Before

for the

and

the

point

birationally

vertical

(i.e.,

along

onto

parallel

the

the

proof

case

let

us

in b o t h

cases,

y-axis,

and

(and h e n c e to

=

draw the

so w e

parametrizing

y-axis)

~0(R,n)

[P]

suggestive

projecting are

by)

center

projecting

the

x-axis

the along

directions.

P

2P

i/

i

I1

i//

~

y

I

/

/ /

/

\

!

/

z = 0/ ./__

/

\

z=O// /

X

/ ---X

/

Y

i

z = 0

Hyperbola:

xy

= z

no(R,~ ) =

[P,Q].

Projection

from

outside the

z = 0

: line

2

at

=.

Parabola: ~o(R,~)

P

is

except

zy =

= x

Projection

from

above

everywhere

outside

226

.

[P}.

integral

origin.

2

P

is z =

integral 0.

3.48

PROOF.

By

(24.12)

Projection

Formula

birational

and

U

=

get

(24.15)

Emdim

(R,P)
O.

be

4.4

H ( A , z ) (I) = The a d d i t i v e

s u b g r o u p of

generated

A[z]

e0

U H (A,z,n) (I) n---0

by We may drop Hz(X), clear

" A "

H ( z , n ) (I)

and

Hz(I)

respectively,

and

simply write

if the r e f e r e n c e

to

A

is

f r o m the c o n t e x t .

(32.3)

Observe

0 g n < ~, Hz(A) homogeneous N o t e that, then

if

z

a homogeneous

Hz(A) and

and

upon defining

z'

that

H z (A)

d i m A = Dim H

of

elements

Hz, (A)

A.

over

i.e.,

A,

there

which makes

for e v e r y

from the d e f i n i t i o n

for

W e say t h a t the

isomorphic,

and

to e a c h o t h e r

= H(z,n) (A)

z-homogenization

are c a n o n i c a l l y

It is c l e a r

it f o l l o w s

domain.

are two t r a n s c e n d e n t a l

Hz, (A)

to c o r r e s p o n d

(32.4)

Hn(Hz(A))

is the n a t u r a l

isomorphism between

az 'n

Thus

that,

becomes

domain

Hz(A)

a unique and

f r o m the a b o v e n o t a t i o n

az

is n

a g Fn(A).

that

~(Hz(A))

= ~(A).

(A). Z

(32.5)

Also

clearly

F 0 (A) = H 0 (H z (A))

vector

(32.6)

For

FI(A)

is i s o m o r p h i c

space and h e n c e

in p a r t i c u l a r It a l s o

~ Hz(~) (32.7) I, we h a v e {Hz(X): x £ I

For that

x e S}. and

degAx

as an

x e A, c l e a r l y w e h a v e

x ~ Hz(X)

follows

gives

HI(Hz(A))

e m d i m A = Emdim H z (A)

D e g H z (A)Hz (x) = d e g A x and

to

that

gives

for a n y

an i n j e c t i v e m a p I c A Hz(1) Further

an ideal

an i n j e c t i v e m a p H z(~)

Hz:

~ Hn(Hz(A)).

in

it is c l e a r

H z(x)

Fn(A) A

and

237

S

ideal

that

It f o l l o w s

e H z(I)

Hz: A - H ( H z ( A ) ) .

• e Fn(A),

is a h o m o g e n e o u s

~ n < ~].

,

~ x e I.

a generating in

Hz(I)

that

¢ H n ( H z(A))

Hz(A)

set for

generated

n H(Hz(A))

for a n y

and

by

= [xz n :

x ~ A, we h a v e

4.5

In p a r t i c u l a r w e have,

for any two ideals

Hz(I) It also f o l l o w s I ~ A

then

in

A,

~ I = I'

I = A, ~ 1 e I ~ z ¢ Hz(I),

so t h a t if

z ~ H z(I).

(32.8). homomorphism. a unique

that,

= Hz(I')

I,I'

Now

let

If

z'

extension

f' (a) = f(a) isomorphism

P e ~(A)

let

f: A ~ A / p

is any t r a n s c e n d e n t a l

f': A[z3

for e v e r y

-

(A/p)[z'3

a ¢ A.

of the h o m o g e n e o u s

In p a r t i c u l a r ,

and

w e m a y take

A/p,

s u c h that

It is t h e n c l e a r domains

Hz(P)

then

f

has

f' (z) = z'

and

that

Hz(A)/Hz(P)

~ = z modulo

= H_(A/P). z In p a r t i c u l a r it follows

over

be the c a n o n i c a l

f'

and

induces Hz, (A/P).

and t h e n w e h a v e

H z ( A ) / H z(P)

~(A)

- n(Hz(A)).

~i(Hz(A))

In v i e w of

and by

an i n j e c t i v e

(32.7),

p - Hz(P)

(32.4),

it follows

gives

it follows that

Hz:

a map

that ~i(A)

Hz:

Hz(~i(A))

c

- ~i(Hz(A))

is

map. §33.

(33.1)

DEFINITION.

0 ~ z ¢ HI(B).

lh

I c B

B

be a h o m o g e n e o u s

domain.

Let

h ¢ H(B), w e d e f i n e

, if

hz -n, for

Dehomogenization. Let

F o r any

F(B,z) (h) =

Also

that

and

if

h c H0(B),

and

h e Hn(B)

for

0 < n