Desingularization Strategies for Three-Dimensional Vector

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Instituto de Matem&tica Pura e Aplicada, Rio de...

Author: Felipe Cano Torres

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Instituto de Matem&tica Pura e Aplicada, Rio de Janeiro Adviser: C. Camacho

1259 Felipe Cano Torres

Desingularization Strategies for Three-Dimensional Vector Fields

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Author

Felipe Cano Torres Departamento de Algebra y Geometrfa Facultad de Ciencias Valladolid 47005, Spain

This volume is being published in a parallel edition by the Instituto de Matem&tica Pura e Aplicada, Rio de Janeiro as volume 43 of the series "Monografias de Matem&tica". Mathematics Subject Classification (1980): 2 4 B 0 5 , 3 2 B 3 0 , 5 8 A 3 0 , 58F 14 ISBN 3-540-17944-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17944-5 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be pa~d.Violations fall under the prosecution act of the German Copyright Law. © Springer-Vertag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 214613140-543210

To

MePcedes

INTROOUCTION

dimensional gularity resp. have

Let D = aa/ax

+ ba/8y

Pin G of power

series

of D at the origin

~(b),

are the orders

no common

In this space,

factor

situation, we obtain

and the order sequence avoided

are

divisor

is given

y' and

lar result

the

has shown tion

v(D')

expect.

by

at the origin blowing-ups

x'=O

x'a/ax'

is one and

that this

i s <1

(see G i r a u d point

of varieties

One can say something a component

to the exceptional

zero

some coefficient

(hence

can not be globalized

(see

If we consider field

number

< ~(D)+I).

181

or

131 f o r

1131).

blowing-up

we have

is

v(D)=1,

under any

This d i f f i c u l t y is " tangent

to the

of such vector

coefficients

of the orders

for D'

of these

blowing-ups

a different

may be

a simi-

we obtain

proof).

191

Giraud

useful for the problem

of reduc-

characteristic. situation:

we can always

is a unit).

result

If we take this approach,

is also

161).

of the ambient

(in General,

field which

of quadratic

in the above

divisor,

by a unit

of the free module

increased.

see

by D'= y ' x ' a / a x ' + ( x ' 2 - y ' 2 ) ~ / a y '

(the m i n i m u m

in positive more

blowing-ups

+ x38/ay

~(a),

that a and b

foliation",

thus the c o r r e s p o n d i n g

of view

But this

obtain

component

by adding,

if necessary,

that the adapted may be formal

order

is

and so it

131 on 1.4.1.4.). the problem

of the reduction

D = ~ai~/~x i from the logarithmic

as for n=2 indicates

a basis

order

after a finite

logarithmic

of s i n g u l a r i t i e s

a/ay',

the adapted

remains

is the sinwhere

applyin G a quadratic

is given

as a vector

it has not been

can be proved:

order

and

moreover

"saturated

if D = ya/ax

of a two-

how "bad"

(see,e.g. , Seidenberg's

increased

the order field

the

v(D) after

has been

b. Assume

of quadratic

point

chart

". In this case

x'2-y '2. Now,

adapted

number

For instance,

in a suitable

(or a derivation

v(D) = min(~(a),v(b)),

of a, resp.

us to c o n s i d e r

of the measure

by considerin G the vector

coefficients)

that

transform

of quadratic

exceptional fields

at the origin

field

k). We can measure

of the number

at every singular

as one could

but the strict

by means

after makin G a finite

v(D)<1

vector

over a field

(this allows

The behaviour not as good

be a plane

that the statement

point for the

of an n-dimensional

of view as above, "global

reduction

vector

the same obstruction result" would

be to

VI

reach a d a p t e d reduction

order

~.

Moreover,

of the a d a p t e d

in higher d i m e n s i o n we can not expect to obtain the

order only by means o£ quadratic

blowing-ups:

introduce the p e r m i s s i b l e centers which will be used in the process.

These centers are

defined from a local point of view and in this way two problems arise duction of s i n g u i a r i t i e s

of varieties):

in each step and the g l o b a l i z a t i o n first

problem

the e x i s t e n c e

o£ this algorithm.

thus we have to

(like in the re-

o£ an a l g o r i t h m defined i o c a l i y These

notes are devoted to the

in the case n = 3.

Technically,

the existence of such an a l g o r i t h m will

existence of a winning

strategy for a reduction

to the general

used in the s e q u e l and to the explicit f o r m u l a t i o n

results

resuit 1.4.2.g.. to c o n t i n u e

If r = adapted order,

so. C h a p t e r s

into the p r e c e d i n g stable case control

(1.4.2.g.).

The control of this case

is made

by means of the Newton

are c o n t a i n e d there.

the c l a s s i f i c a t i o n

versality positions

The

between the e x c e p t i o n a l

singularities

work.

In If, the more

polygon

(see

1101).

III and IV

"type one" and the most com-

is made

d i v i s o r and some

in V.

by c o n s i d e r i n g the t r a n s ideals a s s o c i a t e d

to the

role of the strict tangent space

in the

(see III.I). Aroca for i n t r o d u c i n g me

for his constant aid and for guiding me during the r e a l i z a t i o n

To P r o f e s s o r d. Giraud for his interest and for his t e c h n i c a l aid,

the precise f o r m u l a t i o n

fall

in a very similar way to the

I wish to express my gratitude to Professor J.M. to the subject,

< r-1 and

remaining cases are c o n s i d e r e d

of the various types

vector field which play a role similar to the case of s u r f a c e

of the main

II-V deal with the various types of vector fields which

singularities

plicated c o m p u t a t i o n s

I is d e v o t e d

it is enough to obtain adapted order

deal with the reduction game b e g i n n i n g at the s o - c a l l e d

In general,

Chapter

ones after a finite number of steps of the process.

is treated.

of surface

game

be f o r m u l a t e d as the

of many parts of these

notes.

o£ this

e s s e n t i a l for

To Professor A. C a m p i l l o for his (*)

important suggestions.

(e) This work

Special thanks to Miss A. A r t e r o for

has been p a r t i a l l y supported

by the CAICYT.

her hard typing task.

-CONTENTS-

I.

RESOLUTION STATEMENTS FOR A VECTOR FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. ADAPTED VECTOR FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

1.1.

G e n e r a l h y p o t h e s e s and n o t a t i o n s . . . . . . . . . . . . . . . . . . . . . .

1.2.

Vector fields

1.3.

The a d a p t e d case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.

The f o r m a l

and d i s t r i b u t i o n s

.......................

case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

BLOWING UPS OF VECTOR F I E L D S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Inverse image by a morphism ........................... 2.2,

Adapted

2.3,

Formal

blowing-ups

...................................

biowing-ups

....................................

3. SINGULAR LOCUS AND BLOWING UP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.

II.

3.1.

Adapted o r d e r

3.2.

The d i r e c t r i x

3.3.

Stationary

3.4.

Permissible

o£

a vector

field

.......................

.........................................

sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . centers ...................................

RESOLUTION STATEMENTS. . . . . . . . . . . . . . . .

..........................

4.1.

The g e n e r a l s t a t e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.

R e s o l u t i o n games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A PARTIAL WINNING STRATEGY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O.

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. T Y P E Z E R O S I T U A T I O N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,1,

Description

of

type

zero ..............................

1.2.

Stabiiity

results

1.3.

Type

games .......................................

zero

.....................................

2. A TYPE 0-0 WINNING STRATEGY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.

2.1.

An i n v a r i a n t

of

transversality

........................

2.2.

Polygons for

type 0-0 .................................

2.3.

Preparation ...........................................

2.4.

Main r e s u l t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INVARIANTS ASSOCIATED TO THE TYPE 0 - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.

Polygons f o r

t y p e 0-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.

Good p r e p a r a t i o n .

First

3,3.

Good p r e p a r a t i o n ,

e(E)=l ..............................

3.4.

V e r y good p r e p a r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

cases . . . . . . . . . . . . . . . . . . . . . . . . .

4 . A WINNING STRATEGY FOR TYPE 0 - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.

III.

Good p r e p a r a t i o n

stability

4.2.

V e r y good p r e p a r a t i o n

4.3.

A winning strategy

for

............................

stability

.......................

t y p e 0-1 . . . . . . . . . . . . . . . . . . . . . . .

STANDARD TRANSITIONS FROM TYPE I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. INTRODUCTION. . . . . . . . .

..........................................

7 7 9 13 14 14 17 19 23 28 28 31

35 35 37 37 39 43 44 44 45 49 51 54 54 57 60 66 68 69 71 76

79 79

VIII

1.

CLASSIFICATION 1.1. 1.2. 1.3.

2.

3.

IV.

STANDARD T R A N S I T I O N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 81 87 87 88

2.1.

Definitions

2.2. 2.3. 2.4.

Polygons and invariants ................................... Preparation of ~ .......................................... Standard transitions from the type 1.1.1 ..................

90 92 94

2.5. 2.6.

Good p r e p a r a t i o n . Stability results

95 97

2.7. 2.8.

Very good preparation ..................................... Standard winning strategies ...............................

and

first

reduction

for

good

...........................

' preparation

....................

98 99

REDUCTION OF THE TYPE 1 - 1 - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A WINNING

standard

...................................

101 101

First

cases ........................

104

l--~I'. Case T-I,~ ......................... II--ml' ....................................

108 112

3.1.

No

3.2.

The

transition

I--mI'.

3.3. 3.4.

The The

transition transition

3.5.

Reduction

STRATEGY

transitions

of

the

type

I'-1-1

..............................

114

FOR T Y P E O N E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116

0.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116

1.

THE "NATURAL" T R A N S I T I O N

117

2.

3.

Definition and notations .................................. First reductions ..........................................

117 119

1.3.

Proof

121

of

the

main

result

..................................

NO STANDARD T R A N S I T I O N S

FROM TYPE ONE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..............................................

Introduction

2.2.

Preliminaries

2.3.

The

2.4. 2.5. 2.6.

The c a s e e ( E ( 1 ) ) = 2 ~(1) monoidal with ~(1) monoidal w±th

case

for

e(E(1))=l

the and

case ~(1)

e(E(1))=l

~(1)

quadratic...

127

quadratic

.....................

132

and ~(1) quadratic center (x(0),z(0)) center (y(O),z(O))

..................... ..................... .....................

138 142 145

FROM TYPE I I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction .............................................. The t r a n s f o r m a t i o n T-I,~,~O .............................. The t r a n s f o r m a t i o n T-2 .................................... The t r a n s f o r m a t i o n T-l,0 ......... ; ........................ ~(s+l) monoidal with center (x(s),z(s)) ................... ~(s+l) monoidal with center (y(s),z(s)) ...................

147 147 149 154 156 161 161

A WINNING STRATEGY FOR THE TYPE ONE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. 4.2. 4.3. 4.4.

Introduction .............................................. Standard transitions from the type I' .................. No s t a n d a r d transitions from the type I' .................. A winning strategy for the bridge type ....................

TYPES TWO AND THREE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.

and

126 126

NO STANDARD T R A N S I T I O N S 3.1. 3.2. 3.3. 3.4. 3.5. 3.6.

4.

..........................................

1.1. 1.2.

2.1,

V.

BY TRANSVERSALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideals associated with a vector field ..................... Classification ............................................ Reduction of t h e no t r a n s v e r s a l s types ....................

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

...

162 162 163 165 168

172 172

IX

1.

STANDARD T R A N S I T I O N S 1.1. 1.2.

2.

3.

A winning Invariants

FROM THE TYPES I I

strategy for the

NO STANDARD T R A N S I T I O N S 2.1.

Another

2.2.

Quadratic

2.3.

No

2.4.

A winning

TYPES I I '

no

standard

AND I I I '

FROM I I

invariant

of

....................

AND I I I

standard

for

........................... ......................

transversality

transitions

strategy

AND I I I

if dim Dir(O,E)=I ..................... standard transitions ..................

transitions from

the

II

type

from and

172 172 175 177 177

III ...............

178

III ..................

180

Ill-bridge

...............

................................................

181 183

3,1.

The

case

T=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

3.2.

The

case

~=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

184

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

186

INDEX ........................................................................

187

RESOLUTION

STATEMENTS

FOR A VECTOR FIELD

O. INTRODUCTION

In

this

chapter

that we shall need to

which

ments

proof

we

shall

develop

in the f o r m u l a t i o n

this

work

about r e s o l u t i o n

preliminary

of the main statements

is devoted.

of vector

the

We

fields

shall

also

concepts

and

results

about r e s o l u t i o n

formulate

more

general

games state-

that we shall not prove.

1. A D A P T E D V E C T O R FIELDS

(i.i) General

(1.1.1)

In

separated

hypothesis

the

sequel,

(1.1.2)

Remark.

on

dimension

chapters

X

will

scheme of finite

by n the d i m e n s i o n

the

and n o t a t i o n s

we

denote

a

regular

variety,

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

i.e.

closed

a regular

integral

field k. Let us denote

of X.

Along

shall

n

this nor

on

assume

chapter the that

we

shall

characteristic n

=

3

and

not of

k.

that

make But the

any in

the

assumption following

characteristic

of

k

is

zero. We would like to remark that although the assumption on the dimension

is essential

for the good work of the techniques used, it is not so for the assump-

tion on the characteristic sitive

characteristic

characteristic

(1.1.3) sheaf,

We

and we

shall

most of the results remain

always

indicate

where

true for po-

the hypothesis

on zero

is used.

shall

denote

of X relatively

locally free Ox-modules

(1.1.4)

of k. Actually

We shall

by

~X'

resp.

to k. Because

EX, the cotangent of our assumptions

sheaf,

resp.

on X, both

the tangent

%

and

EX are

of rank n.

say that a closed

iff for each closed point

subscheme

E of X is a "normal crossings divisor"

P of X there is a regular system of parameters

Xl,...,x n

(r.s. of p. for short) of the local ring ~X,P of P in X such that for some s, O<s
(1.1.5)

Let E be a normal crossings divisor

E. We shall

denote by Ex[E ] the

(unique)

and let ~E be the sheaf

subsheaf

of ideals of

of E x such that for every point

P of X one has that

(1.1.5.1) (recall that

~,p[E]

EX,P is

= {D E EX,p; D(JE,p) = ~E,P }

the module of k-derivations

of the local ring Ox,p).

(1.1.6) Let P be a closed point of X, let E be a normal crossings x=(xl,...,x n) be a regular

system of parameters

by x I" ...'Xs. Let ~/2xi, i=l,...,n,

(1.1.6.1)

Then

standard

computations

of OX, P such that E is given at P

be the k-derivations

9/gxi(xj)

show that

of OX, P given by

= ~ij

8/3x i , i=l,...,n,

(Kroned<er

Xl~/~x I .... ,Xs~/~Xs,

symbol).

is a free basis of EX, P

and that

(1.1.6.2)

divisor and let

~/~Xs+l,...,~/~x n

is a free basis of EX,p[E ].

(1.2) Vector fields and distributions

(1.2.1)

Definition.

Let

P be a closed point of X. We shall

P" to any element D of EX,p. unidimensional

(1.2.2) Dp

For

distribution

Any inversible

"vector

field at

of E X will be called "an

over X".

any unidimensional

is generated

Ox-submodule

call

by a vector

distribution

D

and

any

closed

field D. If D' generates D p

point P, the stalk

there is a unit u in OX, P

such that D' = u.D.

(1.2.3) Proposition.

Let us consider the natural pairing

(1.2.3.1)

<,>: ~

and for a given unidimensional

distribution

thogonal of D relatively

Let

P be

~

0x

m let us denote by e(D) the double or-

to <,>. Then

a) e(O) is an unidimensional b)

x ~

any

distribution

over X.

closed point of X and let D be a generator

of Dp such

that

(1.2.3.2)

D :

where x = (Xl,...,Xn)

[ ai~/~xi, i=l,...,n

a i e Ox,p;

is a r.s. of p. of OX, P. Let b be the g.c.d, of ai, i=l,..,n.

Then a(D)p is generated by D/b.

Proof It is enough

(1.2.3.3)

to remark that for a closed point P

~(D)p = {D' e EX,p;

~ ~ ~X,P' <~' D">=O

that both ~X,P and EX, P are free ~,p-modules,

(1.2.3.4)

EX,p = Hom o X,P

W D " ~ Dp

that

(n x p,Ox,p ) '

:~

=O

and that Ox,p is a U.F.D. because it is regular.

(1.2.4)

Definition.

plicatively

We shall

irreducible"

say that an unidimensional

iff D

= e (D)

and we shall

call

distribution e(O)

D is "multi-

the "multiplieative

reduction of ~". (For short, m.i.u.d.= multiplicatively irreducible unidimensional distribution).

(1.3) The adapted case

(1.3.1) Let E be a normal crossings divisor on X.

(1.3.2)

Definition.

D at P is "adapted

Let P be a closed point of X. We shall say that a vector field to E" iff D ~ E X,p[E].

A unidimensional

distribution D

over X

will be called "adapted to E" iff D e E X,p[E].

(1.3.3.) Proposition.

Let D be an unidimensional distribution over X, then

a) The Ox-SUbmodule

(D,E) given by

(i.3.3.1)

(~E)

=

D~E

x[E]

is an adapted to E unidimensional distribution over X. b) Ox,p

Let

P be

a

closed

point

of

X,

x

=

(Xl,...,x n) be a r.s. of p. of

such that E is given at P by x I ... x s and let

(1.3.3.2)

be a generator

D =

[ ai~x i=l,...,n

eI xI

~j = i if aj 4 xi.Ox,p,

Proof It is enough

(1.3.3.4)

i

Of Dp. Then (D,E)p is generated by

(1.3.3.3)

where

let

es ... x

s

e. = 0 otherwise, J

. D

j=l,...,s.

to remark that for a closed p o i n t P of X one has that

( ~ E ) p = {x.D;

~ eI}

5

where I is the principal

(1.3.3.5)

ideal of 0X

,P

given by

I : { k ~ Ox,p; k.D(x I ..... x s) ~ (x I ... X s ~ x , p } .

(1.3.4) Definition.

We shall call the unidimensional

distribution

(~E)

of (1.3.3.1~o be

the "adaptation of D to E".

^

(1.3.5)

Let

D be an adapted

the dual sheaf of ZX[E].

to E unidimensional

L~.~<Us denote by

distribution

a'(D,E)

and let EX[E ]

the double orthogonal

denote

of D with

respect to the natural pairing ^

(1.3.5.l)

<'>E: ~X IS] x ~x[E]

Then computations distribution.

like

Moreover,

--~

(1.2.3) show that ~'(D,E)

0 x.

is an adapted to E unidimensional

if P is a closed point of X, a generator

be obtained from a generator Of D p by dividing it by the g.c.d,

of

e'(D,E)p

can

of its coefficients

in any basis of EX,p[E ]. Now, combining

(1.2.3) b) and (1.3.3) b) one can deduce that

(1.3.5.2)

(1.3.6)

~'(D,E) = (~(D),E),

Definition.

Let D be an adapted

say that O is "multiplicatively

to E unidimensional

irreducible

(1.3.6.1)

D=

relative

Remark. The cesults of this section are

subscheme of X of pure codimension

one".

We shall

to E" iff

(~(O),S)

And we shall call (e(D),E) the "multiplicative

(1.3.7)

distribution.

reduction relative

true

for

the

to E" of D .

case "E is a closed

1.4) The formal case

1.4.1)

Let

(R,M,k)

be

a complete

local

regular

ring of dimension

n containing

actually R is a ring of formal power series over k). Let X ^ be the scheme Spec

k

(R).

^

Let E

be a normal crossings divisor

of X

as in

(1.1.4) given at the closed point

^

of X

by the ideal I c R. Then one has that both

(1.4.1.1)

Derk(R)

and

(1.4.1.2)

Derk(R)[I ] = { D • Derk(R);

are free R-modules

(1.4.2)

D(I) c I}

of rank n.

Definition.

With

notations

as

above,

we

shall

call"formal

vector

fields

^

over the closed point of X " (resp. (resp.

of

Derk(R)[I]).

We shall

"and adapted to E ")

call

"formal

the elements

unidimensional

of Derk(R)

distribution

over the

^

closed point

of X " (resp.

"and adapted

Derk(R)

(resp. of Derk(R)[I]).

(1.4.3)

Let 0

We shall

be a formal

denote

to E ~')

unidimensional

the double

orthogonal

of

the rank-i free R-submodules

distribution 0 in Derk(R)

over

the closed

point

of

of X

by a (0) and we shall denote

^

(0, E

) =

0

n Derk(R)[I]&ike

in

(1.3.6),

we shall

^

irreducible

relatively

say that 0

is "multiplicatively

^

to E " iff 0 =

(a(O),E).

^

(1.4.4) Let P be a closed point of X and R = Ox, P (the completion

of the local ring

Ox,p).

to a k-derivation

D

g

Then

any

Derk(R).

k-derivation

D

If E is a normal

~ EX, P may be uniquely crossings

divisor

extended

of X given at P by I ~ ^

then

any

vector

field

^

I

D

~ EX,p[E ]

be extended

to D ~

Ox,p, ^

Derk(R)II

], where

^

= I.R. We shall call D

cedure

"the associated

6an be used fop an unidimensional

to 0 formal distribution

Op at P".

to D formal vector field".

distribution

The same pro-

in order to obtain t h e " ~ i a Z - e d

2. BLOWING-UPS

OF VECTOR FIELDS

(2.1) Inverse images by a morphism

(2.1.1)

Let ~: X' ---~X be a morphism

k, and let D be an Ox-submodule

between

n-dimensional

regular varieties over

of E X. Let D, be the image of D by the natural mor-

phism of Ox-modules

(2.1.1.1)

~ : Ex

'~

Horno ( ~ X , ~ . % , )

X induced by the structural

morphism

~X over 0 x and the adjoint

z#'. 0 x

property

~ ~.Ox, . Because

of the inverse-direct

of the finiteness

of

image, one has a natural

isomorphism of Ox-modules

.: "omox(aX,~.Ox,)

- ~

(2.1.1.2) * 0 ---+ ~*H°mOX,(~ ~X' X ')"

LetD"

be the image of D

by n. Now, let D ' "

~D"

~

be the image of the natural morphisms

( ~ aX, 0 X, )

"rr n.Hom 0

~,

X'

(2.1.1.3)

HOmOx ' ( n ax,Ox, ) •

Finally,

let us denote

by L)~

the inverse

image of D' ' ' by the natural morphism of

0 X ,-modules

(2.1.1.4)

EX,

~

Hom

(n~x,Ox,) 0x ,

obtained by applying the H o m ~ , ( _

(2.1.1.5)

(2.1.2)

~*~X'

Definition.

of D by ~.

With

,Ox, ) functor to the natural exaCt sequence

---4 ~X

notations

as

~

~X'/X

above, D ~ will

~

O.

be

called

the inverse

image

(2.1.3)

Let us suppose

herent 0 X -submodule

that X' = Spec

of

X" Because

("~"

means

(A) and that

are of finite

type,

Dis

a co-

E X and =

by

Derk(A)~ and Derk(A')~.

"associated

module

by M ~ where M is an A-submodule

(2.1.3.2)

Now, since

that X = Spec

of GX and ~X,

are coherent modules given respectively

(2.1.3.1)

(A'),

over

the

scheme").

of Derk(A).

Derk(A)

Let us suppose

that

D is given

Let M' be the image of the A-morphism

HomA(~A,A').

~A is locally free one has that

H°mA(~A'A')

(2.1.3.2)

= H°mA'(~A ~A A',A')

H°mA'(~A ~A A',A') ~A A'-

Let M''' be the image of M' ~A A' by

(2.1.3.3)

M' ~A A'

~

H°mA'(~A ~A A',A').

And finally, if M wis the inverse image of M''' by the natural morphism of A'-modules

(2.1.3.4)

Derk(A')

---* H°mA'(~A ~A A',A').

Then one has that D w is a coherent Ox,-submodule

(2.1.4)

Proposition.

dimensional

Let ~ : X'

distribution

of EX, given by (M~) ~.

~X be a birrational

morphism and let D be an uni-

over X. Then D r is also an unidimensional

distribution.

Proof Let P' be a closed point of X' and P = w(P'). Then, with notation like in (2.1.1) one has that

(2.1.4.1)

(~*D")p, =D"p ~0 Ox' X,P

It follows that ~ ' i s

'P'"

generated by a sinale element~ince~

is a torsion Ox,,p,-module

and

is birrational, ~X,/X,p,

2.1.4.2)

iS

0

exact.

modules

Now,

of

a single of

rank

n.

element,

,

E X',P'

E x , , p , and Since

the

Ox , p,

right

is

Now t h e r e s u l t

Adapted

blowing-ups

Let

E be a n o r m a l

(2.2.1) X.

"for

Y has

"normal

each c l o s e d

crossings

P of

with

X there

exists

o f O x , P and t w o s e t s

in

fact

where

I E , P and

that

Iy,p

a r.s.

(E,Y)".

If

(2.2.2)

Let

denote

of

p.

P'

(2.1.4.2)

are

thatD~,

is

that

D~ is

free

0X,,p ,

generated

coherent

in

by view

=

E,Y

be

as

above,

blowing-up

with

reduced

structure)

(2.2.3)

Definition. over

E".

following

a regular

condition

is

subscheme satisfied

s y s t e m o£ p a r a m e t e r s in

{1,...,n~

such t h a t

xi).OX, P

[ x i " OX, P i ~ B the

ideals

conditions

let

us

of

E and

(2.2.1.1)

suppose

that

~

X

"n : X'

transform

Y be a c l o s e d

is

Y in

0 X , P.

"suited

for

We s h a l l the

pair

Y has

normal

crossings

with

E,

by

(2.2.2.1)

to

the

let

P 4 E we assume B = ~.

us d e n o t e

bution

X and

the

~ iEA

respectively

satisfying

of

A,B contained

Iy,p

ted

from the

E iff

IE, P = (

the

side

one d e d u c e s

divisor

(2.2.1.1)

let

hand

a U.F,D.

follows

crossings"

point

x = (Xl,...,Xn)

say

EHomOX, (~*ax, 0x, )]

(2.1.3).

(2.2)

of

both

~

X

center given

(resp.

of D by~ ",

Let

Y and

by -l(y

let u E).

us d e n o t e Then E'

D be a m u l t i p l i c a t i v e l y

and resp.

adapted that

to

E).

(~(D~),E')

by

is

the

closed

a normal

the

say that "strict

subscheme

crossings

irreducible

We s h a l l is

E'

divisor

(with of

unidimensional

distri-

a ( 0 ~)

"strict

transform

is

the

o f D by

X'.

~ adap

10

(2.2.4) and by

Remark.

E'mU' the

and

induced

Let U = X-Y, U' E mU

= ~-I(u).

correspond

isomorphism

one

between

Since

~[U':

to a n o t h e r the

U'

----* U

by ~ IU' , one

tangent

sheaves

of

is an

has that ~'

isomorphism

DwIj

and U . T h e n

~DIU one has

that

(2.2.4.1)

(~(D~),E')IU

So, t h e d i f f e r e n c e Let -1

(Y)

at

P'.

( D ,E')p,.

In

be chosen i n

between

(D#,E ' )

and

P'

be

a closed

Let

D'

be a g e n e r a t o r

view

of

point

(1.3.3.3),

such a way t h a t

p does

order

not

depend

on

o£ # on Y a d a p t e d

the

S

Equations.

c S',

x'1'''''x

Let

point

us

of X'

consider such

and

let

f

and

(2.2.4.1)

an i n t e g e r

in

-1(y).

be a l o c a l let

equation

of

D" be a g e n e r a t o r

of

one has t h a t

D' and D" may

P such t h a t

= D".

closed

k is a c o e f f i c i e n t

n of S',

i° ~

8 and

the

point

P'.

We

denote

(2.2.5.1)

field

for S' and

scalars

x

shall

call

it the

"blo-

it by

us c o n s i d e r

(2.2.5.2)

the

hand

side

is

us d e n o t e

thece

i 6 B -

(2.2.1)

exist

and

(2.2.2)

s=O X

a r.s.

{io} , s u c h

and

and s ' = O X

let p,

of p.

that

o

i

+ ~i)x'i

i e B -{i o

o

}

= x' i

natural

a rank-n

= P. Let

~i 6 k,

= (x'

1

¢:

of

= x' i

xi

Let

situation

~(P')

o

left

and

that

xi

The

concentrated

p( ~ E ; Y ) .

P' be a c l o s e d Then

-l(y)

to E" and we s h a l l

(2.2.4.3)

(2.2.5)

is

# , ( a ( D ) , E )p,

of

fP.D'

number

wing-up

of

exists

(2.2.4.2)

This

(e(D~),E')

(1.3.5)

there

JIU,.

' =

mapping

£S ®S S'

free

~

S'-module

£S''

generated

by dx i ® 1,

i=l,...,n

and

11

one has that

i (2.2.5.3)

¢(dx i

~ 1)

= dx i

=

dx' i o

if i = i°

x i+~i)dx'i

+x o

i

dxi

if

dx t i

Thus,

if we denote

by

~i' i=1,...,n

i

~ B -

{i o}

o

if i ~ B

the dual basis of (dx i ® 1), then the induced

morphism

(2.2.5.4)

~ : Der(S')

--e

H°ms'(£S

®S S ' , S ' )

is given by

~io+J~ ~ B.{i ° ~.(xt+~ j j )D.g (2.2.5.5)

@(8lax'

l

) =

i ~ B- ~

x'i " ~i o

idB

a.

1

Now, l e t

us consider

(2.2.5.6)

the

i = io

set

A'

given by

A' = {i o} u {i & A n B; ~i = 0} ~ (A-B).

Then one has that E' is given at P' by

(2.2.5.7)

H

j F=A' Remark t h a t

A', Now,

and so E ' , let

us suppose

(2.2.5.8)

Then, the corresponding

(2.2.5.9)

do n o t

D =

that

x'.

= 0

J

depend on t h e choice Dp i s

generated

[ a.x.~IBx i + ~ i e A i z i~

of

i

o

made i n

by

ai~lax i. A

module im HOms,(~ s ~ S',S') is generated

D" =

[ aixi~ i + [ a. 2. i ~A i ~A I I

by

(2.2.5.1).

12

Now,

in view of (2.2.5.5)

guish two cases:

one can obtain

a generator

io 6 A, i ° # A. If io ~ A, then D~p,

o, = I x i ]~[ai × ' i

(2.2.s.lo)

O

+

In

this

iff

case

D'

there is

o

that io ~ A, then

(2.2.5.11)

is

i

~ B-A,

such

to

E'

and

is

it

is generated

= Ix i

]~

+

that

?

a i_ ~ x ' i

a generator

/x'i

t

).x

0

L . i ~ A ' n B-{IO}

.S' and o (~',E')p,.

of

e = 0 otherwise. Let

us s u p p o s e

(a.-a.

z

z

.8/Sx

,

i

0

/x'

)X'

i

+ 0

i

,;)/Sx'

i

+

o

L~ (ai-a i /x' i )(x'i+~i)~/~x'i+ i 6(A-A') ~ B o o

+

~ = 1 iff

i

0

[ (ai/x' i -(x'i+~i) i e B-A- {io } o

where

+

by

[(ai

0

+

i

+ [ ai~/~x'i]. i4A' vB

a_,z

,

D'

O

(x'i+(i)a i ))l~x' i + o

i

( D ,E')p,

+

O

(ai-a i )(x'i+Ei)~/Sx' i } o

L~ aix'i~/Sx' i 6A'-B

adapted

by

(ai-a i )x i~/~x + ~ B- {io~ o i

+ [ (ai/x' i i eB-A o

¢ = I

~/~×'i

O

L r

+

is generated

,

[ i6A'

i~-(A-A') n B - { i

where

of (D ~ D, ')p,. We shall distin-

there

[ aix'i~/Sx' i 6 A'-B

i

is

(note

ai,

i

6

B-A

+ i[~A '

io

,

D

a i /x i )B/Sx i + o o

uB

~

a'8/~x'i]" 1

B-A)

such

that

ai

~ x'.z

.S'

and

o e=

0 otherwise.

Remark

that

in

this

case

one

can h a v e

that

D'/x'

i

is

a generator

o of ~p ,,

if,

for

instance,

one has t h a t

A = @, B = { 1 , . . . , n }

and t h e

only

a i~

x'i

S' o

is a i . o In both cases, is

generated

(2.2.5.12)

and in view of (2.2.4),

by

Ix' i

] - P.D' o

the strict transform

(~(D~),E')p,

13

where p= p ( ~ ) , E ; Y ) .

(2.3)

Formal b l o w i n g - u p s

(2.3.1)

Let

of

X^

one

p.

suited

formal

adopt

the

can

establish

fop

the

(2.3.2)

closed

Let

point

notatlon the

pair

point

(1.4.1).

of

in

For

"normal

the

irreducible

a

regular

crossings

same way as

relatively

to

in E^

closed

with

subscheme

E^'' and o£ " r . s .

(2.2.1).

Now,

unidimensional

let

D^

Y^ of

be a

distribution

P o f X ^.

~: X' ---~ X^ be t h e

-1 ( Y ^ ) w

of

of

concept

(E^,Y^) "

multiplicatively

over the

sed

us

such

blowing-up

that

~(P')

o£ X^ w i t h

= P.

Let

R'

=

center

0×,

p,

^

Y^ and l e t . We s h a l l

P'

be a c l o -

call

the

mor-

phism

(2.3.2.1)

=^:

"directional

(2.3.3) plete

The morphism regular

R'-module D ^'

blowing-up

local

rings is

free

image o f t h e

0 ^ be t h e

is

inverse

= Spec

form

of~

a formal

(R').

It

Definition. by ~ ^ " .

rank

n and

•

image o£ ~ ^ '

unidimensional

will

be c a l l e d

With If

Y^ i n

to

X^

the

direction

a ring

it

is

of

morphism

k as a c o e f f i c i e n t

Derk(R')

~^

(2.3.4)

of

,

field.

P'"

R + R'

between com-

One has t h a t

isomorphic

to

Derk(R)

the

®R R ' .

Let

morphism

(2.3.3.2)

X^ '

center

O^ ®R R'

and l e t

(R')

corresponds

having

(2.3.3.1)

Then

X^ w i t h

(2.3.2.1)

Derk(R,R')

be t h e

of

Spec

notations

E^ '

Derk(R,R')

by t h e

----~

over the

u Y-),

by r e s t r i c t i o n )

closed

point

o£

image o£ 0 ^ by ~ ^ " .

as a b o v e , we s h a l l

= ~^-I(E^

(given

Derk(R,R' )

distribution "inverse

injection

we s h a l l

call call

a ( ~ ~^) t h e (~(D

"strict

),E ^ ) the

trans "adapted

14

to

E^ s t r i c t

(2.3.5)

transform

Equations.Let X l , . . , x n

Then there note

o f D ^ by ~ ^ " .

by

is 2'

(2.3.3.3)

o f p.

o f p. o f R' X ' l ' ' ' ' ' X ' n

i=l,...,n,

has j u s t

(~(D~^),E ^ ' )

(2.3.6)

a r.s.

be a r . s .

the

the

R'-basis

expression o f

of

R suited

such t h a t

o f Derk(R,R')

the associated t o

Let

X,~,E,Y,P

given by ~ i

be a closed p o i n t

of

~.:

X such t h a t

=

and

~:

X'

(E^,Y^). Let us de-

@/~xi ® 1. Then

( 2 . 2 . 5 . 5 ) and the equations f o r

D formal d i s t r i b u t i o n

corresponding subschemes. Let

the p a i r

( 2 . 2 . 5 . 1 ) holds.

can be deduced e x a c t l y in the same way as in

Proposition.

for

( ~ ^,E ^ ' )

and

(2.2.5).

>X be as

in

(2.2).

L e t D^ = DA be P

at P. Let X^ be Spec (O'X, P) and E^,Y ^ the

X*--*× ^ be the blowing-up w i t h ~(P')

= P and l e t

center Y^. Let P'

P* be the closed p o i n t o f

X~

associated t o P' by the u n i v e r s a l p r o p e r t y o f the blowing-up. Let X~'=Spec(O~p,) = ^

=Spec(OX,,p, ) and

let

~ : X^ '

X^ be the

corresponding morphism.

Then one has

that

D~^ = ( DTr)^p, 1T

(2.3.6.1)

1T

(O ^,E ^ ' ) (a(D=^),E ^'1

Proof.

It

follows

from

,

= (O ,E )^ p, = (~(D~),E')^p,

(2.2.5)

and

(2.3.5).

3. SINGULAR LOCUS AND BLOWING-UP

(3.1)

Adapted order

(3.1.1) Definition. to

a n.Co

divisor

of a vector

field

Let D be &n u n i d i m e n s i o n a i d i s t r i b u t i o n E.

Let

Q be

a

(not

n e c e s s a r i l y closed)

order of D at Q" will be the m a x i m u m of the

(3.1.1.1)

where

q is

DOWn

the

maximal

ideal

of

m

OX,Q "

oven X which

is adapted

point of X. The

"adapted

integers m such that

.2 X,Q[E]

It

will

be d e n o t e d

by v ( D , E , Q ) .

If

15

Z

c X is

a closed

subscheme

and

Q its

generic

= v (D,E,Q).

(For short:

n.c.=

normal crossings).

(3.1.2)

the

case,

one

Spec let

In

(R) E^

where

formal R is

be a n . e .

a complete

divisor

on

proceeds

local

X^

in

we s h a l l

a similar

regular

and l e t

point,

ring

way.

having

Z~ be a f o r m a l

denote

Let

X^

v(D,E,Z)

be t h e

scheme

k as a c o e f f i c i e n t

unidimensional

field,

distribution

over the closed point of X ^ which is adapted to E ^. Let O be a point of X ^ and the corresponding

ideal of R. Then

v(~,E^,Q)

will be the m a x i m u m

=

q

integer m such

that

(3.1.2.1)

D ^ C n m. Der k ( R ) [ I ]

where I

= ideal

(3.1.3).

With

and

let

by

i ~A xi.

of

E.

notations

Xl,...,x

as a b o v e ,

n be a r . s .

Let

us suppose t h a t

(3.1.3.1)

Q

(3.1.3.2)

ding

formal

with nal

generated

is

the

objects,

q-adic

= min

order

of

P.

distribution

Let

for

ai.

of

X such t h a t

(E,{P})

Pe

such t h a t

(q}

E is

given

by

+ i ~ A ai~/Sxi"

that

Moreover,

...,n)

if

Q^,~^,...

are the

correspon-

one has t h a t

one deduces e a s i l y

Theorem.

point

(v q ( a i ) ; i=1

v ( / ~ p , E ^ , Q ^)

center

has that

OX, P s u i t e d

in Ox,p. Then one has

(3.1.3.3)

(3.1.4)

of

Dp i s

v(D,E,Q)

vq (a i )

Finally,

p.

P be a c l o s e d

O = i ~Aaixis/Sxi

Let ~ be the ideal of

where

of

let

an e x p r e s s i o n

P be a c l o s e d

Then, f o r Dover

any a d a p t e d X and f o r

point to

= v(O,E,Q).

like

of

(3.1.3.2)

X and

let

for

7:

E multiplicatively

any c l o s e d

point

P' o f

the

formal

X'--~ X be t h e irreducible

X'

case.

such t h a t

blowing-up

unidimensiow(p')

= P one

16

(3.1.4.1)

v(D,E,P)

Proof. suppose

that

Let

Xl,...,x

Dp i s

n be

genepatd

(3.1.4.2)

tet

denote

(2.2.5.10), If

e:

or

~ =

order from

r-1.

If

the

equations.

follows

(3.1.5)

Let

point

of

of D^

X^

be

X^ .

Let

a

formal

Then

one

exist

P-l,

The

Remark. case.

For

be

same o n e

Stability

as

in

in

the

in

if

ape two

there

order

Let

two

Considering posibilites:

posibilities that

exist

and

let

closed

the e=

let

us

Then

i ~ A such

equations

0 or

e=

fop p = p ( D , E , P ) :

v(a )i = r, w h e r e

o£Ox,p.

~ ^:

point

define~

1.

p = r

~(a i) is the

the m e s u k

follows

that 9 ( a i) : r and the

over

X^ '

~X ^

P of the

X^

and

be t h e in

the

directional direction

of

closed

(3.1.4).

as

D p is

~ the

D'

Definition.

and

> M((~(D~^),E^'),E^',P').

results

(3.1.6.2)

(3.1.7)

ape

filtration

2.3

u.d.

as

instance,

adapted)

(E,P)

X a ~/ax i' i ~A z

there

(3.1.4)

and

generated

by

D = yB/~x

(non

for

that

a n d we made x = x ' , y = x ' y '

the

+

i 6 A such

then

(3.1.6.1)

and

OX, P s u i t e d

way.

m. i .

has

of

9((a(D~),E'),E',P').

O, t h e r e

~(/T,E^,P)

Proof.

adapted

c=

center

(3.1.5.1)

(3.1.6)

p=

X^

with

p.

i

the q p - a d i c

to

in an a n a l o g

Corollary.

blowing-up

If

=

there

respect

of

(2.2.5.12),

If

p= r t h e n

of a i w i t h

r'

and

r = r' = O.

r.s.

[ aixi~ax i 6A

v(D,E,P),

(2.2.5.11)

1 then

result

P'.

r =

a

~((e(D~),E'),E',P').

by

D =

us

>

strict

= y'x'@/ax'

has

D be

a

been

do

not

generated

by

+ x39/~y

tansform

ks

+ (x'2-y'2)~/By

increased

m. ~.

(3.1.5)

u.

d.

'

by a unit,

defined

over

work

for

the

non

17

X,

aqd

Si~ r

adapted

(D,E), >0

to

will

we s h a l l

E."The

be

the

sed to

X.

say

the

Sing

noetherian

(3.1.9)

the

fact

(D,E)

has

let

us d e n o t e

case,

(3.1.3,3)

of

or

X such

to

that

E",

denoted

u(~,E,Q)

>

1.

by For

any

by " ^ " t h e

D

is

there

this

let

(D,E)

multipLicatively

no components o f X,

Sing

> r }.

is

and Sing r

(D,E)

irreducible

codimension

a maximum r

one.

such

is

clo-

equivalent

FinaLly,

that

are

because

Sing r

(D,E)

of

~ ~,

Szngr(D,E).

and

(3.1.8)

X^ = Spec

corresponding

may be s t a b l i s h e d

(O~,p)

objects

where P i s in

X^

in

a similar

a closed

by t h e

way f o r

point

of

X and

morphism X^ --~ X. Then

one deduces t h a t

Sing r

(D^p,E ^)

= Sing r

( D , E ) ^.

The d i r e c t r i x

(3.2.1) lity

Q of

relatively

= { Q ~ X; v ( D , E , Q )

that

(3.1.7)

Moreover,

(3.1.9.1)

(3.2.)

of ~

semicontin{uous.So

of

by Sam ( D , E )

The c o n c e p t s

formal

points

(D,E)

is

properties

denote

the

from

order

Moreover,

that

we s h a l l

of

Sing r

The a d a p t e d

in

set

locus

denote

(3,1,7.1)

(3.1.8)

singular

The aim o£ t h i s

in

(3.1.4.1)

and

strict

tangent

space

(3.2.2)

Let

(R,q)

paragraph

to

estimate

(3.1.5.1).

For

introduced

by H i r o n a k a

be a l o c a l

a ~ R be such t h a t

is

~ q (a) ->1.

regular

this

Ping

L e t H be t h e

necessary conditions

we s h a l l

use t h e

for

concept

of

the

equa-

directrix

(II01,111I).

having

k as a c o e f f i c i e n t

minimum k - v e c t o r

field

subspace o f

and l e t

GPq I ( R )

such

that (3.2.2.1) Let

J(a)

In = H.GP

q

For an r ~ ",)(a),

(R),

the

we s h a l l

(a)

"directrix denote

J

C- k[H] of a"

r

(a)

c Gpn ( R ) .

is

= a(a),

the if

subscheme V ( J ( a ) ) e = ,0(a) a n d j r ( a )

c Spec = O, i f

(GP ( R ) ) . h r>~)(a).

18

(3.2.3.)

Lemma.

wing-up

of

strict

With

Z with

transform

Asstxne t h a t

Y'

is

notations

center

of

on

Y by ~ .

given

&t

P'

(3.2.3.1)

(3.2.4) short,

closed

point

P'

a' E

E Proj

and a d a p t e d t o

r . s . . o£ p.. o f DX, P s u i t e d

for

(3.2.4.1)

(2.2.5.10),

us suppose (2.2.5.11)

(if M = P(~E,P))

P.

0X,p,

that

~nd

(v(J(a)))

(R) and ~:

Let

po±nt

Y = V(a) of

Z'

Z'

and

such

~(a')

--~ Z t h e let

that

Y'

be t h e

~ (P')

= v(a),

blo-

= P.

then

~ -l(p).

irreducible

E and l e t

r = v(t),E,P)

and

(2.2.5.12)

iff

With

there

is

notations

unidimensional

P be a c l o s e d

{E,P) such t h a t

that

: p= r

Definition.

Z = Spec

be a c l o s e d

J(D,E,P)

=

J(D,E,P)

=

o£ X. L e t

(for

X l , . . . , x n be a

+ i~A ai8/Sxi"

> 1.

(see

Now, i n

also

the

view of the

proof

no i ~ A such t h a t

as above l e t

(3.2.5.1)

point

distribution

E i s g i v e n by i ~ A X i . Lett)p be g e n e r a t e d by

D = i EAL~a i x i ~ / a x i

let

(3.2.5)

let

be a m u l t l p l i c a t i v e l y

m.i.u.d.)

Finally,

by

above,

Ilol).

(See

t)

Let

its Let

P'

Proof.

as

of

equations

(3.1.4))

v(ai)=r

one has t h a t

and p = r - 1

otherwise

us d e f i n e

~ L~A

jr(a I)

if

p = r.

~

j r ( a 1)

if

p =

r-1.

i @A The is

"dLrectrix denoted

(3.2.6) the

-up

by Dip

Remark.

generator

(3.2.7) of

o f Z) a t

suppose t h a t

is

the

subscheme

V(J(t),E,P))

fSpec

(GP(Ox,p))

and

it

choice

of

(~E,P).

The

D of

ideal

J (~E,P)

of

GP(Ox, P)

Dp n o r on t h e c h o l f e o f

Proposition. X with

P"

center ~(D,E,P)

Let P,

X,D,E,P Let

P'

the

be as i n

(3.2.4),

be a c l o s e d

= u((~(t)~),E'),E',p

n.s.

')

point then

does of

not p.

let of

depend

suited

~: X'

on t h e

for

(E,P).

X' --~ X be t h e

with

~ (P')

blowing-

= P and Let

us

19

(3.2.7.1)

P' e

Proof. i~A, (i.

is e.

Let

such t h a t

, by t h e

(2.2.5.10),

case

v(a i)

(2.2.5.11)

(Remark t h a t

the

result

directrix

(3.2.7)

(3.2.9)

(2.2.5.12)

ai,

we o n l y

may be d e f i n e d

The b e h a v i o u r

of

the

case

of

monoidalblowlng-ups

even

in

the

analogous

case

by t h e

of

one.

by t h e

given

Spec(Ox,p))

then,

we have t h a t

the

exactly

us make t h e

directrix.

the

directrix

of

of a i

by

equations

If

p = r,

from the

at a closed

in

has

not

blowing

ai, #

of

the

same

lemma ( 3 . 2 . 3 ) .

point).

same way as i n

In t h e

(3.2.5).

formal

Also the

= (x'y')x'a/ax

order

is

the

the if

' + (z,3

= (y,z)

is

-

but

as

not

see

clean

dimension

D p is

blowing-up

transform

same one

As we s h a l l

is

in

as

of

as good as i n (3.4.11), (3.2.7).

the

generated

in

the the

Moreover,

directrix

may be

by

+ z3a/ay + x5a/~z

Here J ( ~ E , P )

quadratic

(3.2.5)

a result

-up,

For i n s t a n c e ,

The s t r i c t

The a d a p t e d has d i m e n s i o n

from the

follows

directrix

assume t h a t

transform

v ( a ' L) = r .

result

for varieties.

quadratic

by x = 0.

D'

(3.3.1)

strict

D = (zy).xa/ax

Let

~et p = r-l,

the

the

one

blowing-up.

(3.2.9.2)

(3.3)

of

concept

(3.2.9.1)

is

define

3.2.4.

&i':~ i s

i E A~ Now, t h e

case

where E i s

of

(P).

remains true.

Remark.

increased

in

-1

(Z~,E,P))~

notations

induced

and

the

the

(Dir

= r and t h a t

blowing-up

~rgument works f o r

(3.2.8)

us a d o p t

Proj

and t h e d i m e n s i o n

in

the

generated

y ' 2z, ) ~ / 3 y '

now t h e

direction

at this

of the

directrix

y = z = 0 indicated

point

by

+ (x ,2 y , x ' 2 ) ~ / ~ z ,

directrix

is

given

by ~ '

= 0 and i t

two.

Stationary s e q u e n c e s

Unless

n ~ 2,

making s u c c e s s i v e

it

quadratic

is

not

possible

blowing-ups.

in

general

to

reduce the (in

adapted

13 I,I 8 I,I131

order the

by

proof

20

for

n =

rate

2

is

made).

stationary

nitely

near

(3.3.2.)

This

paragraph

situations

when

devoted

one m a k e s

to

identify

blowing

-up

those

along

curves

their

which

sequence

of

geneinfi-

points,

Let

be a c l o s e d

Y be a r e g u l a r point.

Let

curve

of

X having

0 be a m . i . u . d ,

(3.3.2.1)

t

is

over

normal X and

crossings

adapted

to

with E.

E and

The

let

P ( Y

sequence

(~(t),X(t),E(t),Y(t),P(t),~(t))

= 0,1,...,

i s obtained i n d u c t i v e l y as f o l l o w s : a) X(o) = X, E(o) = E. Y(o) = Y. P(o) = P. D(o) = D. b)

~(t):

X ( t ) --~ X ( t - 1 )

i s the blowing-up o f X ( t - 1 ) w i t h center in P ( t - 1 ) .

c) Y ( t )

i s the s t r i c t

t r a n s f o r m o£ Y ( t - 1 ) by ~ ( t ) .

d) P ( t )

i s the o n l y closed p o i n t in

e) E ( t )

= ~(t)-1(E(t-1)

£) O ( t )

i s the s t r i c t

U

P(t-1)

(3.3.2.2)

reduced s t r u c t u r e .

> I one has t h a t f o r a component F o f E ( t )

Y(t) ¢

let

) with i t s

n Y(t).

t r a n s f o r m o f /~(t-1) by ~ ( t ) adapted t o E ( t - 1 ) .

Let us observe t h a t f o r t

For the sake o f s i m p l i c i t y

~(t)-1(p(t-1))

F

us assume t h a t

(3.3.2.2)

is also true f o r t

= O.

We

s h a l l denote

(3.3.2.3)

r(t) = v ( D ( t ) , E ( t ) , P ( t ) ) .

(3.3.2.4)

~(t)

(3.3.3)

Definition.

(3.3.4)

Let

and

if

B).

Any

i

o

x =

~ B one

a ~

I ~ INn a n d

if

The

sequence

(Xl,...,xn) has

be

that

0 X , P may be I=(il,.O.,Zn)

•

i

o

= p(O(t),E(t),P(t)).

(3.3,2.1)

a

r.s.

~ A by

expressed ,

xI=

xI

in il

of

is

"stationary"

p.

suited

(3.3.2.2). exactly

i n ...x n . Let

(see

for

iff

(E,Y)

(2.2.1)

one way a s us d e n o t e

r(t)=

at for

a =~a I

r(O)

P.

Then

for

where

t.

# B = n-1

notations xI

all

on A a n d aI

~ k,

21

(3.3.4.1)

Now,

Exp

let

us a s s u m e

that

(a,x)

r = r(O) ~

(3.3.4.2)

D =

A = A° ,

{1,...,n}

- A = A 1.

(3.3.4.3)

1,

aI

~ 0 }

~ n.

p = p (0) and

~ aixi8/8x e A

i Let

= {I;

i

+

for

1 = 0,1

and

let

= rain{

n ) E Exp

ai~/~x i •

hi

/(p+l-

for

y

(ai,x)

y(D,E,Y,P,x) 1

the

k

= y

(D,E,Y,P,x),

c a s e p=

r,

let

1 = 0,1.

= min

Here

~Bhj;

(3.3.5)

The

invariants

us d e n o t e

y

x(0)

(3.3.5.1)

xi

and

6 are

= x and

(t)

= xi

o

Let

(yo,

we assume

h e Exp

6(D,E,Y,P,x)

then

>0}

~ Bhj

1)

(9)

min

= ~ . Finally,

and o n l y

(ai,x)

, i

( ~,

1).

~ AI }

1=0,1

J

(3.3.4.5)

Let

, i ~ A 1, p + l - j

us d e n o t e

61 = m i n { 1 - 1 +

x.

L? Bh J) ; j~

us d e n o t e

(3.3.4.4)

where

by

i 4 A

0 = (hl,...,h

generated

Let

y 1( D , E , Y , P , x )

_h

is

that D p

x(t)

is a

us d e n o t e

r.s.

for

= min

actually

let

x(t)

(t-l);

be o b t a i n e d

xi(t)x

o

of

p.

independent

i

(t)

of

the

choice

of

the

from

= xi(t-1)

i

e B,

o

o£

Ox(t),P(t)

which

is

suited

for

the

pair

(E(t),Y(t)).

short

(3.3.5.2)

y (t)

=y

(D(t),E(t),Y(t),P(t),x(t))

etc.

(3.3.6)

Lemma.

Assume

that

p(O)

= r(O)-l.

a)

I £ y ( O ) > 2,

then

r(1)

= r(O),

b)

I£ y(O)<

then

r(1)

< r(O).

2,

suited

Then ~ (1)

= r(1)-1

and

~(1)

= y(O)-l.

22

c)

If

y(0)

~(I) Proof.

= 2,

then

r(1)

< r(0)

or

r(1)

= r(0),

(3.3.5.1),

( 3 . 3 . 7 ) Lemma. Assume t h a t p (0) = r ( 0 ) . y(0) ~ 2 ,

then r ( 1 )

Y(O) <2,

( 2 . 2 . 5 . 1 1 ) and ( 2 . 2 . 5 . 1 2 ) .

Then

= r(0), #(I)

and ¢ ( 0 ) < y°(0) i f f If

and

the r e s u l t f o l l o w s b y 1ookin 9 a t the monomials

in the c o e f f i c i e n t s in view o f the equations ( 2 . 2 . 5 . 1 0 ) ,

b)

= r(I)-1

= i.

Looking a t

a) I f

p(1)

= r(l~,

6(1) = 6 ( 0 ) , X(1) = y ( 0 ) - I

~(1) < y °(1).

~ ( 0 ) < ~ ( 0 ) and ~(0) = r then r ( 1 )

= r(0),

P(1) = r ( I ) - 1

and y ( 1 ) = ~ . c) (Note that

I f Y (0) < 2 and

in

b) t h e

only

6(0)<

r or

possibility

Y°(O) ~ ~ ( 0 ) , is Y (0)

Proof. S i m i l a r t o the p r o o f o f

(3.3,8)

Proposition.

statements

The

p(O)

= r(O)-I

b) p (0) = r(O)

Y (t)

and and

r(O).

= 1).

(3.3.6).

(3.3.2.I)

#(0)

if

the

c)

holds,

at

Y(O)

Y(O)

C) p (0) = r(O),

is

stationary

1££ one o£ t h e

followin

9

= =.

= ~.

< X°(O)

step

t

and

6(0) = r.

= ~ (0)

one

has

that

p (t)

=

r(t)-1

and

= =.

Proof. I f

(3.3.9) a)

r(1)<

is v e r i f i e d :

a)

Moreover,

sequence

then

or

The b)

respect

(3.3.9.1)

sequence

of to

f o l l o w s from ( 3 . 3 . 6 ) and ( 3 . 3 . 7 ) .

(3.3.8).

the

ideal

(3.3.2.1) For of

Y,

stationary

an a E 0 X , p , then

"Vy(ai)>r-1 there

is

exists

3.3.8.

if

i i

let a)

is

iff

one has r e s i d u a l l y

us d e n o t e

by Vy(a)

equivalent

~ A, V y ( a i ) > r ~ A such t h a t

if v(a i)

to

i

~ A and = r".

the

the order

situations of

a with

23

w h e r e v ( a )z b)

is

denotes the

equivalent

"Vy(a i )

(3.3.10)

with

~ r if

Remark. One can t r y

that

p(t)

= p (0)

adapted order.

respect

to the

maximal

ideal

o f ~nX, P. Now 3 . 3 . 8 .

to

(3.3.9.2)

ring

order

to

for

all

For i n s t a n c e ,

if

(3o3.10.1)

i

~ A.

modify t.

the

But

Dp i s

D = y

2

Vy(a i ) ~

concept

if

i ~ A".

o£ s t a t i o n a r y

p does

not

generated

by

play

in

sequence by r e q u i -

general

the

role

o£ an

) 3 ) 3 .-+ y .y -- + z -~x ~y az

where E i s

given

by y = 0,

one has

~(D,E,P)

blowing-up

given

by

y = x'y'

, z = x'z

x = x',

r+l

:

2 and p(Z],E,P) = 1. I f , then the s t r i c t

we make the

t r a n s f o r m i s gene-

r a t e d a t P' by

y'2x'

a

('3.3.10.2)

D' =

The a d a p t e d

o r d e r remalns t h e

3.4.

Permissible

(3,4.1) sed

adapted to

of

E" i f f a) Y i s b)

If ~:

same,

3_y,2 but

~~

)Y'

y'

+ (x,z,3_y,2z ,)

p(Z]',E',P')

2, ~z''

= 2.

Centers

Definition.

subscheme

;) x~ + ( x ' y '

Let X.

~ be a m . i . u . d ,

We s h a l l

the

following

regular D'

say t h a t

is

X' --~ X o f

Y is

conditions

and i t

the

o v e r X and a d a p t e d t o

a "weakly permissible

transform

X with

center

Y be a c l o -

center

for

D and

are verified:

has n o r m a l c r o s s i n g s

strict

E and l e t

of

Y, t h e n

with

E.

D adapted for

to

any c l o s e d

E by t h e point

blowing-up

P'

of

X'

one

has t h a t

(3.4.1.1)

(3.4.2) dition

v(Z)',E',P')

One can e s t a b l i s h b)

the

<

same d e f i n i t i o n

~(D,E,~(P'))

for

the

formal

case c h a n g i n g

the

con-

by b) ^

If

~^:

X~ '

----~X ^

is

any d i r e c t i o n a l

blowing-up

o£ X^ w i t h

center

Y^

24

(see

(2.3.2))

and i f

(3.4.2.1)

~(/P,,E^,

where

(3.4.3) cate

P^ and P^'

The e x i s t e n c e

the

existence

wing de£inition

Definition.

We s h a l l

say that

a center

a sligth

Let Y is

a)

Y is

b)

Let

closed

be as

"permissible

of

in

(3.4.3) D at

for

(3,3)

that

the

then

o£ X ^ and X ^ ' ,

as i n

blowing-up

o f O ^,

^)

points

sequences the

transform

~D^,E^,p

generalization

regular x =

us suppose

and i t

(Xl,...,x)

that

~p

has n o r m a l

must

conditions

and

a posteriori

one w a n t s t o

let

P adapted

be a r . s .

n

is

crossings

generated

of

use,

(3.3.9.1)

E" i f £

with

E at

the

The f o l i o -

and ( 3 . 3 . 9 . 2 )

P be a c l o s e d

to

indi-

point

o£ Y.

following

con-

one o f

the

two f o l l o w i n g

"Vy(a i)

and t h e r e

(3.4.4.3)

conditions

~ r-1

"Vy(ai) ~ r

if

if

o f OX,

A or

suited

P.

the

pair

(E,Y)

is

so i n

definition

fbr

for

issatis£ied:

~ B. V y ( a i ) ~

i g B-A such t h a t

i (

P

point

+ i ~A ai~laxi"

i g A on i

exist

p.

the

by

D = i ~ Aaixi~/~xi

(3.4.4.2)

i ~ B.

Vy(ai~

~(ai)

~ r+l

r if

i E B-A

= r".

if

i

( B-A"

r = v(D,E,P). We s h a l l

all

the

the

formal

(3.4.5) that

for

D,E,Y

(3.4.4.4)

where

the

strict

areSatis£ied:

and l e t

Then,

are

the

p^,)

o£ s t a t i o n a r y

of

is

(3.4.4)

ditions

/[7", i s

closed

points.

that

Y is

Finally

"permissible one

can

for

D adapted

establish

to

analogously

E" i f f the

it

case.

Remark.

they

say

The

conditions

(3.4.4.2)

do n o t

depend on t h e

On t h e

other

hand,

suited

the

set

and

r.s. of

(3.4.4.3)

o£ p.

closed

are

intrinsic

in

the

sense

x,

polnts

of

Y for

which

Y is

permissi-

25

ble

is

are

not

to

open i n t h e open

in

Zapiski P,

topology.

because

of

Note t h a t

the

fact

the

that

conditions

E must

have

(3.3.9.1)

and ( 3 . 3 . 9 . 2 )

a transversal

component

Y.

(3.4.6)

Proposition.

E then Y is weakly

Proof.

notation

is

to

enough

(3.4.4.2)

(3.4.4.3)

result

(3.4.7)

Remark.

a fixed

P and

to

follows

prove

(3.4.1)

b).

Let

P be a n y c l o s e d

point

o f Y.

p(RE,Y)

= v(D,E,P)-I

N(D,E,Y)

= v(D,E,P).

iff

(3.4.6.2)

Now t h e

for D adapted

iff

(3.4.6.1)

and one has

If Y is permissible

as above.

permissible.

It

For P one has

With

in

v i e w o£ t h e

Although

(3.4.6.2)

p(D,E,Y) for

equations

of

(2.2.5).

does n o t depend on P, one can have

another

closed

point

P1 o f

Y.

In this

(3.4.6.1)

case,

for

necessari-

ly (3.4.7.1)

For

v(D,E,P)

instance

if

X = &3(k)

(3.4.7.1)

where

D is

globally

D = (xy2)x3/ax

E is

P = (0,0,0),

(3.4.8)

and

= v ( O , E , P 1) + 1.

The

given

by x y = O. L e t

generated

+ z3.y~/sy

by

+ z38/3z

Y be y = z = O. One o b t a i n s

the

example

fop

the

poin~

P1 = ( 1 , 0 , 0 ) .

same

kind

of

reasonning

With

notations

proves

the

result

(3.4.6)

for

the

formal

Y is

permi-

case.

(3.4.9) sible

Proposition. for

D and

adapted

to

E at

the

as

in

(3.4.4),

closed

point

let

us s u p p o s e

P and

let

us

that suppose

that

Y

is

26

a Curve. Then,

there is a f i n i t e

(3.4.9.1)

t

sequence

(~(t) ,X(t) ,E(t) ,Y(t),Q(t),D(t) )

= 0,1,...,N, a)

such t h a t X(O)

= X,

E(O)

= E,

Y(O)

= Y,

D(O)

= D

and

Q(O)

is

a closed

point

o£ Y ( O ) .

(i.e.

the

b)

w(t):

c)

E(t)

= ~(t)-l(E(t-1)

d)

Y(t)

is

the

e) Q ( t )

is

a closed

f)

~t)

is

the

g)

Finally

of

First

general (3.4.5)

we can of

are

produce a stationary

(3.4.10)

where

E is

given Let

of Y(t-1)

transform

center

Q(t-1).

by ~ ( t ) .

o£ D ( t - 1 ) for D(N)

by

w(t)

adapted to

and a d a p t e d t o

E(t-1).

E(N).

can be " g l o b a l i z e d " ) .

obtain

that

a finite

Y(t)

and

of

the

E(t)

curves

number o f

sequence and t h e n

result

would (I

points

does n o t w o r k v e r y w e l l

tlol

I I,

to

....

consider.

follows

for

have n o r m a l c r o s s i n g

from

monoidal

).

Now, i n

by

view

But each one w i l l

(3.3.8).

blowing

-up w i t h

permi-

i f D p is generated by @

~x

+ (xzr-1

by y = O. Here t h e Y be g i v e n

with

of Y(t).

permissible

D = (xzr-1)

mension one.

go

point

For instance,

(3.4.10.1)

blowing-up

desingularization

only

The d i r e c t r i x

ssible center.

the

transform

strict is

is

u {Q(t) }).

strict

centers

results there

--~ X ( t - f l )

Y(N)

permissible

Proof. the

X(t)

+ y

2r)

directrix

by y = z = O, w h i c h

@

Y

is

~y + z

given

is

r

)z

by x = z = 0 and i t

permissible

at

has d i -

P. Then two t h i n g s

"wrong": a)

The t a n g e n t

b)

If

we make t h e

order the

space o f Y i s

remains

center

not

blowing-up the

same

contained

x = x',

although

y : the

in y',

the

directrix.

z = z'y',

dimension

of

then the

the

adapted

directrix

and o f

agree.

An e x a m p l e such t h a t

the

tangent

space o f Y i s

contained

in

the

directrix

27

but

b)

holds

may b e g e n e r a t e d

D = ( y z r - 1 )B~x + ( x z r - 1

(3.4.10.2)

and Y b e i n g

(3.4.11

statements a) d i m b) A

Then

one

With

Din

=

then

P,

at

space

of Y at

0'

is

P'

the

b)

in

(3.4.4)

us

since

v ( a i)

least

that

one

of

the

two fo-

strict

with

transform

in Dip

center

of

Y,

(D,E,P). P'

~ adpated

is a c l o s e d to

E and

in

view

point

of

v(D',E'P')

space o f

(~,E,P)

Y at

/TpY)

P.

suppose

that

(3.4.4.3)

v(~,E,P)

=

v(~',E',P'),

= r and

one of t h i s

i & B-A must ai,

have

if a) holds,

holds. then

order

one

Then, the

equal

strict to

of

the

transform

equaof

r = v~,E,P).Since

all there

has t h a t

one has t h a t

(3.4.11.3)

J(O,E,P)

=

jr

[

(ai)

=

i ~ B-A the

result

follows

(3.4.11.4)

(3.4.4.4)

case we do n o t

use t h e

holds

fact

(

jr(ai)

a)

that

~ V(Jr(ai))/TpY). i e B-A

we can a p p l y

assumptions

~ i ~ 8-A,M(ai)=P

from the

P' e P r o j

If

=

~ ~-l(p)

j r ( a i) = J ~ , E , P )

holds,

And them

assume

P is c o n t a i n e d

blowing-up

Proj ( D i r

~

tangent

Let

(2.2.5) that

the

(3.4.11.2) If

as

one has t h a t

Proof.

exist

az

= n-1.

If ~: X' --~ X is the

where Tp Y i s

a i such

+ zr

{1,2, .... ,n}.

(3.4.11.1)

of

notations

(~,E,P)

v B =

~(P')

= ~(~E,P)

tions

~ Y ~y

has that

ii) with

2r)

holds:

i) The t a n g e n t

X'

+ Y

as a b o v e .

Proposition.

llowing

by m a k i n g

the

or b).

same k i n d

of

reasonement,

but

in

this

28

(3.4.12) ments that

As

i)

and

~(~,E,Y)

(3~4.13) formal

4.

a consequence ii)

of

of

(3.4.11)

hold

in

for

the

preceeding

a permissible

proof,

center

the

Y in

state-

the

case

(3.4.10),

(3.4.11)

and

statement,

let

(3.4.12)

may

be

also

formulated

for

the

RESOLUTION STATEMENTS

The g e n e r a l

Before

some p o s s i b l e

(4.1.2)

Let

statement

making

any

pathologies

X = &2(k)

in

where vet,

E is i?

given

by

we b l o w - u p

center

The s t m i c t

(4.1.2.2)

D'

and we have a s i t u a t i o n

~ d~

= x' . x'

the

singular

origin, at this

one.

So~

order.

by t h e

point is is

of only

vector

~ is

field

the

origin.

one c l o s e d

generated

~ ~ , + (y'-(m+l)y'x')

initial

illustrate

8y

there point

a few examples f o r

adapted

+ (y-m,yx)

only

the

transform

like

of the

D be g e n e r a t e d g l o b a l l y

x = O. Then t h e X with

us c o n s i d e r

behaviour

D = x.x

singular.

Moreo-

point

which

by

~ y,

cannot

be d e s i n g u l a r i z e d

by q u a -

blowing-ups,

dratic

(4.1.3) is

the

and l e t

(4.1.2.1)

If

the

crossings tly

always

remark

~(D,E,P).

=

Remark.

(4.1.1)

Y

last

case.

(4.1.)

is

the

the

(4.1.3.1)

Y is

a divisor

identity, with

E,

same t h i n g

but then as ~ .

o? X~ t h e n if

D is

the

strict

the

blowing-up

a m.i.u.d,

over

transform

The d i f f e r e n c e

E'

is

X'

X adapted

~X to

of E and

D' o£ D by ~ a d a p t e d t o

that ~'

= ~-I(E

~:

u y).

must be c o n s i d e r e d

X with

center

Y has

normal

E is

not

exac-

as a d a p t e d t o

29

For instance,

in

the

preceeding example i f

Y is

given by y = O, the s t r i c t

tr&ns-

form i s generated by

(4.1.3.2)

and

it

D'

has no s i n g u l a r

(4.1,4) equal

In to

t&nce,

the

zero

let

case (see

n = 2 it

13 I)

even

with

make

the

given

by

blowing-up

generated

&t

this

x

= O.

given point

(4.1.4.2)

- By'

8 ~-+

D

in

general

transformations Let

= x;

only

one

(y'+m!)x'

reach

&s i n

~ be g l o b a l l y

adapted

(4.1.3).

generated

order

For

ins-

by

8 By

(y-(m!)x-myx)

has

to

singular = y,

point

then

the

P = origin. strict

If

we

transform

is

by

@~ + ( y - ( m + l ) ! . x - ( m + l ) y x ) ax----

&s D.

(4.1.4.3)

of

possibZe

= 0 and

x'

= x'.x'

same f o r m

not

one a d m i t s

Then

by

P'

D'

has t h e

given

if

D = x.x

E is

which

is

char(k)

(4.1.4.1)

where

@

+ (-mx')y'

Bx'

points.

of

X = &2(k)

B

= x',x'

Let

us c o n s i d e r

the

B ~--~

sequence

(~(t),X(t),E(t),P(t),D(t))

by

blowing-up

P(t)

succesively

corresponds

to

the

the

only

sequence

singular

of

point

infinitely

P(t)

near

of

~(t).

points

of

The s e q u e n c e the

algebroid

curve

(4.1.4.4)

y -

~ i[.x

i-m+1

= 0

i > m which

does

steps

&re

a curve let

us

(4.1,4,4) transform

(4.1.4.5)

not

represent

similar,

Y in

such

denote

by

there of

so

any

we s h a l l

a way that D'(t) is

a

curve

the step

Y) and t h u s

in

X.

suppose we s h a l l

succesive t o such

t

Now,

let

us s u p p o s e

= O) one makes t h e O' w i t h

strict

transforms

of

) does

belong

P(t

o

not

one has

D(to)p(

t

) = D'(to)p( o

t

) o

in

any step

bZowing-up

obtain

that

all

that

points D'

being by

#(t).

to

Y(t

o

with

(all

center

regular,

Now

In

of

)

view

(succesive

30

which

is

a contradiction,

it is a s i n g u l a r

(4.1.5)

First

finite

of D ( t

point

statement.

distribution

over

because

X and

D

be

a

multiplicatively

adapted

to

a

normal-crossings

X(O)

b

=(t):

c

E(t)

= X,

X(t) =

d) O ( t )

is

e)

is

f)

E(O)

= E,

Y(t)

(4.1.6)

---> X ( t - 1 )

o

) while

irreducible divisor

E.

unidimensional Then

there

is

a

the

Finally,

for

by

version

"permissible".

characteristic

a fixed

other

schemes i n

(at

(I least

n seems t o

hypersurfaces

each

blowing-up

of~

permissible

each c l o s e d

Remark. A s t r o n g

> 3. M o r e o v e r

(4.1.6)

the

transform

a weakly < n-2.

in

hand,

dimension

Second closed

low

P of

n-1.

statement. point

center

X(t-1)

with

center

Y(t-1).

The

for

D adapted

E(t-1). to

E(t),

such t h a t

< 1.

are true result

dependent

to

X(N) one has t h a t

the techniques

as

adapted

in

by s u b s t i t u t i n g

the

case n = 2 f o r

(4.1.5)

remains

in this

work) the

positive

(4.1.5)

are

result

characteristic

useful

for

an a r -

conjectural

on t h e d e s i n g u l a r i z a t i o n

way i n

"weakly

for

(4.1.5)

results in

hard.

desingularization

for On of

I g I).

The

P of

I).

So t h e

results

n (I 7 1 ,

of

may be o b t a i n e d

versions

view of

n,

is a weakly permissible

by ~ ( t )

center

point

(4.1.5)

I 8

be s t r o n g l y

dimension for

of Both

3 I, in

Y(O)

u Y(t-1)).

strict

dim Y ( t )

and

dim Y ( O ) ~ n - 2 .

is

~(t)-l(E(t-1)

= D

v(D(N),E(N),P)

permissible" bitrary

D(O)

E with

(4.1.5.2)

fop

of D'(t

that

such

for D adapted to

the

must be a r e g u l a r p o i n t

(~(t),X(t),E(t),Y(t),D(t))

a

for

)

sequence

= 0,1,...,N,

n

o

).

Let

(4.1.5.1)

t

o

P(t

sequence

X(N),

the

(4.1.5.1) following

may be chosen statement

is

in

such a way t h a t

verified:

Let

X^

be

31

the

scheme Spec

E(N).

Then,

(~X(N),p)

there

is

a)

E^ ' ~

b)

If

and

let

a normal crossings

#^'

= (~,E ~')

then

with

in

Resolution

(4.2.1) a

us o b s e r v e t h a t

center

In

of

X'

and

such t h a t

D~'

= 0

may be o b t a i n e d

by a sequence o f a d a p t e d

blowing-

hypersurfaces).

games

this

"punctual"

D~'

to ~(N)

one has t h a t

v(D^',E^',P) (Let

(4,2)

divisor

corresponding

E^.

(4.t.6.1)

-ups

~ ^ , E ^ be .the o b j e c t s

paragraph

version

of

we s h a l l

enounce

(4.1.5)

and

where

R is

field.

Let

it

is

the

main

result

formulated

in

in

this

terms

of

work,

which

is

a game between

two p l a y e r s .

(4.2.2)

Let

having

k as

divisor

X ^ =Spec

(R)

a qoeffiGient

on X ^

and

finally

let

~^be

between two p l a y e r s

A and B i s

(4.2.3)

Let

=

X^ ,

t+1"

Definition. D^(O)

for

t

=

Z~,

= 0,1,..., a)

status

b)

If

the

the

(4.2.3.2)

a formal

defined

E^(O)

= E^ ,

local

closed

point

m.i.u.d,

of

o v e r X ^.

ring

of

X ^,

let

dimension

n

E^ be a n . c .

The " r e d u c t i o n

game"

as f o l l o w s :

and assume t h a t

P(O) = P. We s h a l l

inductively

regular

r > 2.

define

Let

us d e n o t e

"status

t"

X^(O)

and "movement

as f o l l o w s

0 = (X^(O),E^(O),~P(O),P(O)). status

t

is

(X^(t),E^(t),Z~(t),P(t)),

"movement

weakly permissible Second t h e

P be t h e

r = £(~,E^,P)

(4.2.3.1)

then

a complete

player

t+l" center

runs Y^(t)

in for

the

following

D^(t)

B chooses a directional

~^(t+l):

way:

adapted to

first E^(t)

blowing-up

X^(t+l)

~

X^(t)

the

player

such t h a t

A chooses

dim Y ^ ( t ) ~ n - 2

a

32

of

X^(t)

with

center

c) to

E^(t).

of

X^(t+l).

Let

Let

Y^(t).

~'(t+1)

E^(t+l)

be t h e -1

= ~(t+l)

Now, t h e

"status

(4.2.3.3)

t+l"

transform

(E^(t) is

u Y^(t))

the

of

D~(t)

and l e t

by

P(t+l)

~^(t+l) be t h e

and a d a p t e d closed

point

4-upla

(X^(t+l),E^(t+l),D^(t+l),P(t+l)).

Finally,

the

player

A "wins

(4.2.3.4)

in

strict

at

the movement

v(LP(t),E^(t),P(t))

this

case,

the

game must

w i n a t a n y movement,

(4.2.4)

i.e.

The movement t

stop

the

at

mov(t) stat(t)

Definition.

if£

< r,

status

t.

The p l a y e r

B wins

iff

A does n o t

game becomes i n f i n i t e .

may be i d e n t i f i e d

(4.2.4.1)

(4.2.5)

the

t"

with

= (Y^(t),

the

pair

(Y^(t),~^(t+J)).

For s h o r t

~(t+1)).

= (X^(t),E^(t),O'(t),P^(t)).

A "realization

of the

reduction

game" i s

a (finite

or

infinite)

sequence

(4.2.5.1)

which to

t

G = {G(t)

respects

the

rules

of

(4.2.3)

(4.2.6)

such

that

~(D(N),E(N),P(N))

Definition. F(t),

t

a) tial

and

t=0,1,...

i£

the

last

element

corresponds

= N, t h e n

(4.2.5.2)

tions

= (mov(t),stat(t))}

A

"winning

= 1,2,...

F(t)

is

in

strategy"

for

< r.

the

player

a sequence

o£ f u n c -

such a way t h a t

defined

over the

set

of

sequences

realizations")

(4.2.6.1)

A is

Git = { G(s) } 0 <s
(which will

be c a l l e d

"par-

33

where G i s

a realization b)

that

F(t)(G)

mov(t-1)

of

is

the

If

G is

all

then If

(4.2.7)

There

finite

is

foLLowing

condition

strategy

is

a

winning

is

a

strongly

o£ t h e

and

strongly

winning

strategy

obtain

status the

the

if

duction

game each

strategy

(4.1.5)

may

for

the

is

of only

game w h i c h

reduction

of

verifies

c),

the property

we s h a l l

the

say t h a t

property

game and t h e

may be o b t a i n e d

(4.2.6.2),

concepts

by p u t t i n g

of wi-

"permissi-

game by p u t t i n g

the

that

then the

defined directional

in

such

so f o r is

of

strategy

is

as i n

for

all

new one.

one

needs

(4.2.3.4),

the

the d)

if

there

i£

there

assure

that

new p o s s i b l e in

order

to

new game.

strategy

a situation possible of

But

to

N as i n

winning

of the ~(t)

or stronglywi-

one and t h a t

the

integers

a strongly

a way t h a t

a winning

the old

game,

0 comes f r o m

biowings-up

is

so f o r

order

number

winning

status

there

reduction

adapted

there

if

is

there

the

a finite

< 1

that

there

one

for

a strongly

true,

time

by c h o o s i n g

old

Strategy

of

be

known f r o m G l t ) .

G verifies

reduction

see e a s i l y

new game, t h e n

reduction

existence

If

obtained

the

generates

(4.2.8)

the

One can

winning the

(note

permissible".

strategy

obtained

E^(t)

N.

version

for

adapted to

strategy":

A strong

(4.2.3.4).

nning

is

instead

v(O(t),E(t),P(t))

o£

starting

d)

N such t h a t

smaLLer t h a n

(4.2.7.1)

once

(t)

reduction

1.

of '~eakly

of D^(t)

exists.

A wins).

winning

integer

Length

center

G(t-J)

= (F(t)(Glt),~^(t+l))

2. One can m o d i f y

instead

the

(so p l a y e r

an

such t h a t

and s t a t

of

a "strongly

of

strategy instead

the

is

Remarks

nning

finite

put

, t=l,2,..,

G is

ble"

G is

we

d) then

(t)

mov(t)

t,

F(t)

star

a realization

(4.2.6.2)

for

game,

a weakly permissible

always produces

c)

reduction

fop the

re-

o v e r X. A c t u a L L y , realizations

(4.1.5.1).

may be

34

(4.2.9)

for

the

The main

result

Theorem.

Let

reduction

n

game,

in this work

=

3

and

let

is the following.

car

k = O,

then

there

exists

a winning

strategy

-

I I

-

A PARTIAL W I N N I N G STRATEGY

O. I N T R O D U C T I O N

(0.0.1)

tegy

In this c h a p t e r we shall begin the proof of the e x i s t e n c e

for the reduction

possibilities zero"

of the

in order

to

game in the case n = 3 and char k = O. We shall status

prove

the

remark that no a s s u m p t i o n

(0.0.2)

Before starting,

0 of the

resolution

existence

(0.0.2.1)

let us c o n s i d e r

stat

status

"^" anymore,

0 o£ the

(0) =

reduction

game

of a winning

on the c h a r a c t e r i s t i c

ties of the status 0 of the reduction

is the

game.

in this

stra

restrict the "of the type case.

Let

us

in any part of the chapter.

for the p o s s i b i l i -

Let us suppose that

(X(0),E(0),O(0),P(0))

game

Dir

those called

some general r e d u c t i o n s

(in

since all the d i s t r i b u t i o n s ,

dim

to

strategy

is made

the

sequel

we

shall

etc. will be "formal").

se that

(0.0.2.2)

of a w i n n i n g

(D(0),E(O))

= O.

not

use

the symbol

And let us suppo-

36

Then, by ( I .

3.2.7)

if

the p l a y e r A chooses the closed point P(0) as c e n t e r , then

he always wins in the f i r s t hold

movement. Thus we s h a l l suppose t h a t

and the reduction game may be modified by saying t h a t

the movement t i f f

( 0 . 0 . 2 . 2 ) does not

the p l a y e r A wins a t

one has

(0.0.2.3)

v(D(t),E(t),P(t)) < r or

dim D i r

(~t),E(t))

= 0.

where r = v ( ~ ( 0 ) , E ( 0 ) , P ( 0 ) ) .

( 0 . 0 . 3 ) On the other hand l e t us remark t h a t one has always E ( t ) ~ e f o r t > 1. Thus if

one

looks

for

a

winning

strategy,

there

is no

loss

of g e n e r a l i t y

in assuming

that

(0.0.3.1)

In the if

E(O)

sequel,

we were

possible

we

looking

status

shall for

make a

this

~ @.

assumption.

strongly winning

I after the first move,

This

strategy.

reduction One

would

would have

not be so easy to

control

the

37

1.

TYPE ZERO SITUATIONS

(1.1)

Description

(1.1.1) three let

Let having

of type

X = Spec

(R),

where

k as a c o e f f i c i e n t

P be t h e c l o s e d p o i n t

irreducible

zero

the

the

assumptions

For a v e c t o r

o£

subspace V o f

(1.1.2.1)

H(V)

We s h a l l

say t h a t

= {f

V and D a r e

(1.1.2.2)

(D i s

in

a generator to

Let (~E)

us

= 2,

of D)

the

tangent

remark by

(1.1.2.2)

if

Definition.

a) dim DiP

for

(E,P)

divisor

on X and

multiplicatively

E such t h a t

v(f)

space TpX,

us c o n s i d e r

the

set

= 1, Tp ( f = O ) ~ V } .

r

exists

to

E transversal

(D(f))

a linear

= r,

position"

iff

= JH(V)

form ~ E JH(V)

some component o f

v(D(f))

let

then

such t h a t

~ = 0 is

E.

jr(D(f))

c J(D,E).

So i f

dim

say t h a t

= J(D,E).

(X,E~)

is

of the

"type

zero"

iff

the

follo-

are verified (D,E)

> 1 and E ~ ~.

(D,E)

(X,E,D,P)

such t h a t

dimension

one has t h a t

We s h a l l

b) D and Dip

Lemma.

of

> 2,

tangent

JH(V)

w i n g two c o n d i t i o n s

(1.1.4)

the

space o f

that

ring

denote a formal

o v e r X and a d a p t e d t o

[f~H(V)J

(1.1.2.3)

(1.1.3)

~ will

an " a d a p t e d

and t h e r e

local

(0.0.3).

~ R;

0 ~

transversal

DiP

sequel

(0.0.2),

regular

E be a n o r m a l c r o s s i n g s

r = v (D,E,P)

all

(1.1.2)

Let

distribution

(1.1.1.1)

with

a complete

field.

of X.In

unidimensional

R is

are

±s o f

in

an a d a p t e d t o

type

zero

iff

E transversal

there

is

a r.s.

position.

of

p.

(x,y,z)

suited

38

i)

E is

ii)

g i v e n by x o r by xy.

(x,y,z)

iii)

~ J(D,E) be a/ ~

Let ~

~ (z).

o r y a/@y a c c o r d i n g t o

E. Then i f

Y (1.1.4.1)

D = ax@/ ~ generates D ,

iv)

If

dim D i r

llowing

one has t h a t ~ ( c ) (~,E)

properties

is satisfied:

J(~E) J(~,E)

(1.1.4.3) (1.1.4.4)

J(D,E)

Proof. (x,y,z)

such

(otherwise his

two

(~x +

from

=

is

if

jr(c)

(y

let

v(c~

In

v ( c ) > r we s h a l l

jr(e)

= (x)

if

is

= (z).

is

where

of

because is

(B,y)

f

dim Din Let

= r

v(c)>

since the

no l i n e a r

4 (0,0)

have

if

We can

three with

choose

components

~(D(f))

= r,

E has one component and

change o f coordinates we may suppose

= 2,

then

J(Z~,E)

dim Dip v(D(f))

two f o l l o w i n g

= (z)

and i i i )

(D,E) > r

= 1.

for

posibilities:

all

If f.

foE has If

E

J(O,E)=(x,z)

by an a d e c u a t e change o f coordinates we have

y and z.

In t h e

> r for

B~O, a change o f form

zero.

E H(Dir(Z),E))

otherwise

~(D(f))

type

E cannot

us suppose t h a t

second c a s e ,

r then if

(D,E)

the

all

first f

case i f

jr(c)=(x,z)

6 H(diP(D,E)).

coordinates

gives

So we may

(1.1.4.4).

Now,

in

t )jr'(D(f ) f ~ H(Dir(D,E) )

transversal

to

the

tangent

Conversely, :

xy

interchange

(1.1.4.5)

J(Z),E)

the

= (ex + Bz),

there

or

us d i s t i n g u i s h

+~ x,z).

no p r o b l e m ,

assume t h a t

D(z)

necessarily

jr(c)

By making an a d e c u a t e

= c.

that

= (~,z)

Since there

the

= 0 and i f there

x

If

components,

J(D,E)

by

and so we have i i ) . fact

jr(c)

(X,E,Z~,P)

e J(RE)

E has t w o c o m p o n e n t s .

= (y,z)

= (x,z);

&11 f ) .

6_y + Yz

:

that

given

> r for

has one c o m p o n e n t , or

suppose

E is

v(D(f))

z E J(Z),E) IZows

us

that

that

y~ 0 i f

Let

= r.

= 1 and E has o n l y one component, then one o f the f o -

(1.1.4.2)

one

+ bB + cal@z Y

(z)

and

from

(1.1.3)

space o f

ii)

i)

and

b)

folZows

E. we have from

iii).

(1.1.3) If

a).

dim Din

If

dim DLr(Z~,E) = 2 t h e n (D,E)

= 1 and E has two

39

components,

then

J(D,E)

component and J ( ~ , E ) (0,0), jr(c)

in

both

= (y,z)

cases i t

= ( x , z ) or j r ( c )

(1.1.5)

= jr(c)

is

sense,

Dir(D,E)

is

transversal

(X,E, ~,P)

= 2 or

b)

follows

or j r ( c )

enough t o consider z

(1.1.4.4)

is a " l e s s

Definition.

(1.1.3)

= (y,z)

[ f ~H(Dir(~,E))

in some

dim

and

then j r ( c )

Remark. The s i t u a t i o n

(1.1.6)

(~)

easily.

= (~

If

E has one

+ 8~) w i t h

E H(Dir(~,E)).

(a,8)

Analogously i f

= (z).

(1.1.5.1)

so,

~

one

has

is

of

the

the

the

only

Jr(f)

one f o r

~ J(D,E)

situation"

"type

property

w h i c h one has t h a t

than

0-1"

iff

(1.1.5.1).

the o t h e r ' s .

it

is

of the

Otherwise

it

type

is

of

z e r o and the

"type

(X,E,D,P)

is of

0-0".

(1.2) S t a b i l i t y

(1.2.1)

the

results

Theorem.

type

Then

one

sider

[X',E',~',P')

of the

view

is

a directional

two f o l l o w i n g

v(D',E',P')

= r and

(X',E',O',P')

that

is

two

Assume

a)

possibilities:

besides

the

blowing-up

in

conditions given

of

one

(I. of

3.2.7),

us s u p p o s e

quadratic

(x,y,z)

is

is

of the

and

blowing-up

of

satisfied :

2~ n o t

type

let

may be chosen

(1.1.4).

that

= 0

1~.

0-0.

(x,y,z) in

be as

in

such a way t h a t

In t h e

first

(1.1.4). J(O,E)

=

case we must c o ~

by

y = x'y';

z = x'z'

If E has two c o m p o n e n t s ,

a generator

has t h a t

(D',E')

not satisfied

x = x';

of

Let

possibilities

b)

coefficients tisfied,

(1.1.1).

< r o r dim D i r

(1.2.1.1)

in

in

~(D',E',P')

are

the

as

a)

Proof.

= (y,~)

notations

0 - 0 and t h a t

(X,E,D,P).

There

With

of ~ ' )

in

view

of

then (I.

c' = c / x ' r - z ' a ' (a',b'

2.2.5),

so, s ~ n c ~

a)

and c'

is not sa ~

40

(1.2.1.2)

(the

In

initial

form being

(c')

= In

(c/x ' r )

r e f e r r e d t o the l o c a l r i n g o f X' a t P ' ) ,

thus,

since a) i s

not s a t i s f i e d

(1.2.1.3)

dim D i r

(0',E') J

(see in

+ Xx'

has

one c o m p o n e n t .

only

(~,B) Dip

as

l e a d s us t o

above.

and

(b'

= O) >

1,

~ (z'

(1.2.1.5)

with

jr(c')

Since a)

us

Jr(c)

suppose

is is

+ Xx')

=

(y,z),

that

of

lemma

or

simetrically

jr(b)

We have b'

=

o f an h y p e r s u r f a c e ) . So a change

= b/x

(1.1.4).

(~Z + B~),

'r

y'a',

-

= (~Z' + B~' + X~') or j r ( b ' )

c'

Let

us suppose t h a t

Jr(b)

= (Z,~),

we s h a l l

jr(c)

=

6~)

= c/x'r-z'a

= ( YZ'

not s a t i s f i e d ,

case

+ 6~'

+

x')

or Jr(c')

the o n l y p o s s i b i l i t i e s

= ( YZ'

+ 6z',x').

are the f i r s t

ones in both cases,

o f the lemma ( 1 . 1 . 4 ) .

one cannot assume J ( O , E ) = ( Z , [ ) ,

t h e r e are o n l y the two f o l l o w i n g po-

~E has two components and J(D,E) = ( y + ~ x , z ) w i t h ~ 0 " .

"E has one component and j r ( c )

(1.2.1.6)

the

(1.2.1.8)

c'

S i n c e dim

enough t o make the change Y ' I = ay'+BzSXx', z ' 1 = Y y ' + G z ' + p x ' , i n o r d e r

(1.2.1.7)

In t h e

t.

with

= (eZ' + B z ' , x ' )

ssibilities:

(1.2.1.6)

(YZ +

E

c':

t o o b t a i n the s i t u a t i o n If

situation

= O) = 1

one has t h a t

jr(b')

and a n a l o g o u s l y

it

Let

If

the

(X, 6) i n d e p e n d e n t s .

(1.2.1.4)

now

(c')

(c'

IZO I v g r . the b e h a v i o u r o f the d i r e c t r i x

z' 1 = z'

reason

r

= dim D i r

= c/x'r-z'a ',

blowing-up

x = x';

dim D i r

is

given

y = (y'-~)x';

(c'=O) = I and

= (x,z)".

by

z = x'z',

41

(1.2.1.9)

jr(c')

now it is enough (1.1.4.2).

to make

In the case

:

(1.2.1.7)

c'

= c/y'r-z'b

' • As a b o v e ,

the

change z ' 1 = z ' + k y ' ,

(1.2.2)

Notation.

quently

used equations

the

We shall

(1.2.3.)

Proposition.

missible

center

a directional

~

jr(c')

from the

fact

(X',E',D',P') following

dim

I101

following

x = x';

y =

(y'-¢)x';

:

x = x';

y = y';

z = x'z'.

(T-4)

:

x = x';

y = y';

z = y'z'.

tangent of

The o n l y

y = y';

(X,E,D,P) to

the

(X,E,~,P)

case w h i c h to

with

the

that

dim D i r

the

Let

(c=O)

By m a k i n g

fre

for the most

is

of the

center

Dip Y then

0-0

(D,E).

and t h a t If

Y is

a per-

(X',E',~',P')

is

one has t h a t

< r.

does n o t

correspond

(1.1.4.3). being

type

suited

But

to

a situation

in this

for

(E,Y).

case

as i n

(x,y,z)

Then t h e

may be t a -

result

follows

= I.

us suppose

is a d i r e c t i o n a l

possibilities

of

= 1.

z = y'z'

directrix

with

case

property

(c'=0)

z = x'z'.

(T-3)

is

Dir

We shall denote:

x = x'y';

which

by

the notation

:

Assume t h a t

Proposition.

and

(T-2)

corresponds

(1.1.4)

is given

obtain

z = y'z'

~(D',E',P')

3.4.12)

p~')

finished.

of a blowing-up.

blowing-up

Proof.

(1.2.4)

is

standarize

(1.2.3.1)

ken i n

y = y';

z ' + XY'

proof

(T-I,~):

z'+

the blowing-up

x = z'y';

and

+ Xx',

Y'I = y' + Xx', z' I = z'+px' , t o

the change

(1.2.1.10)

(I.

(y'

that

(X,E,D,P)

quadratic

is of the type 0-1 and that

blowing-up

of

(X,E,~,P).

is satisfied

a) v(D',E',P')

< r or dim Dir

b) v(D',E',P')

= r and

(O',E'

(X',E',~',P')

= O. is of the type

0-0.

Then

one of the

42

c)

v(D',E',P')

Proof. (1.1.4).

In

Let

view

us

of

= r and

suppose

(I.

(X',E',D',P')

that

3.2.7),

a)

is

we must

is

o£ t h e

type

not satisfied. consider

0-1.

Let

the

(x,y,z)

equations

be as i n

(T-1,G)

lemma

or T-2.

We

have t h a t

(1.2.4.1)

c'

accordingly jr(c') the =

=

to

(x',z')

second (~'+X~')

Dir

or

have

and

then

a

= 2 or

a permissible (D,E)

Dir

(D,E)

If

= 1,

type

0-0.

change

z' I

Y is is

= h and t h e n

de

(b I

(bl)

Let required

in

to

(1.1.4)

a directional

I£ =

of

Y is

(I.

and

shows type

to

by ( x , h )

8/8Yl

in

is the

= 2 one we

change

has

have

z I = z+Xx

that type

jr(c') 0-1

i£

in

= dim

otherwise.

and

(x,y,z)

(I.

with

We can

a

is

O. A c t u a l l y ,

view of

permissibility of

0-1

after

that

0-0

z :

= 1 one has n e c e s s a r i l y

and,

(c'=O)

type

in

too.

(c'=O)

3.4.2).

In(h)

possible

base o b t a i n e d

the

In

x and I n ( h )

suppose not

in

like

in

(1.4.1) t h e n

case dim

the

case dim

independents, = e~

+ 8 ~.

since

I f e~O we

s i n c e Z1 does n o t d i v i from

(x,y,z)).

Thus ~ = 0

z = O. that

if

we make z I

and t h e n

Y is

an

calculation

easy

blowing-up

P'

with

given

center

~ Proj[(~

(1.2.6)

Theorem.

Let

us suppose t h a t

missible

center

and

that

center

Dir

z'+Xx'

given

3.4.2)

the

(1.2.5.1)

with

dim

do

= c / y 'r - z'b'

+ ~ ~')

the

to

c'

dim D i r

always tangent

us o b s e r v e

Moreover, is

is

= coefficient

and Y a r e t a n g e n t

i£

(x',z')

nothing

apply

can make Yl In

=

necessarily

we can

or

(z'+%x',y'

( X , E , D,P)

center

z'a'

Thus,

=

J(D',E')

= 2 there

otherwise

jr(c')

we

Remark.

Dir

or T-2.

case,

(O',E')

(1.2.5)

(T-I,~)

= c / x 'r -

Y. Then one o f

(X',E',D the

three

= h then

have t h e

properties

by ( X , Z l ) . shows t h a t

in

this

Y and v ( D ' , E ' , P ' )

= O) /

is

following

case

= r then

if

(X',E',D'

P')

one has t h a t

(x = z = 0 ) ] .

(X,E,D,P) ',P')

( x , y , z 1)

a

is

of

the

type

directional possibilities

0-1,

that

blowing-up is satisfiel

Y is of

a per-

( X , E , D ,P)

43

a)

v(O',E',P')

< r o r dim D i r

b)

v(O',E',P')

= r and

(X',E',O',P')

is

of the

type

O-O.

c)

v(O',E',P')

= r and

(X',E',O',P')

is

o£ t h e

type

0-1.

that

Proof. the

above

We

shall

suppose

remark)

shows

that

nal property we may

that

to the proof o9

(1.3)

by

The

(4.2.3)

is not satisfied. (x,y,z)

(x,z) or by (y,z).

equations

An

as in (1.1.4) In this

are T-3 or T-4.

easy calculation

The

with

situation

(see

the additio-

(see (1.2.5.1))

rest of the proof

is similar

(1.2.4).

stability beginning

ding

to type

ters

(and not merely

0-I

results

(1.3.2)

and

Definition.

0 is a type tting

0-0

type weakly game

The

(1.3.3)

Definition.

a type

A wins

0-I

at the movement

shall

games

game

correspon-

deal only with permissible

"weakly

cen-

the stro,g

vet

of type 0-0"

is defined

t+l are defined

permissible".

We

as follows.

as in

shall

say

possibilities

Status

(I. 4.2.3) that

pu-

the player

is satisfied

< p. = O.

"reduction Status

tiff

game

one of the two following

t

game o f t y p e

0-1"

and movement t+J

is

defined

are defined

as f o l l o w s .

as i n

one of the three following p o s s i b i l i t i e s

a) v (D(t),E(t),P(t))

c)

into two reduction

and thus we are decomposing

t and movement

of

tiff

The

we

of the reduction

(see I. 4.2.7).

(O(t),E(t)

4-upla.

b) dim Dir

Actually,

permissible)

a) v ~ ) ( t ) , E ( t ) , P ( t ) ) b) dim Dir

0 of type zero

0-0.

Status

instead

A wins at the movement

(3.2) allow a d e c o m p o s i t i o n

"reduction

r-upla.

"permissible"

of

at a status

sion o9 the reduction

0 is

= O.

Type z e r o games

(1.3.1 of

the

a)

we can choose

that Y is giver

suppose

(O',E')

Player

issatSsfied:

< r.

(D(t),E(t))

v(~(t),E(t),P(t))

(1.3.2).

Status

= O. =

r and

the 4 - u p l a

(X(t),E(t),D(t),P(t))

is of the

44

type 0-0.

(1.3.4)

Winning

ned

the

in

can f o u n d

strategies

same a

way as

winning

and s t r o n g l y in

(I.

4.2.6).

strategy

for

of

the

stability

results

tegy

for

the

reduction

game b e g i n n i n g

A TYPE 0 - 0 W I N N I N G

In the

reduction

mlssible wins, only

this

on t h e

section

tangent

otherwise

player

status

The

t

0-0.

to

(2.1.1)

Let a)

type

prove

It

the

and n o t

is

both

we s h a l l

games a r e d e f i prove

that

Thus,

as a c o r o -

two games a b o v e . existence

one

of a winning

stra

zero.

for

used

with

the

quite

directrix,

on t h e

used

or surfaces

(2.1) An invariant

chapter

one deduces t h e

at

A chooses t h e

invariants

of curves

(1.2)

we s h a l l

to the one of the invariants ties

this

fop

STRATEGY

game o£ t y p e

center

of

In

strategies

each o f t h e

llary

2.

winning

history

in

simple: then

closed

the

existence

as

of the

one dimensional

a winning status

certer.

this

t

strategy

there

A choos~ this

realization

of

11oi

I~ I or

at

player

point

control

if

of

This

for the control

of

a per-

center

strategy

of the

game a r e

is

for

and

depends

game. a kind

similar

of the singulari-

directrix.

of t r a n s v e r s a l i t y

(X,E,D,P)

be o f t y p e

Each l i n e a r

0-0.

In

view of

(1.1.4)

we have two p o s s i b i l i t i e s :

form in

(2.1.1.1)

J = [

ar(D(f)) f EH(Dir(D,E))

is b)

For to

transversal each

a fixed

component

of

component o f

E there

is

E.

a Zinear

form

in

J wich

is

tangent

(it

corres-

it.

Moreover, ponds t o

to

(1ol.4.3)).

in

the

case b)

necessarily

E has o n l y

one component

45

(2.1.2)

Definition.

we have

(2.1,1)

b)

(2.1.2)

Remark.

Let

neral,

the

For

a type

and w ( ~ , E )

us

evoiution

0-0 4-upla

= 0 iff

observe

by

(X,E,~,P)

we have

that

(2.1.1)

w = 1 is "less

blowing-ups

goes

we s h a l l

from

define

w(D,E)

= 1 iff

a).

transversal"

"weak"

than

w = O. In ge-

transversaiity

to

"strong"

transversaiity.

(2.1.3)

Lemma.

way t h a t

J(O,E)

Proof.

(2.1.4) perty

(2.1.5)

If

w = 0 the r.s.

See t h e

(2.1.3)

bilities

will

Let

a normalized

(2.1.6)

(2.2)

(2.2.1) ciated

of

(1.1.4)

may be chosen

in

such a

of

(1.2.1).

p.

(x,y,z)

as i n

(1.1.4)

a "normalized

(X,E,D,P)

blowing-up

v(D',E',P')

Proof.

llows

(x,y,z)

be o f

o£

the

and w i t h

system o f

type

(X,E,D,P).

the

parameters

0-0

and

let

Then one o f t h e

additional for

pro-

(X,E,O,P)"

(X',E',D

',P')

two f o l l o w ± n g

be a possi-

is satisfied:

b) w ( O , E )

che

of

be c a l l e d

quadratic

a)

in

proof

A r.s.

Proposition.

directional

p.

# (x,z).

Oefinition. of

of

If

easily

~ w(O',E')

w(~,E)

system

situations

< r on dim D i p

of

of

from that

in

view of

parameters proof

proof.

of

If

In to

order

type

to

a situation

(1.2.1)

w(O ,E)

if

make T - 2

except

= O, t h e

w = O, i n

a)

for proof

order

to

is

not satisfied,

(see 1 , 2 . 2 ) (1.2.1.7) is

find

then

for

and t h e n we a r e

and t h e

result

fo-

easy.

a winning

strategy.

0-0

unify of

(1.1.4.3)

we have t o

Remark. We can suppose t h a t

Polygons for

= O.

= O.

= 1,

the

(O',E')

the the

techniques

type

0-0 with

used,

we s h a l l

introduce

w = 0 and a n o r m a l i z e d

a polygon assosystem o f

parame-

46

ters.

But no f u l l

(2.2.2) f

Let

~ R[y -1]

use

of

p = (x,y,z)

this

idea will

be a r . s .

~ K. Then f

of

may be w r i t t e n

(2.2.2.1)

f =

be made u n t i l

p.

of

in

the

R, K t h e f i e ~

a unique

study

of

of the

fractions

of

type 0-1.

R and

way as

Xfhij xhyizj

where fhij e k. We shall define the "cloud of points of f with respect to p" by

(2.2.2.2)

Exp (f,p) = {(h,i,j);

We shall write Exp(f)

if there is no confussion

(2.2.3)

us d e n o t e

Notation.

sings, d i v i s o r

(2.2.4) =

Let

E at

Definition.

(x,y,z)

nerated

Let

(X,E,D,P)

be a n o r m a l i z e d

be o f

system of

number o f c o m p o n e n t s o f t h e

the

type

0-0

parameters for

D = a x;)/Sx

3

with p.

normal cros

with

w ~ ,E)

(X,E~,P).

= 0 and l e t

p =

Assume t h a t D i s

ge-

by

Y

= 3/ay

D adapted to

or

e(E)

ya/3y

E and w i t h

(2,2.4.2)

if

the

~ ~3

P.

(2.2.4.1)

where

by e ( E )

fhij ~ O}

accordingly

respect

Exp(D,E,p)

to

p"

+

b8

to is

Y

+

cS/;)z

e(E).

The

defined

by:

= Exp(za)

~ Exp(zb/y)

"cloud

of

points

Exp(D,E,p)

of

~ Exp(c)

= 1 and

(2.2.4.3)

Exp(D,E,p)

:

Exp(za) ~ Exp(zb)

w Exp(e)

if e ( E ) = 2.

(2.2.5)

Remark.

Exp(D,E,p)

(2.2.6)

Definition.

With

depends a c t u a l l y

notations

as i n

on t h e

(2.2.4)

generator

the

invariant

D.

m(D,E,p) i s defined by

47

(2.2.6.1)

m(D,E,p)

and m ( D , E , p )

= + ~

(2.2.7)

Remark.

(2.2.8)

Let

@ : IR3 -

if

the

m(D,E,p)

= min{h;

set

on t h e

does n o t

@: ~R2 --+IR 3 be t h e

{(O,O,r)}

o n t o @(IR2).

given

(2.2.8.1)

= IH(~)

(2.2.9)

Definition.

With

is

E Exp(D,E,p)}

empty.

depend on t h e

given

composition

m e ~

IH(m)

rigth

immersion

- ~ I R 2 be t h e

Finally,

(h,-1,r)

o

U {~}

{(u,v); =

u >_0,

{(u,v);

D.

by @ ( u , v )

of

, let

generator

@-I w i t h

= (u,v,r-1) the

IH(m) be t h e

and l e t

projeqtion

from

(O,O,r)

set

u+mv > O} c I R 2

u>

O, v >

0}

n o t a t i o n s as above, " t h e polygon A(D,E,p) o f D adapted to

E and with respect t o p" i s defined by the convex h u l l o f

(2.2.9.1)

[~(Exp(D,E,p)

n {(h,i,j);

j ~ r-l})

+lH(m(D,E,p))]

m {(u,v);

v ~ -1}.

( 2 . 2 . 1 0 ) Remark. A(D,E,p) does not depend on the generator o f D .

(2.2.11) P r o p o s i t i o n . With n o t a t i o n s as above, the curve Y given by ( y , z ) is penmissible

iff

(2.2.10.1)

A(D,E,p) c { ( u , v ) ;

Proof.

(h,i,j)

(2.2.12)

i/(r-j)

Definition.

lowest

t

permissible j

iff

m(D,E,p)

< r-l,

= ~ and f o r

any

one has t h a t @ ( h , i , j )

With

notations

{(u,v),

is no t > 0

= (h/(r-j),i/(r-j))

> 1.

as

above,

the

invariant

6~,E,p)

> 0 such t h a t

(2.2.12.1)

If t h e r e

is

~ Exp(D,E,p) such t h a t

s a t i s f i e s that

the

(y,z)

v > 1}

u+tv = t }

/~ A ( D , E , p ) - { ( 0 , 1 ) } ~ @.

s&tisf3ring ( 2 . 2 . 1 2 . 1 )

we put

6(D,E,p)

= ~.

is

defined

by

48

(2.2.13)

Remarks a) b)

(2.2.14)

6~,E,p)

= ~

6(D,E,p)

> 1.

With

notations

Definition.

"strongly

normalized"

Remark.

There

a permissible

follows

as

from the

above,

we

center.

fact

shall

that

say

v(D,E,D)

that

p

=

"true"

in

and j r ( f )

are

normalized

the

ease

~ (x,z)

of

but not strongly

normalized

regular

is

the

if

hypersurfaces:

v(f)

= r,

dim

Dir

convex hull

Proposition.

ways a s t r o n g l y

Proof. the

concepts. in

e(E)

above

Let

If

of

@(Exp(f,p)

(X,E,D,P)

normalized

If

view of the

= 2,

remark,

definitions,

(2.2.17) is

Proposition.

e(E)

the

strongly

normalized,

Proof.

If

e(E)

enough t o

If

~{(h,i,j);

type

only

:

with and

I.

only

Let

j < r-l})

0-0

and

the

third

"strongly

= 1,

exists

al-

2

+IR o

w~,E)

= 0 there

= ky

• z

is

6(D,E,p)

of = I

for

r-1

+ pz

coefficient normalized"

p = (x,y,z)

possibility

interchange

( X , E , D ,P) then

of

"normalized"

In(b)

is

(f=0)

parameters.

we d e a l

us suppose t h a t

X~ O. Then i t

is

system o f

(2.2.16.1)

with

systems

A (f,p)

--

of

is

A(D,E,p).

(0,1) ~

where A ( f , p )

view

(x,y,z)

then

(2.2.15.2)

(2.2.16)

= r.

D = xr.xB/Bx + (y+x)zr-1)/By + z r. 8/8z.

not

z E jr(f)

is

For instance let D be

(2.2.15.1)

is

This

(0,1) ~

of parameters.

This

(y,z)

iff

(2.2.14.1)

(2.2.15)

iff

are

be a n o r m a l i z e d

(0,1) @ &(D,E,p) r

of

is

D and,

in

equivalent r.s.

of p.,

that

+ x(...)

y and z.

the iff

type J~,E)

0-0,

w ~ ,E)

= (y+kx,z)

= 2 we d e a l o n l y w i t h c and the r e s u l t

= 0 and p : with

(x,y,z)

X~ O.

f o l l o w s from the s t a r

49

dard f a c t s &bout h y p e r s u r f a c e s

([101).

Let us suppose

A(D,E,p) we have t h a t

= 1 iff

m(D,E,p) = 1 o r t h e r e

from

(0,1)

@(Exp(D,E,p) n{j

and t h a t

is

(2.2.18)

Corollary.

is

e

a point different

equivalent

to

If

%r-j

J(D,E)

p is

})

= {y+Xx,

strongly

~ {u+v = 1 }. z) w i t h

normalized,

X = O.

6 = 1

implies

e(E)

= 2.

Preparat±on

(2.3.1)

The main

6 (D,E,p). re

e(E) = 1. Since ( 0 , 1 )

in

(2.2.17.1)

(2.3)

6(D,E,p)

that

in

In t h i s

order

invariant

used

paragraph

the

t o make c o n t r o l a b l e

(2.3.2)

In a l l

this

w ~ ,E)

= 0 and

paragraph

p = (x,y,z)

for

proving

system o f the

existence

of a winning

parameters p will

behaviour

(X,E,D,P)

will

the

of

will

be r e s t r i c t e d

a bit

is mo-

6(D,E,p).

denote a 4-upla

be a s t r o n g l y

strategy

normalized

of

type

regular

0-0 with

system o f

parame-

ters.

(2.3.3)

Lemma.

Let

us suppose t h a t

(2.3.3.1)

or that

zI = z +

e(E)

= (x'Y1'Zl)

Proof. tial of

forms order

which

= 2 and

[ Xixi i > 2

;

Yl = y

= 1 and

(2.3.3.2)

Then Pl

e(E)

Pl

zI

= z + [

is

a stroMgly

is

normalized

by making high

enough.

contributes

to

1>2

z ,

~zl(y

On t h e (0,I)

Yl

normalized

since ~ y l

other

are

X i ix ;

not

= y + [i>2P1

system o f

i

x "

parameters.

one does not change t h e ) and

hand,

one

adds t o

the

one can see t h a t

changed,

actually

expressions old

the

of the

coefficients

points

in

initerms

Exp(O,E,p)

t h e monomiaZs r e m a i n t h e

same

50

ones,

so P l

is

strongly

(2.3.4) Definition.

normalized.

p = (x,y,z) is " p r e p a r e d

"

iff

one

of

the

two

following

possi-

bilities is satisfied a)

6=

6~,E,p)~

E

b)

C or

7Z - { 1 }

6 =

and t h e r e

is

no c h a n g e

of

the

6

(2.3.4.1)

Yl

w i t h p= 0 i f

e(E) = 2, such t h a t

if

type

6

= y + px

(2.3.4.2)

(2.3.5)

I.

; z 1 = z + Xx

Pl = ( x ' Y 1 ' Z l ) one has t h a t

6(~,E,p I) > 6

If p is not prepared,

a change

as in

(2.3.4.1) will be called a "prepara-

tion" change for p. Thus we obtain Pl' if Pl is not prepared, we may repeat. We have two possibilities:

the algorithm stops in a step Pt which is prepared or the al-

gorithm does not stop.

In the last case, by composing all the changes we obtain

y~ = y + [ pi x i ;

Before p r o v i n g a r e s u l t

about

z ~ = z +[ Xixi

( x , y ~ , z ~ ) , we s h a l l need the f o l l o w i n g l e -

mma.

(2.3.6)

Lemma.

Let

n > 6(D,E,p)

and

assume t h a t

n ~

and

n~2.

Let

change

(2.3.6.1)

where

Yl = Y +X i > n p i xi ; z I = z + [ i > n

X zx i

Pi = 0 for all i if e(E) = 2. Then

(2.3.6.2)

6 (Z],E,p I) > 6 (D,E,p)

where Pl = (x'Yl'ZJ)" Moreover one has the equality if n ~ 6(D,E,p).

Proof. Let us suppose that D

(2.3.6.3)

is generated by

D = ax~/~x

+ b@ + c@/Bz Y

us c o n s i d e r

the

51

with

@ = 2/ ~ or y@/By a c c o r d i n g l y t o e(E) = 1 or e(E) = 2. Then Y

(2.3.6.4)

D = alXlB/Bx I + bl~Yl

+ Cl)/BZl

where

a I = a;

(2.3.6.5)

bI = b + [ i > n i~i xi ,a =

c1

Let

us o b s e r v e

that

e + i~ n >

iXi xi

m ( ~ , E , p 1) ~ m i n ( n , m ~ , E , p ) ) .

a,

From t h e s e

behaviour o f the polygon o f an hypersurface ( I I 0 1 )

(2.3.6.5)

equations

and f r o m t h e

one deduces t h a t

A ( ~ E , p 1) ~ A(D,E,p) +IH(n)

( s e e ( 2 . 2 . 8 ) ) and thus the r e s u l t .

( 2 . 3 . 7 ) P r o p o s i t i o n . In the s i t u a t i o n o£ ( 2 . 3 . 5 ) , one has t h a t 6 ~ , E , p ~) = =, where p~ = ( x , y ~ , z ~) and thus p~ i s prepared.

Proof. passage Pt ~

If

6 (D,E,P ~)<-

p~ i s

given

, there

is

t

such

by a change as ( 2 . 3 . 6 . 1 )

that with

6(D,E,Pt)< strict

6 (D,E,p~).

inequality,

The

and t h u s

we have a c o n t r a d i c t i o n .

(2.3.8)

Definition.

The a l g o r i t h m defined

in

(2.3.5) w i l l

be c a l l e d a " p r e p a r a t i o n

o f p".

( 2 . 3 . 9 ) C o r o l l a r y . There i s always a prepared s t r o n g l y normalized system o f r e g u l a r parameters (For s h o r t , we s h a l l weite p . s . n . s . r . p . ) .

(2.4) Main r e s u l t

(2.4.1)

The b e h a v i o u r

of a winning

(2.4.2)

of

6(O,E,p)

where p i s

a p.s.n.s.r.p,

gives

us t h e e x i s t e n c e

strategy.

Proposition.

If

( X , E ~ ,P)

is

of the type 0-0,

w(D,E)

= 0 and

(X',E'~',P')

52

is

a directional

properties

is

quadratic

a)

v(O',E',P')

b)

e(E')

normalized from

the

= O,

follows

If

r.s.of

p.

of

dratic

p =

(O',E')

the

one

of

the

two f o l Z o w i n g

Let

(x,y,z)

p.

.

e(E)=2)

of

(X,E~P).

<

c)

e(E')

= e(E)

(X',E'~

if

that

6(D,E,p)

one has t h a t

that

(X,E,D,P)

dim Dip

and t h e r e

',P')

and l e t

Then one o f

r or

such

(2.4.5.1)

We s h a l l

suppose

is

by

given

exist

e(E')=l.

60,E,p)

= I

for

(This

= 1 for

a s~rongly

also

follows

is

of

the

type

(X',E'~',P') following

0-0,

let

wO ,E)

be a d i r e c t i o n a l

properties

is

=

qua-

satisfied

= O.

a p.s.n.s.r.p,

p'

= (x',y',z')

for

taht

that

(T-l,0):

(x',y',z')

the

(~',E')

6(~',E',p')

p =

(1.2.1).

Conversely,

be a p . s . n . s . r . p ,

< e(E).

blowing-up

of

one has n e c e s s a r i l y

then

us s u p p o s e

e(E')

Proof.

= 0

proof

1 = e(E')

of

(hence

b)

that

then

(I.2.1)).

blowing-up

obtain

from

>

r.s.

a) v ~ ' , E ' , P ' )

the

dim D i r

easily

2 = e(E)

Proposition.

let

< r or

normalized

proof

(2.4.5)

(X,E,D,P),

< e(E).

It

Remark.

each s t r o n g l y

of

satisfied

Proof.

(2.4.3)

blowing-up

works

= 6(O,E,p)

a)

and b)

x = x'; for

c).

y

-

are

1.

not satisfied.

= x'y';

First,

if

z = x'z'.

p'

is

Then we have t h a t It

is

possible

a p.s.n.s.r.p,

then

to one

has t h a t

(2.4.5.2)

where the

o(u,v)

behaviour

(2.4o5.3)

A~',E',p')

= of

(u+v-l,v). the

This

polygon

of

result

=

o (A(D,E,p))

follows

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

m~ ' , E ' , p ' )

from

the

(110])

= m(D,E,p)

-

1.

definition

and t h e

fact

of

the

that

polygon,

53

Now ( 2 . 4 . 5 . 1 ) is

enough t o

follows

from

prove that

p'

(2.4.5.2) is

reasoning is

like

(2.4.5.2)

notsatisfied.

Case A: reasonning

strongly

for

e(E)

like

for

normalized.

and

We s h a l l = 1.

Fact

(2.4.5.3)

Moreover,

if

~

A(/],E,p).

Thus i t

one has t h a t

one w o u l d

distinguish

and

(0,1)

> 2

have ~ ) ' , E ' , P ' )

< r if

two c a s e s :

Now we d i s t i n g u i s h

(2.4.5.2)

that

Moreover,

6~,E,p)

(2.4.5.4)

then

the

a p.s.n.s.r.p.

(2.4.5.4)

since

and

two p o s s i b i l i t i e s .

(2.4.5.3)

If

one d e d u c e s e a s i l y

6' = 6 ( D , E , p ) - I

E /Z and t h e

6~] , E , p )

that

change

p'

> 2,

is 6~ ,

y'l=Y~+~x '

6' z~=z'+Xx'

, increases

6',

(2.4.5.5)

Yl

increases there

is

p'

not

is

then

6~,E,p),

which

no p r o b l e m . strongly

Let

= y +Px6~l is

then

Pl

normalized

=6 ( Z ] , E , p )

and i t Case

is

not

if

we make z I

(2.4.6)

then

is

B:

strongly

continues enough t o

e(E)

= 2.

normalized,

= z + Xx 2,

Definition.

to

Let

If

type

0-0.

6~0 , E , p )

we f i n i s h

(X,E,D,P)

Theorem.

There

= 1.

is

zI

J(D,E)

a p.s.n.s.r.p.

Since

(0,1) ),

If

6' ~

~ A(O',E',p'),

(p,X)~(0,0).

iF If

= z + X x2

such t h a t

2 = 6 ( t ] , E , p 1) =

p by P l " > 2 we F i n i s h =

(y,z)

then

as

above.

we have

If6

' = 1 and p'

(2.4.5.6)

too.

Now,

as a b o v e .

be o f t h e

type

0-0.

If

w(D,E)

= 1,

we s h a l l

put

= O.

put

6(D,E)

(2.4.7.)

;

interchange

since

p'

J(D'E')~(y'+px',z'+Xx'

6 (O,E)

w(Z],E) = O, we s h a l l

6'

be a p . s . n . s . r . p ,

If

(2.4.6.1)

Thus

now t h a t

Yl = Y + #x2

= (x'Yl'Zl)

= z + Xx #+1

a contradiction.

us suppose

(2.4.5.6)

; Zl

= min

{6(D,E,p);

is a strongly

p is

winning

a p.s.n.r.s.p.}.

strategy

for

the

reduction

game

of

54

Proof. The s t r a t e g y i s

defined at the beginning o f the s e c t i o n . I f

status t there i s no permissible curve, then one has 6 ~ , E ) < ~ . I f miss,

then

e' , 6' of

(w',e',

6') ~

(w,e,6-1)

l e x i c o g r a p h i c o r d e r , where w,e, 6 are the i n v a r i a n t s f o r

are

for

(1.2.3),

(X',E',~ ',P'),

(X,ED ,P) and w ' ,

the corresponding quadratic t r a n s f o r m . Now, in

we have a winning s t r a t e g y . On the other hand a c t u a l l y i t

winning s t r a t e g y because there e x i s t s c l e a r l y a

3.

then r e s u l t s

( 2 . 4 . 5 ) shows t h a t

(2.4.7.1)

f o r the

p l a y e r B do not

he must t o choose the closed point given by the d i r e c t r i x ,

above ( 2 . 1 . 5 ) ,

a t the

view

i s a strong~

longest r e a l i z a t i o n o£ the game.

INVARIANTS ASSOCIATED TO THE TYPE 0-1

(3.1) Polygons f o r type 0 - I

(3.1.1)

In the

(3.1.2)

Definition.

p is

suited

sequel

for

we s h a l l

A system

(E,P),

(3.1.3)

Remark.

is

not

true,

in

the

type

(3.1.4)

Let

A s.

thus

Z

o?

=

points" as i n

that

o£ r e g u l a r

and E i s

r.p.

in

as

"normaZized"

has

(X,E,D,P)

parameters

~J(D,E)

given

(1.1.4) a weaker

is

is

the

p = (x,y,z)

by x or

always sense

of

type

is

"normalized"

iff

by x y .

normalized,

than

0-1.

the

but

the

converse

corresponding

concept

0-0.

p = (x,y,z)

be n o r m a l i z e d

(3.1.4.1)

(By

suppose

and l e t

D = ax8/Bx

@lSy i £

E is

Exp ( D , E , p ) (2.2.6).

given is

Finally,

by

x and

defined

as i n

the

polygon

us suppose t h a t 0

is

generated

by

+ bSy + eBl@z

8y = y S I B y i f (2.2.4). &~ ,E,p)

E is

And t h e is

defined

given

invariant as i n

by xy).

The " c l o u d

m~,E,p) (2.2.8).

is

of

defined

55

(3.1.5)

Remark.

The p o i n t s

(3.1.5.1)

(see

o£

¢(Exp

2.2.8.1)

(D,E,p)

n {(h,i,j);

are

contained

in

number o£ v e r t i c e s

and t h e y

satisfy the

(3.1.8)

Definition.

Let

(Z/r!)

us d e n o t e

(3.1.6.1)

the

"main

of

the

vertex".

vertex Let

of

If

(3.1.7)

-1/slope is

only

Remark.

invariants rences,

above property

lowest

us d e n o t e

for first

of

the

segment

one v e r t e x ,

A(D,E,p)

(see

has o n l y

a finite

I I~ ).

~ ~,E,p))

abscissa

of

A~,E,p).

This

vertex

will

be

by

As f o r the

the

the

control

case of

e(E)

(3.1.8)

Let p =

Let Y be given resp.

by

(x,z)

and

type

0-1

and second

vertices

of

6(D,E,p).

=~ .

the

(see are

numbers

IlOI).

a bit

(8, e,~)

But there

will

are

more c o m p l i c a t e d

be t h e main

certain

diffe-

and second t h e

be added as an i n v a r i a n t .

(x,y,z) be a normalized system of regular parameters. let Z be given by (y,z). Then Y is a permissible cen-

Z is a permissible

the line u=l, resp. v=1

first

surfaces,

algorithms

will

the

e(D,E,p)

of

the

preparation

Proposition.

joining

we p u t

number o£ c o m p o n e n t s

ter,

that

c(D,E,p)

value there

follows

by

(3.1.6.2)

the

It

(~(D,E,p),

coordinates

called

2.

j~r-1})

center,

iff the poiygon

A(D,E,p) does not intersect

(u,v being the coordinates in lR2).

Proof. If e(E) = 2 it follows easily from the standard results on hypersurfaces (see

It01).

If e(E) = 1, the only problem is in the fact that

(3.1.8.1)

may

not

$(Exp

imply

that

b

has

(zb/y)

~ (j
multiplicity

r along

m(D,E,p) < m and v=1 must intersect the polygon.

c(v>l)

Z.

But

in this case,

necessarily

56

(3.1.9) Y,

Proposition.

resp.

Z, of

tements

is satisfied:

(X,E~

a)

b)

(x,z),

c)

such

the

are

<

the

lexicographic Proof.

If

by

given

,E' ,p' ),e(E'

o£ p a r a m e t e r s

and l e t

be a d i r e c t i o n a l

0-1 and one o£ t h e

following

blosta-

of

the

blowing-up

and t h e

equations

of

the

blowing-up

and t h e

equations

by T - 3 . center by T - 4 .

or

by the Then

type

, the

(T-l,0)

is n o r m a l i z e d .

(3.1.9.2)

the

system

(X',E',O',P')

center

permissible

be g i v e n

(B ~ '

the

given

are

Let

of the

Z is

given

= (x',y',z')

is

are

not

(3.1.9.1)

for

it

(y,z).

Y is

blowing-up

tions

p'

that

permissible,

the

be a n o r m a l i z e d

resp

blowing-up

Y and Z a r e

that

(x,y,z)

permissible,

Z is of

us s u p p o s e

by

,P)

Y is of

p'

p =

be g i v e n

wing-up

Now, l e t

Let

blowing-up

quadratic

and t h e

equ~

T-2.

equations

one

is

of

the

blowing-up.

Moreover,

let

has that

) , ~(D' , E ' , p ' ) , ~ ( D ' , E ' , p ' ) ) <

(B ~ , E , p ) ,e (E) ,E ( ~ , E , p , ) ,

~ ( ~ , E , p) ) .

order.

e(E)

= 2,

(3.1.9.3)

then

e(E')

= 2 and

A(O',E',p')

= (~(A(Z~,E,p))

where (3.1.9.4)

a(u,v)

= (U+v-l,v)

if

(T-l,0)

= (u,u+v-1)

if

T-2

:

if

T-3

if

T-4

(u-l,v)

= (u,v-1)

then

the

by D,D' I.

result

follows

generators

2.2.5).

in

(110 I)-

Let

o£ ~ and ~ '

obtained

one f r o m a n o t h e r

Now, w i t h

notations

3.1.9.5)

zs t h e

as

image by ~ o£ t h e

Q (Exp

convex

as i n

2.2.7,

(D',E',p')~

hull

of

us s u p p o s e

that

by t h e

one can see t h a t

(j~r-1))+IR

e(E)

2 o

= 1.

Let

equations

the

us d e n o t e of

convex hull

of

57

(3.1.9.6)

~(Exp(D,E,p)

n

(j<

r-l))

2

+IR

--

o

Now, the r e s u l t f o l l o w s from the f a c t t h a t i f 8

decreases.

If

m~0',E',p')

m(D',E',p')

in

(3.2.)

cases

(T-l,0)

and T - 3 one has

Good preparation.

(3.2.1) tain

Here t h e

stability

(3.2.2)

-

I

parameters

Although as

behaviour

o£ t h e

(3.2.3) us

the

change o f

the

anunified

Lemma.

Let

for

suppose

type,

(s,t)

like

are

is

a vertex

ones i n

existence

in

order

to

obtain

a cer-

(3.1.9).

of

significant

a winning

strategy

differences

in

the

type

0-1

technical

= 2.

be a n o r m a l i z e d o£

base and l e t

&(~,E,p)

and

us suppose t h a t

(s,t)

~ ~

2 o

. Let

e(E)=2.

us c o n s i d e r

coordinates

zI

a)

be r e s t r i c t e d the

o£ t h e

there

(3.2.3.1)

X ( k.

will

= 1 and e ( E )

p = (x,y,z)

that

(3.1.9.3).

considered

purposes

cases e(E)

Let

= m(O,E,p),

First cases

under t r a n s f o r m a t i o n s

is treated

where

= m(D,E,p)

T-3

(3.1.9.8)

thus

but

(T-J,O)

(3.1.9.7)

and i f

T-2 or T-4 then m ( O ' , E ' , p ' ) = ~ ,

Let

Pl

= (x'Y'Zl)"

= z + kxSy t

Then one has t h a t

Pl i s n o r m a l i z e d .

b) A ( ~ , E , p 1) c is

Proof.

a vertex

A(~,E,p)

and e v e r y v e r t e x

o£ & ~ , E , p )

of A(D,E,Pl).

a) trivial. b)

Let

us suppose t h a t

~ is

generated

by

different

from

(s,t)

58

(3.2.3.2)

D = axa/Bx + bya/ay + ca/az = alXa/ax

+ blYa/ay

=

+ Cla/az 1

Then one has t h a t

(3.2.3.3)

a = al;

Now, l e t

us f i x

b = bl;

a monomial which

cI

appears

(3.2.3.4)

distinguish

two

cases.

Flrst,

A(~,E,p')

after

suppose

that

M does

not

define

a

fficient Then,

if c.

if

they Now,

contribute let

(s',t')

o f them t o

us

to

If

po±nts which are (s',t').

in

Now, b)

Remark.

explained image

contributes

to a point

2

r

and i t

a£ter

~ (s,t),

it

a m o n o m i a l o£ t h e

to a point

(3.2.3.1)

then

is

is

(s',t')

monomials

~

which

coe-

& (D,E,p). contributes

produced a monomlal

Px h Y i z 1 j

same c o e f f i c i e n t

(3.2.4)

eventually

M contributes

M produces (s',t')

+ IR o

t h e n M = pz

suppose t h a t

(s,t), (s,t).

(s,t),

(3.2.3.7)

the

which

in

(s,t)

Moreover,

and

us

n (a~r-1).

(3.2.3.1),

(3.2.3.6)

in

let

Exp ( b , E , p )

Then M p r o d u c e s m o n o m i a l s ,

all

o£ D.

of

(3.2.3.5)

of

in a coefficient

M = p xhyiz j,

We s h a l l point

= c + xsxSyta + ktxSytb.

The

by saying

by ~

o£ this

the

as M and t h e

rest

segment j o i n i n g

£ollows

fferent monomlals never kili,

and

monomials produced contribute (s',t')

and d l f f e r e n t s

"M contributes to a point

that M defines is

the

from

to

(s,t)

easily.

expression

point

(s,t)

of

(s,t).

a point Let

in Exp

(s,t) of & ~ , E , p ) "

(O,E,p) n ( j ~ r - 1 )

may be

and that the

us observe that since contributions of di-

it ls possible that after

nomials would produce a pair of monomials which kill.

(3.2.3.1) two different mo-

59

(3.2.5) of the

Remark.

A(~E,p)

which

existence

(3.2.6) shall

From t h e

of

may be

following

is

Let

lemma,

one see t h a t

in A (D,E,Pl) in

the

third

and

is

is

the only

a direct

vertex

consequence o f

coefficient.

be n o r m a l i z e d

of

is

A~ ,E,p)

this

(s,t)

and

"well

let

us suppose

prepared"

if

e(E)

= 2.

We

one has one o£ t h e

possibilities: a)

(s,t)Z

b)

(s,t)

z 2 o 2 E and t h e r e o

in

no change as

A ( D , E , p 1) ~ A ( D , E , p )

(3.2.7)

Proposition.

Let

p =

(x,y,z)

(3.2.3.1)

in

such a way t h a t

- { (s,t)}.

be n o r m a l i z e d ,

e(E)

= 2 and

let

us c o n s i d e r

then

one has t h a t

change o£ c o o r d i n a t e s

(3.2.7.1)

zI

such t h a t

if

~st # 0 t h e n

a)

Pl

b)

A(O,E,p)=

Proof.

(3.2.8) tem

the

(x,y,z)

(s,t)

(3.2.6.1)

the

pz r

p =

a vertex

of

not

a monomial

Definition. say t h a t

proof

of

= (x'Y'Zl)

It

is

parameters

p =

~

A(D,E,p)

e(E)

to the

= 2,

(x,y,z)

f r o m an a r b i t r a r y

We b e g i n

(3.2.3.1)

in

each

lies

all

the

that

Definition.

not

s t x y

k st

and i t

is

not a vertex,

normalized.

proof

then

such

that

normalized

with

Let

exists

always a normalized

vertex

by a sequence o f

o£ A ~ , E , p )

is

changes o f t h e

regular prepared.

sysMo-

type

system.

vertex.

are prepared,

us suppose

(3.2.5)

every

a normalized

prepared

vertices

of

there

such a s y s t e m may be o b t a i n e d

Proof.

~ (s,t)

A(Z),E,Pl).

similar

If

(3.2.3.1)

(3.2.9)

is

Corollary.

reover,

(s,t)

= z +

that

system

The l i m i t

and make of this

convergent

as a c o n s e q u e n c e o f

e(E)

= 2 and

successively

p =

changes as

sequencesati~

(3.2.7).

(x,y,z)

is

a normalized

60

base. in

A

"good preparation

(3.2.8).

(3.3)

If

Pl

is

the

Good p r e p a r a t i o n

(3.3.1)

Assume t h a t

(3.3.2)

Before

coordinate

for

p"

obtained

e(E)

is

good

change o f t h e

base,

of

Let

the

o v e r k.

p = (x,y,z)

of D .

For

monomials

We s h a l l

Mon ( D , p )

appear

=Mon

(resp.

b',

resp.

c')=O

let

e(E)

= 1.

us make a s t u d y

of

the

ef?ect

of

a

a

a B y

normalized

base.

Let

in

vgr the

"a",

we s h a l l

expression

of

d e n o t e by Mort ( a , p ) "a"

as

a series

in

the

x,y,z

(a,p)

if

M Mon ( b , p )

~ Mon ( c , p ) .

(disJoint

union)

Let

+ b'8/ay

+ c'~/~z

M g Mon(a,p)

(resp.

Mon(b,p),

resp.

exactly

the

Mon(o,p)).

One has t h a t

(3.3.2.5)

in

be

DM = a ' x ) / a x

and = M o t h e r w i s e .

Fop each

and t h a t

0-I

= z + (x

M £ Mort ( D , p ) .

(3.3.2.4)

a'

type

a coefficient,

which

us suppose now t h a t

where

p ff'P~ Pl"

denote

(3.3.2.3)

Let

write

changes as d e s c r i b e d

D = axa/ax + ba/ay + cB/az

be a g e n e r a t o r of

coordinate

type

(3.3.2.2)

set

the

zI

polygon.

of

we s h a l l

preparation,

(3.3.2.1)

over the

a sequence

= 1

(X,E,D,P)

defining

is

O =

DM, we s h a l l

define

~ DM. M eMon(D,p)

Exp,&,m,

etc...,

formally

in

same way

as

(3.1).

(3.3.3)

Let

a normalized

Pl

=

(x,y,z)

base t o o .

Let

with

z 1 as

us t a k e

in

(3.3.2.1)

and

assume

(a,6)

# (0,0),

Pl

is

61

M = x x h y i z J E Mon (D,p)

(3.3.3.1)

We are interested in the part o f the set

(3.3.3.2)

Mon (DM,p 1)

which g i v e s

i n Exp (DM,E,p 1) p o i n t s

(3.3.3.2)

{(y,6,¢);

and i n t h e p o s s i b l e p r o j e c t i o n s

~< r-1 } k) { ( n , - 1 , r ) ;

to

L e t us suppose now t h a t (3.3.3.3)

in the set

A(DM,E,p 1) and c o n t r i b u t i o n s M 6 Mon(a,p).

Assume f i r s t

that

Then t h e p o i n t s

j>r-1

(i.e.

induced

in

M does n o t produce any p o i n t

(3.3.3.2)

do n o t c o n t r i b u t e

t o "m" and t h e y p r o j e c t

over

+ IRo2] - {(e, 6)} •

r-2.

A~),E,p) a l l

Then M c o n t r i b u t e s to a p o i n t ( ~ ' , 8 ' )

the monomials in Mon (DM,Pl)

induce

c o n t r i b u t e s t o "m". We s h a l l d i s t i n g u i s h two cases: (e,8) = ( e ' , B ' ) then a l l ( s, 8) ~ ( e ' , 6 ' )

= (h/(p-j-1),

points in

i/(r-j-1))

( 3 . 3 . 3 . 2 ) And none

(a, 6) = ( e ' , B ' )

or not. I f

the points c o n t r i b u t e under p r o j e c t i o n to ( s , 8 ) E A(~,E,Pl).

then a l l

in the segment j o i n i n g

the points c o n t r i b u t e t o points in A(DM,E,p 1) placed

(e,8) and ( a ' , B ' ) ,

only one which c o n t r i b u t e s t o ( e ' , 8 ' )

(3.3.3.5)

none o f them c o n t r i b u t e s to (~,8) and the

corresponds t o the monomial

l x h y Z z l J & Mon (DM,Pl)

as m o n o m i a l =

of the polygon&(D,E,p))

in

Assume now j ~

If

+

y i+(s+l )6ZlJ-S~/~z 1].

(3.3.3.4)

in

to m(DM,E,Pl).

One has t h a t

DM = X J ( J ) C [ x h + S a y i + S B z l J - s . a / a x s__L0 s + xh+(s+l~

points

n e ~ }.

of the f i r s t

coefficient.

(The s t u d y of t h i s

2).

L e t us suppose t h a t

M EMon

(b,p).

Then one has

case

is s i m i l a r

fop e(E)

=

62

J (s j ) [xh+Sayi+SB j - s DM =X Z (s zI a/ay + s=O

(3.3.3.6)

+ 6xh+(s+l)c~ y i - l + ( s + l ) B

Let us suppose f i r s t it

Z l J - S a / a z] .

t h a t j > r ( i . e . M does not produce any p o i n t o f A~ ,E,p) and

does not c o n t r i b u t e t o m(D,E,p)).

Assume f i r s t

6 ~ 1.Then,

bution t o "m" and the points in ( 3 . 3 . 3 . 2 ) p r o j e c t over points o f in

(3.3.3.4),

contribute to

except fop ( a, 6).

Let us suppose t h a t

t o "m" and the points in (3.3.3.4). If

(a,B) = ( 0 , 1 ) , M = Xzr ,

there i s no c o n t r i A(DM,E,p 1) placed

in t h i s l a s t case a l l

B= O, i f

t h i s points

1 ~ O, there is no c o n t r i b u t i o n

( 3 . 3 . 3 . 2 ) p r o j e c t oven points o f D(DM,E,p1) placed in

1 = O, then the c o n t r i b u t i o n t o "m" i s given only by the monomial

(3.3.3.7)

x(S(Js)xh+S Zl r - 1 , s = j - r + 1

(if the c o e f f i c i e n t

is nonnull)

(3.3.3.8)

and we

have

m(DM,E,p 1) = h + e ( j - r + l ) ~

Moreover, the points in

(3.3.3.9)

( 3 . 3 . 3 . 2 ) p r o j e c t over points o f A(DM,E,p1) placed in

[(a,B) + l H ( h + ~ ( j - r + l ) ) ]

-

{(%B)}

(see ( 2 . 2 . 7 ) f o r n o t a t i o n ) . Assume now t h a t j = r - l . bution t o "m" and the points in in

( 3 . 3 . 3 . 4 ) except f o r

to

(~,B).

If

~ .

If

i ~ O, there is no c o n t r i -

( 3 . 3 . 3 . 2 ) p r o j e c t over points o f A(DM,E,p1) placed

B= O, M = Xyz

r-1

, in t h i s case a l l t h i s points c o n t r i b u t e

i = O, then the c o n t r i b u t i o n t o "m" i s given by monomial xxhzl r-1 and

(3.3.3.10)

m(DM,E,p 1) = h > m(Z],E,p).

The points in ( 3 . 3 . 3 . 2 ) p r o j e c t over points o f &(DM,E,p 1) placed in

(3.3.3.11)

Finally, A (D,E,p) M eMon

[(~, 6) + I l l ( h ) ]

assume j < r - 2 . induced (a,p).

Let

by M. I n t h i s

(~',

B')

-

{(~,B)}.

= (h/(r-j-1),

(i-1)/(r-j-1))

case we have t h e same r e s u l t

be t h e

as i n t h e case

point

of

63

Let us suppose t h a t M ~ Mon ( c , p ) . "m".

If

Then, there i s never c o n t r i b u t i o n t o

j ~ r , the points in ( 3 . 3 . 3 . 2 ) p r o j e c t over points o f A(DM,E,p1) placed in

( 3 . 3 . 3 . 4 ) , except f o r M = Xzr , which c o n t r i b u t e s only t o ( e , 8 ) . I f

j

< r-1 and

( e ' , 8') i s the corresponding p o i n t in A ~ , E , p ) , we have r e s u l t s as above.

(3.3.4) Remark. To obtain A~,E,P I ) one has t o make the sum o f ( 3 . 3 . 2 . 5 ) and t o co~ sider only those monomials which do not kill in this sum.

(3.3.5)

Lemma. Let p = ( x , y , z )

the f i r s t

t

be a normalized base,

v e r t i c e s o f the polygon A(O,E,p), l e t

let

( a i , Bi) , i

( a t , 8t ) E ~o

2

= 1,...,t

be

and assume t h a t i f

8t = 0 one has the a d d i t i o n a l property

(3.3.5.1)

Let

~i + 8 i ' ~ t

us denote

by i

i

> at

the length of the segment joining

i = I,...,t-I.

(mi,8i) with the next vertex

and iet us denote by - 1 / ¢ i the slope of i i. Let us consider the coordinate change

(3.3.5.2)

z 1 = z + Xx~y~

where ~=at,

8=8t. Then one has that

a) Pl = (x'Y'Zl) is normalized. b) (~i,8i)

, i=l,...,t-1 are the t-1 first vertices of A(O,E,Pl).

c) The monomiais in Pl which contribute to the vertices =

I,...,t-I

and to the

points

the same as for p. Moreover,

in the

segments

one has that

joining

this

(ai,8i) , i =

vertices

are formaiiy

z r is in the third coefficient with res-

pect to p iff it is so with respect to PI"

d)

If t > I, then one has that

(3.3.5.3)

for

the

(¢t_1,-11_1) <

lexicographic e) I f

t

order,

where

¢',1'

( E't_l,-l't_ 1)

means t h e v a l u e s

in

A(~,E,Pl).

= 1, t h e n one has t h a t

(~,e 1) % (a'1,6' 1) for

the

lexicographic

order,

where

(e'l,

8'1)

is

t h e main v e r t e x

of A~,E,Pl).

64

f o l l o w s from ( 3 . 3 . 3 ) .

Proof. I f

(3.3.6)

Lemma.

Let

p =

(x,y,z)

be a n o r m a l i z e d

base and l e t

us c o n s i d e r

the

change

o f coordinates

(3.3.6.1)

z1 = z +

[ (a,B)X

B x y.

6

Let A be the convex h u l l o f

{(~,B);

X 6 ~ 0 }+IR °

2

Then

a)

Pl = ( x ' Y ' Z l )

is

normalized.

b) If w is the first

cissas)

which

is

vertex

of A (looking

not

the

in

polygon

in the sense

A ~ ,E,p),

of increassing

then

w is

a vertex

abs

of

A(D,E,Pl). c) I f

all

the v e r t i c e s o f w are contained in

A(O,E,p)

and not i n the "bor

der"

(3.3.6.2&

B(A(O,E,p)):=

W {segments joining

and t h e

vertex

w = (n,O) o f A( i f o

it

(3.3.6.3)

for

each

vertex

consider

ways e x i s t

(m,6)

of

as A ~ , E , p )

Proof.

If

the

is

0-1,

(3.3.7)

Definition.

be

first

in

and p i s

t

A~ ,E,p) until

enough

sum o f

monomials

ve t y p e

the

finite

vertices

exists)

length of AO ,E,p)}

satisfy t h a t

+ Bn > n

same v e r t i c e s

to

of

to

those

of

B ~0.

ordinate

after produce

the the

change. vertex

Then

AO , E , p I )

strictly

make c o m p u t a t i o n s

(3.3.2.5) D which

such t h a t

as

in For

w after

has e x a c t l y

the

negative.

(3.3.3) b),

note

the

for

(3.3.6.1) that

change,

there since

and al-

we h a -

normalized.

Let vertices

p = (x,y,z) of

the

be a n o r m a l i z e d polygon

A(D,E,p).

base and l e t We s h a l l

say

(ai,6i) that

, i=1,...,t & ~ ,E,p)

is

65

"well

prepared" u n t i l

the v e r t e x

(at,Bt)

iff

for

e a c h (mi,6 i )

one has one o f the

following properties:

a) (~i,8i) ~ ~0 2. b

(~i,8i)

~ ~ 2,

= z + Xx

y

(3.3.7.1)

6i ~ 0 and t h e r e which migth

is

no change o f t h e t y p e

zI =

increase

(a 1,8 I,c 1,-I 1, .... ei_ I,-Ii_ 1, ei,-i i)

f o r the l e x i c o g r a p h i c o r d e r .

(ei,8i) E ~ 2 o

8i '

0 and there exist

(a

=

j'

8.) with j < i

such that

J

C:j + Sj. ei < ei. Z o 2 ' 8i = O, c) is not true and no change z I = z + Xx al may

(~i' 8i) E increase

(3.3.7.1).

We

it is well prepared

shall

say that A ~ , E , p )

is "well prepared"

iff

until the last vertex.

( 3 . 3 . 8 ) Theorem. There e x i s t s always a normalized base p = ( x , y , z ) f o r which A (D,E,P) i s w e l l prepared.

Proof. Let p' = ( x ' , y ' , z ' ) vertex of

A(D,E,p').

If

be a normalized base. Let ( e ' l , 8'1 ) be the f i r s t

A(O,E,p') i s w e l l prepared u n t i l ~'I

If

not, we make the change z' 1 = z'

+ Xx'

y

, 6'1

( e ' l , B'1), we do n o t h i n g .

which increases ( 3 . 3 . 7 . 1 ) the most

We repeat. In t h i s way we o b t a i n a convergent change o f coordinates

(3.3.8.1)

z' 1 = z'

By a p p l y i n g until cond

lemmas ( 3 . 3 . 5 )

the first vertex,

normalized

(3.3.9)

vertex, and so on.

and ( 3 . 3 . 6 )

P'I

= (x''Y''Z'l)"

The c o m p o s i t e

+ ~ XczBx'O~y 'B

we c o n c l u d e

that 6 # ,E,P'l)

Now we r e p e a t of all

is

the algorithm

t h e changes made g i v e s

well

prepared

with

the se-

us t h e d e s i r e d

base.

Remark. I f

p = (x,y,z)

is

a normalized base such t h a t

then i f D = axBl3x + bBIBy + cBl@z

(0,1)

~

&(D,E,p),

66

generates the

D , one

type

vertex,

0-1. one

has

that

Moreover, has

that

(0,1,r-1)

if

p is

(0,1)

~

change

z 1 = z + k y such t h a t

tinues

to

(3.3.10) tion

Very

(3.4.1) gon i s T-2,

of

The

p'

(3.4.2)

~D,E,p),

J r ( c 1) = ( ~ 1 )

type

0-1

is

not

chage

p'

~

p of

is

for

otherwiseD

well

each

(c I = c o e f f ,

could

not

be o f

until

the

first

base

there

is

a / ~ z 1) and i f

(0,1)

con-

prepared

normalized of

a

possible.

it

the

proof w.p

by p'

of

(3.3.8)

is

a "good prepara-

~ p.

see

prepared

or

T-4. the

In

later

(3.5),

have n i c e

But

one

has

singularities

behaviours to

under

ameliorate

under T-I,~

paragraph we s h a l l

this

the systems

parameters

for

transformations

the

withe

suppose

of

choice

of

which

of

the

the

the

type

parameters

poly-

(T-I,0) in

order

~ O.

that

e(E)

= 1 and

( X , E ~ ) ,P)

is

o£ t h e

0-1.

(3.4.3)

Proposition.

coordinate

Let

p =

(x,y,z)

be a n o r m a l i z e d

base

and Z e t

us c o n s i d e r

the

i=l,...,t

and

change

(3.4.3.1)

Let

A(O,E,p) since

denote

since

good p r e p a r a t i o n

well

T-3

such t h a t

and we s h a l l

As we s h a l l

control

type

A(D,E,p),

Definition.

change"

(3.4)

to

be i n

~ Exp ( b , p ) ,

Yl = y + ~ x n '

Pl = ( x , Y l , Z ) .

Then

a)

Pl

b)

A ( D , E , p 1) + I l l ( n )

c)

Let

is

normalized.

(~i,6i),

= A~,E,p)

i=1,...,t

+Ill(n).

be t h e A (D,E,p)

Then, to

~Ek

the

same as

the

monomials

points for

in p.

the

in

Pl

o£

+)H(n). which

segments

Moreover,

vertices

contribute

joining

one has t h a t

to

this #z r

(ei~Bi)

vertices is

in

the

are third

formally

the

coefficient

67

with

respect

to

reference

to a fixed

For

each

j ~ t,

for

&(D,E,p).

d)

e)

If

A(O,E,p)

Mon ( D , p )

for

a) i s a fixed

trivial

is

let

contributes

and t h u s

to

j ~r,

Pl"

(ALL t h i s

is with

prepared

respect

until

to

(mj,Bj)

(~1,61),

iff

it

is

so

t h e n one has t h a t

,E,p)).

L e t M = x x h y i z J be a monomial

in

o?~ :

then there

"m" w i t h

is

no c o n t r i b u t i o n .

Let

j :

r-i,

then

it

po-

t h e monomial

X n~ i + ~ x h + n ( i + l ) z r - 1

(i£

n~ n+l

~ O) one has

(3.4.3.4)

m(DM,E,p 1) = h + n ( l + l )

If

M corresponds

to

points

to a point

(m,B),

then after

(3.4.3.1)

it

contributes

in &(DM,E,p 1)

of the type

(3.4.3.5)

(~,B)

+ p(n,-1)

and t h e monomial w h i c h g i v e s ( ~ , B ) If a contribution

M ~ Mon ( b , p ) to

"m" by

is

and j ~

p~O

formally

r there

%Eixh+n'izr-i

(3.4.3.6)

j~ r-2

to

).

well

prepared with

generator

(3.4.3.4)

in

old

respect

D = ax ~/~x + b S / ~ y + c ~ / S z .

M ~ Mon ( a , p ) ,

(3.4.3.5)

so w i t h

generator

and c) - - ~ d ) .

(3.4.3.3)

If

is

m ( D , E , p 1) ~ m i n ( n + l , m ~

Proof.

ssibly

it

&(D,E,p I)

is well

(3.4.3.2)

If

p iff

is

p & ~.

M. no c o n t r i b u t i o n ,

if

j = r-l,

there

is

we have & r e s u l t

&s

and t h e n

m(DM,E,p 1) = h + n . i .

we have a r e s u l t

as a b o v e .

I £ M ~ Mon

(c,p)

above.

p r o v e s b) and c ) .

(3.4.3.6)

This

one has a l w a y s

and j ~

r,

nothing Finally

i > 2 (see 3 . 3 . 9 ) .

occurs, for

if

e) i t

j~ is

r-1

enough t o o b s e r v e t h a t

68

(3.4.5)

Definition.

prepared,

Let

&~ ,E,p)

is

p = (x,y,z) "very

be a n o r m a l i z e d

well

prepared"

for

base such t h a t

p iff

A(D,E,p)

one has one o f

the

is

well

following

properties:

a) ~= c ( A ( D , E , p ) )

¢/Z

b)

each change

E ~o

and

Pl ~-~ p ' '

for where

o

.

Pl = ( x , ~ , z )

(3.4.5.1)

order,

polygon

and "'" denotes

(3.4.6)

Proposition.

Proof. red,

strictly

one

vertex

(c,-i). which

is

< (~,-1)

i denotes

exists

always

the length

well

a normalized

prepared

= y + ~xE

followed

Repeat.

we do n o t

well

preparation

o£ the first

segment

of the

in &(~,E,p')".

a normalized

make a c h a n g e Y l

ses

"things

There

Take

where

by a good

one has t h a t

(~',-1')

for the lexicographic

= y + ~x E f o l l o w e d

Yl

If

prepared

and

base.

by a

If

it

is

prepared

not

very

good p r e p a r a t i o n

stop

thus

very well

then

the

the

resulting

resulting

base

base.

well

prepa-

which

decre&

polygon is

very

well

has o n l y prepa-

red.

(3.4.7) me

for

Remark. the

The above

result

of

algorithm

each

does not assure

realization

for the proof of the existence

of the

of a winning

that the pair

algorithm,

(E,-I)

is the sa-

but this will

not be used

stability

the

strategy.

4. A W I N N I N G STRATEGY FOR TYPE 0-1

In

this

of

parameters

well

of

the

in

result

section

we

prepared (3oi.9)

of the singularities

of

shall

and

prove

very

in order

welI

the

results

prepared

which

to use the invariant

the type 0-1.

of

wiil

(B,e,~a)

allow

of

systems

us to profite

for the oontroI

69

(4.1) Good preparation

(4.1.1)

We s h a l l

stability

suppose always t h a t

(X,E,D,P) i s o f the type 0-1. Let p = ( x , y , z )

denote a normalized system o f parameters

such t h a t

A(~,E,p)

i s w e l l prepared. Let

us consider a d i r e c t i o n a l blowing-up ( X ' , E ' , ~ ' , P ' ) which i s quadratic only i f and

( y , z ) are not p e r m i s s i b l e and t h a t

p. and l e t us denote by p' paragraph

the phrase

or dim Dip ~ ' , E ' ) In a l l

= (x',y',z')

given by ( T - l , 0 ) ,

= O, or ( X ' , E ' ~ ' , P ' )

T-2,

T-3 or T-4 from

the r e s u l t i n g system o f parameters. In t h i s

" p l a y e r A has won" w i l l

mean t h a t

"v(~',E',P')

< P=v(~,E,P),

i s o f the type 0 - 0 " .

the paragraph we s h a l l suppose t h a t D i s generated

(4.1.1.1)

by

D = axalax + bay + ca/az

where @y means B/@y i f

e(E) = 1 and y@/ay i f

(4.1.1.2)

e(E) = 2, and t h a t D '

i s generated by

D' = a'x'@/@x' + b'@y' + c ' 8 / @ z ' .

Our f i r s t trix

is

(x,z)

r e s u l t concerns

t o the s t a b i l i t y

o f the equation o f the d i r e c -

in a w e l l prepared s i t u a t i o n .

(4.1.2) P r o p o s i t i o n . One has always z E j r ( c ) .

Proof. Let us suppose t h a t

z ~jr(c).

If

e(E) = 2, since we have type 0-1,

necessarily

(4.1.2.1)

If

jr(c)

# ~ O, (0,1) i s the f i r s t

= z + py d i s s o l v e s t h i s zI

v e r t e x o f A(D,E,p) and the coordinate change z I =

vertex. I f

= z + Xx d i s s o l v e s i t .

= (z + X [ + ~ Z ) .

~ = O, X ~ O, then

(1,0) i s the l a s t v e r t e x and

Let us suppose t h a t e(E) = I .

First,

dim D i r (O,E) = 2, then

(4.1.2.2)

where H = H ( D i r ~ , E ) ) ,

Jr(D,E) :

(~ + k ~ + ~ y ) = JH(D,E)

(see ( 1 . 1 . 2 ) ) . One has

let

us suppose t h a t

7O

(4.1.2.3)

If

p~

Thus

JH~,E)

O,

then

(I,0)

there

is

the

s a p p e a r s and t h e

exists

main

a monomial

vertex

sequence

= jr(D(z+py)))

in

of

Moreover, vertices thout

the in

possibility

u+v

touching

dim D i r

= 1,

the

(~,E)

= I.

c)

v~

A~,E,p).

(3.3.7.1)

(4.1.2.3)

of

,

y~

If

will

(3.3.7)

is

not

the

possible,

initial

f o r m o f c o r b.

= z + py,

this

If

X# O, t h e n

p= O,

because t h e r e

change z I = z + X x d i s s o l v e s

contradiction.

Since one has t y p e 0-1 and p i s

(4.1.2.4)

in

be i n c r e a s e d .

= jr(c).

ones,

O,

we ~ake z I

JH(O,E)

O, now, t h e

preceeding

r

y.y

= jr(c+pb).

jr.,E)

= (x,z

Let

vertex

is

this

di-

no o t h e r

vertex

wi-

us suppose t h a t

normalized,

we have t h a t

+ XZ)

and (4.1.2.5)

JH(D,E ) = j r ( D ( z + X y ) )

jr(c+Xb)

now, in

we can

u+v = 1,

(4.1.3)

reason but

Theorem.

as

above

(note

only

that

there

If

e(E)

= 2,

that

is

= jr(c+Xb)

= (z+Xy+px)

this

does r r o t i m p l y t h a t

no v e r t i c e s

then

in

one o f t h e

u+v = 1,

there

is

no v e r t i c e s

v > 0).

two f o l l o w i n g

possibilities

is saris-

lied: a) Player A has won. b)

(X',E',O',P')

is of the type 0-1, p' is normalized

and A(O',E',p')

is

well prepared.

Proof.

have yet there

Let

been

exists

us

proved. a vertex

suppose

Let

that

a)

us suppose

( a ' , B') ~

is

not

true.

The f i r s t

that A (D',E',p')

A(D',E',p')

two s t a t e m e n t s

is not well prepared.

w h i c h may be d i s s o l v e d

by t h e

in

b)

Then,

coordi-

! nate

change

z'

of

(3.1.g),

if

we make z I

1

(see

= z,+Xx,~

y,6'

(3.1.9.4)).

= z+Xx~y B, t h ~

"

Let

(~,B)

Then

( ~ 6) i s

the

hypothesis

= o-l(a' a vertex assures

'

B')

of

where o i s

b~,E,p)

us t h a t

(see

as i n

the

(3.1.9.3))

proof and

71

(4.1.3.1)

A(O,E,p)

and t h a t A(O,E,p I ) i s directional Pl

(also

the

ned f r o m to

Pl

(T-l,0),

property is

Pl j o i n t l y

for

exactly with

(4.1.4) Theorem.

a ( O , E , p 1)

Pl = ( x , Y , Z l ) .

w.p.,

transform

=

T-2,

(x,z)

or

Moreover,

e+8 > 1, t h e n

T-3 or T-4 from p coincides (y,z)

(x',y',z'l).

one has t h a t

of

being

permissible)

The c o n t r a d i c t i o n

appears

the

with the one from

and t h e

base o b t a i -

by applying

(3.1.g.3)

(4.1.3.1).

If e(E) = 1, then one of the following possibilities is satisfied:

a) Player A has won. b) ( X ' , E ' ~ ' , P ' ) A(O',E',p')

is o£ the type 0-1, e(E') = I, p' is normalized and

is well prepared.

c) (X',E',O',p') A(O',E',p')

is of the type 0-1, e(E') = 2, p' is normalized and

is well prepared until the first vertex.

Proof. Let us suppose that a) is not true. First, let us suppose that e(E')

= 1,

i.e.

we can r e a s o n

we make just

as i n

marks:

take

(~',8')

is

true

for

not

= 2,

i.e.

proof of

4.2

type

(4.2.2) 8 of

not

it

is

and

o is

has given

@ood p r e p a m a t i o n

and p = ( x , y , z ) from p is

After

(T-I,~),

t h e main v e r t e x .

blowing-up,

in

the

Let

(4.1.3)

prepared true (~',8')

(m',B')

at

the

above,

vertex

for

= o((s,

end o f

the

proof

but with

the

following

and o b s e r v e

(~,8).

be t h e

by ( 3 . 1 . g . 4 ) ,

(T-l,(),

Let

first

8)),

that

if

us s u p p o s e vertex,

where

as i n

of

c)

(3.1.g), re-

of

now t h a t

then,

( a , 8) i s

now, we can r e a s o n

~ ~o, a r e c o n s i d e r e d .

be a n o r m a l i z e d

same as

(T-l~0)

e(E'):l.

e(E)

and w e l l

e(E')

looking the

(3.3.7)

at

first

(4.1.3)

: the

vertex

above.

= 2.

(X,E,D,P) will

prepared

base.

Let

be o f

the

y1=Y÷~X~ t h e

from p1=(x,y~z).

Thus b y ( 3 . 1 . 9 )

A "virtual"

case

Looking

stability

will

the

of

not

that

The t r a n s f o r m a t i o n s

0-I

(T-I,~)

one

or T-3.

proof

first

(~',8')

(3.1.g)

6(D,E,p)

(4.2.1)

the

the

we make T - 2 o r T - 4 .

of

Very

(T-J,0)

transition

one has o n l y to

the

to

case e ( E ) = l

control is

the

ordinate

made b e f o r e

72

(4•2.3) D=

Proposition•

(~(~),Eo),

relatively

(X,Eo,~,P)

to

Proof.

L e t e ( E ) = 2 • L e t E° be such t h a t is

o f t h e t y p e 0-1 and p i s

I(E o ) = (x) normalized

(then

Eo ~ E ) . T h e n

and w e l l

prepared

(X,Eo,D,P)•

The o n l y

possible

common d i v i s o r

o f t h e new c o e f f i c i e n t s

is

"y"

but this

i s not possible since J(~,E) = (~), t h i s proves ~ = (a(D),Eo). One has type 0-1 since the l a s t coefficient.~RemaiBs unchanged and the middle c o e f f i c i e n t has order r + l • C l e a r l y p i s normalized and i t

i s w e l l prepared since A(~,E,p) = A(D, Eo,P).

(4.2.4) C o r o l l a r y . With n o t a t i o n as above, i f

Pl = ( x ' Y l ' Z ) is given as in ( 4 . 2 • 1 ) ,

then a) Pl i s

normalized and

A(O,Eo,Pl) is w e l l prepared u n t i l the f i r s t

tex (which i s the same as in b) A w e l l preparation o f

A(O,E,p)).

A(D,Eo,Pl) can be made by changes o f the type

= z + x x ~ 1 where a+6~2 (and thus the equations T - l , 0

zI

ver-

are not a f f e c -

ted).

Proof•

(4.2.5)

If

follows

from

Theorem. Let e(E) = I . ,

(3•4•3)•

l e t us suppose t h a t ( x , z ) does not define a permi-

s s i b l e c e n t e r , and l e t z ~-> z I be a goo@preparation o£ the polygon A (D,E,p 1) where Pl = (x'Y1'Z) i s as in

( 4 . 2 . 1 ) . Let us denote P2 = ( x , Y l , Z l ) . Then

B a) The change z ~-~ z I i s obtained from changes z ~-~ z+kx Y l ' where e+B>_2. b) The f i r s t

vertex o£ A(D,E,p 2)

is the same as the f i r s t

vertex o£

A(D,E,p) • c) ¢~O,E,p 2) ~ I

(see (3.1.6) f o r d e f i n i t i o n ) .

Proof. a) and b) f o l l o w s from ( 3 • 4 . 3 ) . I f

E(D,E,p) <1, c) i s t r i v i a l

d e f i n i t i o n o f very good p r e p a r a t i o n . Thus, the only remaining case i s

(4.2.5.1)

~:(D,E,p)

> 1.

by the

73

We

shall

prove

that

in t h i s

case

one

(4.2.5.2)

Let

~ ( D , E , p 2)

us d e n o t e by

ses:

~+B

~ ~

o

the

case.

s+8

segment

(x,z),

then

Second

new

first

2 and

it

with

be m o d i f i e d

case.

(m,B)

is g e n e r a t e d

m (D,E,p)

= 1,

Now, l o o k i n g

at

~ K 2. o

not

(~+8+1,-I) prepared.

First,

contribute

three

ca-

such As e

a vertex

that <

it is

(s+8,O)

I because

(~,B)

and

o£ & ( D , E , p 1) p l a c e d

o£

or

the

it has

ne-

assumption

on

so s ( D , E , p 1) = 1. M o r e o v e r ,

preparation.

let

us m a k e

some

reductions.

Let

us s u p p o -

+ bS/~y + c8/8z.

we can suppose t h a t

clearly

(3.3.3)

e (D,E,p) and i n

segment for

distinguish

by

= 1,

view of

Exp the

We s h a l l

y H-~ y I g e n e r a t e s

by g o o d

(4.2.5.2) L is

A(D,E,p).

~ ~ 2 but e + B E ~ . o o

D = axa/ax

Since one has t y p e 0 - I ,

where

of

= I.

is d i f f e r e n t f r o m

(4.2.5.1)

If

Thus

is w e l l

vertex

cannot

vertex (~,8)

~ ~o" (e,B)

Thus

the

situation

o

joining

ordinate.

se that D

the

EZ

gative

this

(m,B)

, (e,B)

First in

has that

the

joining

Thus, r-1

a

thus the

=

is

no m o n o m i a l s

there

fact

is

Xyz r - 1 e

Mon ( b , p ) .

n o t a monomiall xzr-%Mon ( b , p )

(a,B)

~ (0,1),

we may suppose t h a t

( D , E , p ) ~ L,

(O,O,r)

result.

there

and

(r%rB,O),

since

the

other

m o n o m i a l s do

we may suppose

X

x ~(r-i)y~(r-i)zi-1

i=1 l,a r-1 (4.2.5.3)

b =

~,b x

~(r-i)

Y

8(r-i)+1

i-I z

i=1

c

Let

us suppose

is not w e l l vertex.

Then

that

prepared,

=

r-1 X zr + ~ X xa(r-i)yB(r-i)z r,e i=O I,c

(e+8,O) and

let

is

a vertex

us s u p p o s e

of

& ( D , E , p 1)

that

a change

i

X

,

~0

r,c

(as i n

the

z I = z +Px

first a+6

case) dissolves

but

it

this

74

(4.2.5.4)

X.

since is

=

1,a

otherwise

the

vertex

0

(~+B,O)

i

could

not a v e r t e x o f n e g a t i v e o r d i n a t e ,

Now,

we have

only c. To simplify,

(s+8,O), we must to consider

(4.2.5.6)

have

i

l

eliminated.

Now,

if

there

= 1,...,r-1

l

X

r,c

= 1. In order to eliminate

(-~)S(r-1)x(S+B)(r-i)zi

i,c

after z ~-* Zl, necessarily

(4.2.5.7)

been

only the part of c (after y ~-~ yl ) given by

i=O,...,r-1

(_~)B(r-1) i,c

and t h e n ,

not

let us suppose

zr + [

if (e+6,O) disappears

1,...,r-1

one deduces t h a t

Xi, b = 0

(4.2.5.5)

=

one has

= (r)pr-i i

we c o u l d have e l i m i n a t e d ( ~ , 8 )

i = O,

.,r-1 ""

"a p r i o r i "

w i t h a change

8 ~ B z = z 1 + ( # / ( - { ) ) K y.

Contradiction.

T h i r d case.

(s, 8) ~ ~[ 2 o '

Moreover,

to

obtain

ting,

let

us observe t h a t

fact,

if

0~ 8 = I ,

the

contradiction,

o

. We can reason as above,

we can

necessarily

Once made Y '

(third)

coefficient

and z ~ z

(4.2.5.7)

I

~ > 1 and t h i s

= z + px

a+B

until

too.

(4.2.5.5)

Before s t a r -

by the h y p o t h e s i s .

In

c o n t r a d i c t s the hypo-

, one can assume t h a t the o n l y

is

r-1 Zl r +

~Yl

suppose

e+8 >_ 2 and then B >1

then by the p r o o f o f 4 . 1 . 2 ,

thesis.

(4.2.5.8)

a+B ~ Z

o~8 )i.

( r--i)-1( ~)j (S (r-i)) xe(r-i)+jy8 (r-i)-j (Zl ZXi, c -

[

j

x

i=O j=O From ( 4 . 2 . 5 . 7 ) we deduce field

k,

that

r ~ 0 (mode), where

• is

the c h a r a c t e r i s t i c

o f the

then X r _ l , c ~ 0 and t h u s

(4.2.5.g)

contradiction implies

if

8(r-(r-1))

with

T > 0).

one has that

Let

the h y p o t h e s i s o f i

~ ~

o

such t h a t

this

= B @

case.

T1 d i v i d e

o Thus,

necessarily

r=O

(mod%) ( t h i s

r and TI+1 does not d i v i d e

r.

Then

75

(4.2.5.10)

in view 1

=ST that

r - x

of

• Let I'

(4.2.5.7).

Let

i'

that

be

such

< l and

i

= r-~

o

I' T 1

if Jo = 8t

(

in

i'

1

8(r-io

It

there

a vertex

exist

0 < t
follows

thus w e l l

that in

that

(4.2.6)

Remark.

fact,

ters.

not d i v i d e

8T I. One

has

)~o

(mod T ) .

of

°

the

point

(e+B-1/~l-l',l/tl-l')

segment and

joining

(a,8)

is

and

in A(~,E,P2).

(~+8,0)

of

ordinate

Now, t with

then

8 >1).

the

hypersurface

from

does

)= o

~ ( D , E , p 2) = 1.

(notice

system

TI'+I

8(r-i

B(r-io)-j 0 i 0 z 1 i' 1 (~+8)T - ~ i' r-T x y z

the

prepared

8T 1 and

not divide

tI does

8 ~ l O, t h e n

y

(4.2.5.13)

(4.2.7)

)

Jo

~(r-io)+j x

zs n o n n u l l .

In

. Since

, then

(4.2.5.8)

4.2.5.12)

c ~ 0),

= max ( i , l i ,

divides

- T

(4.2.5.11)

Then, the c o e f f i c i e n t

1

The

p by

case

is

Let

of

is

e(E)

= I,

let

(T-l,0)

or

T-3

and

i£

the

transform

~he f o l l o w i n g a) A has won b)8 c)

much the

a phenomena t y p i c a l

parameters,

that

very

eliminate

regular

Assume

Then one o f

case

possibility

Theorem. of

third

let

(O',E',') 8(D',E',p')

(x,y,z)

(X',E',D',P') p':(x',y',z')

properties (as

vertex

of the

p =

let

easier

is

if

(~+8,0)

characteristic

created

positive

by y ~

well

prepared

be a d i r e c t i o n a l

nor

is Yl

zero. in the

characteristic.

be a v e r y

be t h e

quadratic

the

transform

resultinglystem

(x,z)

~OP ( y , z )

normalized

of

given parame-

:Ls p e r m i s s i b l e .

is satisfied

in 4.1.)

< 8(L),E,p). =

t e m of r e g u l a r

8~,E,p)

and

parameters.

p'

is a v e r y w e l l

prepared

normalized

sys-

76

Let

Proof. B(~,E,p)

us

by l o o k i n g

suppose at

(4.2.7.1)

(this

that

(3.1.9.3).

a)

and b)

not satisfied.

are

Moreover necessarily

~' = E ( ~ ' , E ' , p ' ) = E ~ , E , p ) - I

is

not t r u e

if

B(D,E,p)

~

B(~',E',p')).

e ~,E,p)

By ( 4 . 1 . 4 )

p'

is enough t o prove t h a t no coordinate change Y ' I = y ' + ~ x '

if

it

is followed by a good prep&cation. Now reasoning as in + (x'

followed

6(D',E',p')

>1,

=

and

= ¢-1.

it

can easily show that if Y'I = y '

Then

i s w e l l prepared, so

E t

may increase (E',-19

( 4 . 1 . 3 ) and ( 4 . 1 . 4 ) one

by a good p r e p a r a t i o n

may i n c r e a -

se (e',-l'), then

(4.2.7.2)

Yl = y + ~xe

followed by a good p r e p a r a t i o n may increase

(e= c'+1)

(e,-l)

which i s a c o n t r a d i c t i o n .

(4.3) A winning s t r a t e g y f o r type 0-1

(4.3.1)

Here

strategy

for

lized

well

we s h a l l

put

the

0-1.

type

together This

the

stability

strategy

will

results

above t o

depend on a f i r s t

obtain

choice

a winning

o f a norma-

prepared system o f parameters and o f a c o n s t r u c t i o n in each step by the

p l a y e r A o f a normalized w e l l

prepared system which depends

(not in a unique way)

o f the preceding one and o f the movement of the p l a y e r B.

(4.3.2) first

(Winning

of

all,

parameters

player

p(O)

Now, we s h a l l

= 1)

well

the

otherwise

P(t+l)

corollary

of

a very

by i n d u c t i o n

prepared

or

point

A chooses

(X,E,O,P) be the s t a t u s

if

in

(y(t),z(t))

the

(4.1.2)

the

If

equation

prepared = 2,

normalized

A c h o o s e s p(O)

status

system

permissible

A chooses the transform.

e(E)

be t h e

normalized are

well

0 o f the type 0-1. system well

of

Then,

regular

prepared only.

as f o l l o w s :

(X(t),E(t),D(t),P(t))

(x(t),z(t)) center,

Let

= (x(O),y(O),z(O)),

proceed

Let e(E(t))

strategy).

of

center.

A does n o t w i n the

and l e t

parameters

centers,

quadratic

of

t

in

p(t)

be t h e

which

("very"

if

A has c h o s e n .

If

t h e n A c h o o s e s one o f them f o r Now p l a y e r this

transformation

B chooses a closed

movement, is

then

expressed

as an e a s y

from p(t)

in

77

one o f t h e f o l l o w i n g

ways

(T-l,{),

(4.3.2.1)

Let p ' ( t )

T-2, T-3 or T-4.

be the obtained system o f parameters.

(as in ( 3 . 2 . 1 0 ) and ( 3 . 3 . 1 0 ) ) t o p ' ( t ) (as in

(3.4.6))

(4.3.3)

above by

if

Theorem.

if

e ( E ( t + 1 ) ) = 1. The r e s u l t o f t h i s w i l l

The

strategy

defined

above

G = { G(t) } be a reaiization

stratety

has

been

{p(t)} the sequence

applied.

One

of systems

has

o£ regular

(B(t),e(t),e(t),~(t))

Since

all

the

sy t o

prove),

A(D(t),E(t),p(t))

in

to

(4.3.3.2)

the

prop.

prove that

order.

I£

G is

a(t+l)

theorem

(4.1.3).

(4.2.7).

(4.3.3.2)

<

Let us denote

used by A. We shaIi denote

finite

vertices

it

is

placed

enough t o

in

(I#!)~ °

2

(e~

show t h a t

< (B(t+l),e(t+l),e(t+l),a(t+1))

w(t)

is

given by ( T - I , 0 ) ,

T-2,

T-3 or T-4,

then

instead o£ p ( t + 1 ) . We have always

if

B(t).

I£ e(t)

Then i n e(t)

= e(t+l).

Now, i t

= e ( t + 1 ) = 2 one has t h a t

e(~(t+l),E(t+l),p'(t+l))

= e(t+l)

these

e(t)

= e(~(t+l),E(t+l),p'(t))

6(t+1) =

fl(t+1)

G is finite.

B(t+1) = B ~ ( t + l ) , E ( t + l ) , p ' ( t ) )

(4.3.3.4)

theorem

that

parameters

have t h e i r

in view oF ( 4 . 1 . 3 ) and ( 4 . 1 . 4 ) . Moreover, i f

ve

prove

( 3 . 1 . 9 ) proves ( 4 . 3 . 3 . 2 ) but p u t t i n g p ' ( t )

(4.3.3.3)

by

of the game for which the

~(D(t),E(t),p(t))).

(B(t),e(t), e(t), ~(t))

lexicographic

strategy for the player A.

= (B(O(t),E(t),p(t)),e(E(t)),

polygons order

to

be p ( t + 1 ) .

is a winning

Let

e(D(t),E(t),p(t)),

for

e ( E ( t + 1 ) ) = 2 and a v e r y good p r e p a r a t i o n

Proof.

(4.3.3.1)

Then A a p p l i e s a good p r e p a r a t i o n

= I and

c a s e s one has I£

e(t)

= 1

B(t)

=

(4.3.3.2). < e(t+l)

B(t+l)

we

have

Since B(t+l)

= 2 one has

(see

(4.3.3.4) ~8(t), (3.1.97)

by

we h a that

remains t o consider the case o f ( T - I , ~ ) w i t h ~ ~ O. By t h e o -

78

rems ( 4 . 2 . 5 ) , e(t+l)

(4.2.4)

one has t h a t

= 1 and B(t+1) < B ( t )

but e ( t + 1 ) = I .

(4.3.3.3)

is

true.

i n view o f ( 4 . 2 . 5 ) c ) .

The p r o o f i s f i n i s h e d .

If

Moreover, i f e(t)

= 2, then

e(t)

= 1, then

B ( t + 1 ) ~ B(t)

-

I I I

-

STANDARD TRANSITIONS FROM TYPE I

O. INTRODUCTION

(0.0.1)

In this

stpategy pesults

fop

in

gins

with

nics

of the

ggest

the

if the

tuation"

chapter,

peduction

chaptep

opdep

a bit

to

will

innen contpol This

shall continue game

in the

genepal will

be

case

the

situation,

in the

in "each

situation"

chaptep

is m a i n l y

dinectly

no standapd

tpansitions

victopy.

n = 3 and

called

contnol

will

chap

1. CLASSIFICATION BY TRANSVERSALITY

field

shall

suppose

"type one".

"changes

k = O. In view of the

to Peach a "type

as fop the type of

of a winning

of

In this case, zeno.

zepo si-

that the game bethe tech-

Actually,

situations",

the bi-

while

the

by the polygon.

to the study of the tpansitions

called

ape tpeated.

to a vector

we

be pnovided

devoted

by the polygon,

Ideals associated

Thus,

not be as genepal

found

be contpoled

(1.1)

the ppoof of the existence

II, ~on the playep A it is enough

obtain

mope

polygon

ppoblem

we

"standand

tpansitions".

which may

Also the fipst

80

(1.1.1)

The

ideals which

be introduced

will

serve

here in a general

(1.1.2)

Let

(X,E,Z~,P)

(1.1.3)

As in I I ( 1 . 1 . 2 ) ,

us to define

situations

will

manner.

be such t h a t

let

the t r a n s v e r s a l i t y

e(E) _> 1, dim D i r

(D,E)>1

and r = v ( D , E , P ) > 2 .

us denote

(1.1.3.1

JH

r ~ ,E) = j r

H(Dir(~,E))"

Let

(1.1.3.2)

JP{D,E) = ~ f ~ R J r ( D ( f ) )

where D ms a g e n e r a t o r o f 0 . Finally

one has t h a t

I {1.1.3.3)

J(~E)

Jr',E)

if

JP(~,E) ~ 0

= ~I(E) ~(f) Jr(D(f)/f)

(1.1.4)

The

three

ideals

above

are

generated

by

if

linear

jr

forms

(D,E) = 0

in GP(R),

thus,

one

can consider

r 1 JHW(D,E) = J H(D,E) ~ Gr (R)

(1.1.4.1)

and the same t h i n g f o r JPw(D,E) and Jw(D,E), w i t h o u t l o s t 1 v e c t o r subspaces of G (R), r

(1.1.5)

Let

us denote

over the f i e l d

by J(E)

the ideal

The ideal

(1.1.6.1)

Inr(z),E)

is defined

Inr (L),E) =

They are

k.

of the directrix

J C(E) the i d e a l o f the t a n g e n t cone of E. As above, J * ( E )

(1.1.6)

of i n f o r m a t i o n .

by

Lr Inr(o(f) ).Gr(R) f~R

of E and let us denote

= J(E) ~ G I ( R ) . r

by

81

where

is

D

generatesD.

defined

One

as I n r ( D , E )

that

has

if

p(D,E,P)

In(D,E)

(1.1.6.2)

Inr~),E) = r-1

= (f)

where M(D,E,P)

(1.2)

The i d e a l

In(~),E)

and

[ Inr(D(f)/f).Gr(R) ~I(E)

One has t h a t

JHP(~,E)

we s h a l l

(1.2,2)

give

priority

OefLnition.

to

the

of

is

type

"two"

type

is

@e j e ( ~ E )

< ~ >

the

¢ eJ*(~E)

type

"three"

such t h a t

The

E which

is

±ff

(1.2.2.3)

(1.2.2.4)

(1.2.3)

"one"

which

if

JPH(~,E)

different

from

= 0 ~ Jr(D,E)

zero.

and

= J(D,E).

~ J(D,E)

+ J*(E)

Jr(~,E) is

= O~

verified.

= Grl(R).

(1.2.2.2) Finally

it

is is

JC(E)

= JH(D,E)

or

JC(E)

= Jr(D,E)

and J H ( D , E )

= 0

or

JC(E)

= J

and J P ( ~ , E )

= 0

conditions

tr&nsversal

is

such t h a t

(1.2.2.3)

of

ideals

= O,

JC(E)

and t h e r e

= J(D,E)~

this

~ Jr(D,E)

±£f Jr(~,E)

(1.2.2.2)

is

of

o£ t h e

JC(E)

the

~ Jr(D,E)

sm&llest

(X,E~,P)

(1.2.2.1)

It

= r.

= r.

(1,2.1.1)

is

p(D,E,P)

if?

Classification

(1.2,1)

It

= 0

(1.2.2.1) to

a form

(D,E)

and of

(1.2.2.2)

J(Z~,E).

mean t h a t

verified of

the

there

and type

is

theme

"four"

is

no

±ff

a component

of

82

(1.2.4)

Definition.

Let

(X,E,Z~,P)

be o f

the

type

one.

shall

We

say t h a t

it

is

of

the type:

(1.2.5)

I-1

. If

dim D i r

(D,E)

= 1 and e ( E )

= 1.

I'-1

. If

dim D i r

(D,E)

= I and e ( E )

= 2.

I-2

. If

dim D i r

(D,E)

= 2 and e ( E )

= 1.

I'-2-1

. If d i m

DiP ~O,E)

= 2, e(E)

= 2, and JC(E)

I'-2-2

. If dim

Oir

= 2, e(E)

= 2 and JC(E)

cJr(O,E).

one,

is

Lemma.

I-1,...,I'-2-2 suited

I£

( X , E , D ,P)

above.

for

(E,P)

= 8/3z

that

> r+l

v(c

such t h a t

xz

(D,E)

D is

if

then

a regular

it

of

one o f

s y s t e m o£ p a r a m e t e r s

the

types

p = (x,y,z)

by

e(E)

e(E)

n(b)

= r,

(so J r ( ~ , E ) then jr(b)

I-1,

I'-1,

ii

If

(X,E,O,P)

is

I-2,

I'-2-2,

I?

(X,E,O,P)

is

1'-2-1,

then jr(b)

and

(x,y,z)

be such t h a t

= 2,

Let by

e = I

(1.2.2.1)

=

(Z,~)

or

JP(D,E)

v(c) ~r+l.

(x,z)

Definition. (resp.

jr(~,E)

up t o

Let I'-1-0)

c

a change

(X,E,D,P)

of

= J(D,E)

dim D i r

since order

shuch a way

and

= (z = 0 ) .

E is

(~,E)

# ~ O.

given

= 1,

JH = O,

if

by x = O. I f

~

JP(D,E)

~ Jr(~,E),

so we can

(x,y,z)

otherwise in

in

= (x = z = 0 ) .

J r H ~ O,

so jr

= (~)

and

t h e n we can

suppose

be such t h a t

dim

Jr(~,E)

=

given

by

E is (D,E)

=

(z),

x,z.

be o f

the

type

(I(E))

~ Inr(D,E)

iff

In

= 2,

= jr(b))

= (z + ~ x ) ,

e = 2 and l e t

J(E)

e(E)

so we can suppose t h a t

contradicts

Let

= 1 or

then jr(b)

~ (~),

If

and t h i s

above v ( c ) ~ r + l .

Necessarily

I-1-0

to

is

as

(~ + # ~ )

= 1,

according

(X,E,O,P)

Jr(O,E)

= O.

(1.2.6.1)

is

generated

Bz = z S / ~ z

J H = O, one has

(~,~),

type

or

r

(1.2.6)

there

type

If

Proof.

=

the

i

iii

suppose

of

D = ax;~/;)x + b@/@y + c~z

where ~ z

since

is

Moreover,

(1.2.5.1)

Din

QO,E)

~6JP(~,E).

I-1

or

I'-1.

(X,E~,P)

is

of the

83

and o f the t y p e I - I - I , I(E)

ideal

(1.2.7) (resp.

of

I'-1-I

otherwise.(In(l(E))

i s the i n i t i a l

i d e a l o f the

E).

Remark. I'-1-1)

resp.

If

p = (x,y,z)

i s as in

then x d i v i d e s I n r ( b )

(1.2.8) Definition.

Let

( 1 . 2 . 5 ) and (X,E,O,P) i s o f the type I - I - 1

(resp. x z d i v i d e s I n r ( b ) ) .

(X,E,D,P) be o f the t y p e two. I t

i s o f the t y p e

II-1-1

. If

dim Dip (D,E) = 1, e(E) = 2 and JC(E) ¢ J ( D , E ) .

II'-1-1

. dim D i r

(~,E) = 1, E(E) = 3 and JC(E) ~ J ( D , E ) .

11-1-2

. dim D i r

(~,E) = 1, e(E) = 2 and JC(E) c J ( D , E ) .

II'-I-2

. dim Dip (~,E) = I , n.c.

e(E) = 3, JC(E) c J ~ , E )

and t h e r e i s E' c

E

d i v such t h a t e ( E ' ) = 2 and J C ( E ' ) ~ J ( # , E ) .

II'-1-3

. dim Dip

CA, E) = 1, e(E)

= 3, J C ( E ) c J ( ~ , E )

II-2

. dim

(~,E)

= 2.

II'-2-I

. dim

Dir Dip

= 2, e(E)

(D,E)

= 2,

e(E)

=

3 and for each

and no

E' c

11'-1.2.

H n.c.

div

such

t h a t e ( E ' ) = 2 one has t h a t J ( E ' ) + O ( ~ E ) = Gr(R). II'-2-2

. dim D i r

(D,E) = 2, e(E) = 3, JC(E) ~ J(D,E) and t h e r e i s E' ~ E

n.c~ d i v .

such t h a t e ( E ' ) = 2 such t h a t J(E')

II'-2-3

(1.2.g) the

. dim O i r

If we m a r k

components

of

figures: 11-1-I

:

with E,

(D,E) = 2, e(E) = 3, JC(E) ~ J ( D , E ) .

a pointed

one

+ J(O,E) = Gr|R)

can

contour

represent

the d i r e c t r i x , the

above

and w i t h

classification

a continuous by

the

line

following

84

/

11-1-2:

; II'-1-2:

~Y

; II-2:

II'-1-3:

/ .... l

II'-2-1 :

~"

1

J

; II'-2-2:

II'-2-3:

~_

.

.

.

:/

.

( 1 . 2 . 1 0 ) Lemma. I ?

{X,E,D,P) is

II-1-1,...,II'-2-3

above. Moreover, t h e r e i s

= (x,y,z)

suited for

(E,P) such t h a t

(1.2.10.1)

where if

e(E)

o? the type two,

~ is

B/Bz

or

zB/Bz

according t o

is

one o f t h e types

a r e g u l a r system o? parameters p =

generated by

D = axa/ax +

Bz =

thee i t

yalay

e(E)

+ caz

= 2 o r e(E) = 3,

= 2 End i)

I?

1-2-1,

ii)

I?

II'-1-1,

If

11-1-2,

iv)

I?

11-2,

v)

If

11'-2-1,

J(~,E)

= (z+Xx+~y),

If

II'-2-2,

J(~,E)

= (z+Xy),

iii)

vi)

11'-1-2,

then

then J(D,E) 11'-1-3, II'-2-3,then

then

O(D,E)

= (y+Xx,z),

= (y+Xx,z+~y), J(D,E)

J(~,E)

:

X# O, ~

= (y,z). (z). X~ 0 ~ ~ .

X# O.

X # O. O.

such t h a t ~ ( c ) ~ r

85

Proof. ppose (c)

e(E)

ve,

e =

= 2 and

> r+l

= (x,y),

If

since

we c a n

let

jr

s o we c a n suppose

4,

clearly

(x,y,z)

= O.

be a r . s .

If

dim

suppose J(O,E)

(1.2.5.1)

Dir

that

of

(~,E)

is

not

p.

such

= 2,

that

then

@= z a n d we h a v e

= (ax+By,z)

possible, E is

J(O,E)

II-2.

a n d we h a v e

so

If

given =

dim

II-1-1,

e(E)>

(¢) Dir

11-I-2

2.

by

Let

us

x y = O.

with

su-

Then

@~ J(E)

(D,E)

= 1,

according

as

to

=

abo

aB~ 0

or ~B= O. Assume

that

and up to a change (z+~x+~),

(1.2.11) Then

or

e(E)

of order

(y+lx,z+px)

Definition.

(X,E,D,P)

= 3 and E is given

in the coordinates,

and the resuit

Let

(X,E,D,P)

is of the type

(1.2.12)

II-1-1-1,

Remark•

In the case will

ii)

If

v(f)

3f,

= 1,

~(f)

The first

I(E)

= 1,

c

I(E)

one will

(f)

c

=

easily.

type

11-1

II-1-2-0,

are more

in chapter

between

that J(D,E)

(i.e.

11-I-1

or

11-I-2).

i££

otherwise.

= 3 there

be treated

is trivial

~In(D,E)

II-1-2-I,

e(E)

we shall d i s t i n g u i s h

i)

cond

resp.

the

resp.

J(E)

ding the types which II'-2-3,

of

(1.2.2.3)

we can suppose

foliows

be

II-I-I-0,

(1.2.11.4)

and of the type

by xyz = O, then

useful

possibilities

V. By example,

for divi-

if we have type

the two cases:

and

(f),

correspond

In(f)

In(f)

to

~ J(0,E) ~ J(D,E)

a more

then such

v(O(f)/f)>

that

tangential

v(D(f)/f)

situation

r+l. = r.

than

the

se-

of

the

one.

(1.2.13)

Definition•

Let

(X,E,~,P)

be o f

the

type

3,

we s h a l l

say

that

type III-1-1

• Tf

dim

Din

(D,E)

= 1,

e(E)

= I

and JC(E)

~J(D,E).

III-1-2

• If

dim

DiP

(L3, E)

= 1,

e(E)

= I

and

JC(E)

c J(D,E).

III'-1

• 1£ d i m

DiP

(~,E)

= 1,

e(E)

= 2.

III-2

.

If

dim

Dir

(~,E)

= 2,

e(E)

= q.

II1'-211

.

If

dim

DiP

(~,E)

= 2,

e(E)

= 2 and JC(E)

~ J(D,E).

P is

8B

III'-2-2

( 1 . 2 . 1 4 ) As i n

. If

dim Dip (~,E) = 2, e(E

(1.2.8),

= 2 and JC(E) c J ~ , E ) .

one has the f o l l o w i n g p i c t u r e s : ; III-I-2:

111-I-I:

/

iii,_l:

;iii_2.)___l

III'-2-1 :

;~III '-2-2 :

/ (1.2.15)

Lemma. I f

III-I-I,...,III'-2-2

,/

( X , E , ~ , P ) i s o f the type 3, then i t above. Moreover, t h e r e

is

a r.s.

i s o f one o f the types o f p. p = ( x , y , z )

suited f o r

(E,P) such t h a t ~ i s generated by

(1.2.15.1)

D = axBIBx

+ bBIBy

+ c@ z

where

Bz = BIBz or zBIBz a c c o r d i n g t o e(E) = 1 o r 2, such t h a t

i) If I I I - 1 - 1 , ii) iii) iv)

If

111-I-2,

If 111-2, If

Proof.

If

III'-2-2

such

suppose

= 2 and

take

= (y,£).

then J ( ~ , E ) = ( x , z ) . then

J(D,E)

=

(z).

then J ( ~ , E ) = (z+Xx),

e = I,

(x,y,z)

J(D,E)

III'-I

III'-2-I,

can choose e

then

since j r

that

X~ O.

= 0 and we have t y p e two, so t r a n s v e r s a Z i t y , one

x = 0 gives

(x,y,z)

v ( b ) ~ r + l and

such

that

E and J(~,E) E is

given

= (~), by

xz

(x,z) = O.

or If

(y,z).

dim

Dir

Let (/~,E)

us =

87

=1 since ( 1 . 2 . 2 . 3 ) necessarily

(1.2.16)

the

is

J(O,E)

not

verified,

= (X~+p~)

Definition.

Let

and,

one has t h a t up t o

(X,E,D,P)

J~,E)

a change o f

= (~,~).

order,

be of the type four,

If

dim D i r

( O , E ) 4%

one can s u p p o s e

we shall

that

say that

p~O.

it is of

type:

(1.3)

dr&tic then

iff

4-1,

iff J H ( D , E )

= 0 ~ Jr(D,E).

4-2,

iff

= O.

Reduction

(1.3.1)

J r H(D, E ) ~ O.

4-0,

Jr(~,E)

of the no t r a n s v e r s a l s

Proposition. directional

If

blowing-up

P' is of the type

Proof.

One

rection

tangent

(1.3.2)

Corollary.

nning at type reduction

(X,E,D,P)

zero,

has

e(E)

zero,

one,

that

type

four

v(~',E',P')

and

(X',E',D',P')

= r and dim Dir

i8 a qua-

(~',E',P') ~ I ,

two or three.

= I and the quadratic

to E, so e(E')

If there

is of the

such one,

types

blowing-up

must

be made

in a di-

= 2.

exists

a winning

two or three,

strategy

then there

for the

exists

reduction

a winning

game begi-

strategy

for the

game.

2. STANDARD TRANSITIONS

We

shall

we shall discuss tion,

as w e l l

we

shall as

some

begin

the case

describe results

u n d e r the assumption

with

a situation

in which

the of

I instead

of

1 and, in the next chapter,

the game begins with a situation

transition reduction

of c h a r a c t e r i s t i c

of of

the

zero.

"easy"

control

complexity

of

b y means the

I. In this sec-

of

possible

the

polygon

transitions

88

(2.1)

Definitions

(2.1.1)

Definition.

a directional Dir

and f i r s t

(O,E).

Let

= v(~',E',P')

(X,E,D,P)

ii)

I ~

center

following

of

the

type

I,II

or

and l e t

(X',E',~',P')

(may be q u a d r a t i c )

is

a "standard"

possibilities I and

llI

tangent

transition

be

to

iff

r :

is satisfied:

(X',E',~',P')

is

of

the

type

I (we s h a l l

I).

I ~-~ II.

iii)

II ~-~ II.

iv)

III ~-~ II.

v) III ~

Remark.

III.

The o n l y

(which

is i m p o s s i b l e ) ,

terest

for

us

chapter

type

(X',E',D',P')

the

is

be o f

permissible

say that

write

this

of

and one o f

i)

(2.1.2)

(X,E,D,P)

blowing-up We s h a l l

reduction.

since

II F-~ I and

it

we s h a l l

no s t a n d a r d

wiil

not

be

transitions

between

Ill ~-~ I and,

considered

be i n t e r e s t e d

as

l,II,III

are

the t r a n s i t i o n

a

"victory

I ~-~ I I I

III ~-~ I has

transition". Ill,

no in-

Moreover,

in the t r a n s t i i o n s

II ~

realization

reduction

game

(I.(4.2))

for

in

Ill ~-* II and

III ~-~ III.

(2.1.3) for

Theorem.

each

tions).

t =

1,...,s,

Assume

that

k is zero. II ~

Then

~(t)

stat

for each

be a p a r t i a l is

(0)

a is

standard o£

the

t = 1,...,s,

transition

type

~t)

of the

one

(see

I and

that

is a t r a n s i t i o n

the

I ~

such t h a t the

nota-

characteristic

I,

of

I ~-~ II or

II.

Proof. that

stat

III,

in

Then,

Let G I s + l

can

(t)

is

of

order

to

obtain

the f i r s t

(2.1.3.1)

One

the

assume type a

that

the

first

transition

II for t = 1 , . . . , s - 1

contradiction.

transformation

a(1)x(1)~/~x(1)

is g i v e n

Let

us

and take

is I ~-~ II.

that p =

stat

+ b(1)y(1)~/~y(1)

us s u p p o s e

(s) is of the t y p e

(x,y,z)

by T-2 or T-4 and D ( 1 )

Let

as

in

is g e n e r a t e d

+ c(1)~/~z(1)

(1.2.5). by

89

(taking

the

notation

of

(4.2.5))

where

a(1)

= a/[y(1)]r-l-b/[y(1)] b(1)

(2.1.3.2) c(1) if T - 2 ,

c(1)) > r+l and

= b/[y(1)]

r

and = a/[y(1)]

b(1) c(1)

then

r

= c/[y(1)]r-z(1)b/[y(1)]

a(1)

ifT-4,

r

We

shall

T-4

has

distinguish

nosense,

r-1

= b/[y(1)]

r

= c/[y(1)]r-z(1)b/[y(1)]

two

cases:

stat

(0)

r

is

1-4

or it is I-2.

If P is I-1,

and i f

(2.1.3.3)

¢(_x,z)

= In

(b)

one has t h a t

In

(a(1))

= -¢([(1),[(1))

+ ~(1)(...)

In

(b(1))

:

+ Z(1)(...).

(2.1.3.4)

Moreover, z(1)

since

dim D i r

~-~ z ( 1 ) + X . y ( 1 )

¢([(1),~(1))

(0(I),E(I))

= 1 (never

one can suppose

= 2),

after

an a d e c u a t e

change

that

In

(a(1))=

-¢(~(1)

+

Z(1),~(1))

In

(b(1))=

+¢(~(1)

+

Z(1),_z(1)).

(2.1.3.5)

If

p= 0,

we c o n t i n u e

J(O ( s - l , E ( s - 1 ) ) reach t y p e

llI,

by m a k i n g

T-2

until

t

= s-1.

([(s-1)+py(s-1),~(s-1))

where

For t P~

= s-1

0 since

we m u s t

otherwise

and

In

(a(s-1))

= -(s-1)¢(x(s-1)+#Z(s-1),~(s-1))

(2.1.3.5) In

(b(s-1))

Now we have t o make

(T-l,1/p),

(2.4.3.6)

In

(b(s))

= +¢(~(s-1)+

Z(s-1),~(s-1))

but

=(l~ls.¢(Z(s),E(s))+×(s)(...)

have

~ 0

we

cannont

90

(note

s ~ 0 s/nce

the

characteristic

of

k

is

zero)

and t h u s

it

is not

possible

to

have type III. Assume

that

stat

(0) is I-2.

Then

one can

suppose

that

In

(b) = z r. T h e n

In ( a ( 1 ) ) = ¢ I ( ~ ( 1 ) , Z ( 1 ) , ~ ( I ) )

(2.1.3.7) In ( b ( 1 ) ) = [ ~ ( 1 ) ] r + ¢ I ( ~ ( 1 ) , Z ( 1 ) , ~ ( I ) )

where ~l(O,O,Z) = ¢1(0,0,Z) = O. Moreover, a f t e r and adecuate change z(1) ~

(x,y)

> z ( 1 ) + X y ( 1 ) + p x ( 1 ) one can always suppose t h a t J ( D ( 1 ) , E ( 1 ) ) ) z ( 1 ) . Moreover is

not permissible

andthen the next s-2 t r a n s f o r m a t i o n s are, f o l l o w i n g t h i s

procedure, o f the type ( T - l , 0 ) , p,q ~ ~

o

T-2,

T-3 or T-4 and f o r t = s - l ,

there e x i s t

such that

zn ( a ( s - 1 ) ) = - p [ z _ _ ( s - 1 ] r + ~ s _ l (2.1.3.8)

zn ( b ( s - 1 ) )

= q[_~(s-1)]r+~s_ 1

where ~s_1(O,O,Z) = Cs_l(O,O,Z) = O. One can suppose t h a t the f o l l o w i n g t r a n s f o r m a tion

is

( T - I , ~ ) and then

(2.1.3.8)

with

Tn ( b ( s ) )

¢s(O,O,Z)

= tJ.(p+q)[z(s)]r+¢

= O, then ~ ( b ( s ) ) = r ,

(2.1.4) Corollary.

S

contradiction.

I n a sequence Gls+l

as in

(2.1.3),

the t r a n s i t i o n I I ~-~ I I I

is

not p o s s i b l e .

(2.2)

Polygons and i n v a r i a n t s

(2.2.1)

Definition.

Let

(X,E~,P)

be o f the type I , I I

or I I I ,

a system o f r e g u l a r

parameters p = ( x , y , z ) i s c a l l e d a "normalized base" i f f a) I ( E ) = (x) or I ( E ) = ( x y ) . b) I f If

one has t y p e I I - 1 - 1 - 1 Or 1 1 - 1 - 2 - I or I I I - 2 ,

(X,E,D,P) i s o£ the t y p e I ' ,

then ~ ~ J ( D , E )

p i s c a l l e d a "normalized base" i f f

I ( E ) = (xz) and

91

if

E 2 is

given

by x and E 1 i s

(2.2.1.1)

by z,

In (I(E1)) ~ Inr(D,E)

(a normalized

(2.2.2)

given

then

~

In (I(E2))

~ I n r ~ ,E)

base always exists).

Definition.

be a normalized

Let

(X,E,O,P)

be of the type

I,II or III and let p = (x,y,z)

base. Let

(2.2.2.1)

D = a x ~ l ~ x + b3 + c ~ l ~ z Y

be a g e n e r a t o r

of ~.

Then E x p ( D , E ; p )

is

defined

by

Exp (O,E,p) = Exp (ya) u Exp (b) ~ Exp (yc/z) (2.2.2.2)

Exp ( D , E , p ) Exp ( D , E , p )

for

the types

I,II

and I I I

= Exp (a) = Exp (a)

respectively.

Exp(D,E,p)(for

I),

u Exp (b) u Exp ( c / z ) u Exp ( b / y )

u Exp ( c / z )

Exp+(D,E,p)

is

defined

Exp(a)

Ill)

and

(for

by

(2.2.2.35 Exp(a) u Exp(b)

The i n v a r i a n t m(O,E,p)

(2.2.3)

m (O,E,p)

: ~ if

is

the

minimum

h such

such an h does n o t e x i s t .

Definition.

Let @:{(h,i,j);

(h/(r-j);i/(r-j)).

In

by t h e c o n v e x h u l l

the

{If

j~r-1

situation

(for

}

of

II).

that

(h,-1,r)

E Exp

(D,E,p)

m # = , t h e n one has t y p e

~ I R 2 be g i v e n

(2.2.2),

the

by ¢ ( h , i , j )

polygon

and

1115.

=

A(~,E,p)

is

defined

defined

by

putting

of

[$(Exp

(D,E,p)

(~ { ( h , i , j ) ; j <

r-l})

+

(2.2.3.1) +IH(m(D,E,p))]

where IH(m) i s Exp+(D,E,p)

(2.2.45

like

instead

Remark.

in of

(II(2.2.7)5.

The

n

{(u,v);v>-l}

polygon

& + (0 E,p)

is

Exp ( O , E , p ) .

I n the above situation

(x,z)

is p e r m i s s i b l e

iff

A (O,E,p)c

{(u,v);

u _> 1 }

92

and

(y,z)

(2.2.5)

is

permissible

Lemma.

normalized

Let

is

of

be o f

parameters.

T-2,

permissible).

&(D,E,p)

(X,E,D,P)

system

ven by ( T - l , 0 ) ,

iff

the

Let

Moreover,

let

is

given

as i n

I,ll

or II1

(X',E',D',P')

and l e t

p = (x,y,z)

be a d i r e c t i o n a l

resp.

us usppose t h a t

A(D',E',p')

c

v > 1 }.

types

T-3 or T-4 from p (T-3,

(2.2.5.1)

where

"{(u,v);

it

T-4, is

only

if

blowing-up

(x,z),

a standard

be a

mesp.

transition,

gi-

(y,z), then

= c(A(D,E,p))

(II.(3.1.g.4))

and p'

is

obtained

f r o m p by

(T-I,0),T-2,T-3

or T-4.

Proof. or

T-4

one

(I.(2.2.5)) that

If

has

I

t

gives

or

~ II, the

T-3

II

,

result.

one has I ~

~ II

or

III

~

Let

us

remark

I,

II

>II. that

~-~ I I

or

II1

A computation if

II1

,

~-~ I I I

and i f

over the

>II1,

T-2

equations

(T-l,0)

one

has

in

the

11-1-2-1

and

m' = m-1.

(2.2.6)

Definition.

same way as i n

(2.2.7) let

(T-l,0)

Definition.

(2.2.7.1)

(2.2.8)

Let

(2.3.4)

~(D,E,p),

is

Remarks.

defined

1.

Preparation

Lemma.

be a n o r m a l i z e d

be o f

base.

E(D,E,p),

a,

(x,z)

type

(h,i,j)

by p u t t i n g

6= ~ i f

the

a(D,E,p)

are defined

11-4-1-4

6(D,E,p)

E Exp(D,E,p)

Exp+(D,E,p)

is

1-4-1,

The i n v a r i a n t

= rain { i / ( r - h - j ) ;

2. A + ~

(2.3)

(X,E,D,P)

be a n o r m a l i z e d

6(D,E,p)

6+(~E,p)

invariants

(I1.(3.1.6)).

p = (x,y,z)

and

The

instead

of

is

or

defined

by

and h + j < r } .

Exp ( O , E , p ) .

permissible.

6+ - -> 6 .

o f 6.

Let

(X,E,D,P)

base.

Let

be o f t h e

us c o n s i d e r

type

I1-1-1-1

the coordinate

or

11-1-2-4

change

and l e t

p=(x,y,z)

93

n

(2.3.1.1)

where

z I = z +Z ~ ny n>6

6=6(Z],E,p).

Let

Pl

= ( x , Y , Z l ) . Then Pl i s normalized, one has t h a t

(2.3.1.2)

6(Z],E,Pl)>

and the e q u a l i t y in

( 2 . 3 . 1 . 2 ) occurs always i f 16= 0.

Proof. T r i v i a l l y not c o n t r i b u t e in contribute to

(2.3.2)

6

Pl

is

(2.2.7.1)

normalized. The monomials produced by ( 2 . 3 . 1 . 1 )

t o a p o i n t t<6

do

and i f X 6 = 0, then the monomials which

6 are not a f f e c t e d .

Definition.

"prepared" i f f

Let

6=6~,E,p)

(X,E,D,P) ~ ~o or

(2.3.2.1)

be as above. A normalized base p = ( x , y , z ) i s 6 ~Z

zI

o

and there i s no change

= z + Xy

6

such t h a t 6 ( O , E , P l ) > 6

(Pl = ( x ' Y ' Z l ) ) " From a normalized base p one can always

obtain a prepared base p'

by making a ( f i n i t e

ges

as

or not) sequence o f coordinate chan-

(2.3.2.1).

(2.3.3)

Remark. The lemma ( 2 . 3 . 1 )

is

true also i £ one put

6+

instead o f 6 .

This

a l l o w s us t o s t a b l i s h the f o l l o w i n g :

(2.3.4) D e f i n i t i o n . Let (X,E,D,P) be o f the type 11-1-I-1 or 11-1-2-1. A normalized base p = ( x , y , z ) or

6

e Z

o

and

is

" s t r o n g l y prepared" i f f

there

it

i s prepared and 6 = 6 ~ , E , p ) ~ + +

o

is no c h a n g e

6+ (2.3.4.1)

zI

such t h a t

6+(O,E,p 1)

> 6+ (Pl

obtain a s t r o n g l y prepared b a s e

(2.3.5)

Lemma. Let

(X,E,D,P)

= z +X y

= (x'Y'Zl))" as in

From a prepared base p one can always

(2.3.2).

be o f the type 11-1-I-1 or 11-1-2-1 and l e t p = ( x , y , z )

be a s t r o n g l y prepared base. Then

94

a) J ( D , E )

(X,p) p#

b)

6+ = I iff

Proof.

Since

~ O.

0

= (x,z)

~

k=

X E 0 and

one

6+ > 1 .

(X,E,D,P)

p

If pE O,

if

is

is of the t y p e

normalized:

O, t h e n has

J(O,E)

z I = z + py

type

II-1-1-1.

=

(x+Xy,z+p~).

increases

II-1-1-1.

has

6+ =

6+, c o n t r a d i c t i o n .

Thus

Conversely,

One

if t y p e

II-I-1-1,

1 iff

then X ¢ 0

and 6 + = 1.

(2.4)

Standard

(2.4.1) dratic

transitions

Lemma.

Let

directional

(X',E',~',P')

is

Proof. by T - 2 .

Then

Z'lIn(c')

and t h e

(2.1.3.4),

(2.4.2) quadratic

blowing-up

which

corresponds

the

is

Let

(X,E,D,P)

a

(X',E'~',P')

b

The

is

c

6(D',E',p')

d)

6+(O',E',p')

then

,

one o b t a i n s

the

blowing-up

v(c')

~r+l

= r,

and i n

is

given

then

view of

II-1-2-1.

type

II-1-2-I

II-1-I-1

by T - 2

and l e t

to

(X',E',D',P')

a standard

be a

transition.

Let

(x ' , y ' ,z ' ) i s

the

or II-1-2-I.

from

strongly

~ and

if

p'

=

prepared.

= 6+(O,E,p)-l.

If

a), p'

the is

blowing-up not

is

(strongly)

6

(z' 1 = z'+Xy '6÷

Then

Then

type

is

be a q u a -

transition.

If

v(c')

corresponds

base.

p'

(2.1.3.2). thus

or

the

given

a standard

Necessarily

standard,

the

(X',E',~',P')

= 6(D,E,p)-I.

From 2 . 3 . 5 .

straigthforward.

is

base,

(2.1.3.1)

which

of

and l e t

to

(1.2.5).

be o f

prepared

1-1-1

II-1-2-1.

II-1-1-1

blowing-up

blowing-up

in

not

one has t y p e

type

or

as

as i n

transition

be a s t r o n g l y

Proof.

II-1-1-1

(x,y,z)

generated

resulting

1lows

type

p :

directional

p = (x,y,z)

1.1.1.

the

(2.1.3.5)

Theorem.

type

be o f

of

is

the

(X,E,O,P)

Take D'

from

) may be g i v e n

a contradiction.

back

to

a change

given

b y T - 2 and a ) ,

prepared,

the

change

c) z'

and d) 1

=z'+ky'

fo6'

z 1 = z +Xy 6 o r z I = z + Xy 6+ and

95

(2.4.3) is

Corollary.

of the

ssible

type

curve

I-1-I

dard.

Then G i s

Proof. (since

since

otherwise

( 2 . 5 ) Good

(2.5.1) of

the

quadratic

strategy:

player

Assume t h a t

all

if

there

A chooses this the

is

stat

a permi-

center,

transitions

(0)

other

in G are stan

and

(2.4.2)

>II-1-2-1 the

(one

player

first

transition can

transition

that

all

the

Now it is enough

(I) and to apply

is

is not standard).

assume

A wins).

of a permissible

the

(2.4.2)

c)

1-1-1

J

~ 11-1-2-1

The other transitions

blowing-ups

to choose

(remark that $<=

are

quadratic,

a strongly

prepared

, since 6 = = i m p l i e s

curve).

in

(II.(3.2)),

we s h a l l

study

the

effect

over

the

polygon

of

a change

type

zI

we s h a l l

(2.5.2)

computations ommit o r

Here

or II-1-2-0.

will

s ~ e t c h most o f

(X,E, ~ P ) Remark

= z + ~ x ~ y B.

be q u i t e

will

similar

a situation

that with this assumption

(0,0,r)

each n o r m a l i z e d

Lemma.

a normalized

which are

i n A.

Let

regular

system o f

us c o n s i d e r

system o f Let

to

in

(II.(3,2))

of the types

and

(II.(3.3)),

the

I-1-0,I-2,11-2,11-1-1-0

one has that

~ Exp ( D , E , p )

p a r a m e t e r s p.

a s e t A ~ ~ 2 such t h a t o

parameters.

us c o n s i d e r

those

them.

denote

(2.5.2.1)

is

the

game such t h a t

preparation

As

Since the

(2.5.3)

reduction

following

then

center.

(2.5.1.1)

for

the

directrix,

the following

base p(1) for stat the existence

the

(2.4.1)

By

,

to

follows

of the

finite.

if II-1-1-1

II-1-2-1

G be a r e a l i z a t i o n

and w h i c h

tangent

he c h o o s e s t h e

wise

are

Let

L e t A' c

coordinate

A be t h e change

A~&(D,E,p), set

of

where p = ( x , y , z )

vertices

of A (D,E,p)

96

B

(2.5.3.1)

z = zI +

[

y

eAkaBx

( ~, B) Then one has t h a t : a)

Pl

= (x'Y'Zl)

b)

A ( D , E , p 1) c

c)

The

is

A(D,E,p).

vertices

of

the monomials ge o f

normalized.

A(D,E,p)

which

contribute

Definition.

v = (~,B)

of

i n A' to

are

also

them a r e

vertices

exactly

of

the

A(O,E,p I)

same up t o

and chan

z by z I .

Proof. This p r o o f i s s i m i l a r

(2.5.4)

not

to the p r o o f o f ( I I . 3 . 2 . 3 ) .

Let p = ( x , y , z ) be a normalized system o f parameters. A v e r t e x

A(~,E,p)

is

" w e l l prepared" i f f

one o f the f o l l o w i n g p o s s i b i l i t i e s

is

verified: a)

(a,B) ~ l

b) There

is

2

o no change o f the type z I

the v e r t e x " ) We s h a l l are well

(2.5.5)

such that

say t h a t

A(D,E,Pl)

AO ,E,p)

(or p)

~

= z +kx~

A(~,E,p)

is

(called

"preparation of

- {v}.

well-prepared i f f

all

the v e r t i c e s

prepared

P r o p o s i t i o n . With

n o t a t i o n s as above, beginning w i t h

p = (x,y,z),

one can

reach a normalized system o f parameters by a sequence o f changes o f v e r t e x preparations,

such t h a t the corr@@POnding polygon i~ weLL prepared.

Proof. I t

(2.5.6) A(D,E,p).

(2.5.7) of

Remark. This

is

The lemma ( 2 . 5 . 3 ) allowsus

Definition.

A(Z],E,p)

a d i r e c t c o r o l l a r y o f lemma ( 2 . 5 . 3 ) .

which

to

Let is

stablish

p =

well

is

(x,y,z)

prepared.

true the

also

if

one p u t A + ( D , E , p )

instead

of

following:

be a n o r m a l l z e d The

vertex

base

(m,B)

is

and said

let to

(a,6)

be a v e r t e x

be " s t r o n g l y

well

97

prepared"

iff

one o f

the

following

possibilities

is satisfied

a) (~,B) & A+(D,E,p) b)

(~, B) ~ ~ 2 .

c

(%6)

~ C2,

(~,6)

= z+Xx~y B The base p zs

said

to

~A+(~,E,p)

such t h a t be

and t h e r e

is

no change o f

A + ( D , E , p 1) c A+ ( ~ , E , p )

"strongZy

well

prepared"

iff

-

all

the

{(~,6)}. the

type

z1 =

(pl=(x,Y,Zl)).

vertices

of

A(D,E,p)

from

a norma-

are strongly well prepared.

(2.5.8)

One can

obtain

(2.6)

S t a b i l i t y results f o r g o o d preparation.

Proposition.

prepared

wise,

i£

Let

base,

Proof.

If

same way as i n

prepared

sistem

well

the

well

Lized

(2.6.1)

p in

a strongly

(X,E,D,P)

then

parameters

(2.5.5).

be as

in

(2.5.2).

If

p = (x,y,z)

is

a strongly

z ~ J(D,E).

A+(D,E,p)

~ + X ~ + p~

system of

n

{u+v = I }

& J(D,E)

= ~,

and v g r .

then

the

k # O, t h e

result

follows

easily.

Other-

change z1= z + k x d i s s o l v e s

the

vertex (1,0) of A +(D,E,p), c o n t r a d i c t i o n .

(2.6°2)

Theorem.

prepared it

is

ble).

Let

base,

let

centered

at

Assume

transition

is

that

(X,E,D,P) (X',E',D',P')

(2.5.2),

or

(y,z),

(the

last

in

quadratic

ease

it

the

standard,

p = (x,y,z) blowing-up

cases is

only

given

if

be a s t r o n g l y

which the

by T - l , 0

is

center or

T-2

quadratic is

well or

permissi-

and

that

the

Then

In the monoidal

b)

(X',E',D',P

c)

I£

p'

p'

is

=

let

be a d i r e c t i o n a l

(x,z),

a)

Proof.

be as i n

')

cases,

is

o£ t h e

(x',y',z')

strongly

a)

follows

b)

One has

is

well

from type

the

transformation

type

is

given

by T - 3 o r T - 4 .

I-2,I-1-0,II-2,II-1-1-0

obtained

from

or

p by ( T - I , 0 ) , T - 2 , T - 3

II-1-2-0. or T-4,

then

prepared.

(2.6.1). I

or

II

and c o m p u t a t i o n s

as i n

the

proof

of

(2.1.3)

98

show t h a t

the

fact

for

Inr(o,E)

of

being

or

not

in J(E)

is

preserved

(remark that

for type I, J(E) = In(l(E)). c) Similar to (11.(4.1.3)

(2.7) V e r y good p r e p a r a t i o n .

(2.7.1)

As

in

(II.(3.4))

= y +X x o v e r t h e

we a r e

polygon,

in

gozng

order

to

to

investigate

the

have a c o n t r o l

effect

of the

of

a change

transformation

Yl

=

(T-I,~)

~ 0. The simpler

that

results those

dy o f t r a n s i t i o n s in

chapter

II

stability

(II.(4.1)). ~

>III,

(X,E,~,P)

will

for

We s h a l l III

~

) II,

find

the

"controled

results

I 1 1 ~--~ I I I ,

situations"

as i n

(II.(4.1))

II

II

~

that

in

will

be

the

stu

we s h a l l

make

be o f

the type

I-2

or of the

type

I-I-0.

(but

the

one w o u l d be i r r e l e v a n t ) .

(2.7.2)

Lemma.

rameters.

Let

Let

p = (x,y,z)

us c o n s i d e r

the

be a s t r o n g l y coordinate

(2.7.2.1)

Let

needed

V. Here

last

of

of

Yl

A' = [ A ( ~ , E , p )

a)

+;H(n)]

n IR 2. o

Pl = (x'Yl'Z)

well

prepared

system o f

pa-

change

= y +

[ Xi x i " i>n

Then

is normalized.

b) A' I = [A(~,E,PI)

+IH(n)] m IR 2 = A, o

c) All

A(D,E,p 1) which are vertices

vertices

normalized

of

of A'1 a r e s t r o n g l y

well

prepared.

Proof.

(2.7.3) tisfies

Similar

Definition. exactly

"strongy

well

the

to

the

We s h a l l

proof

say t h a t

same c o n d i t i o n s

prepared".

of

of

(II.(3.4.3))

p is

(quite

"strongly

(II.3.4.5),

simpler).

very well

changing

"well

prepared"

iff

prepared"

it by

sa-

99

(2.7.4) Proposition. There exist always a strongly very well prepared system of parameters for (X,E,D,P) of type 11-2.

Proof. See

(II.(3.4.6)).

The proof of the foIlowing result is very simiiar to the proof of (II.(4.2.5))

(so it serves aiso for positive characteristic).

(2.7.5) Theorem.

Let (X,E,O,P) be of the type I-2 and p = (x,y,z) be a strongiy ve-

ry weli prepared base. Let us suppose that ter. Let Yl = y + (x, ~

0, and let z ~

(x,z) does not define a permissible cen-

z I be a strongIy good preparation of Pl =

= (X,Yl,Z). Let us denote P2 = (x'Yl'Zl)" Then a) The first

vertex of 6 ( D , E , p 2)

is the same as the first

vertex of

A(O,E,p). b) ~(D,E,p 2) ~ 1. Proof. Similar to the proof of (II.(4.2.5)).

(2.8) Standard winning strategies.

(2.8.1) Here one obtains

some winning

strategies

for the

reduction

game beginning

at the type I-I-0 or I-2 and with the assumption that alI the transitions are standard. First

of ail,

iet

us fix

stat

(0) =

(X,E,D,P)

of the type

I-1-0 or I-2

and let p(0) be a strongly very well prepared base.

(2.8.2) Let G be a realization of the reduction game beginning at stat that

aii the transitions

are

standard.

In order to construct

(0)

and such

inductively re-

gular systems of parameters p(t), t = 0,1,..., assume that G satisfies the following property:

for each t = 1,2,...,

the center Y(t-1)

oF ~(t) is the quadratic center

or it is given by (x(t-1),z(t-1)) or (y(t-1),z(t-1)), where p(t-1) = = (x(t-l,y(t-1),z(t-1)); is strongiy

("very"

moreover,

if stat

(t-l)

Y(t-1) is always permissibie. Assume that p(t-1) is of the type

I) weiI prepared. Then the above

100

property

implies

that

obtained

regular

system

if

stat

(t)

(2.8.3)

is

of

the

Definition.

beginning iff:

~(t) of

type

Let (0).

is

given

by

parameters. I)

(T-I,(), Then p ( t )

good p r e p a r a t i o n

G be a r e a l i z a t i o n

at

stat

a)

#(t) is standard t = 1,...,s.

b) Let p(t) Y(t)

G follows

the

be as above.

is given

(y(t),z(t))

by

T-2, is

of

or

defined

to

Let

p'(t)

be t h e

be a s t r o n g

("very"

p'(t).

the

"standard

T-3 or T-4.

reduction

winning

game o f

strategy"

length~

until

the

s+1, step

s

If (x(t),z(t)) gives a permissible center, then

(x(t),z(t)).

is permissible,

If

(x(t),z(t))

is not permissible,

but

then Y(t) is given by (y(t),z(t)). Y(t) =

= P(t) otherwise. O < t < s.

(2.8.4)

Definition.

Let G be as above and let us fix I < O. G follows the "l-retar-

ded standard winning strategy" until the step s iff a)

=(t) is standard t = J,...,s.

b) If (x(t),z(t))

is permissible,

then Y(t) is given by (x t),z(t)),

o < 5 < s. c) If (x(t),z(t))

is not permissible,

= A(D(t),E(t);p(t)) t-t'~l

has only one

one has that

vertex and

#(t'+1)

(y(t),z(t)) vertex and

(y(t'),z(t'))

is permissible, for

~(t) =

each t' < t such that

is permissible,

A(t') has only one

is given by (T-I,0) or T-3 from p(t'), then Y(t) is

given by (y(t), z(t)), 0 < t <s. d) If the assumptions of b) and c) are not verified,

(y(t),z(t)) is perm~

ssible and ~(t) has only one vertex, then Y(t) is quadratic or Y(t) is given by (y(t),z(t)). 0 < t < s. e) Y(t) = quadratic center,

(2.8.5) such until

Theorem.

that s for

for

otherwise.

L e t G be a r e a l i z a t i o n

a fixed

strategy

each s ~ l e n g t h

of

of

(2.8.3)

G. Then G i s

the or

finite.

0 < t < s.

reduction (2.8.4),

game b e g i n n i n g then

G follows

at this

stat

(0)

strategy

101

Proof. strategy tly

of

It

follows

(2.8.3)

each t i m e .

and

For t h e

from for

(2.2.5),

(2.~.2)

strategy

0-retarded,

case,

may be ( 8 , ~ , e )

the

1-retarded

and

(2.7.5). then

Remark t h a t

(8,e,a)

for

the

decreases stric-

does n o t d e c r e a s e ,

but 8 r e -

mains stable and after 1-times, 8 decreases.

3.

REDUCTION OF THE TYPE 1 - 1 - 1 .

In t h i s the

standard

duction

section

ones i n

order

game when i t

existence

we s h a l l to

begins

consider

prove the

with

all

existence

a situation

of

a winning

strategy:

a)

The p l a y e r

A wins.

b)

The p l a y e r

A obtains

possible

we s h a l l

transitions

of a winning

I-1-1. prove

We s h a l l

that

one

and n o t m e r e l y

strategy

not

for

the

re-

prove directly

o£ t h e

the

two p o s s i b i l i t i e s

holds:

In the

the

following

reduction

chapter,

a type

we s h a l l

game b e g i n n i n g

at

prove

type

I'-2

I'-2

or I'-1-0.

that or

there

exists

a winning

strategy

for

1'-1-0.

(3.1) No standard transitions.

(3.1.1)

Here

ones t h a t will gly

will

shall

we must t o

be o f

the

prepared

(3.1.2)

we

type

I-i-l,

We have J ( D , E ) center

So, we can suppose t h ~ quadratic

the

consider.

system o f

chose t h e

define

special

In a l l

the

and p = ( x , y , z )

transitions paragraph

different

from

the

standard

(X,E,O,P)=(X(O),E(0),O(0),P(0))

= (x(0),y(0),z(O))

:

p(0)

a fixed

stron-

parameters.

=

(x,z).

(x,z)~if (x,z)

Thus i n it

is

is not

the

first

permissible, permissible

blowing-up.

(3.1.3) Let us suppose that ~ is generated by

step then

of the the

and t h a t

game t h e

player

the

A will

player

player

A

always win.

A choses the

102

(3.1.3.1)

a(0)x(0)

where

necessarily

first

movement,

T-2.

v(c(0))

@/@x(0) + b ( 0 ) a / B y ( 0 )

~r+l,

necessarily

the

Thus we can suppose t h a t

(3.1.3.2)

v(b(0))

= r.

player

0(1)

a(1)x(1)@~ x(1)

is

+ c(0)~/~(0)

I£

B must

generated

the

player

choose

the

A does

not win

transformation

in

the

given

by

by

+ b(1)y(1)@ ~ y ( 1 )

+ c(1)B ~ z ( 1 )

w h e r e one has t h a t

a(1)

= a(0)/y(1)

(3.1.3.3)

b(1) c(1)

Now,

there

not

> r+l

I-1-1-I won,

Last

or

one

and the

player

II-I-2-1

(see

has

transition

p(1)

type

I',

A has

forget

the

if

r

r

r

that

notation =

v(c(1))

not won,

(2.4.2.)),

and

then if

and the t r a n s i t i o n

= r.

is

(X(1),E(1),0(1),P(1))

v(c(1))

is not

=

r and

standard.

We

the

of the

player

shaii

A has

denote

this

by

in

(x(1),y(1),z(1))

J(0(1),E(1)) king

or

I

Assume

=

= b(O)/y(1)

> r+l

(3.1.3.5)

(3.1.4)

b(0)/y(1)

= c(O)/y(1) r - z(1)b(O)/y(1)

v(c(1))

v(C(1))

type

-

are two p o s s i b i l i t i e s

(3.1.3.4)

If

r-1

(3.1.3)

(3.1.3)).

(x(1),~(1))

iff

n e c e s s a r y a change z I

I'

one has had a s t a n d a r d

a strongly

of

~

prepared

Then 0 ( 1 )

base

is

we have t y p e

transition.

for

us d e n o t e

(X(1),E(1),0(1),P(1))

generated II-1-2.

Let

If

as i n

(3.1.3.2),

we have t y p e

by

(so we and

II-1-1,

by ma-

= z + px we can assume J ( 0 ( 1 ) , E ( 1 ) ) = ( ~ ( 1 ) + X ~ ( 1 ) , ~ ( 1 ) )

w h e r e X ~ 0 (see 2 . 3 . 5 ) .

missibie,

Let

us suppose f i r s t

the

piayer

(x(1),z(1))

is

Now,

player

if

the

not

A wins

that

we have t y p e

by c h o o s i n g

permissibie

and

the

A does

win

in

not

this

center.

piayer this

II-1-2.

(x(1),z(1))

is

pe~

So we can suppose t h a t

A chooses

movement,

Then i £

the

the

quadratic

player

B must

blowing-up. choose t h e

103

t r a n s f o r m a t i o n T-2 and one has t h a t

a(2) (3.1.4.1) c(2)

=

(a{~)-b(1))/y(2)

b(2)

= b(1)/y(2)

r

r

= c(1)/y(2)r+l-z(2)b(1)/y(2)r.

We have o n c e more two p o s s i b i l i t i e s

(3.1.4.2)

If

the

type

v(C(2))

player

II-1-2

strongly

A has or

n o t wo~ and

II-1-1

prepared

> r+l

and

system

p(2)

of

or

v(c(2))

v(c(2))~r+l,

then

=

(x(2),y(2),z(2))

parameters

(see

= r.

(X(2),E(2)~(2),P(2)) obtained

(2.4.2)).

If

from

is

p(1)

one has

of

by T - 2

v(c(2))

the is

= r,

a

then

one has t h a t

(3.1.4.3)

and

if

y(2)

the

player

A has

not

won,

I In

one

has

(c(2))

type

I'.

We s h a l l

denote this

transition

by

(3.1.4.3)

II

In ced, of

then

the

(3.1.5)

after

type

res that

II-1-1

if

Theorem.

With

and

for

t

A has :

if

a finite

case

player

bilities

way,

in t h i s

the

rectional

this

not

the

the

player

notations

e we have t y p e

quadratic

transform

of s t e p s

not won,

won and

I'

transitions

number

A has

~

we have a s i t u a t i o n

in v i e w of th.

A wins

as

I ~-~ I' or II ~-* I'

in the

above

and

of

Then i f

if

char

(k) t

C)

(X(s+I),E(s+I),O(S+I),P(s+I)

one o f

= 0.

is

of

produ-

nexttheorem

assu-

the

type

zero.

us s u p p o s e

are

(X(s+l),E(s+l)#~(s+l),P(s+l))

(X(s),E(s),D(s),P(s)),

(O(S+I),E(s+I))

The

= 1,..,s-1

r > v (O(s+l),E(s+l),P(s+l)).

b) d i m Dir

been

(X(s),E(s),D(s),P(s))

= O, l e t

is s a t i s f i e d

a)

not

next m o v e m e n t .

(X(t),E(t),D(t),P(t)),

I-1-I.

(2.4.2).

have

the

of

type

that I-1-2

is

any di-

following

possi-

I04

Proof. the

We

shall

suppose

t r a n s f o r m a t i o n s are

that

quadratic

a)

and b)

are not s a t i s f i e d . N e c e s s a r i l y a l l

and i n t r i n s i c a l l y

defined,

does not depend on the p a r t i c u l a r choice o f parameters Actually, if t > 1,

(which

meters,

as in

so our

reasonement

(3.1.4) and ( 3 . 1 . 3 ) .

there i s a permissible curve tangent t o the d i r e c t r i x in some step is

easily verified t

one has E= ~

and i t

as above the type I - 1 - 1

is

= 1), f o r some s t r o n g l y prepared system o f para-

not possible t o reach by quadratic t r a n s f o r m a t i o n s

(see ( 2 . 2 . 8 ) and ( 2 . 4 . 2 ) ) .

Let p = ( x , y , z ) = p(O) be as in

(3.1.5.2)

In

( 3 . 1 . 1 ) , one has t h a t

(b) = ¢ ( x , z )

where ¢ i s not the power o f a l i n e a r form. Now one can proceed as in theorem (2.1.3) just

to

obtain

that

for

a certain

p(s+l)

=

( x ( s + l ) , y ( s + l ) , z ( s + l ) ) one has t h a t

w i t h n o t a t i o n evident

(3.1.5.3)

and so,

I n r ( b ( s + l ) ) = X. ¢ ( y ( s + l ) , z ( s + l ) ) + x ( . . . )

a f t e r an adecuate

change, we can suppose t h a t _ y ( s + l ) , z ( s + l ) ~ j r ( b ( s + l ) )

and thus we have type zero.

(3.1.6) not trol

Remark.

occur,

The w i n n i n g

is canonical

and

strategy does

for A, if transitions

not depend

of the chosen

l ~--~ I' and ll~--~ I' do coordinates

for the con-

of the processms.

(3.1.7)

So we have only t o

prove t h a t

there e x i s t s a winning s t r a t e g y i f

in

some

step o f the above processus (before reaching I I - 1 - 1 ) one has the t r a n s i t i o n s I ~-~ I ' or I I ~-~ I ' .

(3.2) The t r a n s i t i o n I ~

(3.2.1)

I'.

F i r s t cases.

Let us take the n o t a t i o n s o f

(3.1.3)

and assume t h a t a f t e r the f i r s t

d r a t i c bZowing-up given by T-2 from p = ( x , y , z ) , one has t h a t

(3.2.1.1)

~(c(1))

= r.

qua-

105

Now, i n

v i e w o£ t h e

equations

(3.1.3.3)

(3.2.1.2)

one has t h a t

Z(1)

I In ( c ( 1 ) ) .

and Z(1)

(3.2.1.3)

In

order

to

following

simplify

the

expressions

x'(1)

Then D ( 1 )

is

(3.2.1.5)

generated

D :

Where a ' ( 1 )

= x(1);

the

future

transformations,

Let

us make t h e

a'(1)x'(1)~/

= a(1);

y'(1)

= z(1);

z'(1)

= y(1)

by

b'(1)

~x'(1)

+ b'(1)~/Sy'(1)

= c(1);

c'(1)

+ c'(1)z'(1)~/~z'(1).

= b(1).

One has t h a t

(3.2.2.1)

where

of

= J~ (1),E(1)).

change o£ c o o r d i n a t e s

(3.2.1.4)

(3.2.2)

E jr(c(1))

In

¢ is

not

a power o f

& linear

(b)

form.

a'(1)(x'(1),y'(1),O) (3.2.2.2)

b'(1)(x'(1),y'(1),O)

a'(1

,...

(3.2.3) Proposition.

as s e r i e s

in

By ( 3 . 1 . 3 . 2 )

one has t h a t

= -¢(x'(1),y'(1)). = -y'(1)¢(x'(1),y'(1)).

c'(1)(x'(1),y'(1),O)

(we c o n s i d e r

= ¢(~,~)

= ¢(x'(1),y'(1)).

x'(1),y'(1),z'(1)).

I£ there is a permissible curve tangent to Oir ~ (1),E(1)) for

(X(1),E(1), D(1),P(1)),

then after the corresponding monoidai blowing-up,

the adap-

ted order drops in aii the points.

Proof. der curves

Since

z'(1)

(z'(1),x'(1))

(x'(1),z'(1))

gives

E J(O(1),E(1)) or

and

(z'(1),y'(1)+¢(x'(1)))

a permissible

curve.

Then,

(z'(1)

:

0) ~ E ( 1 ) ,

one has t o

w h e r e ~ ( ¢ ) > 1 . Assume f i r s t necessarily

consithat

106

(3.2.3.1)

r

~x,(1),z,(1))(a'(1))~

and t h i s

contradicts

(3.2.2.2)

since

¢(x'(1),y'(1))

# x ' ( 1 ) r.

Thus t h i s curve can-

not be p e r m i s s i b l e . Let

us suppose t h a t

(y'(1)+ ¢(x'(1)),z'(1))

is

a permissible

curve.

Let

us make the change

(3.2.3.2)

and l e t

x"(1)

x'(1);

us suppose t h a t

(3.2.3.3)

Let

:

y"(1)

~x'(1))

:

y'(1)+t(x'(1));

= k x'(1)+ ....

a"(1)(x"(1),y"(1),O)

:

z"(1)

= z'

1)

Then one has t h a t

- ¢(~"(1),Z"(1)-k~"(1))

us p u t

(3.2.3.4)

Since

@(x,y) = ¢ ( x , y - ~ ) .

V(y)(~) _ > r - l ,

one has

p r-1 @( x , y ) = ~y + cxy

(3.2.3.5)

where

o ~ O. On t h e

other

way

(3.2.3.6)

In

(because one has type I ' by ( x , z ) ) . rect

and the d i r e c t r i x

Now the r e s u l t

testing

(b"(1))

follows

from

= z

r

being tangent t o

(3.2.3.5)

( y " , z " ) cannot be given

and ( 3 . 2 . 3 . 6 )

since r ~ 2 ,

by d i -

over the equations

(3.2.3.7)

x"(1) = x"(2); y"(1) = y"(2);

z"(1) = (z"(2)+X)y"(2)

x"(1) = x " ( 2 ) ; y"(1) = y " ( 2 ) z " ( 2 ) ; z"(2) = z " ( 1 ) .

(3.2.4) missible

Remark.

In

view

£0 t h e

curves

in

D(1)

tangents

of

a winning

tic

center.

p'(1)

=

strategy. Then,

the

(x'(1),y'(1),z'(1))

above r e s u l t , to

Thus we s h a l l equations by

of (T-l,()

the

directrix

assume the

one can suppose t h a t

that

in

order

the player

transformation or T-2 if

player

are

to

there

prove the

is

existence

A chooses the necessarily

A does n o t w i n .

no p e r -

quadra-

given

from

The £ o l l o -

107

wing theorem shows that T-2 is not a good choice for the player B.

I3.2.5)

Theorem.

directional

With

quadratic

notations

as

blowing-up

above, by

given

Then one of the following possibiiities

assume T-2

that

from

(X(2),E(2),O(2),P(2)) p'(1)

is

a

(x'(1),y'(1),z'(1)).

is satisfied:

at r > v(D(2),E(2),P(2)). b) dim D i r

Proof.

(D(2),E(2),P(2))

Assume t h a t

(3.2.5.17

D(2)

a)

is

= 0.

notsatisfied.

= a'(2)x'(2)8/ax'(2)

Then D ( 2 )

is

+ b'(2)y'(2)3/~y

generated

by

2) +

+ c'(2)z'(2)8/~z'(2) where a'(2)

= a'(1)/y'(2)r-l-b'(1)/y'(2)

(3.2.5.2)

b'(2) c'(2)

In

view of

(3.2.2.2)

= b'(1)/y'(2)

one has

(3.2.5.3)

b'(2)(x'(2),y'(2),O)

initial

form

(3.2.5.4)

of

b'(1)

In ( b ' ( 1 ) )

= 0

= -y'(2)¢(x'(2),l

c'(2)(x'(2),y'(2),O)

the

r

= c'(1)/y'(2)r-l-b'(1)/y'(2)r.

a'(2)(x'(2),y'(2),O)

Now, s i n c e

r

= 2.y'(2)¢(x'(2),l

is

= ~'(1)

. f(['(1),['(1))

one has that

(3.2.5.5)

(see

(3.2.5.3)).

In

(a'(2))

= -~'(1).f(~'(1),~'(1))

So

(3.2.5.6)

In the other

+ ~'(1)Z'(1)(...)

z'(1)

hand,

since

¢

e jr(a'(2)).

is not a power of a linear form, and the order has not

108

dropped, one has that

¢ ( x ' ( 2 ) . 1 ) = y x ' ( 2 ) r-1 + 6x'(2) r

(3.2.5.7)

where X ~ O. Then

(3.2.5.8)

In

(b'(2))

= ~'(2).f(~'(2),£'(2))

+ ~'(2)Z'(2)(...)

+ yZ'(2)~,(2)

Thus,

jr(b'(2))

~ (~ + X~ + p Z ) ,

pear in the i n i t i a l

form.

(3.2.5.9)

dim Dir

jr(b'(2))

Moreover,

X 6 - p~ = O, s i n c e

(3.2.5.10)

if

r-1.

this

is

true

X~ 0 and ~ ' ( 2 ) r does not a p -

( D ( 2 ) , E ( 2 ) ) ~1 one must have t h a t

= ( £ + X~ + PZ , e~ + BZ)

otherwise

J ( D ( 2 ) , E ( 2 ) ) ~ (z,z+Xx+py,ex+By) = ( z , x , y )

So, we can suppose

that

(3.2.5.11)

where

If

since

+

jr(b'(2))

~ ~ 0 in

= (~'(2),

v i e w o£ ( 3 . 2 . 5 . 8 ) .

(3.2.5.12)

In

(b'(2))

This

a~'(2)

implies

+ 6y'(2))

that

= ~(z'(2),x'(2)

+ 6y'(2))

Let us suppose t h a t

~(u,v)

=

[

Piju

ij

v

i+j=r then,

in view of (3.2.5.8),

visible

by

contradiction.

z'(2).

But

in this

(3.3.1)

Pot ~ 0 because the initial form is not di-

situation,

x'(2) r must appear

in the

initial

form:

Then

(3.2.5.13)

(3°3)

necessarily

dim D i p

The t r a n s i t i o n

As we have

I

~

seen

I'.

in

(Z)(2),E(2))

= O.

Case T-1

the

precedent

paragraph,

it

is

enough

to

consider

the

10g

case in which Dir

(X(1),~(1),E(1),P(1))

has not permissibie curves tangents to

(~(1),E(1)) and in the following quadratic transformation,

the piayer B chooses

one of the equations

(3.3.1.1)

(T-I,c)

form p'(1)

(3.3.2)

= (x'(1),y'(1),z'(1))

Let

= (x,y,z)

us i n t r o d u c e

be a r e g u l a r

some n o t a t i o n ,

De(n;P)

for

Let

by t h e

vector

= - x ~

us d e n o t e

before

(3.2)).

starting

let

to

study

case.

Let

p =

us d e n o t e

+ (n-1)yB~y + (n+l)zS~

by A ~ * ( r ; p )

this

the

subset

z

o£ D e r k ( R )

xz=01

composed

fields

(3.3.2.2)

D~* = a x ~

such t h a t

as i n

system o f p a r a m e t e r s ,

(3.3.2.1)

n = 0,1,2, ....

(notation

~D**,

xz = O, P) :

r-1

x + b ~ / B y + cz ~

and v ( b )

= r-1.

The f o l l o w i n g

lemma w i l l

simplify

our task:

(3.3.3)

Lemma.

Let p =

(x,y,z) be a regular system of parameters of X at P and let

E be given by xz = 0. Assume that ~ is generated by

(3.3.3.1)

D = xyr-l(l+xxn-ly)DW(n;p)

w h e r e Dee ~ A e # ( r ; p ) . me

Dir is

that

the

strict

(D',E',P') obtained

by

(3.3.3.2)

> 1.

L e t ~ : X' --> X be a q u a d r a t i c transform

(X',E',D',P')

Then

given

(T-l,0)

~ is

from

p,

by

then D'

that

f r o m p.

generated

(n ~ 1 ) .

directional

satisfies

(T-l,0) is

+ x . D ew

r

blowing-up =

Moreover,

v(O',E',p') if

p'

generated

by

by

D' = x ' y ' r - l ( l + X x ' ( n + l ) - l y ' ) D { ( n + l ; p ' ) + z ' D ' W *

where D' * ~ e A e # ( r ; p ' ) .

Proof. (3.3.3.3)

z e J(~,E).If

~ is

g i v e n by T - 2 ,

then ~'

is

x'y'(l+xxn-lyn)(-nx'B/ax'+(n-1)y'a/ay'+2z'a/az')+z'D

''

and assu and

dim

= (x',y',z')

110

If

r>2,v(O',E',P')~2
If

~ is given by T - I , { ,

(3.3.3.4)

r=2, a c o e f f i c i e n t o f O" has order 1 and dim D i r ( D , E ' ) = 0 .

~0,

then D' i s generated by

x'(y'-~)r-l(l+xn(y'-~))(-x'@/3x'+n(y'-~)@/By'+(n+2)z'@/Bz')+z'D"

and v ( D ' . E ' , P ' ) <1 < r .

The s e c o n d p a r t o f t h e

lemma i s t r i v i a l .

(3.3.4) Returning t o the s i t u a t i o n o f ( 3 . 3 . 1 ) , l e t us prove f i r s t pose

{ = 0 without loss o f g e n e r a l i t y . With notations as in

t h a t one can sup-

(3.2),

one has t h a t

D (1) i s generated by

(3.3.4.1)

D(1)

= ¢(x'(1),y'(1))D*(0,p'(1))

+ z ' ( 1 ) D ~*

where D** ~ A * * ( r , p ' ( 1 ) ) . Let p'

(1) = ( x ' ( 1 ) , y ' { ( 1 ) , z ' ( 1 ) ) ,

where y ' { ( 1 ) = y ' ( 1 ) + ~ x ' ( 1 ) . One has

that

(3.3.4.2)

D*(0,p'(1)) = D*(0,p'{(1))

(3.3.4.3)

A * * ( r , p ' ( 1 ) ) = A**(r,p~ (1)).

And so, i f

¢ ~ x , y ) = @(x,y+~x)

(3.3.4.4)

D(1) =

where D**

~ A**(r,p'

needs,

us assume

let

one has t h a t

¢~x ' ( 1 ) , y '

(1))D*(0,p'

(1)) + z ' ( 1 ) D * *

( 1 ) ) . ~ n c e t h i s decompos±tion {=

is

the

only property t h a t one

0.

(3.3.5) Let p(2) = ( x ( 2 ) , y ( 2 ) , z ( 2 ) ) be obtained from p ' ( 1 )

by ( T - l , 0 ) . Then E(2) i s

given by x ( 2 ) . z ( 2 ) = 0 and D(2) i s c l e a r l y generated by

(3.3.5.1)

D(2) = x ( 2 ) ¢ ( 1 , y ( 2 ) ) D * ( 1 , p ( 2 ) ) + z ( 2 ) D * * ( 2 ) .

I f the player A has not won, n e c e s s a r i l y one has t h a t (except f o r a constant factor)

(3.3.5.2)

~ x , y ) = xy r-1 + X y r

(note t h a t ¢ is not a power o f a l i n e a r form). Moreover, one has

that

111

(3.3.5.3)

So

we

D**(2)

can

write

(3.3.5.4)

D(2)

and we can a p p l y

(3.3.6)

i) case

player

if

lemma ( 3 . 3 . 3 ) ,

the

notations

player

A wins

in

permissible

curve

center,

~

X(2)

v (D(3),E(3),P(3)) cessarily

then is

is

(y',z')

the

dim

on

lemma

if

the

curve

One

curve for

player

(~ ( 3 ) , E ( 3 ) )

with = 0.

and i n

always the

this

case t h e

quadratic

center.

has t h a t

(y,z)

D'.

by t h e

Thus,

is

a

re-

A does n o t won by c h o o s i n g (T-l,0)

Now, one has t o

blowing-up

and

he w i n s .

(3.3.3).

a permissible

curve.

dim D i n

J(D(1),E(1)) the

(X(2),E(2),~(2),P(2))

by 1emma ( 3 . 3 . 3 )

prove that

center

if

(y(2),z(2)),

For t h i s ,

let

then

us o b s e r v e t h a t

ne-

one has t h a t

(3.3.6.1)

and

DiP

for

B has a l w a y s chosen

a permissible

or

then

by c h o o s i n g

(I.(3.3)), player

curve

a permissible

of

is

following.

ape t w o p o s s i b i l i t i e s :

center

steps

a directional

< r

there

define

notation

sequences

and t h e n ~ ( 2 ) , z ( 2 ) ) : X(3)

not

the

the

a permissible

number o f

D iff

on s t a t i o n n a r y

quadratic

does

us t a k e for

obtain

A chooses t h i s

a finite

Let

to

as a b o v e ,

defines

(y(2),z(2))

Proof.

the

the

(y(2),z(2))

ii)

suits

= x(2)y(2)r-1(l+A(2)-ly(2))O*(1,p(2))+z(2)D**(2)

Theorem. With

in this

~ A**(r,p(2)).

=

(z'(1)),

adapted

order

has

since

z(2) r

is

(3.3.5.4)

since dropped

(D(1),E(I))

otherwise

one w o u l d

by making

a monomial

which

= 2

(T-l,().

have

Now,

appears

in

then

there

the

type

the

zero

result

coefficient

(or

better)

follows of

from

8/~y(2)

in

D(2).

(3.3.7) after der

Remarks. the

I.

quadratic

has n o t d r o p p e d . 2.

By

(3.3.5.2),

blowing If

r > 2,

up w h i c h

r = 2 there

Although

if

does n o t is

correspond

is

only

to

one p o i n t

T - 2 and t h e

o v e r P(1)

adapted or-

a t most two such p o i n t s .

now we a r e

only

interested

in

the

reduction

game b e -

112

ginning

at

so f o r

(3°4)

the

the

type

type

1-1-I,

above

II

Assume now that

of

also

valids

for

the

type

1-1-0

and

one has a s e q u e n c e

(X(t),E(t),D(t),P(t))

= 0,1,...,s+1,

-up

are

~--> I '

3.4.1.1)

t

results

I-1.

The t r a n s i t i o n

3.4.1)

the

the

with

s >1,

precedent

such t h a t

each s t e p

(X(O),E(O),D(O),P(O))

d i m Dir

I£

In

that

this

in

the

case

yet

A as

(3.3)

and

the

simpler

one

is

tid

way f o r

the

player

t

player

the

blowing-

analogous A to

to

will

win.

also

not

type

II-1-2.

i£

I'.

the

resolution

be a d i r e c t

situation

the

Vt.

we have t y p e

A can w i n

The

Vt.

> I

we have

= s+l

result

(3.2).

= r

(D(t),E(t))

If t = 1 , . . . , s

feature.

directional

= (X,E,D,P).

v(D(t),E(t),P(t)

prove

a quadratic

one and

(3.4.1o2)

One has t o

is

will

transition

The o t h e r

way f o r

be d i v i d e d

I ~

I'

one w i l l

game has had t h i s the

in

victory

two

possibilities,

and one can g i v e allow

us t o

o£ the p l ~

obtain

an e x p l i c i a type

I'-2

or I ' - 1 - 0 .

(3.4.2) player

Remark. B.

One

directrix

in

(3.4.3)

Lemma.

...,s+l,

The can

sequence also

(3.4.1.2)

suppose

that

is

unique

there

is

no

and

it

is

permissible

the

only

curve

choice

for

the

tangent

to

the

any s t e p .

p = (x,y,z)

= p(O)

may be c h o s e n

in

such

the t r a n s f o r m a t i o n

~(t):

X(t)

---~

X(t-1)

a way t h a t

for

each t = l , . . .

113

is

given by T-2 from p ( t - 1 ) =

(x(t-1),y(t-1),z(t-l))

and p ( t )

= (x(t),y(t),z(t))

is

the p.s. o f p. obtained.

Proof.

It

is

enough

t o make a strong p r e p a r a t i o n o f z(q) and "go back" t o

z in the usual way.

( 3 . 4 . 5 ) Assume t h a t

p = (x,y,z)

has the above p r o p e r t y , and t h a t D(O) i s generated

by

(3.4.5.1)

where

D(O) = ax@/~x + b ~ ~

In(b)

(3,4.5.2)

=

¢(x,z)

D(1)

and v ( c )

>r+l.

Then 0 ( 1 )

= ¢(x(1),z(1))(-x(1)@/ax(1)

+ c@/@z

is

generated

+ y(1)@/ay(q)

by

-

z(1)a/@z(1))

+

+ y(1)D**(1)

where De*(1)

E Derk(R(1)lE(1)l.

(3.4.5.3)

then

D*'(t;p)

D(s)

is

generated

(3.4.5.4)

Slnce

I £ we d e n o t e

(3.4.5.5)

= ¢(x(s),z(s))DW'(x;p(s))

= (~(s),z(s))by

(3,4.3),

v(DWW(s)(x(s))/x(s))

Obviously:

D**(s)(z(s)))

~ r.

(3.4.5.6)

Let

is

r;

v (D**(s)(y(s))/y(s))

us d i s t i n g u i s h

~(D**(s)(z(s)))

Let

generated

(3.4.6.1)

~

us c o n s z d e r

the

first

+ y(s)DWW(s)

one has t h a t

v(D**(s)(z(s)))

(3.4.5.7)

(3.4.6)

x + yB4 y - tzB/az)

by

D(s)

J(D(s),E(s))

= (-tx~

by

possibility

>

the

~ r

two following

posibilities:

r+l

= r.

(3.4.5.6).

Then

one

by

D ( s + I )= ¢(x(s+l ) ,z ( s + l ) ) D * ' ( s + l ; p ( s + q ) ) + y ( s + l ) D * * ( s + l )

has

that

L]( s + l )

114

where

if

p'(s+l)

b**(s+l)

=

(x'(s+l),y'(s+l),z'(s+l))

~ A**(r,p'(s+l))

for

the

transition

and

2 by s+2,

the

same way. Then,

(3.4.7)

Let

I

(see t

prop.

~I'

this

us c o n s i d e r

(3.4o7.15

Now,

studied

(3.2.3), in

3.3.2).

th.

in

O(s+l)

~ is

better)

homogeneus

of

degree

r.

If

1 by s + l

proved e x a c t l y

an e x p l i c i t

Then ~ ( s + l )

as

in

manner.

is

generated

by

+

+ y(s+l)O**(s+l)

~(x,z)

~ ~(x,0),

then

one has t y p e

zero

(or

and so n e c e s s a r i l y

if

ces

in

(3.4.5.7).

has t h a t

changing

may be

= ¢(x(s+l),z(s+l))O*'(s+l;p(s+l))

(3.4.7.25

Now,

Actually,

(3.3.6)

A wins

one

case we can r e a s o n e x a c t l y

(3.3).

th.

+ ¢(x(s+l),z(s+l)).~/Sz(s+l)

where

this

and

and

player

possibility

(x(s+l),z(s+l),y(s+l))

for

(3.2)

(3.2.5)

case t h e

the

=

~(x,z)

one makes a c o o r d i n a t e

that

= Xx r ,

change

x = y(s+1),

(X(s+l),E(s+1),O(s+1),P(s+1))

v (DW~(s+l), E(s+l))

=

r-1

or

r).

)# 0.

is

This

y = z(s+1),

o£ t h e

proves

type

the

I'-2

result

z = x(s+l), or

in

I'-I-0

the

one d e d u -

(depending

beginning

of

of

this

section.

(3.5)

Reduction

(3.5.1) then, the

of

Theorem.

the type

If

by c h o o s i n g

directrix

the

the

and

I'-I-I.

reduction

quadratic

such

a curve

game b e g i n s

center if

if

it

there

exists,

at

a situation

is

no p e r m i s s i b l e

the

player

of

A wins

the

type

curve

1'-1-1,

tangent

o r he o b t a i n s

to

type

I'-2 or I'-I-0.

Proof. be such transform the

that of

Let I(E) E2,

computations

P(s)

=

~

(xz),

there of

Assume such t h a t

(X,E~,P)

is

(2.45

now t h a t strict

be t h e

status (x),

(0).

I ( E 1)

=

never

a transition

can be a p p l i e d G is

a

transform

I(E25

in

E = EI U E2 and l e t

= z

I£

like order

realization of

Let

o£

one

I ~-~ I ' to

obtain

the

is

always

or

II

the

game

E2 (one can assume t h a t

~

p = (x,y,z) in

I'

the

strict

o£ ( 3 . 1 )

and

victory.

beginning ~(t)

is

at

(X,E,O,P)

quadratic

for

115

t > 2).

Then t h e

above

computations

stat'(t)

where

E(t)

mark t h a t the is the

= E'(t)

the

player

blowing-up

type

11'-I-2

A wins

in

= strict

follow

by c h o o s i n g

there this

the

transform

the winning

some s t e p .

and

to

realization

G'

defined

by

= (X(t),E'(t),~(t),P(t))

(E2(t)

G' may be does n o t

monoidal of

U E2(t)

may be a p p l i e d

is

But

in

strategy this

a permissible

center.

of in

E2). the

It

is

enough t o

sense o f

ease,

6 =

~,

center

tangent

not making

and t h e n to

the

re-

star

(s)

directrix:

-

I V -

A WINNING STRATEGY FOR TYPE ONE

O. INTRODUCTION

In begins

with

stratety

this type

for

ALL general,

this

chapter one,

it

in

is

order

continued to

complete

the

study

the

o£ t h e

proof

reduction

of the

existence

game when

it

of a winning

case.

possible

no standard

transitions

from the type

one cannot obtain directly the victory,

I will be studied.

In

but a speciai type, called "bridge

type". But the study o? the cases which give the victory is aiso usefui for the st~ dy of

the bridge type. The end part of the chapter is devoted to the study of the reduction game

beginning I'.

at the bridge

In this way,

the

type and aiso

existence

of the

reduction

game

beginning

at the type

of a winning strategy for the reduction game begi~

ning at the type one is proved. In

view

of

the

results

o£ the chapter

III,

are not considered as a beginning of the reduction game.

the

types

I-I-I,

or I'-I-I,

117

1. THE "NATURAL" TRANSITION

(1.1) Definition and notations

(1.1.1)

In t h i s

section,

(X,E,~,P) w i l l

c e s s a r i l y o f the type I - 2 ) . p = ( x , y , z ) such t h a t i t

be o f the type I

Let us f i x

once f o r a l l

(and, as we s h a l l see, ne-

a r e g u l a r system o f parameters

i s s t r o n g l y w e l l prepared.

Moreover, assume t h a t x = z = 0 i s not a permissible curve f o r Let us f i x

(1.1.2)

(X,E,D,P).

1> 0 as "index o f r e t a r d n e s s " .

A

Definition.

"model

for

the

n a t u r a l t r a n s i t i o n " beginning at

(X,E,D,P), p

is a sequence

(1.1.2.1)

H = {H(t) = ( G ( t ) , p ( t ) ) } t = O , . . . , s + l

with s >1, such t h a t : a) Gls+l

is a partial

(X,E,D,P), stat

realization

such t h a t

~(t),

t=l,...,s

(1) i s o f the type I f ) ,

T-2,

T-3 or T-4.

p(t)

And p ( t )

reduction

game,

are standard,

and s t a t

b) p(O) = p, f o r each t = l , . . . , s + l , (T-l,0),

of the

beginning

at

e(E(1)) = 2 ( i . e .

( s + l ) i s o f the type I . i s obtained from p ( t - 1 ) by

is

s t r o n g l y w e l l prepared f o r t

=

= 1,...~s. c)

Gls+l p(t)

(1.1.3)

f o l l o w s the (see I I I

1-retarded standard winning s t r a t e g y r e l a t i v e l y t o

(2.8.4)).

Remarks. Before showing t h a t

the above model i s

u s e f u l f o r the c o n t r o l o f

c e r t a i n s t r a n s i t i o n s in the reduction game, l e t us make some e v i d e n t remarks: i)

#(1) i s

given by T-2 or T-4.

i s a permissible curve f o r ii) iii)

~ ( s + l ) i s given by ( T - l , { ) ,

If

we begin

is

given by T-4, then y = z = 0

(X,E,~,P) and A(~,E,p) ~

has only one v e r t e x .

O.

(X,E,D,P) must be o f the type I - 2 , necessarily i f

#(I)

since as a consequence o f (III.(3;1)),

with type I - 1 ,

the no-standard t r a n s i t i o n g i -

118

ven by ~(s+l) must t o end i n

Definition. Let

(1.1.4) tional that

quadratic

~is

ii)

(1.1.5)

(and not type I ) .

(X,E,D,P) be o f the type I f ,

blowing-up,

a "natural

i)

type I '

and ( X ' , E ' , D ' , P ' )

transition"

r = v(D',E',P')= (X',E',~',P')

Remark. A c t u a l l y ,

and Let ~: X' --) X be a d i r e c -

the s t r i c t

iff

v(D,E,P)

i s 0£ the type I .

we are i n t e r e s t e d i n

the f i r s t

a sequence o f standard t r a n s i t i o n s begining a t type I winning s t r a t e g y has been a p p l i e d .

The main r e s u l t

nal s i t u a t i o n i s

than the i n i t i a l

will

better

transform. We s h a l l say

(strictly)

natural t r a n s i t i o n a f t e r

when the 1-retarded standard

in t h i s section i s t h a t the f i one.

The f o l l o w i n g p r o p o s i t i o n

a l l o w us t o work w i t h a model.

( 1 . 1 . 6 ) P r o p o s i t i o n . Let G be a r e a l i z a t i o n o£ the game beginning a t

(X,E,D,P) and

assume t h a t the p l a y e r A has f o l l o w e d in G the 1-retarded standard winning s t r a t e g y w i t h respect t o p u n t i l the step s. Let us suppose too t h a t s t a t II

and t h a t

(1) i s o f the type

~(s+l) i s a n a t u r a l t r a n s i t i o n . Then, there e x i s t s p' = ( x ' , y ' , z ' )

s t r o n g l y w e l l prepared such t h a t

(1.1.6.1)

and

~D,E,p')

there

exists

a

model

H for

the

= A(D,E,p)

natural

transition

beginning

at

(X,E,D,P),p'

such t h a t

H(t) = ( G ( t ) , p ' ( t ) )

(1.1.6.2)

t

= 0,1,...,s+1.

Proof. I t

f o l l o w s from ( I I I .

( 1 . 1 . 7 ) The r e s t o f t h i s

( 2 . 6 . 1 ) ) and ( I I I .

(2.6.2)).

section i s devoted t o prove the f o l l o w i n g theorem:

Theorem. Let H be a model f o r the n a t u r a l t r a n s i t i o n beginning at (X,E,D,P),p.

119

Then

there

is

a

strongly

well

(X(s+I),E(s+I),D(S+I),P(s+J))

-(1.1.7.1)

(1.2)

zed

system

of

regular

parameters

< B (D,E,p).

Lemma.

Let

(X,E,D,P)

be o f

the

type

I-2

(or

1-1-0)

and p = ( x , y , z )

a norma]i

base such that O is generated by

Assume

D = ax)/Sx

that

prepared.

(~,B) Then

e

A(D,E,p)

(~,B)

(Remark

~

is

&r(b;p)

. Ar(b;p)

(1.2.1.2)

denotes

Ar(b;p)

+ b~

y + cS/az.

a vertex, and i t

the

= convex

is

(~,B)

E Z

not well

hulk

of

2 o

and t h a t

prepared

characteristic

(h,i,j)

(G,B)

is

not well

as a v e r t e x

of

Ar(b;p).

polygon

{(h/(r-j);i/(r-j));

~ Exp(b;p),j

< r } + IR 2 ) . o

Proof. A coordinate change z I = z + X x ~ y B m u s t eliminate Pl = ( x ' Y ' Z l )

where

D = alX)/3x

a 1 = a, b I

since

change z I

(O,O,r) ~

= b,

cI

~ Exp

+ bl)$

= c + X(~#/a (b;p),

y + c1~

+Bx~yB-lb).

necessarily

from A ( D , E , p )

(e,B)

z

Necessarily ~

Ar(b;p)

(%B)

(by making

¢ &r(b;p'), the

z).

(1.2.2) Corollary. With notations as above, A(D,E,p)

(G,B)

then

(1.2.1.3)

but

if Ar(b;p)

is well prepared,

then

is well prepared.

(1.2.3) Lemma. Let p = (x,y,z) be a r.s. of p. and f E R, let us suppose that

(1.2.3.1)

Then

for

reductions

(1.2.1.1)

Let

p'(s+l)

such t h a t

B(D(s+I),E(s+I),p'(s+I))

First

(1.2.1)

prepared

~(f;p)

{(h,i,r-1);

is w@ll prepared

(h,i) ~ IR2}

~Exp

(f;p) = @.

(characteristic zero!).

inverse

120

Proof. I£

(O,O,r)~

If

(O,O,r)

(It

is

the

Exp ( £ ; p ) E Exp

= O,l,...,r

known " g o o d

the

(f;p)

result

and

the

By t h e regular

the

expansion

parameters is

&t

= G',

6(~,E,p)

Let

(X,E,D,P),p.

Let

(1.2.5.2)

respect

by i n d u c t i o n

H

to

Then

=

there

there

exists

Clearly

A cA(O(s),E(s);p(s)).

After

(1.2.5.4)

z'(s)

if

is

that

w(t)

a(t)(0,0,z)

one has t h a t ,

of

the

then

for

each j

=

zero!).

i n an e a s y way t h e

theorem

(1.1.7).

be a m o d e l

another

is

The

D(t)

is

for the

model

weii

H'

well

prepared

sys

characteristic

zero

=

natural

{G',

transition

p'(t)}

such

begi-

that

G =

prepared.

generated

(x(t),y(t),z(t)),

= 2,o..,s

that

obtain

and p ' ( s + l )

b(s)-a(s).

(1.2.5.5)

vertex,

prep&red).

here.

is

p'(s)

t

given

by

= =

> O, m ( t )

m(t)z r + (higher

~ Exp

a change

of

= z(s)

+

Since

T-2,

terms)

degree

terms)

Let

E IR~

given

A = A r(£;p(s)).

coordinates

= (x(s),y(s),z'(s)),

is

by T-2

T - 3 o r T - 4 one can deduce

degree

(£;p(s)).

[

~(1)

> 0 such t h a t

n(t)z r + (higher

(O,O,r)

{(h,i,r-1);(h,i)

= 1,...,so

by ( T - l , 0 ) ,

X ~ O, n ( t )

b(t)(O,O,z)

=

prepared

always well

= a(t)x(t)a~x(t)+b(t)y(t)a/ay(t)+c(t)~/az(t)

p(t)

(1.2.5.3)

£

is

transformation)

~ Exp ( £ ; p )

= { G, p(t)}

us s u p p o s e

D(t)

o r T - 4 and F o r t

Let

well

polygon

(z+kxay6ir(characteristic

p'(s+1)

= A(~,E,p')

Proof.

of

importante

Proposition.

nning

with

(the

a not

a b o v e lemmas one w i l l

base f i e l d

(1.2.5)

is

(m],Bj,r-j)

(1.2.4)

of

trivial

o£ t h e T e h i r n h a u s e n

one has

by c o n s i d e r i n g

of

is

(m,B)

(1.2.3.2)

tem

property"

k Bx(s)

y(s)

then

m Exp ( £ ; p ' ( s ) )

= ~.

One has

121

Since

A c convex h u l l

of

Ar(a(s);p(s))

u

A r ( b ( s ) ; p ( s ) ) , one deduces t h a t p ' ( s )

i s s t r o n g l y w e l l prepared. The change ( 1 . 2 . 5 . 4 ) may be (X(s-1),E(s-1), O(s-1),P(s-1)) induction

just

to

in

"given

the usual way, changing

back" t o

p(s-1)

by p ' ( s - 1 )

(X(O),E(O),D(O),P(O)). Moreover, the polygon i s

and by

not modified in

each step. Let us observe t h a t

(1.2.5.6)

where a ' ( s )

D(s) = a ' ( s ) x ( s ) a / a x ( s ) + b ' ( s ) y ( s ) a / B y ( s ) + c ' ( s ) a / a z ' ( s )

= a(s),

b'(s)

= b(s).

Now, i f

we make T - I , ~

from p ' ( s )

t o obtain

(s+l),

we have (1.2.5.7) D @+1) = a ' ( s + l ) x ' ( s + l ) ~ / @ x ' ( s + l ) + b ' ( s + l ) a / a y ' ( s + l ) + c ' ( s + l ) B / @ z ' ( s + l )

Clearly p'(s+l)

(1.2.5.8)

is

normalized and, s i n c e

b'(s+l)

= (b(s)-a(s))(y'(s+l)+()/x'(s+l)

r

one has that

(1.2.5.g)

{(h,i,r-1)}

and by a p p l y i n g

lemmas ( 1 . 2 . 1 )

n Exp ( b ' ( s + l ) ; p ' ( s + l ) )

and ( 1 . 2 . 3 ) ,

p'(s+l)

=

is well

prepared.

(1.3) Proof of the main result

(1.3.1) enough

Here

the theorem

(1.1.7)

to take a fixed model

(1.3.1.1)

Let us s i m p l i f y

(1.3.1.2)

is proved.

H such that

p(s+1)

6(O(s+l),E(s+l),p(s+l))

the n o t a t i o n

In view of the above reductions,

it is

is well prepared and to prove that

< B(D,E,p).

by w ~ i t t i n g

B(t)

= B(O(t),E(t),p(t))

A(t) = A(O(t),E(t),p(t)).

(1.3.2) Proposition. 6(s)
t=O,l,...,s+l

122

(1.3.2.1)

A(s)

Proof. the

Since

~(1)

first assertion NOW,

: ~(A(O)),

given

us

proceed

by T - 2

q(u,v) = (u,u+v-1),

~ &(O).

But

T-4,

one

this

is

has

that

6(1)

<

8(0).

Then

t : 1,...s.

If ~(1)

n {(u,O)}

(1,0) ~ D ( O ) since

w e l l prepared (Remark t h a t given by T-4,

~).

is given

by T-2,

then

&(1)

=

then

A(1)

(1,0)

or

by induction.

(1.3.2.2)

iff

~ e IR } =

is satisfied since 6(t) < 8(t-1),

let

where

is

m {(u,O);

~

(X,E,D,P) i s o f the type I - 2 and p(O) i s

not true

in

general f o r type I I - 2 ) .

then A (0) must have only one v e r t e x ; D ( I )

= ( u , v - 1 ) . Then one has ( 1 . 3 . 2 . 2 ) i f f

= o(A (0))

If

~(I) is

where o ( u , v ) =

(1,1) i s the only v e r t e x o f A(O), but in t h i s

case we must t o make T-3 instead o f T-4. Let us suppose t h a t

(1.3.2.3)

~(t) n {(u,o)}

where 1 < t < s .

If

clearly.

If

(1.3.3)

~(s+l)

D(s+l)

{t+l)

~(t+l)

is

is

are given

is

given

given

given

(T-l,0)

by

(T-I,~),

~

O,

and

then

&(t+l)

b(s+l) c(s+l)

=

r e a s o n as a b o v e .

the

coefficients

= c(s)/x(s+l)

a(s),b(s),c(s)

are

Yl(S)

= y(s)

and P l ( S )

+ ~x(s)

= a(s)/x(s+l)

of

a generator

of

the

the "convex

r

= (y(s+l)-~)(b(s)-a(s))/x(s+l)

where

r+l

coefficients

-

z(s+l)a(s)/x(s+l)

o£

a

generator

= (x(s),Yl(S),Z(S)).

hull",

Let

let us introduce

A*(i) = [[Ar(a(s);Pi(S))

u

r r

of

D (s).

Po(S)

= p(s)

A*'(1)

= o(A*(1)),

Let

us d e n o t e

and l e t

us d e -

the auxiliary polygons

Ar(b(s)-a(s);Pi(S))O

u An+l(c(s)-z(s)a(s);Pi(S))]],

(1.3.3.3)

~ {(u,O)}

by

a(s+l)

nots by "[[...]]"

or T-3,

by T - 2 o r T - 4 we s h a l l

(1.3.3.1)

(1.3.3.2)

by

=

where o ( u , v )

= (u+v-l,v)

i=0,1

123

Easy c o m p u t a t i o n s

show t h a t :

(1.3.3.4) A*(1)

(the of

last

assertion

follows

from

A*(O)

= A(s)

+IH(1)

= a(s)

the

effect

of

+IH(1)

a change

= y+kx

Yl

in

the

polygon

a surface).

(1.3.4) &*'(1)

Lemma. and

&:~'(1)

A(s+l)

Proof.

(1.3.4.1)

there

vertex

A(s+l)

c &*'(1).

~ g(s+l),

=

(1.3.4.2)

the

U

main

vertices

Ar((b(s)-a(s))/x(s+l)r;p(s+l))

Ar+l (Y(s+l)[(c(s)-z(s)a(s)]/x(s+l)r+l;p(s+l)]]

(p,q)

(p,q)

Moreover,

of

since

= [[Ar(y(s+l)a(s+l);p(s+l))

exists

then

c

same a b s c i s s a .

(p',q')

U

then

(0,1)

have t h e

Let

A(s+l)

+

e

A(s+l)

(p',q'))

and

such there

(p(r-j),q(r-j),])

that

(p',q')

exist

•

e (p,q)

j < r such

u

+ IR o

2

(if

(p',q')

is

that

~ Exp(y(s+l)a(s+l);p(s+l))

U

U Exp((b(s))/x(s+l)r;P(s+l))

u

uExp(y(s+l)[c(s)-z(s)a(s)]/x(s+l)r+lz(s+l);p(s+l)).

Now, l o o k i n g

at

(1.3.3.2)

(1,3.4.3)

This is

(P,q)

proves

obtained

(1.3.5)

the

Let A~IR

Let

A-

2 o

(1.3.3.3)

E A*'(1)

assertion

by r e v e r s i n g

(1.3.5.1)

( ~=

last

and

the

or

and

us d e n o t e

by

(p,q-1/(r-j))

A(s+l) ~

A*'(1).

that

E A*'(1).

The p r o o f

of A*'(1)+(0,1)

arguments.

be a p o l y g o n .

Let

us d e f i n e

~n{u+v

~(a)).

one d e d u c e s

= ~(~)}

5(A)

=

the

number such t h a t

cA(S+1)

a

124

(1.3.5,2) resp.

A of smallest

the vertex of

v*(A)

= (G*(A),B*(A)),

vW'(A)

= (~*'(A),B*'(A))

(resp. biggest) abscissa in

(1.3.5.3)

a(A) n {utv = ¢(A)}

(Remark that i t i s possible to have ve(A) = v e ' ( A ) ) .

(1.3.6)

Lemma.

Let

= y+~x,

a;# O. L e t

f e k[['x,y,z]], A1 = A r ( f ; (

i) v*(A) ii)

~ ( f ) _>r.

x,y~z ) ) .

Let

A= A r ( f ; ( x , y , z ) )

and l e t

Yl

=

Then one has t h a t

= v*(A 1)

B*'(A 1) < B*(A) - 6*'(A),

Proof.

i)

See

(TI.

3.4)

ii) Let us suppose that

_

(1.3.6.1)

f

= ~)fhijX

h i j y z .

First, let us observe that we can suppose that

(1.3.6o2)

(we

can

fhij

forget

the

on r and r e m a r k i n g

others that

the

E 0

~

(h+i)/(r-j)

monomials power o f

for

= 6(A)

our

and j < r .

purposes).

x which divides

f

Now,

working

is

not

affected

is

notsatisfied

by

induction

by y ~-~ Y l '

we can suppose t h a t

(1.3.6.3)

f

=

[ a
where a = r . B * ' ( A ) , s = 0,1,...,b-a,

b :

r. ~(A).

In t h i s

[ a
the

above

O, f b l

~

situation

ii)

iff

for

each

indeterminates

fl'

since

one has

(1.3,6.4)

Thus

flxb-ly
system

fb ~ O. To obtain

has

a

no

contradiction

fl(2)~ l-s~

= O.

solution

it is enough

is precisely b-a+1 = number of indeterminates.

in

to prove

the

that

the

rank of its matrix

The matrix of (1.3.6.4)

is

125

(1.3.6.5)

A = (a

nm

)

o ~n,m~b-a

where (1.3.6.6~

anm

= (a+m) a+m-n n

Then

(1.3.6.7)

(1.3.7)

det A = ~

In

order

xed ( h , i , j )

to

finish

the

proof

(b-a+l).a

of

the

~ O.

theorem

h/(r-j)

(1.3.7.2)

(h,i,j)

= ~*(A*(0)),

Now, by t h e lemma ( 1 . 3 . 6 ) B*'(A*(O))

(h+l,i-l,j)

= 6*(A*(0))

U Exp ( b ( s ) - & ( s ) ; P o ( S ) ) U

([c(s)-z(s)a(s)]/z(s);

and by t h e p r o p o s i t i o n

~ O) one has t h a t

(1.3.7.3)

i/(r-j)

6 Exp ( a ( s ) ; P o ( S ) ) u~xp

there

exist

Po(S)).

(1.3.2)

i > O, I ~

~ Exp(a(s);Pl(S))

implies

(which

U Exp(b(s)-a(s);Pl(S)

) u

Pl(S)).

that

(1.3.7.4)

(h+i+j-r,i-l,j)

4 Exp(a(s)/x(s+l)r;

vExp((b(s)-a(s))/x(s+l)e;p(s+l))

p(s+l)) U r+l

OExp((c(s)-z(s)a(s))/z(s+l)x(s+l)

looking

at

(1.3.4.2),

(1.3.7.5)

one has t h a t (h+i+j-r r-j

( h+i+j--r r-j

or

, i - 1 .-) ( A ( s + l ) r-j i-i+1) E r-j

A(s+l)

But

(1.3.7.5)

implies

such t h a t

U Exp([c(s)-z(s)a(s)]/z(s);

And t h u s ,

L e t us c o n s i d e r

such t h a t

(1.3.7.1)

This

(1.1.9).

h+i+j-r r-j

6(~*(0))-1,

(i-l+l)/(r-j)

i-1 r-j

_ B(s)

- 1 r-j

= B(s) - (l-1)/(r-j).

;p(s+l))

that

a fi.

126

On t h e

other

hand,

t h e main

(1.3.7.6)

vertex

(~'(A*(1))

of

A~'(1)

looking

that

the

at

(1.3.7.4)

ordinate

(1.3.7.7)

and the

2.

of

(1.1.7)

(1.3.7.5)

t h e main

B(s+l)

p r o o f of

-

1,

by

~'(A*(1)))

=

6*'(&*(1))).

and a p p l y i n g

vertex

of

the

&(s+l)

< 8(s) - ( l - l ) / ( r - j ) <

lemma

(1.3.4)

one

deduces

verifies

B (s)

< 6(0)

is

o£ t h e

type

prepared

system

of

is f i n i s h e d ,

NO STANDARD TRANSITIONS FROM TYPE I

(2.1)

Introduction

(2.1.1) p(0) we

and

B(s+l)

given

+ 6''(A*(1)-1), (6(~(0))

So,

is

Here we s h a l l

suppose

= (x(0),y(0),z(0)) shall

standard

suppose winning i)

If

ii)

If

is

that

(X,E,D,P)

a very

well

player

A make

strategy,

i.e.:

(x(0),z(0))

is

(x(O),z(O))

is

not

center

oP t h e

(y(0),z(0))

(2.1.2)

Let

w(1): X ( 1 )

sen

the

player

strict

transform

dard.

One has t h e

B with of

only

A chooses the

~

X(0) the

one

four

designed

= 1 and w ( 1 )

quadratic

b)

e(E(1))

= 2 and w ( 1 )

quadratic

c)

~(1)

the

and t h a t

parameters.

following

(y(0),z(0))

blowing-up

by A.

Assume t h a t

e(E(1))

I-1-0

Now

a L-retarded

center. is

permissible

player

and

A chooses

the

center.

possibilites:

center

regular

always

then

or

center.

directional

a)

monoidai with

vertex,

quadratic

center

and

quadratic

be t h e

I-2

then A chooses t h i s

(X(0),E(0),~(0),P(0)).

following

choice

permissible

has

Otherwise

its

permissible,

A(D(0),E(0),p(0))

ill)

by

the

that

(x(0),z(O))

Let

which

has been cho-

(X(1),E(1),#(1),P(1))

this

transition

is

not

be t h e stan-

127

c)

(2.1.3)

This

b)

d)

and

obtain

~(1)

monoidal

section

above.

a "bridge

is

with

devoted

Moreover,

Preliminaries

(2.2.1)

lemma. i)

in

Without

substitue gives

p(0)

is

well

T-2

that

the

case

c),

section

the

player

the

will

A can w i n

player

A

be a l s o

D is

not

(if

in

the

cases

he does

not

for

study

a),

win)

can

useful

the

of

the

p(0),

and T - 2 of

Now i i i )

let

us

then

remark

J(D,E)

if is

making follows

that

necessarily

the

very

well

prepared).

not

player

A has

not

possible

since

e(E(1))

a change Yl from the

won,

~(1) = I.

= y + ~x and a good

fact

must

and t h e

adapted

that

the

abscissa

is

given

by

Now, one can preparation: of

the

main

be o f order

the

type

drops

1-2,

after

since T-l,0.

if

one

Assume

by

D = ax~/Sx

In(b)

= r

(2,2,2.2)

Then,

D(1)

player

not

(X,E,D,P)

= (x,z)

(2.2.2.1)

the

p(0).

unchanged.

generated

where v(c)~r+l,

quadratic

one can assume t h a t

from

(but

~(I)

permissible.

from

and 1 i ) .

I-I-O,

(T-l,0)

prepared

result

= I and

generality,

~ J(~,E),

i)

First,

of

e(E(1))

z(0)

by t h e

(2.2.2)

case

Since

p(0)

remains

If

prove

this

by

(x(0),z(0))is

vertex

has t y p e

loss given

Proof. or

the

is

iii)

(T-I,~)

for

= (1)

ii)

that

to

type.

(2.2)

this

in

(y(0),z(0)).

type".

The c o m p u t a t i o n s bridge

center

after

= a*(1)x(1)8/)y(1)

A has

not

then

+ cS/az

T-I,O,D(1)

= z(1)~/)y(1)

D*(1)

won,

+ b~/ay

is

generated

by

+ x(1)D*(1)

+ b*(1)~/)y(1)

v(D*(1),E(1))

> r-1.

+ c*(1)Y)z(1),

I£

v(c*(1))

> r and t h e

pla-

128

yer

A

has

that

not

then

one

has

a

standard

transition

I --> I. Thus,

one

can a s s u m e

= r-1.

~(Ce(1))

(2.2.3)

won,

One has t h a t J H ( ~ ( 1 ) , E ( 1 ) )

~ x(1)

and then or the p l a y e r A has won, or one

has type zero ( p l a y e r A wins too) or

(2.2.3.1)

and

JH(D(1),E(1))

one

has

type

has t h i s

case.

(2.2.4)

Lemma.

directrix

in

4-0

We

(i.e.

can

not

suppose

= (~(1))

transversal).

that

(X(1),E(1),D(1),P(1))

there since

In

is

no

the

sequel

let

permissible

otherwise

the

us

assume

centers

player

that

tangents

A w i l l win

one

to the

in the f o -

llowing movement.

Proof. Sing

J(O(1),E(1))

(#(1),E(1))

Since

by

tangent

(2.2.3.1)

D(1)

In

(c(1))

permissible,

(~(1),~(1)).

to the d i r e c t r i x

In

view

of

(2.2.2.2),

and c o n t a i n e d

in E(1)

the

only

is x(1)

curve

: z(1)

:

in O.

one has

(2.2.4.1)

with

=

= a(1)x(1)B/ax(1)

= Xx(1) r

then

there

(X~0), is

no

In point

+ b(1)a/ay(1)

(b(1)) over

+ c(1)@/@z(1)

= z(1) r + x(1)(...) P of

adapted

if

order

x(1)

= z(1)

= r after

= 0 is

the m o n o i d a l

blowing-up.

(2.2.5) player

Remark. A,

purposes,

The

although it

is

above it

is

"vertical" not

interesting

curve

permissible, to

remark

x(1)

= z(1)

as we s h a l l

that

this

curve

= 0 will see l a t e r . is

maybe chosen

by

For g l o b a l i z a t i o n

permissible

if

it

is

r-

-fold.

(2.2.6) wing-up,

(2.2.6.1)

Since then

J(O(1),E(1)) the

D(2)

player

= (~(1),~(1))

if

B must choose t h e

= z(2)r[-x(2)a/Bx(2)

the

player

transformation

+ y(2)B/By(2)

A chooses the

quadratic

T-2 from p(1).

- z(2)@/Bz(2)]

Then

+ x(2)D*(2)

blo-

129

D*(2) = a * ( 2 ) x ( 2 ) B / a x ( 2 ) + b*(2)y(2)@/ay(2) + c * ( 2 ) B / a z ( 2 ) .

If

the p l a y e r A has not won, then

v(D*(2).E(2))

(1,0,0)

Moreover

If me

> r-1

~ Exp ( x ( 2 ) c * ( 2 ) ; p ( 2 ) ) .

(X(2),E(2),D(2),P(2))

is

not

of the

type

zero

(or

better)

one can assu

that

(2.2.6.2)

Thus

In

(X(2),E(2),D(2),P(2))

(c*(2))

must

= ~(2).~(~(2),£(2)).

be o f

the

type

one,

type

I'-2-2

or

I'-I-0.

Let

us

denote

(2.2.6.3)

p'(2)

(2.2.7) of

ments

time se

Theorem. the

the the

the

D(2).

write

(2.2.7.1)

game

the

from

let

us c o n s i d e r

0 < t < r-1

that

r

=

and l e t

r-1

following

(X(2),E(2),D(2),P(2)).

center,

necessarily

inductively

D(t+2)

by

be t h e

v(D(t+2),E(t+2)).

= (x'(t+2),y'(t+2),z'(t+2)).

D(t+2)

= utvr[u~/au

D*'(t+2)

(T-l,0)

(t+l)w~/~w] = a*'(t+2)ua~

In

result

order

From ( 2 . 2 . 6 )

-

(t+1)v~/~v

the

move

Then,

player

from

if

each

B must c h o o -

p'(2);

applying

simplify

one has

-

+ wD*'(t+2) u + b*'(t+2)@/Bv

where v(D*'(t+2),E(t+2)) (O,O,r)

of

to

+ c*'(t+2)w@/@~

(2.2.7.2)

the

otherwise

win.

Let

Assume

glven

= (y(2),z(2),x(2)).

status

quadratic

blowing-u p

A will

(u,v,w)

as a b o v e ,

A chooses the

directional

player

notations

reduction

player

Proof. to

With

= (x'(2),y'(2),z'(2))

= r-1

~ Exp ( w b * ' ( t + 2 ) ; p ' ( t + 2 ) ) .

+

t-times

notation,

T-l,0 let

us

130

Then,

w E J(~(t+2),E(t+2))

p'(t+2) t±on

by

(T-l,0),

continues.

chooses

(T-I,~),

First

(T-I,~),

~0,

Then

one

to

~0,

or T-2.

case:

(2.2.7.3)

and t h e n e x t q u a d r a t i c

has

or T-2. prove

transformation

D(t+2)

If

it

that

(T-l,{),

transformation is

the

{~0.

is

given

g±ven by ( T - l , 0 ) pLayer

A can win

L e t v i = v+~u,

from

then the

if

the

induc-

pLayer

B

then

= ut(vl-~u)r[u3/Su+(~u-(t+l)(Vl-~U))3/3Vl

-

-(t+l)w~/~w]+wD*'(t+2).

If

one makes ( T - l , 0 )

(2.2.7.4)

from

D(t+3)

(U,Vl,W) , then

= u't+l(v

' 1-~) r [u'a/~u-(t+2)(v

, 1-~)~/~v'1

-

-(t+2)w'~/~w']+w'D*'(t+3).

and s i n c e

t+1 ~ r - 1 ,

Second center.

Let

the adapted order

case:

T-2.

Here

Uo,Sl,Ul,...,Sl,U

I

the

has d r o p p e d .

pLayer

A wins

be n o n e g a t i v e

integers

±=I,...,L.

L e t n = U o + S l + U l + . . . + u L and h = t + 2 + n .

to

Uo=l t r a n s f o r m a t i o n s

D(t+2),

T-2,...~n

L transformations Let

(u,v,w)

D(h)

(2.2.7.5)

Where e ( Z ) , f ( 1 )

T-2,

T-2,

inductively

are obtained

inductively

(e(l),f(l))

inductively

(2.2.7.7)

Uo = 1,

be t h e

(T-l,0),

result

the

quadratic

(si,u±)~(0,O), o£

applying

uI transformations

from p'(t+2). If

~(~(h),E(h))

= r,

then

by

(e(0),f(0))

~(x,y)

D(h)

always

ue(L)vf(L)[XlU~/~u+~Lv~/~V+YLW~/~w]+wD*'(h)

(2.2.7.6)

where

Let

with,

s I transformations

= (x'(h),y'(h),z'(h)).

=

by c h o o s i n g

= (x+y-n,y),

= (t,t+l)

= o2u1 ~ S l ( e ( l - 1 ) , f ( 1 - 1 ) )

o2(x,y)

= (x,x+y-r).

The numbers

by

X = t+2, o

%

= -(t+l),

X1 = ~ L - I

-

Ul~L

Yo = o

kl,#l,y

I are obtained

131

Pl = YI =

Let

PI-1

YI-1

- SlXl-1

- SlXl-1

- Ul~l

us denote

OW'(h)

(2.2.7.8)

= aW'(h)u~/Su

+ b~'(h)vS/~v

In(b*'(2))

z'(2) r-1

+ c~'(h)w~/3w

Now, s i n c e

(2.2.7.9)

(see

(2.2.6)),

=

one d e d u c e s that

(2.2.7.10)

where ¢ i ( O , O , w )

= O, i = 1 , 2 , 3

(2.2.7.11)

Inr-l(a*'(h))

= X61~r-1

+ ¢1

Inr-l(b~'(h))

= Icl~r-1

+ ¢2

Inr-l(c*'(h))

= l(l~r-1

+ ¢3

and 6 1 , e l , ( l

are given

inductively

60

=

=

-1

; e°

61 =

1 ; (o

61_ 1 -

el = ~i-1 ~i =

One can p r o v e t h a t

(XI,WI)

(2.2.7.12)

e(1)

+ f(1)

tion)

1 ~e(1)

e(1-1)

(2.2.7.14),

that

in contradiction As

a corollary

(2.2.7.15)

and so n < 2 t + 1 .

e(1)

with of

~ r-l;

(2.2.7.12)

-

Ul~l"

and t h a t

+ f(l-1) f(1)

-

_< r - 1 .

e(1)

+ f(1)

v(D(h),E(h))

= r.

one has t h a t

r < e(1)+f(1)

(Sl+U I )

~ f(1-1)

1 ~ f(1)

= 0 implies

the fact

-1

u I eI

Sl 61-1

~ (61,E I)

!

=

by

- Sl ~i-1

e(1) ~ e(1-1);

(2.2.7.14)

for

~*1-1 -

~ (0,0)

(2.2.7,13)

Note,

+ x'(2)(...)

< 2t+l-n

< r

(may be p r o v e d by i n d u c

132

Now,

let

us

prove

(X(h),E(h),D(h),P(h)), and

i£

choose

he c h o o s e s always

by c h o o s i n g

then (T-I,~),

(T-l,0)

the

let

~ #0,

then

in

view of

2£.

+ f(1)

us p r o v e

= r,

61 ~ 0 ~ P I '

Xl

one = 0.

has In

Inr(

(2.2.7.17)

player

B must the

A chooses

choose

player

n < 2t+I,

the

quadratic

(T-I,0),

T-2

A wins,

Since

then

player

the

or

the

center

(T-I,~),

player

A will

in ~0

B cannot

always win

center, that

one can suppose

e(1)

e(1)

the

player

(2.2.7.16)

If

if

the

or T-2

quadratic

First,

that

two

the

+ f(1)

> r+l.

possibilities

first

that

(unless

symmetry):

I£.

6

0 ~I,

1

case

~ue(1)vf(1)

+ wa*'(h))

=

XSlwr +

e(1) f ( l ) +~¢1(2,Z,2 ) + Xl!

and

since

e(1)>

1,

f(1)

> 1 the

directrix

of

Z

this

coefficient

is

of

dimension

zero.

In the second case,

w ~ jr(wa*'(h))

(2.2.7.18) r

In (pl u

and

since

e(1)~

1,

f(1)

e(1) f(1) e(1) f(1) V + wb*'(h)) = pl~

~1,

dim

Oir

(D(h),E(h))

= O. Then

(mod w)

(2.2.7.16)

may be s u p p o -

sed t r u e . Now, b y ( 2 . 2 . 7 . 1 6 ) and

the

player

B must

( u , v 1 = v+ ~ u , w )

choose

(T-l,0),

(2.3)

(2.3.1)

e(l)

+ f(l)<

The case

Let

us

T-2

= u'e(1)+f(1)-P(v'

+ ( ~1- ~ ) ( V ' l - ( ) ) /

and s i n c e

at

(2.2.7.5) or

one d e d u c e s t h a t w#JO~(h),E(h))

(T-I,~),

¢~0.

Now,

applying

T-l,0

to

one has

D(h+l)

(2.2.7.19)

and l o o k i n g

e(E(1))

assume

2P-t

along

8V'l+(Xl-Xl)W'

by ( 2 . 2 . 7 . 1 4 )

= 1 and ~ ( 1 )

this

1-~) f

the

(1)

(klU'a/~u'

)/)w')+w'D*'

order

+ (h+l)

has d r o p p e d .

quadratic

paragraph that

(X(2),E(2),D(2),P(2))

is

obtained

133

?rom (X,E,D,P) that i f

as in

(2.2),

that

the p l a y e r A has not won in t h i s

t r a n s i t i o n and

the p l a y e r A chooses r-1 times the quadratic center, then he does not win.

More p r e c i s e l y ,

l e t us denote by

(2.3.1.1)

(X(r+1),E(r+l),D(r+l),P(r+l))

the r e s u l t o£ appZying r-1 times the t r a n s f o r m a t i o n ( T - I , 0 ) from p ' ( 2 ) t o ( X ( 2 ) , E ( 2 ) , 0 ( 2 ) , P ( 2 ) ) . Since Dir

the

p l a y e r A has not won,

r

= v ( O ( r + 1 , E ( r + l ) ) , dim

( ~ ( r + 1 ) , E ( r + l ) > I and one has not type zero (then one has type one I ' ) . The

main r e s u l t

is

that

with

the above hypothesis the p l a y e r A can win

by making a monoidal t r a n s f o r m a t i o n centered a t ( x ( 1 ) , z ( 1 ) ) in

(X(1),E(1),O(1),P(1))

and thus he can avoid the above s i t u a t i o n .

(2.3.2) (E,P)

Definition. in

a

Let

situation

p =

(x,y,z)

(X,E, D,P)

be a r e g u l a r

such

that

Exp

system

(D,E;p)

of

is

parameters

defined.

suited

Given

for

a subset

A C ~ , 2 , 3 } , the " o r d e r _vA(~,E,p)" i s

(2.3.2.1)

~A(~,E,p)

(2.3.3)

Remark.

If

not

generic

order

the

riant

which

ssible

says

vgr.

= inf

Y is

{

given

by

a l o n g Y. I n t h i s

"how f a r

( h l , h 2 , h 3) ~ Exp ( ~ , E , p ) } .

(y,z)

and A = ( 2 , 3 ) ,

case Z A ( D , E , p )

is Y from being

in general~A(D,E,p)

may be c o n s i d e r e d

permissible".

Mope p r e c i s e l y ,

(2,3.4)

Y is

permi-

ZA(D,E,P ) = r.

Let

us r e c a l l

(X(2),E(2),#(2),P(2)),

(2.3.4.1)

that

in

then

if

view of

(2.2.6),

~(2)

generated

O(2) = a ' ( 2 ) x ' ( 2 ) @ / B x ' ( 2 )

is

one has t y p e

one I ' - 2 - 2

by

+ b'(2)B/Sy'(2)

+ c'(2)z'(2)B/Bz'(2)

us c o n s i d e r

(2.3.4.2)

Exp ( D ( 2 ) , E ( 2 ) , p ' ( 2 ) ) vExp

(b'(2);p'(2))

= Exp ( y ' ( 2 ) a ' ( 2 ) ; p ' ( 2 ) ) v V Exp ( y ' ( 2 ) c ' ( 2 ) ; p ' ( 2 ) ) ,

is

as an i n v a -

iff

(2.3.3.1)

Let

~ hi; i EA

or I'-l-Ofor

134

The

Exp i s

logously, in

the

defined the

usual

(2.3.5)

for

D(2)

r@latively

polygon A (D(2),E(2);p(2))

to

x(2)

= 0 if

may be d e f i n e d

it

by t h e

Lemma. T h e r e

¢:

is

of

Exp ( D ( 0 ) , E ( 0 ) ; p ( 0 ) )

Exp ( D ( 2 ) , E ( 2 ) ; p ( 2 ) )

by

¢(h,i,j)

= (h+2(i+j-r),j+l,h+i+j-r)

¢-l(h',i',j')

Proof.

It

(2.3,5.3)

follows

= (2j'-h',r+h'-i'-j'+l,i'-l).

from

the

foilowing

equations:

x(1)

= x(0)

x(2)

= x(1)y(1)

x'(2)

= y(2)

y(1)

= y(0)x(0)

y(2)

= y(1)

y'(2)

= z(2)

z(1)

= z(O)x(O)

z(2)

= z(1)y(1)

z'(2)

= x(2)

and a(1)

(2.3.5.4)

= a(O)/x(O) r-1

b(1)

= b(1)/x(o)r-y(1)a(O)/x(O)

r-1

c(1)

= c(1)/x(o)r-z(1)a(O)/x(O)

r-1

a(2)

= a(1)/y(1)n-l-b(1)/y(1)r b(2)

c(2) a'(2)

(2.3.6)

Theorem.

there

(2.3.6.2)

= b(2);

b'(2)

= c(2);

c'(2)

. Assume t h a t

Z A(D(2),E(2);P

is

r

= c(1)/y(1)r-z(2)b(1)/y(!)

Let A = {2,3}

(2,3.6.1)

= b(1)/y(1)

' 2))

~ 1.

a vertex

w =

(u,v)

~A (D(2),E(2);p'(2))

such t h a t (2.3.6.3)

v

+

rv

<

r

and

v

<

1.

r = a(2).

the

projection

a bijection

(2.3.5.2)

Then,

is

way.

(2.3.5,1) given

as

type from

I.

Ana(O,O,r)

135

Moreover,

if

r_>3,

one can t a k e w such t h a t

(2.3.6.4)

u + rv

Proof.

It m u s t

to exist

(2.3.6.5)

such

(h',,i'

that

i'+j' ~ 1 .

(2.3.6.6)

w'

Obviously,

it

is

One

has

that

(see Lemma ( 2 . 3 . 5 ) ) .

(Remark t h a t

[et

(2.3.6.3)

< 13

Now, l e t

> 2r+1

0 < r,

i'/(r-j'))e

= ¢-l(h',i',j')

i'+j'

i

then

there

exists

a point

A(D(2),E(2);p'(2)).

and

(2.3.6.4)

for

this

point

w'.

Let

~ Exp ( D ( O ) , E ( O ) ; p ( O ) )

One has t h a t

(2.3.6.8)

(2.3.6.9)

E Exp ( ~ ( 2 ) , E ( 2 ) ; o ' ( 2 ) )

j ' ~ 1, and

prove

(h,i,j)

( b y lemma ( 2 . 3 . 5 ) ) .

j')

= (h'/(r-j');

enough t o

(2.3.6.7)

< r.

-1

r > 3.

j+l+h+i+j-r

us p r o v e t h a t

~ j'

= h+i+j-r

< j)

but

this

i

< l{=~h'

< i-r.

-

If

r < r.

> 2r+l+h+j-r

contradicts

~ j'

i'+j'

i-r

> r+l,

>

r+l+h+j

< 1.

So,

>

h' <

one has t h a t

r

r

One has t h a t

(2.3.6.10)

h'/(r-j')

+ ri'/(r-j')

<: r

iff (2.3.6.11)

but

h'

this

first

is

part

of

always

true

(2.3.6.3)

(2.3.6.12)

Then i '

for

r->

r = 2.

3 and

i'+j' <

Finally,

let

The

same

argument

us s u p p o s e

that

r = 2 and

(h'/(r-j'),i'/(r-j'))

= j'

= 1 and h'

(2.3.6.13)

and t h e n

since

< r(r-(i'+j')).

= O. But t h i s

(0,1,1)

one has t y p e

zero for

implies

:

1.

(0,1)

that

~ Exp ( b ' ( 2 ) ; p ' ( 2 ) )

(X(2),E(2),D(2),P(2)),

contradiction.

proves

the

136

(2.3.7)

Proposition.

Assume t h a t

(2.3.7.1)

Then

if

~A(D(2),E(2);p'(2))

the

player

A chooses

wins in t h e sense t h a t

Proof.

Let r '

the

center

=~A(D(2),E(2);p'(2))

(2.3.7.2)

looking

Vy(a(1)) ~

and

moreover

r'

is the

let

us suppose t h a t

the player

r'-I

v y,(b'(2))

~ r'

Vy,(C'(2))

~ r'-I

(2.3.5)

vy(b(1))

number

(X(1),E(1),D(1),P(1))

each p o i n t

Y' ~ X(2) be g i v e n by ( y ' ( 2 ) , z ' ( 2 }

~

one deduces t h a t

r'-l;

Vy(C(1)) ~

which satisfies

r',

the above

inequalities.

x(1) = x(2) y(1) z(1)

strict

(2.3.7.5)

transformation

= y(2)

= (z(2)+~)x(2)

i s g i v e n by

D(2) = ( 1 / x ( 2 ) ) r ' - 1 ( a ( 1 ) x ( 2 ) a / s x ( 2 ) + b ( 1 ) ~ / 3 y ( 2 ) + (c(1)/x(2)

-

(z(2)-~)a(1))8/8z(2)),

And t h e a d a p t e d o r d e r d r o p s s i n c e

(2.3.7.6)

V(c(1)Ix(2) r') < r-r' < r

because (2.3.7.7)

(see ( 2 . 2 . 3 . 1 ) ) .

he

over P(1).

B chooses the t r a n s f o r m a t i o n

(2.3.7.4)

th~the

and l e t

Vy,(a'(2))~

r'-l;

maximum

drop for

for

One has t h a t

at the equations in the proof of

(2,3.7.3)

(x(1),z(1))

the adapted order will

and Y c X(1) be g i v e n by ( x ( 1 ) , z ( 1 ) ) .

~ 2.

Inr(c(1))

= x(1) r

+

First

137

If the

(2.3.7.8)

then

B chooses

player

x(1)

the

strict

D(2)

=

is

given

order

y(1)

= y(2);

z(1)

= z(2)

by

(1/z(2))r'-l((a(1)-c(1)/z(2))x(2)8/~x(2)

+ b(1)@/~y(2)

And t h e a d a p t e d

transformation

= x(2)z(2);

transform

(2.3.7.9)

the

drops

+ (c(2)/z(2))z(2)~/@z(2)).

since

v(b(1)/z(2)

(2.3.7.10)

+

r

'-1

) < r-r'+l

< r

because b(1)

(2.3.7,11) (see

= z(1) r + x(1)(,..)

(2.2.2.2)).

(2.3.8)

Proposition.

With

notations

(2.3.8.1)

~{2,3}

as a b o v e ,

if

(X(2),E(2),D(2),P(2))

~ 1

then a)

If

r >3 then

most b)

If

r=2,

most

Proof. r-1

times

us d e n o t e

the

in

in

In

the

the

following

then the

view

player

the

directional

r-1

player

following

of

A wins

the

movements f r o m

A wins r+l

theorem

blowing-up

by c h o o s i n g

(2.2.7), given

from

one

inductively

quadratic

center

at

(X(2),E(2),~(2),P(2)).

by c h o o s i n g

movements

the

the

quadratic

center

at

(X(2),E(2),D(2),P(2)).

can

suppose by

(T-l,0)

that

8 has

from

chosen

p'(2).

Let

by

A (i) = a ( ~ ( i ) , E ( i ) ; p ' ( i ) )

(2.3.8.2)

i=2,3,...,r+I.

It is c l e a r

(2.3.8.3)

where c(x,y)=(x+y-l,y)

that

A(i+l)

= a (&(i))

(always under the hypothesis that

v(D(i),E(i),P(i)

:

r,

2 < i _
138

Now,

in

view of

the

(2.3.8.14)

A(r+l)

and t h i s

the

not

that

us suppose

vertex

o£ A ( 2 )

true,

not

appear

by t h e

~ 8(3).

computations

by J(D(3),E(3))

the

are

the

follows

(2.4)

T-2

then

the

0,Y # ~

yer

to win

o?

more.

is

the

in

v i e w o£ ( 2 . 2 . 6 . 1 )

Then,

This

is

a).

Moreover,

zero.

~A(2).

o£

part

that

(0,3/2).

type

(0,2)

directrix

T-2,

A(4)

Then

(b'(5))

and

i£

the

vertex

(2.3.8.4)

implies

divisible

is

that

by z'(3)

(X(3),E(3),D(3),P(3))

(~,~)

after

making T-2,

easy computations is

given

by

and

(0,1)

are

= XZ'(5)~'(5)

0. So dim Dir

(make

is

given

one has t h a t

(~,1)

and

from

(2.2.6.1)

(x'(4),z'(4)). the

only

show t h a t

So t h e vertices

of

player 8(5).

B I£

we shall

of

(III.

weakly

here the case that

(2.4.2)

the

In

generated

treat the possibility

situation

o£

b) of the

introduction

= I, we shall give a direct

permissible

(3.2) and

to c o n s i d e r

= 0 and the player A has won.

quadratic

in the case o9 e(E(1))

by using

+ rE'(5) 2 + yi'(5)~'(5)

(~(5),E(5))

= 2 and ~ ( 1 )

section,

done

o? the results

(2.4.2.1)

the

proves

Form o£ b ( 3 )

(X(4),E(4),D(4),P(4)) once

In this

is

initial

This

us o b s e r v e

has

one has t h a t

B must choose

vertices

In

have

D(O)

let

one

(2.3.6)

> 3:

easily that

As we A

First,

otherwise

(2.2.6.1)),

The case e ( E ( 1 ) )

(2.4.1)

< r.

since

since

only

(2.3.8.7)

where ~

r = 2.

r

u+v < 1 }~

coordinates

player

of

choose

that

i£

= (~'(3},£'(3)).

directrix

must

{(u,v);

no e n t i r e

from

So t h e (0,1)

one has t h a t

with

theorem

But

~

(2.3.6)

m(D(P+I),E(P+I),P(P+I))

Let

does

(0,1) the

implies

only

(0,1)

theorem

curves

way for the pla-

which are not permissible.

(3.3))

and the remark

(III.(3.3.7)),

(X,E,D,P)

is o? the type

I-2.

(2.1),

~(1)

is

given

by

(T-2)

from

by

D = a(O)x(O)B/ax(O)

+ b(O)@/~y(O)

+ c(O)8/az(O),

(2,12

p(O).

In view

it is enough

Assume

that

t 39

then,

D(1)

is g e n e r a t e d

by

D(1)

(2.4.2.2)

= z(1)r(-x(1)8/~x(1) -

z(1}9/gz(1))

= a(1)x(1)~/~x(1)

(recall

that

In(b(O))

= z(o)r).

Since

I In

(c(1))

one has t h e

(2.4.2.4)

the

case second

must choose

the

has

dim

case,

y(2)

(2.4.2.2))

us a s s u m e

(2.4.3.1)

given

by

one

has

that

for

the

case

is

directrix:

= (y(1))

. . . . . . .

(Z(1),£(1),~(1)

(D(1),E(1))

= 0 and

A chooses

= y(1)z(1);

x(2)

the

+x£(1) )

the

third

quadratic

= x(1)z(1);

center,

of

the

then

the

type

zero.

player

B

z(2)

= z(1)

<

r

that

J(D(1),E(1))

Lemma. L e t

is a

standard,

one has t h a t

(X(1),E(1),D(1),P(1)) re

possibilities

(Z(1),£(1)

(2.4.2.7)

(2.4.3)

not

= r.

v (D(2),E(2),P(2))

let

is

. . . . . . .

(2.4.2.6)

Thus,

transition

transformation

(2.4.2.5)

(see

+ c(1))/~z(1),

(Z(1),~(1))

player

-

=

. . . . . .

Dip the

y(1)O*(1)

following

J(D(1),E(1))

last

and

the

v(c(1))

Since Z(1)

In

+

+ b(1)y(1)~/gy(1)

(2.4,2.3)

The

+ y(1)~/)y(1)

p'(1) is

= (Z(1)).

= (x'(1),y'(1),z'(1)) of

the

type

= (x(1),z(1),y(1)).

I'-2-2

and p ' ( 1 )

is

normalized.

bijection

¢ : Exp(D(O),E(O),p(O))

)

Then

Exp(D(1),E(1),p'(1))

Moreover,

the-

140

(2.4.3.2)

~(h,i,j) ¢

Proof. at

the

-1

The f i r s t

= (h,j+l,h+i+j-r)

(h',i',j')

part

= (h',j'-h'-i'+n+l,i'-l)

is

trivial.

For

the

second

part

it

is

enough t o

look

equations:

(2.4.3.3)

D(1)

= a'(1)x'(1)8/~x'(1)

(2.4.3.4)

x(1)

= x(0)y(0);

(2.4.3.5)

(2.4.3.6)

+ b'(1)8/~y'(1) y(1)

= y(0);

a(1)

= a(O)/y(O) r-1

b(1)

= b(0)/y(0)

r

c(1)

= e(O)/y(O)

r

a'(1)

= a(1);

b'(1)

-

-

+ c'(1)z'(1)~/Sz'(1) z(1)

= z(0)y(0)

b(O)/y(O) r

z(1)b(O)/y(O)

= c(1);

c'(1)

r

= b(1).

(Recall that

(2.4.3.7)

Exp ( D ( 1 ) , E ( 1 ) , p ' ( 1 ) ) uExp

since

one has t y p e

I'-2).

Theorem.

the

(2.4.4)

a)

In

= Exp ( a ' ( 1 ) y ' ( 1 ) ; p ' ( 1 ) ) ~

(b'(1);p'(1))

u Exp ( c ' ( 1 ) y ' ( 1 ) ; p ' ( 1 ) )

above s i t u a t i o n ,

there

a r e two p o s s i b i l i t i e s :

The p l a y e r

A wins

by c h o o s i n g

the

curve

(y'(1),z'(1))

b) The p l a y e r

A wins

by c h o o s i n g

a permissible

center

as c e n t e r . in

less

than

three

steps.

Proof, The p r o p o s i t i o n p'(1).

So i f

a)

is

not true

(2.4.4.1)

Then,

(2,3,7)

can

be a p p l i e d

exist

(h',i',j')

e Exp ( D ( 1 ) , E ( 1 ) , p ' ( 1 ) )

(2.4.4.2)

i'

First,

let

( X ( 1 ) , E ( 1 ) , D ( 1 ) , P ( 1 ) ) and

we can suppose t h a t

Z{2,3 }(D(1) ,E(1) ,p' (1))

there

to

us o b s e r v e

that

+ j'

in

~1.

such t h a t

< 1.

view

of

(2.4.2.2),

one

has

the

same

pro-

141

perty

that

the

in

theorem

the q u a d r a t i c c e n t e r ,

(2.2.7)

and one can assure t h a t

then the p l a y e r B must choose

(T-I,0)

if

the

p l a y e r A chooses

at

least

r-1

t i m e s from

p'(1).

there

are

Let

(h,i,j)

=

@-l(h',i',j').

two

possibilities

(2.4.4.3)

Assume

j

j

= O,

then

i'

= i'-1

= 1 and

By

= o

j'

= O.

or

Then

(2.4.4.4)

(2.4.3.7),

j =

one

i'-I

0 = j'

has

that

0

< i',j'.

Then

=-1.

= h+i+j-r

and

h = r-i.

Let us observe t h a t

(2.4.4.5)

h' + h'

since

one

i = O.

has

this

type

+

I'-2.

implies

i'

+

So

h'

j'

=

+

the f i r s t

zero

or

= (O,O,r)

> r+1 and h'

> r.

N e c e s s a r i l y h'

= h = r,

= (r,l,0)

E A(D(1),E(1);p'(1)).

T-I,0)

one has t h a t

(~,~) ~ A(D(2),E(2);p'(2))

adapted o r d e r dim

Oir

Now, then

i'

transformation T-I,0,

(2.4.4.7)

the

+

r

r > 3, the o r d e r drops i n t h e n e x t b l o w i n g - u p (which may be supposed

r = 2, a f t e r

(if

j'

>

(h',i',j')

~

(h',i',j') (1,1/r)

If

r

+ j'

that

(2.4,4,8)

Now, i f

i'

h = r+1-1.

has

not

(D(2),E(2)) assume But

j

since

= O, = -1. j

since Then

= -1,

(2.4.4.8)

(2.4.4.5).

= h+i+j-r,

but

this

i'

one

= O. has

that

If

= r+2-i

~ j'

1,

easily

that

we have t y p e

(~'(2),~'(2)).

= O,

i ~

thus

we

have

h < r,

0

=

this

j'

= h+i+j-r,

implies

= (h,O,O)

Then, n e c e s s a r i l y j '

so h'

implies

J(D(2),E(2))

(h',i',j')

which c o n t r a d i c t s = h < r and I

dropped),

= I.

and s i n c e I < i

As above, one has t h a t one has t h a t

h'

h ' = r+1 o r h'

= =

142

= r.

Then

(2.4.4.9)

2 (1 +-r--1

,0)

1 (1 + T - 1

,0) e A(D(1),E(1);p'(1))

e A(D(1),E(1);p'(1))

or (2.4.4.10)

If

r >

3,

in

both

cases

the

adapted

order

drops

in

the

two

first

transformations

(T-l,0). Assume t h a t If

(h',i',j')

r = 2 and t h a t

= (r,O,1),

then,

(2.4.4.11)

the

initial

(~'(2),~'(2)). ble

after

(1,0)

Moreover,

point

form

The o t h e r

with

no e n t i r e

(h',i',j')

of

making

= (r,O,1) (T-l,0),

or

(h',i',j')

= (r+1,0,1)

one has t h a t

~ A(D(2),E(2);p'(2))

b'(2)

vertex

is

divisible

by [ ' ( 2 ) ,

so J ( D ( 2 ) , E ( 2 ) )

of A(D(2),E(2);p'(2))

coordinates).

The

next

is

two

(~,3/2)

(the

transformations

= only

must

possi-

be g i v e n

by T - 2 and

(2.4.4.12)

which

(~,~)

implies

type

If point ly

of

no

vertices

If

the

after

the

formation with

(2.5.1) graph

T-2,

with then

the

monoidal

t h e main

(r+1,0,1) of

and t h e

blowing-up

First,

in

the

let result

us

is

then

Exp

(0,3/2), and

the

adapted

= 0).

(O(1),E(1);p'(1)), then

(3,0).

(x'(3),z'(3))

becomes

order

center

~

of A(D(2),E(2);p'(2))

center,

adapted

(~(4),E(4))

(0,3/2)

(T-I,~),

this

one has t h a t

After are

making

(~,3/2)

becomes

order

permissible

drops, and

the

if

after

the

(2,0).

permissible

the

on-

(T-l,0),

and

the

only

next

and

trams-

blowing-up

drops.

(x(@,z(0)).

introduce assures

polygon

vertices

is

dim D i r (r,0,1)

are

(y'(3),z'(3))

with

fact and

A(D(1),E(1);p'(1))

center,

~(1)

better

coordinates

transformation

is

this

(2.5)

entire

=

= (z(2))

next

(or

(h',i',j')

of

J(D(2),E(2))

zero

E A(D(4),E(4);p'(4))

that

the if

definition the

player

of

the

"bridge

A does n o t w i n ,

type". then

In

this

para-

he can o b t a i n

143

a bridge

type. Let

ce

if

ter

one

us assume t h a t

has

type

I-1-0,

the

then

initial the

situation

player

o£ ( 2 . 1 )

A wins

a£ter

is

the

of

the

type

blowing-up

I-2,

with

sin

the

cen-

(1.2.5)

and

(x,z).

(2.5.2)

Definition. a)

It

We s h a l l is

of

the

say that type

(X,E,D,P)

one

I'-2

or

is

of

the

I'-1-0.

"bridge

(See

type"

chapter

III,

iff

(3.5.3)). b) T h e r e that

(2.5.3)

Remark.

is

a normalized

(0,1 + l/r)

In terms

to the existence

the

of p such that

= (z + I x )

or Jr(b)

(r+l,0,O)

1emma. Assume t h a t

(see c h a p t e r

if ~

E Exp ( b ; p )

& Exp ( y a ; p )

u Exp ( b ; p )

player

such

a)

and

b) above

is generated

are equivalent

by

and

(O,O,r)

the

(3.5.9)

of A(D,E;p).

conditions and

III,

+ b~/~y + cz~/~z

= (z,x)

(2.5.3.2) (2.5.3.3)

vertex

E = (xz=O)

D = ax~/ax

jr(b)

(2.5.4)

main

of coordinates,

(2.5.3.1)

then

is

base p = ( x , y , z )

u Exp ( y c ; p ) .

A has n o t won i n t h e

status(X(1),E(1),D(1),P(1)),

then a)

(X(1),E(1),D(1),P(1))

b)

p(O)

may be c h o o s e n

by T - 3 ,

(2.5.4.1)

is

D(1)

then D(1)

is

in

o£ t h e such

generated

= a(1)x(1)~/)x(1)

type

4-0.

a way t h a t

if

Jr(b(1))

+ b(1)~/ay(1)

= (z(1)) jr(c(1))

(O,O,r)

E Exp ( b ( 1 ) ; p ( 1 ) ) .

is

by

+ c(1)a/az(1)

with (2.5.4.2)

p(1)

or

(z(1),x(1))

= (x(1))

obtained

from

p(O)

144

Proof.

Assume t h a t

(2.5.4.3)

If

0(0)

is

generated

D(O) = a ( O ) x ( O ) a ~

A has

not

transition

is

won, not

necessarily standard,

x(O)

+ b(O)a ~ y ( O )

+ c(O)a ~ z ( O ) ,

is

given

by T - 3

from

= r.

Since

jr(b(0))

~(1)

v(c(1))

by

p(O).

Moreover,

= (z(0))

since

the

and b ( 1 ) = b ( 0 ) / y ( l )

r

then

jr(b(1))

(2.5.4.4) jr(b(1))

or

In

both

fy

the

cases,

making

hypothesis

(2.5.4.5)

(Recall

= (~(1)+~(1)+6#(1),y[(1)+6Z(1))

a change

on p(O)

nor

if

dim D i r

Exp ( b ( 1 ) ; p ( 1 ) ) . In

this

In

z ( O ) ~--~ z ( O )

the

jr(b(1))

that

= (~(1)+~(1)+6Z(1))

polygon

= (z(1))

(D(1),E(Q)) (2.5.4.5)

situation,

(2.5.4.6)

there

are

cases

are

of

if

the

6 Z O, type

then

zero

too.

(2.5.5)

b) and a)

Theorem.

the

player

the

quadratic

type.

With

is

the

following

+rex(l)

= (~x(1)

(or

jr(c(1))

proves

automatic

from

hypothesis

necessarily

that

type

(O,O,r)

zero,

so

possibilities

6 = O.

for

jr(c(1)):

+ 6_y(1)

+ 6y(1)) + B_y(1))

better).

Moreover,

the

first

and

third

of

= (x(1))

b).

(2.1).

A has n o t won a n d he d o e s n o t w i n center,

Moreover

~ ~ O, t h e n

the

zero

one can suppose

So n e c e s s a r i l y

(2.5.4.7)

thls

A wins).

= (z(1),mx_(1)

type

does not modi-

(z(1),ax(1)+6_y(1))

if

= (z(1)

jr(c(1))

case,

or

above,

jr(c(1))

each

&(~(O),E(O);p(O))

= 0 then

jr(c(1))

In

+ mx(O)y(O)+By(O) 2 which

the

status

If in

in

the

the

status

(X(1),E(1)~(1),P(1))

next movement,

(X(2),E(2),D(2),P(2))

then ls

of

by choosing the

bridge

145

Proof.

The a b o v e

lemma i m p l i e s

(2.5.5.1)

So

i£

that

J(~(1),E(1))

A chooses

the

quadratic

center,

= (x(1),z(1))

the

player

B must

choose

T-2

from

p(1).

Let

us suppose t h a t

(2.5.5.2)

with

In

¢(0,z)

= z r.

Then,

(2.5.5.3)

D(2)

Now i f

is r

given

z(2)B/Bz(2)]

by x ( 2 ) . y ( 2 ) .

= v(D(2),E(2),P(2))

and p ' ( 2 ) s a t i s f i e s

the

2.6.

with

~(1)

(2.6.1

ble

monoidaL

In

vement

this

he w i l l

monoidal

case win

one.

situation

(2.6.2) from

If

p(0).

the

where a(1) ce t h e Dim

(O(1),E(1))

Jr(c(1))

proof one

generated

not

+ y(2)~/3y(2)

-

+ y(2).D*(2)

Dim ( D ( 2 ) , E ( 2 ) )

and b) o f

= (y(2),z(2),x(2)).

then

(X(2),E(2),D(2),P(2))

(2.5.2).

one

of

wins.

If

by c h o o s i n g

this

it

is

vertex.

ALso

won t h e

player

he does n o t w i n a quadratic

important this

case

in

center

the

fact

has

sense

the

first

mo-

or a permissi-

that only

for

the

ini-

I-2.

b(1)

zero,

B must c h o o s e

the

transformation

T-4

by

+ b(1)y(1)~/~y(1)

= b(O)/y(1)

standard,

= 0 or type

by

= (x'(2),y'(2),z'(2))

A always

generated

r-l,

= J(D(1),E(1)):

is

(y(O),z(O))

A has n o t is

p'(2)

a)

= a(1)×(1)~/~x(1)

is

~(2)

1 < dim

second

type

= a(O)/y(1)

transition

the

only

player

O(1)

and

player

For t h e

Then D ( 1 )

(2.6.2.1)

the

the

Let

eenSer

has of

T-2,

+ x(2)r@/@z(2)

conditions

in

A(D(0),E(0),p(0)) tial

making

= @(~(1),~(1))

= ¢(x(2),z(2))[-x(2)~/~x(2)

-

and E ( 2 )

after

(b(1))

r,

c(1)

necessarily there

are

the

+ c(1)~/~z(1)

= c(O)/y(1)r-z(1)b(O)/y(1)

v(c(1))

= r.

following

I£

r.

one has n o t dim

possibilities

for

Sin

146

(2.6.2.2)

jr(c(1))

= (Z(1)

jr(c(1))

= (x(1))

jr(c(1))

In the third case,

if A c h o o s e s

+ GS(1))

= (~(1),Z(1))

the q u a d r a t i c

center,

he wins always

because of the

fact that

(2.6.2.3)

(the

(O,O,r)

adapted

(2.6.3)

Let

order

drops),

us s u p p o s e

that

tex

of

(0,2-1/(P+1))

A(O)

= 6(D(O),E(O);p(O))

drop

that

i+2j

or jr(c(1))

us r e m a r k Now,

Ly one

vertex.

that

> 2r

or

(O,i,j)

~ (£(1)+¢~(1)).

in

B(O) view

This

Moreover,

implies

easily

that

this

one must

be t h e

=

= Exp ( 0 )

(0,2r+i,-1)

(otherwise

the

adapted

6(0)

= 2-1/(r+1)

general

hypothesis

> 1+1/r. of

the

implies

that

(2.6.3.4)

for

(2.6.3.5)

i ~

if

j~-1,

every

of

(h,i,j)

(2 - ~

1

(2.1) ~ Exp

the

polygon &(O)

(O(O),E(O);p(O))

one has i+2j > 2r and

Exp (1)

= Exp ( a ( 1 ) ; p ( 1 ) )

) (r-j)

if j=-I one has i > 2r+1.

u Exp ( b ( 1 ) ; p ( 1 ) )

Exp ( c ( 1 ) / z ( 1 ) ; p ( 1 ) ) It

is

easy to

show t h a t

order

Thus

that

In particular,

main v e r

if

~ Exp ( D ( O ) , E ( O ) ; p ( O ) )

(2.6.3.3)

Let

This

E Exp ( y ( O ) c ( O ) / z ( O ) ; p ( O ) )

since

(O,i,j)

deduces

will

= (Z(1)+~x(1)).

~ A(D(O),E(O);p(O)).

(2.6.3.2)

one

jr(c(1))

(0,2r+1,-1)

(2.6.3.1)

and t h e n

~ Exp ( b ( 1 ) ; p ( 1 ) )

there

is

a bijection

u

Let

have o n one

has

147

(2.3.6.3)

¢:

given

by

curve

given

ble.

@( h , i , j )

Now,

by

if

=

Exp (0)

(h,i+j-r,j).

(y(1),z(1))

A chooses

By

is

this

Exp ( 1 )

(2.6.3.4)

and

contained

center

(2.6.3.7)

~

in

he w i n s

(O,O,r)

that

necessarilym

(2.6.4)

Assume t h a t

and =

if

(r,r+l,-1)

llows

= (x(1)).

(r,r+l,-1)

since

that

(h,i,j)

e Exp

(remark

that

(1-I/(r+1),1)

and t h u s

that

the

is

permissi-

or

(h,i,j)

since

= (y(1))

= 0).

jr(c(1))

(2.6.4.1)

Singr(o(1),E(1))

one d e d u c e s

~ Exp ( b ( 1 ) ; p ( 1 ) )

jr(c(1))

(remark

(2.6.3.6)

is

implies

that

~ Exp ( c ( O ) . y ( O ) / z ( O ) ; p ( O ) )

(0) in

This

one

e(1)

the

deduces the

only

only

vertex

easily

that

monomial of A(O).

h+i+2j-r

>

r

is

x(1)r),

of

order

r

Then,

for

each

(h,i,j)

it

= fo-

eExp(O)

one has that

(2,6.4.2)

In

particular,

that

3.

if

j~-l,

(x(1),z(1))

adapted

1 - -~-~-~ ) ( r - j ) .

h >(1

order

is

h+j ~

permissible

r

and

if

j=-I

then

and

if

the

player

h >

r.

As

A chooses

in

(2.6.3)

this

it

center

follows then

the

drops.

NO STANDARD TRANSITIONS FROM TYPE II

(3.1)

Introduction

(3.1.1)

In

this

section

we shall c o n s i d e r

pe I but after a few standard ones.

More

of

a

the

ters.

type

I-2

or

I-I-0

and

Let us fix a r e a l i z a t i o n

p(O)

of the

no standard t r a n s i t i o n s

precisely,

very well

let us fix

prepared

produced

from ty-

(X(O),E(O),D(O),P(O))

system

of

regular

parame-

reduction game of length bigger than s+l:

148

(3.1.1.1)

G = { G(t)

such

that

the

with

respect

player

A

to p(O)

has

= (mov ( t ) , s t a t

followed in G the

until the step s (see

(3.1.1.2)

~(s+l):

defines

a

no

standard

nor

(t))

natural

}

t=0,1,...

1-retarded

(1.1.4)).

X(s+l)

---~

transition

standard

Moreover,

winning

assume

strategy

that

X(s)

and

that

star

(t)

is of

the

type

II

p(t)

,

for t = 1,2,...,s.

(3.1.2) t

Without

= 0,1,...,s+I

for

red

(see

of

generality,

o~ r e g u l a r

such a way t h a t and

loss

t=s+1,

p(t) may

is

systems

obtained

be

of

parameters

inductively

(T-I,~),

(1.1.6),vgr.).

one can assume t h a t

~

Thus,

for

can

stat

from p(t-1)

0 and moreover,

one

there

by

the

a sequence

(t),

t

= 0,1,...,s+1

(T-I,0),T-2,T-3

each p(t)

distinguish

is

is strongly

five

or T-4, well

followings

in

prepa-

possibili-

ties:

a)

~(s+l)

quadratic

by

(T-I,0),

from p(s). ~E 0,

b) ~ ( s + l )

,~

,~

,

(T-l,(),

c)

~(s+l)

,,

,,

.

T-2,

d)

~(s+l)

monoidal with

e) ~ ( s + l )

(3.1.3)

given

Let

n o t won i n

~,

us d e n o t e stat

Theorem.

by

(s+l).

8(t)

This

a)

If

b)

If

=

(x(s),z(s))

~'

(y(s),z(s)).

is

is

devoted to

quadratic

and

.

.

.

8(&(~(t),E(t),p(t)).

section

w(s+l)

from p(s)

center

,,

from p(s)

Assume prove the

B(0)

~

that

the

following

1+1/r

then

player

A has

theorem.

the

player

A can

always win.

obtain

a bridge

type

If

a bridge d)

is

quadratic

and 8 ( 0 ) >

1+1/r,

then

the

player

A can

(or win).

c) A can o b t a i n

~(s+l)

If

~ (s+l) type

is

monoidal

with

center

(x(s),z(s)),

then

the

player

with

center

(y(s),z(s)),

then

the

player

(or win).

~ (s+1)

is

monoidal

149

A can always win.

(3.2)

The t r a n s f o r m a t i o n

(3.2.1)

r

T-I,~

~

In this paragraph

we shall

Theorem.

3,

= 2 and

6(0)

If

r >

a

of the theorem

prove

then

the

= 8(&(~(0),E(0),p(0)))

se, the player A wins As

0

or obtain

Corollary

the following

player

wins

in this

<1+1/r then the player

a bridge

of this,

A always

one

situation.

A always

wins.

If

Otherwi-

type. obtains

the

corresponding

part

of the

proof

(3.1.3)

(3.2.2)

Lemma.

Without

if D(s)

is generatd

(3.2.2.1)

loss

o£ generality , one

can

assume

that

p(s) satisfies that

by

D(s) = a ( s ) x ( s )

3 / ~ x ( s ) + b ( s ) y ( s ) 9 / ~ y ( s ) + c ( s ) 9/ ~z(s)

then (h,i,j)

(3.2.2.2)

~ Exp ( b ( s ) - a ( s ) ; p ( s ) )

P r o o f . The same p r o o f o f

(3.2.3)

Lemma. ( O , O , r )

Proof.

(3.2.4)

It

Assume that D ( s + l )

(3.2.4.1)

D(s+l)

(1.2.5).

~ Exp ( b ( s ) ; p ( s ) )

follows

-~j # r-1.

and a l s o

from the p r o o f o f

(0,O,r)

6 Exp ( b ( s ) - a ( s ) ; p ( s ) ) .

(chapter III.(2.1.3)).

is g e n e r a t e d by

= a(s+l)x(s+l)~/~x(s+l)

+ b(s+l)~/Sy(s+l)

+ c(s+l)~/~z(s+l) with a(s+l)

(3.2.4.2) b(s+l)

c(s+l)

= a(s]/x(s+1) r

= (y(s+1)-¢)(b(s)-a(s))/x(s+1) r

= c(s)/x(s+l)r+l-z(s+l)a(s)/x(s+l)

r.

+

150

Then,

s i n c e ~ (s+l)

is

nor

standard

nor

(3.2.4.3)

First

V(c(s+l))

let

us o b s e r v e

that

in

view of

(3.2.4.4)

So i f

player

A has n o t

won i n

(3.2.4.5)

otherwise

type

(3.2.4.5)

and

is

is

(3.2.3).

quadratic

~(s+2)

or better.

jr(b(s+l))

(x(s~l),z(s+l))

the

zero

center,

generated

(3.2.4.7)

D(s+2)

and

(3.2,3)

movement,

= (x(s+l))

this

implies

+ X[(s+l))

or

permissible

the

A wins

Assume

it

then

that

the

player is

player

one has

one can assume t h a t

(By t h e way,

= (~(s+l)

has

~ z(s+l).

Jr(c(s+l))

(3.2.4.6)

If

this

one

= r.

(3.2.2)

jr(b(s+l))

the

since

natural

not

B must

that

([(s+l),[(s+l))).

by

permissible. choose

choosing I£

T-2

the

£rom

this

center

piayer

p(s+l).

by

A chooses Assume

that

by

= a(s+2)x(s+2)~/~x(s+2)

+ b(s+2)y(s+2)8/~y(s+2)

+

+ c(s+2)~/~z(s+2)

I£ the

piayer

p'(s+2)

A has n o t won,

= (x'(s+2),y'(s+2),z'(s+2))

(3.2.4.7)

then

the

then

~,3

player

A wins

as

in

one has t y p e

I'

(in

fact

a bridge

= (y(s+2),z(s+2),x(s+2)).

}(D(s+2),E(s+2);p'(s+2))

(2.3.7)

by'choosing

type).

Let

I£

~ 2

the

center

(x(s+l),z(s+l)).

Thus

assume t h a t

(3.2.4.8)

(3.2.5) in

(2.1)

has t h a t

Z{2,3~(0(s+2),E(s+2);p'(s+2))

The s i t u a t i o n and

(2.2).

will Looking

be r e d u c e d at

to

(3.2.4.2)

a situation and i n

~

1.

which

has been a l r e a d y

v i e w o£ ( 3 . 2 . 4 . 5 )

and

studied

(3.2.4.6)

one

151

(3.2.5.1)

where

¢ and

p'(s+2)

¢ are

a(s+l)

= ¢(y(s+l),z(s+l)+x(s+l)(...)

b(s+l)

= Xz(s+l)r+x(s+l)(...),

c(s+l)

= @(y(s+l),z(s+l)+x(s+l)(...)

homogeneus

= (x',y',z')

in

(3.2.5.2)

order

D(s+2)

I ~ 0

of

degrees

r and r + l

respectively.

to

simplify

notation.

Let

us make

One has t h a t

= x'(~(1,y')@/@y'+¢(1,y')z'B/Bz')

+

+ x y ' r ( x ' ~ / a x , - y , ~ / @ y , - z , ~ / a z ,) + z ' D e ( s + 2 ) .

S±nce the adapted order has not dropped, one has t h a t

(3.2.5.3)

¢(1,y)

= ~yr+l

¢(1,y)

Moreover,

if

Y4 O, one d e d u c e s

easily

(3.2.5.4)

(remark sion

that

c(s+2)

is

the

+ yyr-1

r

r-1

+ cy

that

jr(c(s+2))

coefficient

= (x',y',z')

of

D/By')

and t h e n

the

directrix

has d i m e n -

zero.

(3.2.6) A wins

Proposition. by c h o o s i n g

Proof. T-l,0.

If

plies

that

the

= 6y

+ 6yr

the

Let

always

the the

us p r o v e

player

A does

(y'(s+2),z'(s+2))

transformation

(3.2.6.1)

With

T-l,0

above

notations,

quadratic

that not is

(and t h e

D(s+2+t) =

(B,~)

~

(0,0),

then

player

win

this

B must choose contradicts

permissible.

adapted

order

If

the the

has n o t

always

(t+l-

player

fact

player

the transformation (3.2.4.8)

B has chosen

dropped)

one has

x"(By"r@/By"+Ey"r-lz"B/Bz '') +

+ X x " t y " r ( x " 2 / a x " - ( t + l - ( ~ / 1 ) x " t + 1 ) y " @ / @ y "_ -

the

center.

the

not

if

( 6 / X ) x " t + l ) z " B / a z " ) + z"D,e(s+2+t)

where p' (s+2+t) = ( x " , y " , z " )

in order t o s i m p l i f y the n o t a t i o n . Obviously

(3.2.6.2)

z" ~ J(Z)(s+2+t),E(s+2+t))

which t

im-

times

152

and the

next q u a d r a t i c

easy computation

over

t r a n s f o r m a t i o n must be given by ( T - l , 0 ) , (3.2.6.1)

shows t h a t

if

the

(T-I,~)

p l a y e r B chooses

or T-2. An

(T-I,~),

( ~ 0

or T-2, then the adapted order drops.

(3.2.7) then

Proposition.

the

player

With

B must

the

hypothesis

choose

at

least

of

(3.2.4),

r-1

times

if

the

the

player

A does n o t w i n ,

transformation

(T-l,0)

from

p'(s+2).

Proof.

By ( 3 . 2 . 6 )

one can assume B =E = 0 and t o apply the p r o o f o f ( 2 . 2 . 7 )

to

(3.2.7.1)

D(s+2)

= x y'r(x'B/Bx'-(1-(~/X)x')y'B/By

- ( 1 - ( 6 /X ) x ' ) z ' ~

(3.2.8)

Let

us denote Yl(S)

= y(s)+~x(s)

'-

z')+z'De(s+2).

and Pl(S) = ( x ( s ) , Y l ( S ) , Z ( S ) ) . Let us de-

note

(3.2.8.1)

Exp* = Exp ( Y l ( S ) a ( s ) / x ( s ) ; P l ( S ) ) U V Exp ( ( Y l ( S ) - ( x ( s ) ) ( b ( s ) - a ( s ) / x ( s ) ) ; P l ( S ) ) u uExp

Looking a t

(Yl(S)(C(S)-Z(s)a(s))/x(s)z(s);Pl(S))

( 3 . 2 . 4 . 2 ) , there i s

(3.2.8.2)

@: Exp *

,

a bijection

---->

Exp ( D ( s + 2 ) , E ( s + 2 ) ; p ' ( s + 2 ) )

given by

(3.2.8.3)

¢ (h,i,j)

(compare w i t h

= (h+2(i+j-r),j+i,h+i+j-r)

(2.3.5)).

( 3 . 2 . 9 ) P r o p o s i t i o n . With the hypothesis o f ( 3 . 2 . 4 ) , i f

r > 3 then the p l a y e r A wins

in less than r-1 movements by choosing always the q u a d r a t i c c e n t e r .

Proof. obtain

i-r <

One can r+l and

reason thus

as

in

h' < r+l.

(2.3.6). But

Since

(2.3.6.11)

now -1 < r , j ,

from

(2.3.6.9)

we

is s a t i s f i e d with this condition

153

also.

Now i t

is

enough t o

(3.2.10)

Proposition.

then

player

the

quadratic

apply

With

A will

the

the

win

in

above p r o p o s i t i o n

hypothesis less

that

(3.2.4),

r+l

if

(2.3.8)

B(O)

= 3 movements

<

a).

1+1/r

and

by c h o o s i n g

r

= 2,

always

the

center.

Proof.

By

reasonning

a vertex w = (u,v) ~ &(s+2)

as

in

(2.3.8)

b),

it

is enough

= &(~(s+2),E(s+2);p'(s+2))

(3.2.10.1)

(if

of

as i n

to prove that

there

is

such that

u + 2v < 2

v = 1 one has t y p e First,

since

z e r o as i n

B(O) < 1 + 1 / r

the

final

part

and ~ ( 1 )

is

6(s)

< 6(1)

of the

given

proof

of

by T - 2 o r T - 4 ,

(2.3.6)). one deduces e a s i l y

that

(3.2.10.2)

(remark thus

that

it

a

vertex

of

& (0)

is

given

may n e v e r be ( 1 - 1 / ( r + 1 ) , 1 + 1 / r ) :

< 1

by

only

(h/(r-j),i/(r-j)) possibility

for in

order

some

(h,i,j)

and

t o make (3.2.10.2)

false). Assume (2.3.6)

and

( h ' , i' ,J' )

the

that

(3.2.10.1)

definition

is

not

o£ Exp ~ i n

Then,

(3.2.8.1),

E Exp ( D ( s + 2 ) , E ( s + 2 ) ; p ' ( s + 2 ) )

(3.2.10.3

true.

such t h a t

h'/(2-j')

+ 2i'/(2-j')

by l o o k i n g

the

only

i'+j'

~1

at

possibility

the

proof

for

of

finding

and

< 3

is that (3.2.10.4)

(h,i,j)

(where ¢ is

as i n

(3.2.8.2)).

(3.2.10.5)

Then

the

=(0,1). since 1/r

or

= ¢-l(h',i',j')

But t h i s

(0,4,0)

main This

&(1)

vertex

(a(s),B(s))

is a c o n t r a d i c t i o n

= q (&(O)),

I/(r+1));

since

with

implies

&(s)

in the

the ordinate

easily

that

~ Exp ( c ( s ) ; p ( s ) )

of

q(u,v)

= (-1,5,-1)

=

has a ( s ) = O .

following

(u,u+v-1), of the main

way:

So

B(s)=l,

by

(3.2.10.2),

necessarily vertex

a(1)

is always

and

(~(s),B(s))= 6(1)

= ~ or one~

=

I and

1/3

(i.e.

the transfor-

154

mations

~(t),

t=2,...,s

possible

that the main vertex of & (s) would be (0,1).

(3.2.11)

The

above

are

necessarily

proposition

is of the bridge type

ends

given

by T-I,0

and

in this way

the proof of the theorem

(3.2.1),

T-2

(3.3.1) Here the corresponding

(3.3.27 Assume that D(s+l)

part of the theorem

is generated

(3.1.3) will be proved.

by

(3.3.2.1)

D(s+I) =a(s+l)x(s+l))/)x(s+l)+b(s+l)y(s+l)~/)y(s+l)+c(s+l)~/Sz(s+l)

with

= (a(s)-b(s))/y(s+l)

z(s+l)b(s)/y(s+l)

-

won,

then

r,

one has t h e

One

r,

can

the

case,

(3.3.3)

Assume f i r s t

deduced

that

(3.3.3.1)

c(s+l)

v(c(s+l))

= r.

If

possibilities

for

jr(c(s+l))

jr(c(s+l))

= (~(s+l),Z(s+l))

A wins

from

the

by the

(3.2.3)

Jr(c(s+l))

fact

or

(III.(2.1.3)).

= (Z(s+l)+X[(s+l)).

This

E Exp ( D ( s ) , E ( s ) ; p ( s ) )

This

is

t h e main v e r t e x

of

the

~ &(O(s),E(s);p(s))

polygon

&(s).

Then

= &(s)

implies

that

r+l

A has

= J(~(s+l),E(s+l)):

and (0,2-1/(r+1))

player

~ Exp ( b ( s + l ) ; p ( s + l ) )

(0,2r+1,-1)

(3.3.3.2)

= c(s)/y(s+l)

= (x(s+l))

= (Z(s+l)+X[(s+l))

(0,0,r)

can be e a s i l y

r,

Jr(c(s+l))

the player

(3,3.2.3)

= b(s)/y(s+l)

that

jr(c(s+l))

third

which

b(s+l)

assume

following

(3.3.2.2)

In

since D(s+2)

(see (3.2.4.7)).

(3.3) The transformation

a(s+l)

it is not

not

155

(3.3.3.3)

B(s)

Now, s i n c e

6(0)

(3.3.3.4)

p'(s+l)

then the

(3.3.4)

> 6(s)

part

degree

the

theorem

Assume now t h a t

given

c(s+l)

in

by

=

by m a k i n g

:

is

(~(s+l),z(s+l),y(s+l))

proved

in this

Let

{(h,i,j);]>

f(h,i,j)

case.

us d e n o t e

r }

-~

by

IR2

(h/(r-j),i/(r-j)).

Since

the

only

monomial

of

one has t h a t

(r,1,-1)

~ Exp ( D ( s ) , E ( s ) ; p ( s ) )

= Exp ( s )

that

(3.3.4.3)

f(Exp

Now, =

type,

(x(s+l)).

IR3 -

x(s+l) r,

is

(3.3.4.2) and

(3.1.3)

jr(c(s+l))=

f:

r

(r+l).

= (x'(s+l),y'(s+l),z'(s+l))

b) o f

projection

I /

-

and one has a b r i d g e

(3.3.4.1)

the

= 2

if

6(0)

(s)

> 1+1/r

(x'(s+l),y'(s+l),z'(s+l))

ce ~ ( 1 )

is

given

which

by T - 2 o r T - 4

vertex

the

that

6(s)

only

obstruction

must be o f

the

< 1,

form

By c o m b i n i n g (e(s),B(s))

is

has

2u+v> 2}.

a type

bridge

as a b o v e ,

since to

< 6(1)

the

by T - 2 )

< B(0)

main

(h/(r-j),i/r-j))

(3.3.4.3)

t h e main v e r t e x

and t h e of

&(s),

us now c o n s i d e r

(3.3.4.6)

¢ : Exp ( s )

given

by

the

last

assertion,

let

a certain

Sin-

6(s)

~ 1,

be ( 1 - 1 / ( r + l ) , 1 + l / r ) , us o b s e r v e

(h,i,j)). one d e d u c e s

then

> 1/2.

bijection

~

<1+1/r.

=

< 1+1/r

cannot

fact

6(0)

p'(s+l)

one has t h a t

of A(0)

for

by m a k i n g

Assume t h a t

vertex

prove the

a (s)

Let

is

{(u,v);

(actually

(3.3.4.5)

which

) }c

(y(s+l),z(s+l),x(s+l)).

6(s)

implies is

{(r,1,-1)

one

=

(3.3.4.4)

This

-

Exp ( D ( s + I ) , E ( s + I ) ; p ' ( s + I ) )

that

if

that

the

156

(3.3.4°7)

Now, ¢

-1

@ (h,i,j)

let

(h',i',j')

(h',i',j')

= (h+i+j-r,j+1,h)

e Exp ( D ( s + l ) , E ( s + l ) ; p ' ( s + l ) )

= (h,i,j).

By ( 3 . 3 . 4 . 5 )

(3.3.4.8)

2h + j ~

Then

j'+i' >

that

@ 2j'+i'

~ j'+i'/2

~

r+l

_-~

~ (r+1)/2

> 1

2 and

(3.3.4,9)

By

us suppose

one has t h a t

r

j'+i'

and l e t

~{2,3}(D(s+l),E(s+l);p'(s+l))

(3.3.2.3)

if

the

player

A

chooses

the

> 2.

center

(y'(s+l),z'(s+l))

then

he a l w a y s

wins.

Thus,

(3.4)

theorem

The t r a n s f o r m a t i o n

(3.4.1) will

the

In

this

(3.1.3)

is

proved

T-l,0.

paragraph

Assume t h a t

the

D(s+l)

(3.4.2.1)

corresponding

is

generated

D(s+l)

a(s+l)

= a(s)/x(s+l)

z(s+l)a(s+l),

wing

and

r,

of

the

= a(s+l)x{s+l)~/~x(s+l)

b(s+l)

v(c(s+l))

proof

of

the

theorem

(3.1.3)

If

the

player

the

third

case,

the

= (x(s+l))

Jr(c(s+l))

= (Z(s+I)+x~(s+I))

Jr(e(s+l))

= (~(s+l),z(s+l))

player

A wins

since

r,

c(s+l)

= c(s)/x(s+l)

A has n o t won,

possibilities

Jr(c(s+l))

+

+ c(s+l))/~z(s+l)

= (b(s)-a(s)Yx(s+l)

= r.

(3.4.2.2)

In

part

by

+ b(s+l)y(s+l)~/ay(s+l)

-

t h i s case.

be made.

(3.4.2)

with

in

one has t h e

r+l

-

folio-

157

(3.4.2.3)

which

(0,0,r)

can be deduced

(3.4.3)

from

Proposition.

Then no one o£ t h e

Proof.

If

(3.2.3)

Assume

transformations

any

(i)

is

given

(111.

both

is

not T-3,

eases the

Assume t h a t

won, one has t h e

the

< 1+1/r

~(i),

1
by T-4,

JP(c(s+l))

bridge type

order

that

the

given

pZayep A has

not

won.

by T - 4 .

then

< 1/r

of A(s)

drops

with

= (x(s+l)), by p u t t i n g

and

<s i s

< B(O)-I

main v e r t e x

adapted

(2.1.3)).

6(0)

6(s)

Since ~(s+l)

(3.4.4)

or

that

(3.4.3.1)

and i n

e Exp ( b ( s + l ) ; p ( s + l ) )

is

(1-1/r,1/r)

or (1-1/(r+1),l/(r+l))

T-l,0.

if

B(0)

p'(s+l)

> 1+1/r

and t h e

player

= (x'(s+l),y'(s+l),z'(s+l))

A has n o t =

= (y(s+l),z(s+l),x(s+l)). Assume t h a t

B(O) < 1 + 1 / r .

First,

one has

(3.4.4.1)

(2r+1,0,-1)

(3.4.4.2)

(2-1/(r+1),0)

On t h e

other

hand,

A(t)

t=l,...,s,

w h e r e 01 ( u , v )

or or

the

contradicts

theorem

(3.4.5)

(3.1.3)

Proposition.

(s).

= ~t(A(t--1))

= (u,v+v-1)

(3.4.4.4)

and t h i s

e&

by ( 3 . 4 . 3 ) ,

(3.4.4.3)

fop

E Exp ( s )

and f o r

a t(u,v)

= (u+v-l,v)

at(u,v)

= (u,u+v-1)

t(u,v)

(3.4.4.2) is

proved

If

B (0)

= (u-l,v)

(may be p r o v e d in

t ~2,

this

> 1+1/r,

case

for

then

by i n d u c t i o n Jr(c(s+l))

the

player

since

(1,0) ~A(O)).

Thus

= (x(s+l)).

A can

always

win

or

obtain

158

the

bridge

type.

Proof. =

In the

case j r ( c ( s + l ) )

(x'(s+l),y'(s+l),z'(s+l))

we have t h e

(3.4.6)

bridge

=

type

(join

Assume now t h a t

= (Z(s+l)+X~(s+l)),

(x(s+l),z(s+l),y(s+l)) to

(3.4.2.3)),

Jr(c(s+l))

if

(1,r,-1)

(3.4.6.2)

(1/(r+1),1-1/(r+1))

implies

B(s)

Lemma.

If

or

T-l,0

for

is

giyeq

by T-2,

6(0)

each

i,

~

1+1/r

1 < i < s.

and

t h e main

> 1 -

see

easily

that

A has n o t won,

Then

EA (s).

(r+l)

= (Z(s+l)+X[(s+l)),

let

t be

the b i g g e s t

then index

~(i)

such

is

T-2

that

~(t)

B(1),

~(1)

then

Let

by T - 2 , vertex

1 /

jr(c(s+l))

s-t

Proof. given

to

~ Exp ( s )

Moreover,

(3.4.7.1)

is

player

us

=

that

(3.4.6.3)

(3.4.7)

allow

= (y(s+l)+x~(s+l)).

(3.4.6.1)

This

the

a change p ' ( s + l )

(e ( 0 ) , 6 ( 0 ) ) ~(0)

of

< 1 and

be t h e B(O)

> r-1.

main

vertex

~ 1+1/r,

there

of

A(O).

are only

Since B (s) ~ three

possibilities

for

A(O):

(3.4.7.2)

(~(0),6(0))

=

(1-1/r,1+1/r) (1-1/(r+1),1) (1-1/(r+1),1+1/(r+1)).

then,

the main

vertex

(3.4.7.3)

of A ( 1 )

verifies

(a(1),B(1))

=

(1-1/r,1)

or

(1-1/(r+l),l-1/(r+1))

or

(1-1/(r+1),1)

Now T - 3 w i l l

never

be a p p l i e d ,

this

proves

the

first

part.

For the

second

part,

let

159

us d i s t i n g u i s h

two c a s e s :

t=l

or t >1.

(3.4.7.4)

where if

If

A(s)

o(u,v)

(e,B)

6(t-I)

<

ched)

and

=

(u+v-l,v)

is

a vertex

I,

then so

and

o£

A(1)

necessarily

(~(t),

the

6(t))

= os-l(A(1))

result

with

follows

from

B= I - I / ( r + I ) ,

e(t-l)

=

t=l

(3.4.6.2)

and

the

then ~ > I-I/(r+I).

= 1-I/(r+I)

(otherwise

(1-1/(r+1),1-1/(r+1)),

now,

If

(3.4.6.2)

one

can

fact

t > I, is

that since

never rea-

reason

as a b o v e ,

since

A(s)

(3.4.7.5)

(3.4.8)

Lemma.

(h,i,j)

~ Exp ( s )

Proof. and

thus

form

of

First,

will

of

(3.4.7),

ordinate

this

is

us

E

A(s)

and

be d i f f e r e n t

or

then

this

as i n

(h,i,j)

X = O.

it

is

the

vertex

since

f : IR3 - { ( h , i , j )

(3.3.4),

; j~r

fop

each

If

X ~ O,

a vertex

adapted

of A (s)

order

comes f r o m

~(t)

is

given

then

will

a vertex by T - 2 ,

(2r+1,0~-1)

E

Exp

(s)

(otherwise

the

drop).

us t a k e t h e

Let

(~,6)

and

of

A(t-1),

one c a n n o t

initial

where

obtain

G Exp ( s ) .

~ + 2 B > 2.

If

(s)

] ~

(~) Let

}

~

-

{(1,P,-1)})

(3.3.4.1),

r or

(h,i,j)

c {(u,v);u+2v

= (1,P,-I)

= (h/(r-j),i/(r-j))

the

s i n c e X = O, r e a s o

o(u,v) '(u,v)

there

is

(~',6')

E A (s).

= (u+v-l,v) = (u,u+v-1).

~ &(t-l)

> 2}.

one has t h a t

us d e n o t e

(3.4.8.3)

lemma ( 3 . 4 . 7 ) ,

IR2 be as i n

one has t h a t

f(Exp

(3.4.8.2)

By t h e

then

>I.

see t h a t

impossible

(3.4.8.1)

then

let

i+j

= (Z(s+I)+x~(s+I))

zero. Let

Let

1+1/P and j r ( c ( s + l ) )

one has t h a t

c(s+l)

but

nning

6(0) ~

(2-1/(P+1),0)

notations ~> I ,

If

= o s-t(a(t)).

such t h a t

i

+ j

> I.

Let

160

(3.4.8.4)

os-t(o'(m',B'))

End s - t > r - l .

Let

(m",fl")

= o'(a',6'),

(3.4.8.5)

(~,B)

and s i n c e

if

~" = ~'

~"~

> l(r+1)/(r+2)l.2

(3.4.8.8)

< 0 then

(i,j)

Proposition.

Then one h i s

the

Let

. 2 -

= B = 6" > 1 / 2 ~

or

(0,-1)

I£

~

1+1/r

6(0) the

p'(s+l)

bridge

(3.4.9.1)

type,

monoidal

=

or

(0,0)

and

r 1 = r+----2

2i+j

> r

End n e v e r

jr(c(s+l))

center

=

u n l e s s A has won,

2i+j

> r,

contradiction.

(Z(s+I)+x[(s+I))

then

the

(y(s+l),z(s+l)).

(x'(s+l),y'(s+l),z'(s+l))

Z {2,3}(O(s+I),E(s+I);P

and i t

'(s+l))

= is

(x(s+l),z(s+l),y(s+l)).

enough t o

prove

th&t

~ 2.

a bijection

(3.4.9o2)

given

r+l + (~'-1 > ~

= (1,-1)

by c h o o s i n g

Proo£.

ls

B" ~ r / ( r + 2 )

then

i/(r-j)

pL&yer A wins

There

2~

r > 2,

(3.4.8.9)

(3.4.9)

I(r+l)/(r+2)l.

B" = 6'

And s i n c e

> r+l.

implies

(3.4.8.7)

i+j

= (a"+(s-t)(B"-l),B")

a" + (r+1)6"

Now ( 3 . 4 . 8 . 6 )

If

then

a+2B > 2,

(3.4.8.6)

If

= (e,fl)

by

mma ( 3 . 4 . 8 )

¢:

~(h,i,j)

Exp ( s )

= (h+i+j-r,j+1,i).

~

Exp ( D ( s + I ) , E ( s + I ) ; p ' ( s + I ) )

Then

(3.4.9.1)

follows

immedLatly

from

the

le-

161

(3.5)

~(s+l)

monoidal with

(3.5.1)

If

(3.5.2)

Since J(~(s),E(s))

p(s)

the

and i f

player

D(s+I)

(3.5.2.1)

(x(s),z(s))

A does n o t w i n

is

D(s+l)

center

~ ~(s),

genenated

he w i l l

obtain

always a bridge

one can suppose t h a t

~(s+l)

is

type.

given

by T - 3 f r o m

by

= a(s+l)x(s+l)~/~x(s+l)

+ b(s+l)y(s+l)~/~y(s+l)

+

+ c(s+l)~/3z(s+l)

where then

a(s+l)

=

v(c(s+l))

Jr(c(s+l))

a(s)/x(s) = r.

If

r

the

b(s+l) player

= J(~(s+l),E(s+l))

possibilities

suppose t h a t

= (Z(s+l)+X[(s+l))

jr(c(s+l))

= ([(s+l),~(s+l))

type

z e r o o r dim D i r

(D(s+l)),E(s+l))

= 0).

I n any

one has t h a t

(0,0,r)

by the same se o f

reason

(3.5.2.2)

the

In the

as

(3.2.3)

player

first

~(s+1)

(3.6.1) similar

monoidal with

The p l a y e r to the

Since

A will

(III.(2.1.3)). win

is

center

of

This

by c h o o s i n g (3.5.2.2)

of the

bridge

implies

the

that

quadratic

in

the

third

c~

center.

one can see t h a t type

in

view of

(3.5.2.3).

(y(s),z(s))

A can a l w a y s w i n

proof

z(s)

or

e Exp ( b ( s + l ) ; p ( s + l ) )

and second o f

(X(s+I),E(s+I),D(s+I),P(s+I))

(3.6.2)

= c(s)/x(s)r-z(s+l)a(s+l)

= (x(s+l))

Jr(c(s+l))

produce

(3.5.2.3)

(3.6)

c(s+l)

A has not won, one can a l s o

jn(c(s+l))

(The o t h e r

r

satisfies

(3.5.2.2)

case,

= b(s)/x(s)

in

this

one

can

case.

The

proof

o£

this

cesutt

is

very

(2.6.1).

~ J(D(s),E(s))

suppose t h a t

~(s+l)

is

given

by T - 4 f r o m

162

p(s). =

We o b t a i n

b(s)/y(s+l)

standard, ty

r,

In

the

third

Since

the

possibilities

of

the

player

X= 0 ) .

A wins

this

t h e main v e r t e x

Then one d e d u c e s

that

b(s+l)

the

as i n

=

transition

(3.5.2.2)

implies

£ Exp ( D ( s ) , E ( s ) ; p ( S ) )

is

r,

z(s+l)b(s+l).

= (Z(s+l)+X[(s+l)),

(0,2-1/(r+1))

partiouZar

a(s+l)=a(s)/y(s+l)

-

possibility

(0,2r+1,-1)

and n e c e s s a r i l y

with r

We have a l s o

Assume j r ( c ( s + l ) )

(in

(3.5.2.1)

= c(s)/y(s+l)

= r.

(3.6.3.1)

tex

as i n

c(s+l)

v(c(s+l))

(3.5.2.3).

(3.6.3)

D(s+l)

and t h e

is

not

proper_

(3.5.2).

that

= Exp ( s )

of &(s),

thus

for

each

(h,i,j)

(if

j

it

is

the

"only"

E Exp ( s ) ,

vet

one has

that

(3.6.3.2)

and t h e n

i/(r-j)

one d e d u c e s

(3.6.3.3)

and

the

wins

if

I = (y(s+l),z(s+l))

~i(a(s+l))~

curve

given

by c h o o s i n g

(3.6.4)

that

by

this

I

> 2-I/(r-1)

r;

is

permissible.

Assume j r ( c ( s + l ) )

= (x s + l ) ) .

permissible

(3.6.5)

4.

and t h e

This

£ r;

Vl(C(s+l))

By t h e

property

~ r

(3.5.2.3),

the

player

A

Then

(r,r+l,-1)

(1-1/(r+1),1)

then

center.

(3.6.4.1)

and

~l(b(s+l))

< r)

is

the

only

player

ends t h e

proof

vertex

A wins

of

of

eExp(s)

A(s).

by choosing

the

theorem

One d e d u c e s this

center.

that

(x(s+l),z(s+l))

(See a l s o

is

(2.6)).

(3.1.3)

A W I N N I N G STRATEGY FOR THE TYPE ONE

(4.1)

Introduction In

this

section

we

shall

establish

a

winning

strategy

for

the

player

A

163

when t h e

reduction All

ter

III

is

a

a bridge

us t o

state

If

the

strategy

the

for

the

And

i£

to obtain

the

victory

o r by t h e

(4.1.2)

there

tain

a

is

If

the

player

reduction the

reduction

winning

strategy

for

the

player

Standard

of

a

this

chapter

and i n

t h e chap_

I-1-1

a type

or

A in

the

I'-I-1,

begins

or

victory

there

by a t y p e

prove the

begins

player

obtain

one l - 2

is

l-1-0, or

to

then obtain

a strategy

in

I'-2.

we s h a l l

game

any t y p e

with

following

any

order

one I ' - 1 - 0

type

to obtain

or

I'-2,

two t h e o r e m s :

one

I'-I-0

the

victory

type,

then

or

I'-2,

or to

ob-

type.

the

tence

to

game when i t

section

If

Let

with

order

type

obtain

the

Theorem.

(4.2.1)

A in

with

(4.1.3)

(4.2)

have been made i n

game b e g i n s

or to

for

one.

following:

In t h i s

a strategy

bridge

which

begins

study

type.

Theorem.

then

it

remains to

bridge

any t y p e

reduction

type.

It

with

computations

Theorem.

there

order

the

allow

(4.1.1)

game b e g i n s

transitions

(X,E,D,P) winning

game b e g i n s

strategy

a bridge

I'-I-0.

In

there

is

a

A.

from the

be o f

with

the

type

type

for

the

I'

I'-2

or

player

A for

the

this

paragraph,

"standard"

the

transitions

exiswill

be p r o v e d .

(4.2.2) iff

Definition.

there

(6(A),-l(A))

is

Let

no change Yl <

p =

(x,y,z)

be a n o r m a l i z e d

= y + ~xn such t h a t

(E(Afl),-l(A1))

for

the

(4.2.3)

Definition.

a l w a y s an

order,

called

I'-prepared

then

where

A1 = A ( O , E , P l ) .

I'-prepared

L e t G be a r e a l i z a t i o n

p is

Pl = ( x , y l , z )

lexicographic

& = A(O,E,p);

From p one can o b t a i n

if

base.

base by a sequence o f changes yl=Y+~X n.

of the

reduction

game b e g i n n i n g

at (X,E~,F~

164 and l e t

p = (x,y,z) a)

be a n o r m a l i z e d

G is

"standard"

form of b)

Assume

step

until

(z=O)

for

that

p

strategy"

the

step

t:O,1 ....

is

until

s and f o r

base: s < Length

O
step

Then

s <

<s,

iff

P(t)

~ strict

trans-

,s.

I'-prepared.

the

(G)

G follows

length

there

is

(G)

iff

a regular

the

"standard

G is

standard

system o f

winning until

the

parameters

p(t)

such t h a t b-1.

•

p(O)

. b-2.

If

= p

e(E(t))

T-3 or T-4• re

is

is

Z'-prepaeed

• b-3.

p'(t)

Remarks.

situation 1'-2,

follows the

, then,in

1'-I-0,

strategy

for the

the

then

player

winning

Theorem.

G ~llows

the

result

follows

from

of

chapter

sier

to

since

the there

has

(see

"natural" is

(2•4.1)).

transition

each

standard

step

type

(T-I,0),T-2, Z')

then

the

strategy.

of the of

control of

no good p r e p a r a t i o n s

p(t)

the

for

the

(see Z Z I . 2 . 8 . 4 ) .

considered is

On t h e

always

other

realization until

assumptions

the

as a v i c t o r y of

hand,

o¢ t h e step

the

types

there

is

a

game G w h i c h

s then

G follows

s.

of the

control

to

<s

G is

are

respect

0 < t

until

to

by

T - 3 o r T - 4 and p ( t )

usual

(t),

III.

the

the

stat

strategy

remains

the

T-2,

is

if

It

(T-l,(),

zero

that

usual

of

type

verifies

transitions

is

that

a way t h a t

the

ImI),

with

that

winning

from p(t-1)

(t)

verifies

such

standard

I£ the

sim±lar

A in

by

0 < t < s,

strategy

II'-1.

obtained

from p'(t)•

L e t G be a r e a l i z a t i o n

Proof•

the

Y(t),

one

is

p(t-1)

us assume t a c i t l y

or

strategy

from

obtained

standard

p(t)

= 2 (hence s t a t

obtained

(4.2.3)

11'-2,

standard

(4.2.5)

Let

then

e(E(t))

The c e n t e r

O-retarded

(4.2.8)

If

= 3,

game such t h a t Then G i s

type

the

I'

the

z ~-* z l ) ,

(see

transition

section

I

each s < length

(G)

finite.

~--> I f '

polygon

for

or

II'

~-~ I I ' ,

the

standard

II'

'

(actually,

,I', it

is

then

the

transition but

this

slightly

is ea-

165

(4.3) No standard t r a n s i t i o n s from the type I '

( 4 . 3 . 1 ) Here the p r o o f o f the theorem

(4.3.2)

Proposition.

Let

be an I ' - p r e p a r e d base.

(X,E,D,P)

a)

If

be f i n i s h e d .

be o f the type I ' - 1 - 0

Let G be a r e a l i z a t i o n o f the

(X,E,D,P) such t h a t G folZows the G is not standard u n t i l

(4.1.2) w i l l

or I ' - 2

and l e t

p=(x,y,z)

reduction game beginning a t

standard winning s t r a t e g y u n t i l

the step s - I and

the step s < length (G) (with respect t o p). Then:

E1 i s

given

by x = O, then

D = (~,E 1) and D is m u l t i p l i c a t i v e l y

i r r e d u c i b l e and adapted t o E1. b) Let G' be obtained by p u t t i n g m o v ' ( t ) ned i n d u c t i v e l y from

stat'(O)

= mov(t) and s t a t ' ( t )

= (X,E1,D,P). Then G'

o f the r e d u c t i o n game o£ length ~ s+l and s t a t ' ( s )

is

be o b t a i -

a realization

= s t a r ( s ) . Moreover

for t=O,l,...,s-1

(4.3.2.1)

stat(t)

i s o f the type I '

star'(t)

(resp. I I ' ) < ~

i s o f the type I

(resp I I

).

Proof. Assume t h a t

(4.3.2.2)

D(t)

where p ( t )

= (x(t),y(t),z(t))

y ( t ) @/ay(t), v(b(t))

= a(t)x(t)B~ x(t)

i s obtained as in

depending on e ( E ( t ) ) .

= ~ 0 < t <s-1 and t h a t z ( t )

d i v i d e s simultaneausly a ( t - 1 ) b(s-1), then,

then P(s)

for

~ strict

t=O,...,s-1,

z(t)

is

multiplicatively

: ~(t)

as in

~ = @~ y ( t )

(III.

(2.1.3))

d i v i d e s simultaneously a ( t ) t=l,...,s-1.

If

or one has t h a t

and b ( t )

iff

z(t-1)

z ( s - 1 ) d i v i d e s a ( s - 1 ) and

transform o f z = 0 (otherwise the adapted order drops), does not d i v i d e aCt)

irreducible

...

~(I),

and b ( t ) .

If

E2(t)

is

the s t r i c t

U E2(t) , from the above r e s u l t one deduces t h a t and adapted t o E ' ( t ) .

ved. Let f ( t )

( 4 . 2 . 3 ) and

By arguments

and b ( t - 1 ) ,

transform o f z = 0 and E ( t ) = E ' ( t ) D(t)

+ b(t)~ t + c(t)z(t)@/@z(t)

then

In p a r t i c u l a r a) i s p r o -

166

(4.3.2.3)

stat(t)

= (X(t),E(t),(e(Df(t)),E(t)),P(t))

stat'(t)

(see

I.(J.2)

and

ted to E'(t),

I.(1.3)).

But since ~ ( t )

is m u l t i p l i c a t i v e l y

(e(Df(t)),E'(t)) = (e(D(t)),E'(t))

t=O,1,...,s.

irreducible

The proof of b) is finished

since v(b(t))

= r.

(the other

by remarking

statements

follows

= D(t)

that v ( ~ ( t ) , E ' ( t ) ) = r ,

Remarks. From the above p r o o f one deduces t h a t t h e c e n t e r s Y ( t ) ,

...,s-1

are

11-I-2-0

always

tangents

(III.(2.4.1))

to

Oir

({)(t),E(t))

one o b t a i n s t h a t

= Dir

stat'(t)

t=0,1..

straigthforward).

(4.3.3)

looking at

and adap-

one has

(4.3.2.4)

..,s,

= (X(t),E'(t),(~(Df(t)),E'(t)),p(t))

(D(t),E'(t)).

t=0,1,... M o re o ve r,

is of the type I - 1 - 0 ,

I-2,

by

II-2,

or II-I-I-0.

(4.3.4)

Definition.

G'

above

is called

the

I-equivalent

realization

of G with

res-

pect to p.

(4.3.5) tion

Theorem.

of the

missible

or there

to

(G)). Then

Two

of the type

cases: II (i.e.

In p=(x,y,z)

Assume

the

weli

stat(s)

prepared. a bridge

then

one

choice

Y(O)

is

one

Then

for

situation,

type

is

or o£ the type the bridge

not

is perstandard

I-I-0 or I-2, type.

nor of the type

I-I-0

I-J-O or I-2) or stat(s-1)

can

assume

s=1.

If w (I)

is

quadratic,

one can apply the computations is monoidai

p such

I(Y(O))

=

one

Y(t)

is

or II-1-2-0).

If ~(1)

E(O),

~ (s)

stuation,

I (i.e.

type.

to

and

A in order to obtain

the

let G be a realiza-

for t=1,...,s-1,

t=O,...,s-J

is not a victory

is of

case

I-I-0 or I-2 and

is standard

is a victory

II-2 or I-1-1-0

first

transversal

directrix

stat(s-i)

type

w(t)

for the player

that

der to obtain can

the

star(s)

is a strategy

I-2.

be of the

game such that

tangent

Proof. nor

(X,E,O,P)

reduction

and

(s < length

Let

can

(x,z) assume

and

of

(2.2),

with center Y(O) the

I(Y(O))

result =

and

us fix

(2.4)

in or-

contained

follows

(y,z)

let

from

the

in E(O) (2.5).

If

computations

167

of

(2.6)

may be u s e d . In

us

prove

D(s-1)

the

that

is

second

there

is

generated

(4.3.5.1)

case,

one

a strongly

can

assume

well

that

stat(1)

prepared

r.s.

is

of

o£ p.

the

type

p(s-1)

II.

Let

such t h a t

by

D(s-i)

= a(s-i)x(s-1)8/~x(s-1)

+ b(s-1)y(s-1)~/Sy(s-1)

+

+ c(s-1)~/~z(s-1)

in

such ~ way t h a t

there

(4.3.5.2)

Let

us p r o c e e d

p(O)

well

tion

stat(t)

(if

forward.

= m.X~(s-1) r

(mod ( [ ( s - 1 ) , Z ( s - 1 ) ) )

is

the

(4.1.2).

Let

us f i x

realization

of

Y(t).

GIt

there

that

strategy

of

l-equivalent

(4.2.3).

a) G ' I t

is in

composed o f the

inductively

~ (t+l)

in

Let

and t h e

the

obtain

strategy.

G'I

t

us d i s t i n g u i s h

is

it

section

is

3,

type

in

that

until

the

given

by

straigh-

since

or to

win.

order

to

GIt

define

transi-

touching

follows

defined

standard

is the

prove

a partial following

by the

standard

t' < t.

Let

G'I

t

two p o s s i b i l i t i e s :

or natural

transitions.

usual

way f r o m

&(O(t),E(t),p(t)),

from

p(t')

(T-I,~),

by

the

Assume

is

then

a strategy

Y(t)

For the

result

One has t o

then

one can choose

without

a bridge

(x,y,z).

(4.2.3),

standard

of

us d e f i n e

p =

p(t)

monoidal

the computations

Assume now t h a t let

--> s t & t ( 1 ) by T - 2 o r T - 4 .

no ppobZem)

base

as i n

realization,

given

one needs t o

follows

standard

is

if

(4.1.2)).

I'-prepared

stat(O)

one can m o d i f y

is

from

theorem

is

~ (1)

t+l~s-1,

follows

an

transition

a way t h a t

game w h i c h

If

ced

such

property

the

the

1,

quadratic

only

of

the

a way t h a t t~

result

(Proof

for

such

in is

(4.3.6)

be t h e

Inr(b(s-1))

it

Now~ t h e

(4.3.5.2)

winning

(mod ( ~ ( s - 1 ) , Z ( s - 1 ) ) )

-~stat(t+l)

T-3

center

= 1.X~(s-1) r

in

(4.3.5.2)

m> O , X ~ 0 w i t h

Inr(a(s-1))

prepared

T-4

1~0,

by i n d u c t i o n :

conditions or

is

T-2,

where T-3

or

T-4

Then Y ( t )

is

cho~

p'(t)

is

obtained

and

strong

(very)

good p r e p a r a t i o n . b) No a ) . The a b o v e

Then one a p p l i e s strategy

is

the

enough

strategy for

of the

proving

the

theorem

(4.3.5).

theorem

(4.1.2).

If

one has

168

b,

then

dard,

the

the

result

result

follows follows

from from

the

the

theorem

usual

(4.3.5),

control

of

the

if

one

has

a)

or

GI t

is

stan-

polygon.

(4.4) A winning s t r a t e g y f o r the bridge type

(4.4.1)

Definition.

Let

( X , E , ~ , P ) be o f the type I - 1 - 0 or I - 2

and l e t

p = (x,y,z)

bea v e r y w e l l prepared base. Let G be a r e a l i z a t i o n o f the reduction game beginning at

( X , E , ~ , P ) . G f o l l o w s the " i - r e t a r d e d general winning s t r a t e g y " w i t h respect t o

p

iff a) ~ ( t )

i s standard or n a t u r a l t < length (g)

b) Assume t h a t then Glt I

w(tl),~(t2),...

follows

the

,

t I < t 2< . . .

l-retarded

are the n a t u r a l t r a n s i t i o n s ,

standard winning s t r a t e g y w i t h

res-

pect t o p. c) Let p ( t 1) be obtained as in the section 1, v e r y w e l l prepared, and l e t G1 be obtained in a n a t u r a l way from G as a r e a l i z a t i o n beginning a t s t a r (tl).

Then a),

b) and c) are t r u e i f

one begins w i t h G1, P ( t l ) .

(Remark t h a t one has has a r e c u r s i v e d e f i n i t i o n ) .

( 4 . 4 . 2 ) Remark. The r e s u l t s o f the chapter I l l show t h a t i f

(4.4.3) us f i x

and o f the s e c t i o n 1 o f t h i s chapter

G f o l l o w s the above s t r a t e g y , then G i s f i n i t e .

Theorem. Let

(X,E,D,P) and p = ( x , y , z )

be as in the above d e f i n i t i o n .

Let

1. Let (~,B) be the main v e r t e x o f the polygon 6(O,E,p) and assume t h a t

(4.4.3.1)

fl < 1 + 1 / r

Then,if G is any realization

,

~ < 1.

of the reduction game beginning

at (X,E,D,P)

such that

Gls follows the i-retarded

generai winning strategy and stat(s+1)

is not

situation,

a winning

game beginning

stat(s+l).

then

there

is

strategy for

the

reduction

a

victory at

16g

Proof.

In

nor

natural.

standard the

result

the

natural

(T-I,~),

from

one

8(s-1)

(~ = 0 o r

not)

is

quadratic and

the

or

main

and,

is

stat(t)

or

is

assume t h a t

centered

at

that

~(s+l)

is

of

8 <

(x(s),z(s))

is

the

A(s+l)

is

type

~(s)

must

permissible,

= (1-1/r,1+1/r)

nor

then

centered

I for

Moreover,

Now,

is

(y(s),z(s)),

1, a < 1 ( r e m a r k

preserved).

1 + 1/r+1.

(~(s-1),B(s-1))

can

Assume

situation"

abscissa

since

one it

(2.1.3).

"stable

= 1 + 1/n

(4.4.3.2)

induction,

one can deduce t h a t

the

thus:

by

(3.1.3)

obtains

transition,

>1,

(4.4.2),

~(s+l)

By ( 4 . 4 . 3 . 1 )

otherwise

8(s-1)

of If

follows

(x(s),z(s)). since

view

0
~(s-1)

at < s,

after

< 1 and

be g i v e n

by

one has t h a t

or

(1-1/(r+1),1+1/(r+1)).

This

implies

that

(4.4.3.3)

(n,r+l,0) or

~ Exp ( D ( s ) , E ( s ) , p ( s ) )

(r+l,r+2,-1)

E Exp ( D ( s ) , E ( s ) , p ( s ) ) .

Assume t h a t (4.4.3.4)

D(s+I)

= a(s+l)x(s+l))

+ b(s+l)8/~y(s+l)

Then f r o m

(4.4.3.3)

on d e d u c e s

(4.4.3.5)

~ x(s+l)

+

+ c(s+l)8/)z(s+l)

that

(O,r+l,0)

or

(0,r+2,-1)

Exp ( a ( s + l ) y ( s + l ) )

v Exp ( b ( s + l ) )

V

Exp ( c ( s + l ) y ( s + l ) / z ( s + l ) ) .

Now,

by

(2.5.4),

if

the

dratic)

transformatdon

(4.4.4)

Theorem.

(X,E,#,P)

of

parameters

system

a partial

realization

llows

standard

and such t h a t

P(s+I)

A has n o t won i n

must be T - 2 ,

Let

-prepared

the

player

of

winning

the

be o f

the

verifying reduction

strategy

# strict

and t h e n

with

transform

stat(s+1),

the

player

bridge

type,

the

respect of

A wins

and

assumptions

game b e g i n n i n g to

then

at

let of

the

following

by ( 4 . 4 . 3 . 4 ) .

p=(x,y,z) (2.5.2).

(X,E,~,P)

p until

the

z=O and s t a t ( s + l )

is

(qua-

step

be an l ' -

Let

Gls+l

such t h a t

it

be fo-

s (see(4.2.6))

not a victory

situa

170

tion.

Let

p'

(x',y',z')

=

the

G'Is+i

be t h e be

(s+1)-retarded

Proof. parameters p'(O) ration

obtained general

For

each

obtained

= p'). of

by

p(t).

I-equivalent

as

in

one

can

If

obtained

the

may be d e d u c e d , is

the

tegy

main

of

since

in

vertex

this

as i n

main

vely

then

the (1,1)

that

or

have

e(E(t))

and

by

too)

then

of

follows

p'(t))

(4.4.1)) p'(t)

be a r e g u l a r

from

is

p(O)

a strong

= p

system of (resp.

(very)

from

good p r e p a -

(4.3.2). of

and

= & '(t)

(~(t),6(t)) the

is

&(t)

is

well

if

T-4 is

prepared,

by b o t h

(s+1)-retarded

then

strategies

made b y t h e

all

have

l=l,...,t

is

standard

general

(x(t),z(t))

~ (t+l)

is

as

given

of

< 1,

the

thesis

((~(t),8(t)) winning

winning

stra-

stmategy,

above;

p(t))

p(1-1)).

t=s,

(recall in

both

by T - 3 and t h e

Since

that

of

<1+1/r,

or

ordinate

type then

if

~(t+l) the

order

If

respecti-

one d e d u c e s

~(l)

for is

center is

(~(t),6(t))=

is

(~(t),B(t))

zero

p'(t)

then

then

Moreover

strategies. adapted

> 1,

(1,1),

clearly that

(m(O),B(O))=(O,l+1/r)

one d e d u c e s

implies

and t < s,

if

2 o

respectively

(0,1),

this

= (1,1)

(x'(t),z'(t))

preparation

0 < 1
from

because

(~(t),B(t))

=

A(l),

ordinate

+IR

prepared.

possible

(always

obtained

not well

(~(t),B(t))=

a contradiction If

= (~(t),B(t))

polygons

Since

> 1-1/r,

= 2).

v e r y good

t < s,

proof

may be made by t h e

(1,0).

a contradiction given

G'Is+1

= & (D(t),E'(t);p'(t))

it

movement o f

abscissa

would

Then

E2 and l e t

case

~) o r T - 3 f o r

one

in

that

A'(t)

the movement o f t h e polygon t h a t i f (T-l,

prove

to

p'.

(resp.

as

(m(t),8(t))

& (t)

of

p(t)

of

Assume t h a t

=(0,1),

from

preparation.

T - 3 must be made s i m u l t a n e o u s l y

(4.4.4.2)

following

good

GIs+ 1 relatively

= &(D(t),E(t);p(t))

the

vertex

since

(4.2.3)

of

us d e n o t e

& '(t)

is

let

(resp.

&(t)

E'(t)

very

strategy

(4.2.3)

(4.4.4.1)

where

p by

0 < t < s,

induction, Let

from

winning

t,

realization

must

a strong (~(t),B

decreases,

by

= (0,1),

star(t) given

given

by

(remark

by T - 3 and be t h e

curve

normalization

(t))

=

if

t=s,

(1,0)

and

one can

171

reason

as a b o v e .

(4.4.5) the

Corollary.

bridge

rem,

P(s+1)

One ~

can

strict

one can apply

(4.4.6)

The a b o v e

for

reduction

the

is

a winning

strategy

for

the

reduction

game b e g i n n i n g

at

type.

Proof. point

There

follow

the

transform

of

standard

winning

E 2 is obtained.

strategy

Now,

of

(4.2.3)

until

in view of the above

theo-

(4.4.3).

corollary

ends

game b e g i n n i n g

the at

the

proof type

of

the

one.

existence

of

a winning

a

strategy

-

V

-

TYPES TWO AND THREE

O. I N T R O D U C T I O N

In when i t

view

begins

(I.(4.2.9))

than

r,

or

o r o£ t h e

I¢

Can

obtain

tains

the

1.

sequel

with

the

chapters,

two or t h r e e .

In t h i s

by s t u d y i n g

"victory

dimension

it

the

to

study

the

chapter

the

proof

of

reduction t h e main

game

result

these types.

situation"

of

remains

means

directrix

situation

equal

to

of

zero,

adapted or of

the

order type

less zero,

t y p e one.

one

the

above

be c o m p l e t e d

The IV.

the

by a t y p e

will In

o£

structure

begins a

type

with

of

this

type

II

"bridge",

chapter or

which

Ill

is

quite

and t h e

will

similar

transition

be a s p e c i a l

to is

type

the not

III',

chapters standard, for

which

III

and

then

one

one o b -

victory.

STANDARD TRANSITIONS FROM THE TYPES II AND ZZZ

(1.1)

(1.1.1)

A winning

Theorem.

strategy

Let

if

dim D i r

(X,E,D,P)

(~,E)

be o f

the

= 1

type

III-1-1

and

let

(X',E',D',P')

be a

173

quadratic Then in

directional

(X',E',O',P')

order

to

is

obtain

Proof.

blowing-up the

then

= a/x 'r,

one

one

has t y p e

has

jr(c)

type

=

(x).

One has T - l , 0

and D'

b'

III-1-1

must

choose

then

a

=

can

T-l, ~ or

victory Then

T-2

from

situation.

the

strict

If

and l e t

T-I,~, 0"

A

D be as i n

+ c'B/Bz'

Assume

a change

A

chooses

T-2 one

one

player

by

V(b')

that

in

I n r ( a ') = ¢ ( y ' , z ' ) that

I£

the

jr(0,,E,)

y,z,

can

obtains

assume

is g e n e r a t e d

can

.) *

Put

assume p'

that

= 1

center,

# 0, e(E")

= 0 without

r+l,

~ 0. S i n c e

"

quadratic

JP(0",E")

that~

one

+ x'(.

the

V ( c ' ) _>

and

loss

then

B

= 2 and of g e n e -

by

a" I = c , /x I',r , b,,1 = b'/x I ,,r_yl.a1,, ' c" 1 = a , /x . I ,,r-1-a I ,,. I£ ~(b I'') = r, one as above.

Assume

that

¢(y,x)

=

Theorem.

assume

~(bl")

> r+l,

then

inr(al,, ) = Zl,,r + Xl-(... ) In

and

P'I"

transform

(1.1.1.3)

(1.1.2)

Assume

for

situation.

D" = a " l X " l ~ / ~ x " 1 + b " 1 8 / S y " 1 + C"lZ"18 /S z" 1

reason

where

then

victory

a strategy

= c/x' r+1 - z'a' . If

, c'

a

-

(z',y',x').

(1.1.1.2)

where

generated

+ b'B/By'

4-I),

not

-

(x'1,y'1,z' I)

rality.

is

up to

= @(y,z),

is

(III.(1.2.15))

4-0

or

is

(X',E',D',P').

situation.

(hence

it

there

or a v i c t o r y

--

=

a'x'@/Sx'

b/x'r+1-y'a'

=

Inr(a)

:

that

or

from

be as i n

four Let

III-l-1

situation

D'

a'

assume

p = (x,y,z)

(1.1.1.1)

where

type

a victory

Let

(ITI.(1.2.15.1)).

of

and

that

a)

b)

If

y

r

+ X

Let

r

. "El r-1 ,,r ,. (c 1 ) = Xx I " -z I +x 1' [ . . . )

xyrml.

(X',E',~',P')

(X,E,~,P)

is

or t h e r e

victory

from

(X,E,O,P)

o£

one deduces t h a t

be a q u a d r a t i c

it is not a v i c t o r y

III-I-1

If

From ( 1 . 1 . 1 . 3 )

the

situation.

type

is a s t r a t e g y

X~ 0

directional

dim D i r

(~",E")

= O.

b l o w i n g up o f (X,E,D,P)

Then:

II-1-I, for the

then

(X',E',D',P')

piayer

A in o r d e r

is of the t y p e to o b t a i n

the

(X',E',D',P'). is of

the

type

II-I-2,

then

(X',E',~',P')

is of the t y p e

174

11-1-2 c)

If

or

(X,E,~,P)

11-I-2

Proof.

a)

and 0 '

Assume t h a t

is

generated

= a/x 'r,

b'

In(a)

Assume f i r s t

¢-4

(1.1.2.2)

z' 1 = z'

+

X y',

(1.1.1),

by T - 2 one o b t a i n s a victory

~ ~ one

+ b'a/ay'

c'

Jr(D',E')

~ 0.

(Q-~)(y',z')

a change

y,r since

can

see

victory

then

situation

since

Corollary.

(X,E,O,P)

of the

Proof. -up wins.

one

(III.(2.4.2))

c)

(1.1.3)

p

centered Assume

=

Assume

(x,y,z)

has

type

or

a

e(E')

type

II

victory

is or

that

a

a winning III

permissible there

' . If

v(b')

and

(remark that

one has t y p e

v(e')>

r+l,

Inr(a ')

=

four,

then

x

zero.

Then,

up to

that

= (x)._ Now, r e a s o n n i n g

as i n

as

jr(c')

has

type

and by ( T - I , ~ )

in or

one

= 0,

then

one o b t a i n s

~ is

(Ili.(1.2.10)). II-1-2

situation.

(i.e.,

=

not a power o f a l i -

One the

If J r ( D ' , E ' )

V(bl")

has

(T-l,0).

transition ~ O,

If

is s t a n -

then

one

has

a

= 2.

One can see t h a t in

One has

+ x'(...)

now ¢ - 9

11-1-1,

One can r e a s o n as i n

There

Since

situation

be

Let

D be as i n

= @(y+(x,z).

situation

otherwise

assume

a victory

situation.

= c/x'r+l-z'a

(¢-~)(y',z')

b)

dard,

type

Then

(1.1.1)

= O,

of the

+ c'~/~z'

# 0.

n e a r f o r m and one can r e a s o n as i n

Jr(~',E')

is

and l e t

and I n ( b )

or a victory

Inr(b ') =

hence

(X',E',O',P')

(III.(1.2.10))

= ¢{y+(x,z)

'r,

111-1-I

Assume t h a t

necessarily

then

by

¢= ~ and one has t y p e

Jr(D,E).

111-1-2,

be as i n

= (b-a)(y'-~)/x

¢(y',z')+x'(...)).

r,

type

O' = a ' x ' a / a x '

where a'

and

of the

p = (x,y,z)

(1.1.2.1)

then

is

or 11-1-1.

Let

(III.(1.2.10.1)). T-I,~

11-1-1.

with

in

strategy

dim D i r

tangent

a curve

for

(~,E)

each o f t h e

curve

is no such

b).

the :

reduction

the

in each

at

1.

above c a s e s , to

game b e g i n n i n g

after

directrix,

step

making a b l o w i n g then

the

of a r e a l i z a t i o n

player

of the

A

ga-

175

me.

If the

realization

stabilished III-l-1

4

is not finite,

in an infinite ~ III-l-1,

then

sequence the

correspond

to the

(1.2)

Invariants

for the standard

Remark.

zed b a s e .

Let

infinitely

(X,E,D,P)

Then D i s

generated

in view of

of transitions

permissible

tions

(1.2.1)

then,

II-1-2 ~

curve

near points

(1.1.1)

must

to

and

(1.1.2)

it is

II-1-2 or

exist,

of some regular

since

curve

this

transi-

(see(l.(3.3)).

transitions

be o f t h e

type

II-2

and l e t

p = (x,y,z)

be a n o r m a l i -

by

(1.2.1.1)

D = axal@x + b@/ay + cB/Bz

and (1.2.1.2)

Exp ( D , p )

compare

with

like

(II.(3.3))

in

(1.2.2) prepared

(II.(2.2.4.2))

base. a)

for

and good

Definition.

Let

= Exp ( a )

the

type

preparation

(X,E,D,P)

The base p i s

u Exp ( b / y )

zero

and

e(E)

may be d e f i n e d

be o£ the type

"strongly

For each v e r t e x (~,B)

u Exp ( c / z )

well

as i n

III-2

prepared"

E A+(D,E,p)

= 1.

One can deduce (II.(3.3.7)).

and l e t

p = ( x , y , z ) be a well

iff

such t h a t

is no change

b)

For

each

that

(1.2.3)

Lemma.

a)

well

Zl=Z+~x~yB

vertex

(n,O)

~

A+(D,E,p)

is not modified

and

If

well

then

p

is

prepared

prepared,

and B > 1

0

which may dissolve

A(~,E,p)

2

(~,B) E Z

'

re

results

there

this

is

no change

(n,O) disappears

one can

vertex

obtain

--

the '

Z l = Z + k X n such

in A (#,E,p). +

a base

by a sequence o f changes Z l = Z + k x a y f l , such t h a t

p'

strongly A(~,E,p)=

= A(D,E,p'). b)

If

Proof.

p is

strongly

a) It follows b) If (0,1)

well

prepared,

from the

results

E A+(D,E,p),

= z + Xy d i s s o l v e s

this

then J(~,E)

in

= (z+Xx).

(II.(3.3)).

since dim Dir(~,E) vertex.

--

of A + ~ , E , p )

= 2, then a change

zI

176

(1.2.4)

Remark.

"strongly

good

strongly

very

Very

good

preparation

preparation" well

instead

prepared

base

is

of

defined

"good

from

as

in

(II.(3.4.5))

preparation".

a strongly

well

Also

by

one

prepared

putting

can

base

obtain

in

the

a

usual

way (see II.(3.4.6)).

(1.2.5) a

Theorem.

strongly

(very

directional ning

Let if

(X,E,O,P)

be o f

type

well

blowing-up

strategy

(see

III)

such

that

the

prepared

the

III.(2.8.4)).

type

or III-2

base.

center

Then

II-2

Assume t h a t

follows

one o£ t h e

and l e t

the

p = (x,y,z)

(X',E',D',P')

O-retarded

following

is

standard

possibilities

be

is

a

winsatis-

fied: a) The t r a n s i t i o n

is

not

standard.

b) dim Dip (D',E') = 1. c)

The transition

is standard

and there

is a s t r o n g l y

(very if type

III)

well prepared base p' = (x',y',z') such that the invariant (B',e(E'),~',a') B(A(D,E,P)),

Proof. is

of

the

(T-I,~),

~#

11ows f r o m (see is

11-2.

O,

T-2,

11.(4.4.3))

T-3

that

from

served after

and b)

or

T-4.

and if

,

is

v e r y good

then

w h e r e X~ O. S i n c e

prepared

(i.e.

p is

tex,

in this

but

follows

if

one

modify

last has

p is the

prepared case, T-2,

T-4

not

main until

(T-3

o£

vertex

is

has

is

the

of

first

Let

of

prepared,

(T-l,0),

result

fo-

one deduces , where it

P'I

is

pre-

that

z :

vgr.). Assume

nor the

vertex)

by

then

first

us assume t h a t

A(~,E,p)

(X,E,D,P)

the

~ ~ O,

one can see t h a t

possible).

given

A(O',E',P'I)

111-2.

unless

have a s t a n d a r d

not

is

(T-I,~),

welt

type

that

5~ O, t h e n

vertex

and i t

prepared,

one c a n n o t

one

main

(II.(4.3.3)).

the

(T-I,~),

(see 1 1 . ( 3 . 4 . 3 ) )

is

well

Assume f i r s t

transformation

If

the

preparation

:

not

has

8 =

etc.

the

(~+B-l,B)

Then one can r e a s o n as i n

= z + X x does

and

one

(X,E,D,P)

(B,e(E),c,~) where

not s a t i s f i e d .

(Zli.(2.2.5)).

B'~ B

now t h a t

If

= J(~,E). (~+X~),

are

z = J(D,E)

p by T - I , ~ ,

strong

Assume

a)

Then

(III.(2.6.2))

obtained

B' = B ( A ( D ' , E ' , p ' ) )

Assume t h a t

type

is strictly smaller than

=

a change z I fact

(1,0)

transition:

Assume

J(D,E)

that is

the

then

now t h a t

it

is well

main v e r -

the

one

=

result

has

177

(T-l,{),

and t h a t

(1,O)

is

Yl = y+ ~x, z I = z+ Xx t h i s

(1.2.6) Corollary.

not the main v e r t e x o f A ( ~ , E , p ) ,

then

e~l,

and a f t e r

p r o p e r t y holds and one obtains 8 '< 8 .

Let G be a realization of the reduction game beginning

at

(X,E,D,P) of the type II or III and let us assume that all the transitions in G are standard

( unless the

last one) and that fop each s~Gls follows the 1-retarded stan-

dard winning strategy

Proof.

(definition as in (III.(2.8.4))). Then G is finite.

One can r e a s o n as i n

(111.(2.8.5)).

2. NO STANDARD T R A N S I T I O N S FROM IZ AND IZI

(2.1) A n o t h e r

(2.1.1)

of t r a n s v e r s a l i t y

invariant

An i n v a r i a n t o f t r a n s v e r s a l i t y f o r

the types I I '

and I I I '

will

be i n t r o d u -

ced in order t o be able f o r d e f i n i n g the bridge type mentioned in the i n t r o d u c t i o n . This i n v a r i a n t is also u s e f u l f o r the study o f the type I I '

(2.1.2)

Definition.

Let

(X,E,D,P)

be o f the type I I '

T=T(X,E,~,P) i s defined t o be zero i f f mal crossings d i v i s o r ,

and I I l '

(resp.

III').

The i n v a r i a n t

there i s a decomposition E = EI U E2, Ei

i = 1,2, e(E2) = 1 such t h a t there is

nor

@ ~ R, w i t h I ( E 1) c@ .R

in such a way t h a t

(2.1.2.1)

jr(D(~)/¢)

¢ J(E 1)

Otherwise ~ = 1.

(2.1.3) o f p.

Remark. suited

(2.1.3.1)

for

(E,P)

that

(X,E,D,P)

such t h a t

D is

is

of

the

generated

type

If'

and p = ( x , y , z )

is

a r.s.

If

• = 1,

by

D = a x B / B x + by@/@y + ezB/Bz

Then ~= 0 i f £ then j r ( a )

Assume

p may be chosen in

# 0 (resp. j r ( b ) , j r ( c ) )

such a way t h a t implies j r ( a )

jr(a)

= (~),

+ jr(b)

~ (~,Z).

(resp. j r ( b )

= (Z), j r ( c )

=

178

= (z)). Assume I(E)

= (xz),

that

(X,E,D,P)

and D i s

generated

(2.1.3.2)

Then

0 (resp.

be

p =

Let

(x,y,z)

T= 0 one

(hence j r ( c ~

Lemma.

type

or

III

With (if

Proof.

(2.1.6)

~ = 0 and E2 i s

b) c)

dratic ponds

Let

there the

of the

that

jr(a)

resp.

jr(c)

the

type

iff

of

as is

p is

(2.1.3)

in

p =

(x,y,z)

is

such

that

~

(x).

If ~ = 1 , t h e n

III',

a regular

jr(a)

= (z).

II'

or

suited

and

(2.1.2),

respectively

(X,E,D,P)

is

if

for

(E,P),

• = 1 one

I(E)

system o f ~ (x,z)

has t h a t

and

jr(c)

~ 0

if

T = O,

II'

or

then

III')

(X,E1,D,P)

is

the

of

and n = v ( ~ , E 1 , P ) .

blowing-up

of

a standard

of

the

type

system

of

(2.1.2),

III'.

It

parameters

then

of the

is

type

p = (x,y,z)

lll-bridge such t h a t

if

one has t h a t

(D, E1) = 2. ~ Exp

(D,EI,P) , j < r ~

h~l.

£ Exp ( D , E I , P ) .

no s t a n d a r d

that

be o f

a normalized

decomposition

(1,r,0)

Assume

to

and

See ( 2 . 1 . 3 )

(h,i,j)

Quadratic

(2.2.1)

be o f

property

(X,E,~,P)

a) dim DiP

(2.2)

a way

"normalized"

notations

Definition.

E = E1 U

III'

+ b~/@y + c z B l ~ z

= (x),

(X,E,D,P)

the

type

= (z)).

(2.1.5) II

has

in s u c h

jr(a)

is

the

by

chosen

implies

Definition.

if

iff

p may

jr(c)),

parameters and

of

D = axSl~x

• = 0 iff

(2.1.4)

is

transitions

(X,E,#,P)

is

of

(X,E,D,P)

such

transition.

Let

lemma ( I I I . ( 1 . 2 . 1 5 ) ) .

Then t h e

from III

the that

type it

III-2 is

p = (x,y,z)

and t h a t

not

a victory

be a base

transformation

(X',E',O',P')

is

situation

is

a qua-

nor c o r r e s -

verifying t h e p r o p e r t i e s

given

by

(T-I,()

o r by T - 2 .

179

If

it

i s given by T-2,

tion

or

a standard

one has e ( E ' ) = 2 and i f

transition

and

if

Jr(~',E')

Jr(D',E')

Then one can assume t h a t

the t r a n s f o r m a t i o n i s

of generality,

Then,

D'

by T - I , 0 .

if

is

~ 0 one has a v i c t o r y given by

This

•

1)

a'

implies

a/x'r;

=

b'

a'

b' c'

(2.2.2)

Assume now t h a t

JH(D',E')

=

(x').

be given

by T-2

+ c"~/@z",

where

(2.2.2.1)

= r,

this

tter). lar,

This (see

a"

v(c")

=

z 'r

= ¢(y',z') = ~(y',z')

;

' =

C

where

c/x

'r+l

- z'a'

x'(...)

+

+ x'(...)-y'a' + x'(...)

- z'a'

implies that (2.2.1.2)).

one o b t a i n s

the

T h e n Z]"

b"

a victory

Z + Xx + ~

following is

generated by D " = a " x " B / B x " + b " y ' ~ /~ y"+

= b'/y"r;

c"

situation.

6 Jr(D',E')

= c'/y"r-z"b

Then v ( c " )

= jr(b')

(2.2.3.1)

implies that ¢ and ~ are h o m o g e n e o u s o f

Put

Pl

= (xl'Yl'Zl)

= (y",z",x").

aI

then

( q u a d r a t i c ) t r a n s f o r m a t i o n , must

''.

> r+l,

From

= b" = x z l r + X l ( ¢ ( 1 , Y l ) - Y l

degree

= (~).

In p a r t i c u -

r+l.

(2.2.2.1) r

hence v ( c ' ) > r + l .

and one has t y p e zero ( o r be-

T h e n one can assume J H ( D ' , E ' ) ~ 0. This i m p l i e s J r ( D ' , E ' )

(2.2.3)

without loss

J H ( Z ) ' , E ' ) # O. Since one has not a v i c t o r y s i t u a t i o n ,

= a'/y"r-l-b";

implies that

this

and,

that

(2.2.1.2)

But

b/x'r+l-y ' a

=

(T-I,~)

situation.

generated by D=ax~/@x + bB/8y + c~/@z, then

i s generated by D'= a ' x ' B / @ x ' + b'@/@y' + c ' ~ / ~ z ' ,

(2.2.1

If

~

= 0 one has v i c t o r y s i t u a

and

(2.2.1.2)

one has t h a t

)+XlZl(...)

b I = c" = -Xylz l r + x l y l ( ( ~ ( l , y l ) / y l ) - 2 y l r - @ ( l , y l ) ) + x l z l ( . . . ) cI

where

X ~ O. Assume t h a t

= a"

= -xzlr-Xl{(1,Yl)+XlZl(...)

@(1,y)

o t h e r w i s e the dimension o f neously zero, from ( 2 . 2 . 3 . 1 )

the

= py

r-1

directrix

r r+l +yy + y . One can assume t h a t p =0, since is

zero.

Since

one can deduce t h a t and i f

y and y - 1 are not s i m u l t a -

j r ( a 1) = ( Z l ) , then one has

180

a type I l l - b r i d g e .

( 2 . 2 . 4 ) P r o p o s i t i o n . With n o t a t i o n s as above, there i s a b i j e c t i o n Q : Exp (D,E,p)

~

Exp ( D " , E " I , P l ) given by

(2.2.4.1)

where

$(h,i,j)

= (h+2(i+j-r)+l,j,h+i+j-r)

I ( E " 1) = ( X l ) .

Proof.

It

follows

from

(2.2.1.1)

(2.3) No standard t r a n s i t i o n s from I I l

( 2 . 3 . 1 ) Assume t h a t duces

a

that

directional

(X,E,~,P)

blowing-up

the t r a n s i t i o n i s

Then J r ( O ' , E ' )

is

and

(2.2.2.1).

and I I

o f the type 111-2 or 11-2 and t h a t with

permissible center

not standard and ( X ' , E ' , ~ ' , P ' )

~ 0 and e ( E ' )

= I.

In

is

tangent

#: X' ~

to

the

X in-

directrix

not a v i c t o r y s i t u a t i o n .

view o f the above paragraph, one has one o f

the f o l l o w i n g p o s s i b i l i t i e s .

a)

# quadratic and (X,E,D,P) i s o f the type I I - 2 .

b) ~ monoidal.

( 2 . 3 . 2 ) Assume t h a t t e r s as in Dis

= in

# is

(III.(I.2.8)).

q u a d r a t i c . Let p = ( x , y , z ) be a r e g u l a r system o f parameSince e ( E ' ) , # must be given by ( T - I , ~ ) , ~ 0 .

Assume t h a t

generated by D = axB/Bx + by@/@y + c B/@z. There are two p o s s i b i l i t e s : J r ( a ) =

(z)

or

the

jr(a)

= O.

precedent

If

jr(a)

paragraph

= ~,

then

after

and one o b t a i n s

making

a bridge

Yl = y + ~ x ,

type

in

the

one can r e a s o n as

following

quadratic

blowing-up.

(2.3.2)

Assume t h a t

= ( x , y , z ) as in with or

by

# i s monoidal. Without

(III.(1.2.8))

or as

the a d d i t i o n n a l p r o p e r t y t h a t (y,z).

Then the

in

loss o f g e n e r a l i t y , one can choose p =

(III.(1.2.15)),

depending on type I I

the center o f the blowing-up i s

t r a n s f o r m a t i o n must be given

by T-3 or

or I I I ,

given by ( x , z )

T-4 from p,

since

181

jr(O,E) tory

=

(z).

Moreover,

situation),

Assume

that

O'

T - 3 ,

necessarily

D is

is

generated

Jr(#',E')

= ( qx ' )

as in

v(¢) ~r.

dratic

= (y",z",x")

order

in

= ( _x ' )

and

to

III-2

v(c')

and ~ i s

given

where jr(a)

by T - 3 .

= (z),

a£ter

where

'

that

JH(D',E')

= 0.

> r+l

(see

(2.2.2)).

Remark t h a t

_

Then

+ z' r .y'(...)

+ x'(...)

= kx'r+x'¢(x',y',z')+z'r(...)

(2.2.2),

one has a v i c -

prove

b'

(2.2.3).

(2.3.2.3)

type

(otherwise

= c/x'r+l-z'a

= z 'r

Let p" = ( x " , y " , z " )

as i n

c'

a'

Reasonning as i n

center).

o5 t h e

b'=b/x 'r,

(2.2.2)

and j r ( b ' )

(2.3.2.2)

is

= I

= a'x'B/@x'+b'B/~y'+c'B/Bz'

= a/x'r;

a'

have e ( E ' )

by D = a x B / B x + b ~ / B y + c@/Bz,

by D'

Now, one can r e a s o n

one must t o

(X,E,D,P)

generated

(2.3.2.1)

where

since

X~0

one has t o make T-2 ( i f

A chooses the qua-

be the o b t a i n e d base and put Pl = ( x l ' Y 1 ' Z l )

=

Then D" i s generated by

D" = a l X l ; ) / @ x I + b l ; ) / B y I + C l Z l B / B Z l

where

aI :

b" = b'/y ''r

= xzlr+XlZl(...)+x12ylr(..,)

b I = c" = c ' / y " r - z " b '' = - X Y l Z l r

c I = a" = a ' / y " r - l - b "

Then, Dir

one

has a t y p e

(D",E")

tion

(aLways T - 2 )

then

(2.3.3) is

Theorem.

a strategy

(2.4)

A winning

(2.4.1)

(if

I?

for

the

the

strategy

Theorem.

theorem

Assume

?or

that

with

resumes

A in

the

the

r+l

Dir

(D",E")

• = O, and a £ t e r

the

?oilowing

situation

type

(the

the computations

game b e g i n s order

and

)

dim

a victory

reduction

player

v ( b 1) ~

IIl'

one o b t a i n s

The f o L L o w i n g

)

= -Xz I +xlY 1 + X l Z l ( . . . ) + x 1 2 y l r ( . . .

Ill-bridge

= 1 one has a t y p e

+ Xl(...

r

to win

at or to

the

order

= 2).

obtain

II

trans£orma-

drops).

made up t i l l

type

I ? dim

now.

or

a type

Ill,

then,

there

Ill-bridge.

III-bridge

reduction

game b e g i n s

at (X,E,~,P)

which

is

of

the

182

type

III-2

ritying

and

such

there

is

a normalized

system o f

parameters

p = (x,y,z)

ve-

that a)

V (h,i,j) then

b) Then t h e r e

is

modify

p

(which

does

such

strategy

that

a way t h a t

transformation not

and

i

~

r+l,

~ Exp ( D , E , p ) .

Assume f i r s t

Then t h e

j=-1

h > 2.

a winning

is

j < r ~ h ~ 1. M o r e o v e r , i f

6 Exp ( D , E , p ) ,

(1,r,O)

Proof.

b).

that

modify

a)

for

the

the

player

J(D,E)

=

reduction

A chooses the

(z),

may be g i v e n and

b))

one

game b e g i n n i n g

without

by T - 2

can

quadratic

touching

on T - I , ~

assume t h a t

at

the

.

the

(X,E~,P).

center.

One can

conditions

By making

Yl

a)

and

= y + ~ x

transformation

is

given

by T - 2 o r T - l , 0 . If is

not

one has T - 2 ,

standard

and s i n c e

Assume tion

is

not

n o t won,

that

by

e(E')

= 2,

one

(2.2.4),

(2.4.1.1)

and i f

(I/r,1)

the

standard,

then

since

deduces

the

that

dimension

of

player

after the

strict transform

the

then

the is

can a p p l y

results

for

given

the

transition

A wins. by T - l , 0 .

of

(2.2)

First, and

if

if the

the

transi-

player

A has

(2.4.1.1)

one

same base Pl one has t h a t

= (1,0,1)

A has won.

If

following

directrix

the

one deduces t h a t

player

transformation

(1,r,O)

r >3,

6 A(D,E,p),

~ Exp ( D " , E " l , P l )

r = 2,

looking

quadratic

becomes

zero.

at

(2.3.1),

transformation

If the transition

from

(necessarily is standard,

T-2),

the

then the

(X',E',~',P') satisfies once more the properties a) and b). One can

repeat. If the pmocessus does not stop, after a change YJ = y ~ ~i x i ' Zl = z+[ ~ x i which quence

of

A(~,E,p).

not modify

transformations

a),

T-I,0.

Then the player A wins

transformation

(2.4.2)

does

b), This

one can assume implies

by choosing

that

that one (1/r,1)

has an infinite se-

is the only vertex of

the center given

by

(y,z),

since the

is not standard and e(E') = 2.

Theorem.

There

is

a

winning

strategy

for

the

reduction

game b e g i n n i n g

at

183

type

a

Ill-bridge.

Proof. =

(x,y,z)

be as

formation, a)

(2.4.3) II

TYPES I I '

as

Assume

(X',E',~',P')

(2.4,1).

is

type

that

is

the

a victory

Now one can r e a s o n

There

lll-bridge

a winning

and l e t

player

A chooses

situation,

in

the

strategy

E = E1 U the

a bridge

same way as i n

for

the

E2 and p

quadratic

type

or it

=

transverifies

(2.4.1).

reduction

game b e g i n n i n g

at

III.

AND I I l '

The case • = 0

(3.1.1)

In this

"victory

(3.1.2) =

or

be o f t h e

(2.1.6).

Corollary.

a type

(3.1)

(X,E,~,P)

in

then

and b) o f

3.

Let

situations"

Theorem.

(x,y,z)

Let

with

as

(X,E,D,P)

possibilities

(X',E',D',P') b)

or If

is is

it the

of

the

type

III-bridge

will

be c o n s i d e r e d

"victory

situations"of

the

type

or

• = 0 and

II'

parameters.

center

tangent

to

IIl'

with

Let

(X',E',~',P')

the

directrix.

introduction.

let

p =

be a d i r e c t i o n a l Then one o f t h e

fo-

satisfied strategy

a victory

is

and t h e

the

be o f

of

the

for

the

reduction

game b e g i n n i n g

at

situation.

transformation

(X',E',~',P')

Ill

as

system

a winning

is

II,

well

a permissible

a) T h e r e

f r o m p,

types

be a n o r m a l i z e d

blowing-up llowing

paragraph,

is

types

quadratic, II'

or

then

III',~=

it

is

given

0 and t h e

by ( T - I , ~ )

obtained

or T-2

base i s

nor

malized.

c) the

If

case I I l ' )

it

%= O, and t h e

Proof. assume t h a t

the

the

transformation

is

given

obtained

Assume

monoidal,

by T - 3 o r T - 4 ,

base i s

that

is

(up

(X',E',D',P')

is

to

a change y l = y + [ ~ i x i

of the

types

II'

or

m

in

III'

normalized.

(X',E',D',P')

transformation

then

is

is

quadratic.

not

a victory

There

are

situation.

First,

two p o s s i b i l i t i @ s :

P'

let

us

E stric

184

transform

of

z = 0 or not.

(X',E',D',P') =(x)

or

result

may be c o n s i d e r e d

I(E1)=(xy) a)

follows

, depending from the

Assume t h a t ven

by

Assume t h a t

(T-I,~)

or

as t h e

strict

on e ( E )

strict

from

p.

If

e(E)

of the

transform D is

transform

transform

= 2 or

computations

P ~

T-2

P' ~ s t r i c t

of

= 3.

of z=0.

Then,

( X , E 1 , D , P ) , where I ( E 1 ) =

Now, i n

view of

(2.1.5),

the

above s e c t i o n s .

of

z = 0.

generated

Then t h e

transformation

by D = a x S / ~ x + b~

is

+ czg/az

gi-

where

Y =

Y

a/~y

on

= a'x'~ /~ x' F r o m the

+ b'~y,

equations

~

+ c'z'8

that

e(E)

/8 z '

= 2 on 3.

necessarily

(resp.

jr(a)+jr(b))

jr(a')

(resp.

jr(a')+Jr(b'))

T = 0,

II

or

better

(X',E',D',P')

JP(D',E')

~ (x)

of

the

of

type

D'

= 0,

generated

otherwise

(x,y)

(resp.

llI') If'

is

by D ' =

one has a ) .

that

(resp.

~(~')

(instead

is

Assume t h a t

one can d e d u c e

jr(a)

one has n o t a t y p e

deduces

on

of the t r a n s f o r m a t i o n ,

(3.1.2.1)

And i f

depending

y~/~y

in

or

=>

(x',y'))

the

III'

case e ( E ' )

and p'

= 2,

one

= (x',y',z')

is

normalized. Now p

is

Yl

let Y be a p e r m i s s i b l e

normalized,

is as in c).

(3.1.3) a type

Yl

ll'

or

usual = y+(x

wise

Ill'

=

.

one

for

can can

There

is

with ~=

The above the

The s t r a t e g y

I(Y)

tangent

(x,y)

and

~

reason

a winning

to the d i r e c t r i x ,

I(Y)

=

(x,z)

or

I(Y)

s i n c e T = 0 and =

(yl,Z

where

as above.

strategy

for

the

reduction

game b e g i n n z n g

at

O.

theorem case is

a l l o w s us t o

Ill' as

one usual:

if it is p e r m i s s i b l e ,

can to

otherwise

use t h e

prepare choose

techniques

the

polygon

(x,z)

if

the q u a d r a t i c

it

is

of the

polygons

in

by means o f

changes

permissible,

other-

center.

The case ~= 1

(2.3.1) p

way: n

(y,z)

(3.2)

Now,

Corollary.

Proof. the

necessarily

curve

Definition.

(x,y,z)

jr(D(z)/z)

be

a

Let

(X,E,D,P)

normalized

~ 0 (hence j r ( D ( z ) / z )

be o f

system

of

= (z)) u

the

type

parameters

II' p

or

III'

is

"strongly

where D g e n e r a t e s D

with

• = 1 and

let

normalized"

iff

185

(3.2.2) =

Theorem.

(x,y,z)

Let

(X,E,D,P)

be a s t r o n g l y

rectional

blowing-up

be o f t h e

normalized

with

type

system o f

a permissible

II'

or

Ill'

with

parameters.

center

tangent

T= ~ and l e t

Let to

p =

(X',E',D',P')

the

directrix.

be a d i Then one

of the following possibilities is satisfied.

a) There is a winning strategy for the reduction game beginning at (X',E',D',P')

or it is a victory situation.

b) (X'~E',D',P')

is of the type II' or III' with T= 1, the transformation

is given by (T-l,(), T-2, T-3 or T-4 (up to a change yl=y+[~i xi in the last

case)

from

p and

the

obtained

base p' =

(x',y',z')

One has t h a t

J(~,E)

9 z and one can deduce t h a t

is strongly

normalized.

Proof. must

be

given

equations, a)).

Moreover, or

that

ly,

then

p'

(3.2.3)

(T-I,~),

since

~(~)

the

by

e(E')

p'

is

types

II'

normalized.

or

Proof.

As i n

assured

or

T-4

if

a)

is

to

have t y p e s

II'

jr(D'(z')/z') ~ (x,y)

not satisfied. or

III'

= 0 this in

the

transformation Looking

withT

implies

cases I I I '

at

the

= 1 (otherwise that

and I I '

Jr(D(x)/x)~ respective-

normalized.

There III'

If

+ Jr(D(y)/y)

strongly

Corollary.

T-3

~ 2 one must

jr(O(x)/x) is

T-2,

the

is

with

a winning

is

until

(3.2.4)

The above c o r o l l a r y

for

the

reduction

game b e g i n n i n g

at

• = I.

(3.1.3)

control

strategy

one can t a k e

one o b t a i n s

ends t h e

the

a victory

proof

usual

strategy

over the

polygon.

situation.

o f t h e main

result

(1.(4.2.9)).

The

-REFERENCES-

Ill

ABHYANKAR, S . S . . " D e s i n g u l a r i z a t i o n Pure M a t h . , A . M . S . v o l . 40. A r c a t a

121

CANO, F . . " T e o r l a de d i s t r i b u c i o n e s sobre variedades Mort. I n s t . J o r g e J u a n , C . S . I . C . , M a d r i d • 1983•

131

• "Desingularization V o l . 296, 1, pp.

141 Proc.

la

t51 trois".

of plane curves". 1981.

of plane 83/93.

vector

Proc.

o f Symp.

algebraicas".

fields".

Transac.

of

in

Mem. y

the A.M.S.

• "Techniques pour l a d 6 s i n g u l a r i s a t i o n des champs de v e c t e u r s " . R ~ b i d a ( 1 9 8 4 ) . H u e l v a . To a p p e a r i n " T r a v a u x en c o u p s " . Hermann. • "Jeux de m 4 s o l u t i o n p o u r Publ. E c o l e P o l y t e c h n i q u e .

les c h a m p s Palaiseau.

de v e c t e u r s 1985.

en d i m e n s i o n

161

CERVEAU, D. - MATTEI, J . F . • A s t e r i s q u e 97. 1982.

I zl

COSSART, V • . "Forme n o r m a l e d ' u n e f o n c t i o n en d i m e n s i o n R ~ b i d a ( 1 9 8 4 ) . m u e l v a . To a p p e a r i n " T r a v a u x en c o u p s " .

181

GIRAUD, J • . "Forme n o r m a l e d ' u n e f o n c t i o n sup une s u r f a c e de c a r a c t @ r i s t i q u e positive"• B u l l . Soc. M a t h . F r a n c e , 111, 1983, p. 1 0 9 - 1 2 4 .

191

" C o n d i t i o n de Jung p o u r l e s r e v ~ t e m e n t s P r o c . A l g e b r a i c G e o m e t r y , T o k y o / K y o t o 1982. L e c t u r e S p r i n g e r V e r l a g . 1983. p• 3 1 3 - 3 3 3 .

Ilol

HIRONAKA, H•. " D e s i n g u l a r i z a t i o n of excellent surfaces"• Adv. S c i • Sem. i n A l g . Geom. b o w d o i n C o l l e g e ( 1 9 6 7 ) . A p p e a r e d i n L e c t u r e N o t e s i n M a t h . n~1101 Springer Verlag (1984)•

Ill] a Field

• "Resolution of c h a r a c t e r i s t i c

"Formes h o l o m o r p h e s

int&grables

singuli&res".

trois"• Proc. Hermann.

radiciels de h a u t e u r un" N o t e s i n M a t h . n ° 1016.

of the s i n g u l a r i t i e s of an a l g e b r a i c v a r i e t y zero". Ann. of M a t h . 79, 1 0 9 / 3 2 6 . 1964.

1121

SANCHEZ-GIRALDA, face algebroide".

T.. "Caract~risation C•R• Ac• S c i • P a r i s

1131

SEIDENBERG, A • • " A d y = B d x " . Am. J .

R e d u c t i o n oF t h e s i n f u l a r i t i e s o f Math• 1968, p. 2 4 8 / 2 6 9 .

la

des v a r i ~ t ~ s p e r m i s e s L• 285. 1977. oF t h e

d'une

differential

over

hypersur-

equation

-INDEX-

Adaptation

(of

an u n i d i m e n s i o n a l

distribution)

5

Adapted

blowing-ups

9

Adapted

order

14

Adapted

strict

Adapted

unidimensional

Adapted

vector

9,

transform

4

field

Associated

formal

distribution

6

Associated

formal

vector

6

A

field

109

(r;p)

Bridge

143

type (p(D,E,Y))

10

Cloud of

points

( Exp(D,E,p))

46,

Cloud of

points

(Exp+(D,E,p)

91

Cloud

points

( Exp(f,p))

46

Blowing-up

of

order

Cotangent

13

blowing-up

17,

Directrix Formal

unidimensional

Formal

vector

General

distribution

18

6

fields

resolution

91

2

sheaf ~X

Directional

6 statements

3O 60,

Good p r e p a r a t i o n

66,

47

IH(m )

Ideals

J r ( D E)

1-equivalent I'-prepared Infinitely Initial

JP(D,E)

and J ( D , E }

8O 166

realization base near

ideals

163 points

InP(D,E)

24 and I n ( D , E )

81

Invariant

m(D,E,p)

46

Invar/ant

6 (t~,E,p)

47,

Invariant

w(D,E)

45

Invariant

6(D,E)

53

Invariant

6+ ( t ~ , E , p )

92

Invariant

6(A)

123

Invariant

~(X,E,D,P)

Inverse

7,

image

for the natural

Movement

irreducible

31, unidimensional

92 13

117

transition

t ( mov t)

Multiplicatively

92

177 55,

Invariants ~ ,B,e

Model

13

4

distribution

distribution

4

43

96,

175

188

Multiplicative

reduction

Multiplicative

reduction

4 relatively

to

5,

E

Natural transition

118

Non adapted order

16

Normal crossings

9,

Normal

2

crossings divisor

Normalized system of parameters

45,

O r d e r ~A(D,E,p)

133

Permissible center

24

Polygon

A(D,E,p)

47,

Polygon

&+(D,E,p)

91

Polygon

A(b;p)

119

Preparation Prepared

regular

Realization Reduction Regular

of

system of the

parameters

reduction

game

parameters

( r.

54,

91

50,

51

50,

93

31, s.

of

p.

)

43

2

Retarded general winning strategy

168

Retarded standard winning strategy

IO0

Singular locus

17

Standard realization

164

Standard transitions

88

winning

13

32

game

system of

Standard

6

100,

strategy

164

Stationary s e q u e n c e

20,

23

Status t (stat t )

31,

43

Strict

9,

transform

Strongly

normalized

regular

system

of

13

parameters

(s. n. r. s. of p. )

48,

184

Strongly prepared base

93,

175

Strongly well prepared base

98

Strongly well prepared vertex

96

Strongly winning strategy

33,

Suited regular system of parameters

9,

Tangent sheaf = -X Transformations (T-l,(), T-2, T-3, T-4

2 41

Transition I--~ I'

102

Transition II--~ I'

103

Type zero

37

Type 0-1

39

Type 0-0

39

Type III-bridge

178

44 13

90,

178

189

Types 1-1-0, 1-1-I, 1'-1-0, I'-1-1

82

Types 11-1-1, II'-1-1, 11-1-2, 11'-1-2, 11'-1-3, 11-2, 11'-2-1, 11'-2-2 and 11'-2-3

83

Types 11-1-1-0, 11-I-2-0, 11-1-1-1 and 11-1-2-1

85

Types 111-1-1, 111-1-2, III'-1~ 111-2, 111'-2-1, 111'-2-2

85

Types 4 - 0 ,

87

4-1,

Types I - 1 ~ I ' - 1 ,

4-2 I-2,

I'-2-1,

I'-2-2

82

Types o n e , t w o , t h r e e and f o u r

81

Unidimensional distribution

3

Vector field

3

Very good p r e p a r a t i o n

68, 98,

Weakly p e r m i s s i b l e

23

center

176

Well prepared v e r t e x

59, 65,

96

Winning s t r a t e g y

32, 44,

76

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Beer Johnston VECTOR MECHANICS FOR ENGINEERS VECTOR MECHANICS FOR ENGINEERS MD DALIM #999860 12/18/08 CYAN MAG YELO BL...

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Vector by Robin Cook Jack Stapleton and Laurie Montgomery, the dueling forensic pathologists who bounced off each other...

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