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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 =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)
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)
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 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 ~)
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 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
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)
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.
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
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
-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
(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 $ 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 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)
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)
r+l,
>
r+l+h+j
< 1.
So,
>
h'
r = 2.
3 and
i'+j'
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 _ 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
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
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
~
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)
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
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
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 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 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