Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
813 Antonio Campiilo
Algebroid Curves in Positive Charac...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
813 Antonio Campiilo
Algebroid Curves in Positive Characteristic
Springer-Verlag Berlin Heidelberg New York 1980
Author Antonio Campillo Departamento de Algebra y Fundamentos, Facultad de Ciencias, Universidad de Valladolid Valladolid/Spain
AMS Subject Classifications (1980): 14 B05, 14 H 20
ISBN 3-540-10022-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10022-9 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by SpringeroVerlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
INTRODUCTION
A since
number
Zariski
(the of
are
over
equivalent
that
more
recently
of
the
in
any
field
which
attempt
is
the
being
r a
chain
an of
algebroid
arbitrary
of
an
case and
( 18 ) , a n d
curves
characteristic, instead in
Salamanticiensis any
irreducible
longer.
first
as
main
appeared
algebraic in
curves
a work
(Universidad Essentially
algebroid
an
zero.
expansion
completely
over
the
Puiseux
characteristic
plane
of
using
o.f t h e
developed
was
available
the
development
It
it
curve
[]
by
G.
de is
based
on
= k({x,y~]
type
=
x(z
y = y(z
by
a systematic
of
x
z
give
parametrizations
not
zero
3 ) .
employed
Acta
algebroid
attention
Moh
obtain
and
of
( 15)9
to
in
plane
extensive
Hamburger-Noether
a parametrization k
received
,
Those
However
called
published
appeared
characteristic
so
characteristic.
Salamanca)
of
of
initially).
expansion
usually
case
field
irreducible
of
have
Equisingularity"
Lejeune
to
Hamburger-Noether
Ancochea,
over
not
[
of
closed
in
particular
available"
equisingularity
The an
has
intend
expansion
as
are
equisingularity
considered
notes
algebraically tool
the
closed
AngermBIler
These theory
in
p > 0
papers
of
"Studies
Zariski
characteristic a few
his
a-n a l g e b r a i c a l l y
situation
only
definitions
published
definitions curves
of
element relations
of
r r
the
)
)
,
quotient
field
of
[]
,
obtained
from
x~y
IV
2 y
=
a0t
x +
h
aO2x
+
. . .
+
2 x
=
h
a]2zl
+
...
h
aOh x 1+
+ alhlZl
z
+ h 1
z
1
x
z
2
2
Zr_1
where
a.. jl
C
expansion
of
a plane
Puiseux
expansion
and
are
determined
the
curve,
by
the
I
well
contains
Puiseux
expansion
we
over
introduce
these
an
the
Hamburger-Noether
calculated
from
given
the by
the
exponents
from
singularity
ring,
etc...
and
results
and
of
on
resolution
Chapter
I I
of
is
of it w i t h
and
of t h e maximal iV. and
of
From compute
Newton
exponents
model
field
and
in c h a p t e r
of
I would his
in c h a p t e r irreducible like
comments
V
we
and
study
twisted
to e x p r e s s
the
determine
devoted
the
for
any
the
singularity
characteristic,
this
model
them
we
in t e r m s
my
local
ring
contact We the
also
of t h e
or
from
find
Newton
the
curve
of
several
relationship
coefficients
criteria
sincere
thanks
is
the
for
curves.
suggestions.
compare*
polygons.
Lejeune.
equisingularity
for
local
a complex
ones,
of t h e
characteristic
the
comparison
exponents.
of v a l u e s
derived
zero.
closed
usual
values
for
curve.
characteristic
exponents
definitions
and
using
expansions
Finally,
Aroca
the
semigroup
characterisitic between
expansion
of
that
parametrizations,
algebraically
with
to
its
al.gebroid
I I I , by
system
These
of
known
characteristic
exponents
The
of
a
process
in c h a r a c t e r i s t i c
In c h a p t e r of a c u r v e
equivalent
values
irreducible
to t h e H a m b u r g e r - N o e t h e r
define
zero.
of
an
is
resolution
existence
of
to
which
semigroup
curves,
us
characteristic
by
the
singularities
enables
curve in
Chapter algebroid
..... " ....
k.
This exponents
+
= a r 2 Zr
to P r o f e s s o r
TABLE
Chapter
I .
Parametrizations
of
1. 1.
Preliminary
concepts
1.2.
The
cone
] .3.
Local
1 .4.
Transversal
1 .5.
Resolution
Chap'ter
tangent
of
2.2.
Hamburger-Noether
2.3.
Intersection
2.4.
Hamburger-Noether
.
9
Multiplicity
expansions.
1t
..................
21
............................ expansions
of
27
algebr0id
Discussion
........................
expansions
of
multiplicity
of
of
45 curves.
53
...............
twisted plane
Theory
curves.
algebroid
curves
for
exponents
on Equisi'ngularity
plane
plane
expansions
Characteristic
Report
l
.................................
singularities
Puiseux
curves.
.....................................
parameters,
2.1.
3.1
algebroid
Hamburger-Noether
III.
CONTENTS
.................................
parametrization
I I .
Chapter
OF
60
curves
.......
alqebroid
62
curves.
......................
69
3.2.
Characteristic
exponents
3.3.
Characteristic
exponents
and
Hamburger-Noether
expansions
81
3.4,
Characteristic
exponents
and
the
........
88
3. 5.
Characteristic
exponents
and
Puiseux
.........
10'2
Chapt
er
plane
I V.
Other
algebroid
systems
Newton
4.2.
M aximal
4.3.
T he
semigroup
4.4.
T he
degree
V .
Newton
invariants
for
'26
polygon
series... the
equisin,qularity
of
curves.
4. 1 .
Chapter
of
..............................
coefficients contact
of of
of
Twisted
................................... higher values
the
genus
of
[]
P reliminary
5.2.
E quisingularity
E.s.1
5.3.
Equi
E.s.2.
Space
5.4.
Equisingularity
E.s.3.
Coincidence
arity
121
in
[]
12'2 ...............
135
cMrves.
5.1.
s ingul
concepts
........................
...............................
conductor
algebroid
I 1"2
and .
notations Generic
..................... plane
projections
quadratic of
141 .......
transformations. semigroups
of ....
t46 15B 161
values. References
Index
..................................................
.......................................................
Symbols ..........................................................
164 16? 168
CHAPTER
PARAMETRIZAT1ONS
This
chapter
concept
of
over
algebraically
an
Although teristic
local
k
shall
series
ring
function
denote in
order
used
direct
it
{X i }
1-.< i ( N
by
k((_X))
indeterminates
on
k((_X))
be
be found
1 . 1 .1 . ( W . P . T . ) . -
Let
monic
polynomial coefficients
order unit
s, U(X)
P((_X'),X N) in
k((X_'))
f(~)
=
any
closed
field
the
curves
characteristic.
with treat
is
a set
with
denoted
Preparation
may
a unique
algebroid
to
X
consequences
exist
of
= k (~X I .....
will
Weierstrass
there
field
algebraically
_X =
of
irreducible
of
the
case
this
of
case
characin
detail.
CONCEPTS.
the
XN
CURVES
systematization
useful
work.
in
the
differences
Jr1 t h i s
regular"
with
essential
frecuently
Theorem is
to
ground
thought
an
If
The be
have
be
characteristic. we
not
ALGEBROID
of
PRELIMINARY
Let
k,
devoted
closed
are we
1.
OF
parametrization
there zero,
is
I
U(X)
is
in
XN) ~
the
by
i.e., in
(where
k.
(W.P.T.) down
E: k ( ( _ X ) )
k((X))
in
power
and ,
be
will its
proof
a series
which
__U(f(0,...,0,XN)). and
a unique
that
.
and
~29 ) .
Then,
degree
_X'= (X 1 ..... X N _ I ) )
P((~'),XN)
The
_..U
stated
s =
over
formal
coefficients
Zariski-Samuel
f(X)
arb.itrary
indeterminates
Theorem
It
such
of
of
in
s
XN
Definition curve
1.1 .2.-
if t h e r e such
that
is
An no
irreducible
confusion)
alg..ebroid over
k
is
curve
a
(or
noetherian
simply
a
local
domain
3)
means
that:
k
1)
[]
is
2)
[]
has
3)
k
is
a coefficient
tf
rn
is
the
is
complete.
contained
Krull
dimension field
maximal in
] .
[]
for
ideal and
[]
of
is
[]
. , the
isomorphic
property
to
the
field
Ft/ m
by
the
canorical
epimorphism
[]
>
El/ m
Remark
1 .1 . 3 . -
Since
F7
is
noetherian,
the
vector
space
m/ --
over
k
called
is the
finite
dimensional.
embeddin For
The
9 dimension
every
basis
number
of
0
B = {x.}
t
Emb( [] ) = dimk(
m/ --
m
.
2 m
2)--
is
-
1.
of
N
the
maximal
ideal
m ,
2 S = { xi + m
is
a set
of
generators
of
the
k-
}1~ i (N
vector
space
m/
2 .
--
becomes
a basis
basis
the
of
of
this
ideal
Let
I
of
space
if
and
only
[]
1~;i
we
a basis
{N
find
of
a natural
m .
X
Thus,
there
prime
ideal
is
1
i
exists of
if
B
Using
surjective
S
is
a minimal
>
an
and
the
{x }
x.
isomorphism
k((X_)).
3)
k-homomorphism
t-I
(I)
set
m.
B = {x.}
completeness
vector
The
m
The
k((X))/p~
condition
2)
[] means
,
where
that
the
p
is
a
depth
of
fact,
if
p__
I . We m a y
we s e t
×., : ×.,
N
that
such
these
identify
[]
wecan isomorphisms
with
write
the
ring
[]
=k(Cx
exist,
is
k((X))/p_
1
.....
exactly
.
×.)).
In
The
Emb(O).
m
nimuo
When
an
identification embedded
as in
give
a prime
ring
of
C.
theory: the
an
ideal
The
induces
is
allows 13 i s we
shall
by
the
form
depth
it
for
1 .1 . 4 . -
Let
sets
of
indeterminates
There
exists
k((Y]]
onto
1 ~ i ~< N , the
an
over
integer
k((X]]
following
by
over
a t~ A
= (0)
-
2)
,
There
shall First,
k((Y(]
(t)(0 u--(fN
)))
of
non
~< N
....
a (])
construct
, with
,0,y(1))) N prove .
If
, such
the
proof.
an
Therefore,
series
ring.
~(Yi
the
be
two
a'~
isomorphism
(0),(1).
~)
from
) = Li(__X) ,
ideal
by
Y (])
a = ~ (a')
has
m + l --< i
;
this,
0 ~ i4:
fi(Xl
N.
' " " " ' x ' ) A E: ' -a' ~l
i
that
u(f
(x 1 ....
--
induction
i
x )) '
the
i
isomorphism
isomorphism
= { Y "( 1 ) } 1 <
i~< N
~(I)
new
from
variables,
k((Y]) such
I
, then =
,
series
=
an
,
t ..... Xi] )
I
let us c o n s t r u c t
= ~(1 )(a_')
....
f N'
that
I
zero
0,x.)) '
- -
To form
assumptions
a P, A . #~ ( 0 )
N-m
I
- -
if
that
A i = k((X
are
u(f.(0 --
that
the
X = {X.} -~ 14 i~
-
m + l "-<< i
onto
Spec(k[[X}]). which
power
, and
( 1 )
theorem
relations
such
m
where
We
.~
,
1 4i
scheme
homomorphism
affect
a formal
local
Spec(k[[X]]),
normalization
not
to
the
in
' s..c.h e m e the
C is
called
meaning
13. N o t i c e
linear
k,
is
is
propertiesi
1)
Proof:
}
i
curve curve
Spec(F1)
0 ~< m 4 : N - 1
defined
independent
do
k
m,
and
ideal
a of
{Y
[]
affine
following
ideal
y =
the
the
embedded
Then,
Spec(E])
the
-
that
h a s .a R c e c i . . s e
which
any
I .
schemes
of
I,
say an
depth
with
give
of
give
definit.!...gn
shall the
prove
of
of
we
and
Theorem
is
shall
to
embedding
to simplify prime
Thus,
identified
a closed
we
"embedding"
N-space
Now
done,
p C k((X)]
word
E1
is
N-space.
The
curve
above
U -take
f N ,q ( 0 . . . . .
there
(fN(1)(Y1 fN
f(])
exists
0,Y N ) 4 0
verifying
Y(
"'" E: -a ' '
E: a ( ] )
f~4 we
#~ 0 . set
Let
y (I1 ) =
fN,q Y .t
,
the 1 ~ i
leading ~N,
~(1)
the
we p i c k
identity a (i 1 )
out
(Hilbert~s
k((Y))
and
-
E: k ,
L(1) N lineaRy
y(1)
=:
t
. i
=
1 --
+
(1)
a.
Y
I
series
f (N 1)
Now,
p
over
~(P)
: k((Y])
such
that,
> (p)
a
t) ( f ( iP )
Assume
take
above
rise if
be k.
(Y1
(P) +'
to
<~ N - 1
<:i
and the
>
L!I)(Y
1
0,Y
)=0
(1P
" " " 'aN-1
'
'
1)~0
N, ) E:: _a (1)
an i n t e g e r
there
" " "
:'
Y
))
our
1 4 P < N, there
lineary
exist
non
=
~ i
u --
(
conditions and
exists
by
, N-(p-1)
(P) i
vet ify
,
that given
isomorphism
(1))
I
Assume
find
Y(P)
_
independent
~
..
forms
series
z e r o
(0
i
{Yi(P)}I
=
an i s o m o r p h i s m
"-
f(P)
'Y(P)N-p ) )
in
an
a (p)
. , .
~
that
0
, Y(P)) L
).
there
exist
1
(0).
k ((Y(lP,) . . . , Y" N(_pp) J ~x,,l a n d u s e linear
kC(z))
..
u (f(p+l)(y(p+l)
:~
appropriate
~(P+]) then
+ o(p+tg
l
q
,
1
k((V(~)))
series
~(P+l)(a'),_
--
k
k ( ( Y ( 1 p) . . . .
to an i s o m o r p h i s m
,!p+l>
fN
,aq( 1 )
that
a non zero
a (p+I)=
(0 . . . . . (
forms
N
Y(P~)•I
a(P)Pt
procedure
f
linear
>
k((Y(P))) (p) = (~ (a'),
k((Y(lP)' ''''
--
Then,
I
= ~(1)(f
let
indeteRminates
(1)
that
(1)
over
Yi
f!P)l -C --a(P)f'~
the
if
N
y(1) N
independent
if
= f'"
' such
Then,
~(+) : k((Z))
and the
fN
'
Nullstellensatz). L(.1)
ape
in
> non
k((z(P++>))
zero
.-pc, y(p+l))) ,
change
which
gives
such that
series
°rify+n+
= u(f(p+l)(0 ,
0 y(p+l))). I
the
N
Since an I~
integer
a'
p
= {(P)
such
and
trivially
;Z ( 1 )
fi
we
that
have
(1)
= f(i p )
a'r~
does
the
k = (0),
not
hold.
conditions
hence
Setting
stated
in
1 .I . S . -
ducible
series
In
the
which
U
(g.(0
--
if
above
theorem
Z
[]
is
its
of i
}
For
1~
~
if
given
by
d# ( Z . )
Krull
over
finite
there
is
g. E: A .
are
.
a
denotes
an
irre-
I
.....
X.)).
be
chosen
may
i
local
ring
{zi}
I
(for
Suppose
set
its
that
1~< i { N
to
k C
be
irreducible.
m-adic
topology,
is
a coefficient
rn
and
continuous
for
indeterminates
homomorphism
>
= z.,
[]
I
..< i 4
N,
which
We
say
that
is
their
respective
I
k
t . 1 .6.if
Theorem
the
1.1
above
an
ideal
.7.-
of
Let
k((X]]
(a)
X
={
X
(b)
m
= (0)
,
There
1~i
is
injective.
integer,
a r~ A
non
formally
independent
indeterminates
over
k,
N
0
:~L ( 0 )
--
exist
~
be I
an
m
are
i
} ~
and
a ~ A --
{z.}
homemorphism
--
-.< m ,
-.
m+l
..< i
~N
that:
.
1
zero
series
f
C
i
a
-
PI A . ,
m+l
denote
by
i
-{ i~:
N,
that
u
(f
--
Set m)
theorem
(p)
topologies.
Definition
such
f
ideal)
any
I
U (gi(X1
--
a complete
k((Z))
:
a
X = Y
then
series
maximal
I-'-I.
,
I
=
I
the
be
f
0,X.))
prime,
Let m
= { Z
is
a'
divides
.....
1
-
where
--
m=N-p,
the
I
field
exists
true.
Remark
Thus
there
the
maximal
i
[]
(o ....
,o,x
))
=
u(f
i
= ideal
k((X_))/ of
--
a
k((X~)
,
(x
1 ....
x.))
i
'
xi = X i
(resp.
D
+ -a , ).
.
I
and
Then
the
following
M (resp.
--
statements i)
If
are
m
true:
>0,
i)
{xi } I -< i4 m
ii)
[]
iii)
[]
is an
iv)
The
height
are
formally
integral of
a
extension is
m = 0
if
and
m>
0.
only
if
a
a
of
N-m~
and
m = dim
is
over
k.
.... ~ XN ) "
= k((x I ~ . . . ,Xm) ) (Xm+1
Particulary 2)
independent
k((x I .... Xm) ) , hence
its
depth
is
m.
(l--l).
M-primary
ideal.
Proof: 1)
Case
i) given
by
The
canonical
qb(X.)
=
x,
I
ii)
f
we
may
injective
because
a hA
apply
1 .....
((×]
f(X)
Xm ) ) =
> []
(0).
m
k((x
k
if
is
¢ :k(['X 1 .....
I
As
Conversely,
homomorphism
Xm ) )
' " " " 'Xm
is
a subring
)) (× m + l .....
E: k((__X))
, by
division
algorithm
the
of
using
the
× N)
[[] ,
c
we
have
[]
properties
(Zariski-Samuel
of
the
,
r29)),
series and
i
write N f(X I .....
X N)
q N -1 +
and
U.(X i
q
1J
N .
= _U ( f i ( X 1
Ui(X
1 .....
X N)
fi(X1
q m + 1 -I
=0
" " ~
X N)
....
(
£: k ( ( _ X ) ) ;
...X#
, . . ,Xm)Xm+lm+l
R
IN;
" "
.
'
i
m+]
(X1 ,..
I
~
X
m
)Ck((X
1 .... Xm) )
We o b t a i n
qm+l - 1 > i
X.I
m+l
m+l
X.)).l
' . " ' . X N ). = . 7 " i =0 N
.....
i RiN ' .
i
q N -1 f(xl
) _ i=m+l
. . . i
where
=
m+l
i Ri
=0
.... N
i
(x 1 , . .,x m+l
k((x I . . . . . Xm) ) ( X m ÷ 1 . . . . . X N )
i
m ) x m+lm+t" . . x #
E:
iii)
Now,
By
xi))
W.P.T.
applied
fi+l(Xl
V
an
k ( ( x 1 . . .•.
the
where
using
.....
is a unit
argument
k((x 1 .....
=
to
in
Xi+ I
is
integral
coefficients over
if
k((X
Xm ) ) '
m+1
x< i -<-< N .
=
m
m=O~
we
have
and
> 0,
An
depth(a)
we
I , .
X.,+I)
and
may
conclude
.. ,x i) ,
give
us
gi*l
((X1
gi
m+1
,..
that
Corollary Then,
that
if
a
1 .....
Xi) ) .
= k((Xl
....
each
polynomial
follows
that
It
(Xm+l
is
integral
1 .....
not
.... ( x ' r ) ~ /
....
fi
xi+ 1
xi)(since over
Xm ) ) )
= m.
M-primary.
r > O.
= dim(k((x])))
Let
[]
be
there
and
an
exists
if we
set
Conversely
Thus
= dim(
irreducible a prime
x,
p
ties
k((X1))/(
r) X1
= 0
= X I
algebroid ideal
+p I
p
, the
curve
(:::k((X))
over such
following
proper-
--
hold:
-C
1)
pn
2)
There
p_ p~ k ( ( X
1 .....
1)
= N-m.
is c l e a r l y
kI'(X1))"'"
>i E m b ( F l )
k ((_X)) /
17 ~
, Xi),Xi+
M-primary.
I .I . 8 . N
~
a monic
Xm))
x.l
dim(k((x
depth(a)
a
= dim(B)
is
i
+1
-
implies
<
that
(iii)
= N -
depth(a)
~::
xi))
dim ([])=
height(a)
Therefore
If
in
= 0). Hence,
(i)
( Xm+
+1
I+1)
By
"))
'
, . . ,x.),xl
depth(a)_
2)
"
I .... "
iv)
(ii)
V (X 1 , ....
k((x
k((x 1 .....
Xm))
.. X '
with
gi+t((x]
=
k((X I
,
in
f i +1 , m + l . ~< i --< N ,
X.,+l)
in
as
k((X 1 ..... exist Xi)),
Xi))
=~ ( 0 ) ,
non zero 2 4 i 4 N,
2 ..< i 4 N ;
(irreducible) such
that
_p N k ( ( X 1 ) series
)
= (0).
k.
u (f.(X
--
~
X )) =
k((x 1 .....
XN) )
5)
k((x 1 .....
XN) )
Therefore
o(f
i
4)
sequel.
is
~
x1
1 .1 . 9 . -
2,~i~<
.
''"
3)
Remark
formally
1
--
formally
The
five
over
By
is
....
'
0
independent = k((Xl) ) is
an
above
we shall
independent
(0
i
i
(x 2 .....
XN) extension
will
= k((xl)
each
"
k.
integral
[]
and
X ))
over
properties
write k.
'
x.,
of
k((xl)).
be assumed
) (x 2 . . . . .
in
the
XN) , x 1 b e i n c l
integral
over
k((Xl))~
and
hence
algebraic
N.
over
(5)
k((Xl)).
over
Every
k((Xl))
having
g((x t)
This for
El
~
0,
= Z
S
is
k[(Xl)),
a zero
is
actually
if
use
the
, in
order
i
of
in
+ b s - 1 ( x 1) Z
Indeed,
we may
over
coefficients
polynomial
0 4 j ~ s-1 .
b.(0)
z E: m its
Z)
'
integral
is
polynomial
k([Xl}}, s-1
+ . . . + b o (x 1).
distinguished,
the
W.P.T.
an i r r e d u c i b l e
smallest
, applied
i.e.
, b (0) = 0 J for which
integer to
the
two
variable
i
series
g((X1),
Z)
coefficients
in
k((X1) )
not
and
a unit
a unit)
g((xl),Z)
Hence was
g~((Xl),Z)
the
polynomial
Since
(Zariski-Samuel, irreducible
Remark
as t w o
I .1 o 1 0 o -
and
to
U(X 1 ,Z)
U(x 1 ,Z).
= 0
which
with
z
k(~'Xl,])
i < s (note
e
k((X 1 ,Z))
(Z)
is
that
such
a contradiction,
i > 0
Z)
since
with g is
that
which
had
because the
/(g((Xl), Z) )
, P. 1 4 6 ) ,
Emb(r-t)
g>~((X]),
g~((xl),Z)
as a z e r o
variable
If
a new polynomial
degree
=
[2'9]
find
the
polynomial
minimum
k((X 1
g((×l),Z) degree.
,Z)~ /(g(XI ,Z))
g((X1),Z)
is
also
series.
,< N
the
curve
E1 c a n
be embedded
in
an
for
N-space.
When
an e m b e d d i n g
confusion),the if
X=X
1.1 .8.
in
the
curve
a 2-space
ideal
p
I , I , 1 1 .-
is
said
(=algebroid
s actually
does not divide 1 are trivially sat
Proposition is
N=2
the
to
plane
principal,
leading
be
form
plane.
In
or plane
if
f,
the
five
case,
there
p = (f(X,Y)).
of
this
is
no
Furthermore properties
in
sfied.
[]
is
a regular
domain
if and only
if
Emb(E~])
one.
This can
also
be
is
a well
obtained
Moreover,
[]
to
power
a formal
2.
is
THE
] n this irreducible
known
from
the
regular
if
series
algebroid
in
commutative
normalizatian
and
only
rin 9 in
one
TANGENT
section
result
if
algebra,
theorem
[]
is
as
a
isomorphic
indeterminate
over
but
it
corollary.
as
k-algebra
k.
CONE.
we
curve
shall
study
from
an
the
tangent
algebraic
cone
and
of
an
geometric
view-
point , Let cally
[]
closed
be
Field
an
k.
irreducible Consider
oo
gr
([])
=
m
Definition the
1 .2.1
affine
.-
algebraic
algebroid
the
graded
curve
over
the
a
gebrai-
ring
.
@ m'/ i+I n=o -- m
The
tangent
variety
cone
to the
Spec(gr
curve
[]
is
defined
(F~)). m
If
a basis
{x.} i
9r
(1--~)
is
generated
of I~
as
m
~
k-algebra
is
--
the
graded
by
the
classes
{x +rn 2} i
there
is a c a n o n i c a l
k(X 1 ..... X
i
epimorphism
XN) I
~
gr
>
x +m
m i
ring
'
m
Then
given
([[])
--
2
I 4i(N
to
be
10
( ix.}
t-
being
~
indeterminates
over
k),
and
therefore
an
isomorphism
gr
where to
an
a
(F~)
m
is
a
in
Remark
of
k N
k(X 1 .....
homogeneous
embedding
variety
~
the
ideal
tangent
defined
1 . 2.2.-
XN)
by
If
of
a
k(X 1 .....
cone
the
[]
/
in
ideal
kN
:
XN).
This
It
the
is
gives affine
rise algebraic
a.
= k((X))
/
is
the
curve
defined
by
the
ideal
E p_
and
if
as
above,
in
p.
a
is
then
Proposition of
type
homogeneous
a
is
a X
m
1
,
There
a C
k,
series
f2
C _p
b x2q
+ X 1 g(X 1 ,X2)
,
way,
by
fi
replacing
would
obtain
dim
gr
([~])
X
is
a ~
= dim
the
no
series
defining
leading
forms
in
instead
0,
we
p
of
with
would
1 .1 . 8 . )
would
E; V'-~a .
i
k(X)
by
m>
(corollary
we
of
its
tangent
all
the
cone
series
a leading
form
O.
X 1m E; _a ,
Otherwise
the
ideal
generated
1.2.3.-
Proof: as
the
conclude of
f2
has
have a
leading
that
' i>2,
X 1 C
X 2 C
and
using
V'~a ._
~ -
" Now,
form
of
In
the
type same
induction,
we
Then,
kCXl . . . . .
xN)/~_ = dim k(Xl . . . . .
XN)/(Xl
.....
XN)=0
m
would
be
a contradiction,
since
dim
gr
(E])
=
1 (see
Zariski-Samuel,
m
p.
[29),
235).
Corollary
1 • 2 . 4 .-
Let
g((x
1) , Z )
=
zS + b
b.(x ) ~ k[[x ~) the irreducible polynomial j 1 1 ' z G m. Consider g as a two variable series. of
g
iS
a power
In
particular,
of the
a linear series
fi
form
and
in
1 . 1 .8.
(x 1 ) Z s-1 +...+ s-1 over k((Xl)) of Then,
~(g((xl),Z)) may
be
taken
b
an
o
leading
= ~
(g((O),Z))= be
)
I element
the
to
(x
~form
g((XI)~X')'I
s.
11
Proof:
First,
leading
form
we f
prove
(X~Y)
that
is
if
a series
a power
of
f(X,Y)
a linear
is
irreducible
form.
In
fact,
then
its
making
a
r
linear order of
change
of
and
hence,
r
k[(X]~[Y]
monic
of
variables, by
the
W.P.T.,
r.
Then
degree
polynomial
f may
of
k[[X))[Y'),
with
aC:k,
be considered we
may
to
assume
be
that
ills
f'(X,Y')=f(X,XY')/y,r and
hence
by
regular
is the
in
Y of
a polynomial an
Hensel's
irreducible temma
f (1,Y')= r
f'(0,Y')=(Y'+a)
r
irreducible is
as r
(aXl+bZ)
g is
two
variable
with
Proof:
we
1 .2.5.-
Choose
the
By
have
The
so
fr(X'Y)=(Y+ax)r
series
a,b~k.
distinguished
Lemma
and
the
r=s.
tangent
series
(see
] .1 . 9 . )
previous Hence
cone
gi((XI),X.)
and
so
proof
g((×l),Z)
its
proposition
the
to
Now,
a curve
is
in the
above
leading
b/0
follows
, and
fi(Xt hen,
, ....
X i )"
since
dim
_
t follows
X2 +
the
quotient
a basis
for
( X2+
the
~2 X1
[]
XN]
=
form
" " "
XN+
cone
is
=
XN +
the
since
a straight
line.
corollary
instead r
of
gi
/V-aa =1
c~2X1 ~ . . . ,
tangent
LOCAL
Let
leading
k(X 1 .....
=
that
3.
The
form
easily.
i
of
is
is
, we
of
type
must
( X i + . 0~iX] )
i
have
O.N X 1 ) .
straight
o. N X 1
line
defi~ned
by
0.
PARAMETRIZATION.
be
field which
an
irreducible
of
r].
Choose
the
conditions
algebroid a normalized
of
I . I ,8.
curve
over
basis hold)
{x
k. i
of the
}
Let
I k
maximal
F
(i.e, ~ ideal
m.
Since
13
=
k((xl))(x
k((xl))
2 .....
×N)'
(x 2 . . . . . XN)
we h a v e :
is a s u b f i e l d
of
F
be
containing
12
= k((x]))
F
and
therefore
the
field
(x 2 .....
extension
XN),
F /k((x
))
is
finite.
1 On t h e the
ring
k((Xl)
)
Furthermore, relative
other is
its
hand,
of
I--],
which
1 .3. 1 .-
m
ndeterminate
is
k([Xl)
-
If
is
the
over
k,
formally
independent
valuation
closure
ring
in
F
is
over
of
the
k,
k((Xl)). same
as
the
i.e.
[]
dominates
is
discrete
integral
k((xl) ) Theorem
x1
a complete
relative
closure
as
=
1--1.
a complete
discrete
valuation
ring
of
F
) . maximal the
ideal
[] , t C m m 2 , and
of
homomorphism
given
T
is
an
by
I
h : k[(T)) T
s a
I
:,
[]
>
t
k-isomorphism.
Proof:
Since
associated
F /
is
k((xl))
valuation
to
finite
k([Xl)~,
and there
k((x is
]
))
only
is one
complete valuation
for
the
ring
A
I
of
F
of
all
which the
Finatiy, which
that
over
valuation
k((Xl) ~.
rings
the valuation is
[]
injective
lies
a discrete
lying
ring
over
ring
In
order
has
k
as
a coefficient
Let
q
be
the
field
prove
[]
= k([Xl)
the
maximal
~
k((xl~ ) , it
is d i s c r e t e
valuation to
As
s
noe
is is
[]
--- k ( ( x l ) )
intersection__
trivial
that
lies
over
first
we
O = A.
k((×])),
itself.
second
statement,
shall
prove
field. ideal
of
k((Xl)
) . We m a y
homomorphism
k
the
/
> 9_
El / -m
construCt
an
13
which
is
also
onto,
since
[]
/-
is
algebraic
over
k
and
k is
m
algebraically
closed. The
the
topology
topology
homomorphism over
of
we
diction.
by
of
Tm
would
for
Definitions for
the
the
curve
h the
Let
z = s(t)
an
having
Remark Thus
be v
s(T)
value
is
parameter
Proposition the d e g r e e
defined,
valuati:on
is
1
t
for
We c l a i m
ideal
t = 0,
because
t E: m a n d
actually
the
is
that is
and
=
that and
we
since
t
t,
= k((
is
m-adic
it
will
fact,
9et
if
a contra-
a uniformizin9 [-I.
parameter 9 para,rneter
shall
t for
write
t )).
valuation
valuation
In
would
ring
we
natural
we
hence
m = 0.
a uniformizin
F
associated ,
k('(T)),
a uniformizing
a such
t ))
of
valuation
normalized
the
to
El.
of
k((t)).
If
z c
We
[3,
have
u (s(T)).
parameters
are
exactly
those
elements
in
v.
The
intrinsic
m.
[]
for
E: k ( ( T ) ' )
in
an
say
ring
= k((
the
be
discrete
uniformizing
1 .3.3,it
its
thus
We s h a l l
v (z)
F
well
epimorphism,
Henceforth,
that
The
= 0,
complete
v
with
some
valuation
[~1.
say
by
will
m
is
[]
shall
t
1 .3.2.discrete
h
for
have
Actually
parameter
induced
kernel
generated
m > 0,
is
1:~.
The be
[]
h
natural of
[]
valuation and
it
may
of be
[]
does
defined
not
using
depend any
on
x
1" uniformizing
"~.
I .3./4.-
The
of the e x t e n s i o n
ramification F / k((Xl))"
index
of
[]
over
k((Xl))
is
14
Proof:
Let
by
v
the
any
formal
implies
e
be
the
ramification
associated power
valuation
series
i:n
index to
x1,
of
[] we
If
[]
over
k((Xl)).
v(x 1 ) = h , and
have
v(s(x]))
= h
Denote
if
s(x 1 ) is
U_(s),
which
that
e = ( Z : h Z ) = h.
Now,
X]
= xl(t)
may
if
t
, with
check
denotes
U ( x 1)
the
= e.
--
vector
F over
shows
(1)
z
ii)
is v
Fop
iii)
is
an
the
valuation
independent of
IF U
}
is
a basis
: k((Xl)) is
z t
a
of
the
] = e = V(Xl).
k((x]))-module
0 , and
over
the
ramification
/
is
The
v(z)
v(z)
>/ 1 , t h e
k.
natural
index
k((z~).
integral
module
t
is
is
[7,
valuation
of
k((z))
[]
is
v(z).
rank
algebraic
z is of v
extension of
formally
ideas then
free an
>/ I , t h u s
maximal of
an
k((z))
element
ideal
of
that
e-1
I
with
maximal
that
z C El,
extension
/ k((z~
k((t))
i) T h e
ii)
proves
,t
_
[]
iv)
k((t))
{1 , t ~ . . .
follows
any
formally
k((z~)-
z g
we
hold:
with _
Proof,
k((t))
have
e.
statements i)
It
also
I .3.5.-
following
in
we
t
that
k((xl)).
that
rank
Proposition
LJ,
i
,'19=o k ( [ ' X l ] )
=
s equality
space
of
r-l
for
su,ccessive divisi.ons
e-1
[]
Note free
Using
parameter
formuIa:
(1
Th
a uniformizing
extension
formally
an
of
over
independent is
a
v(z).
independent
k((z)) is
, and
trivially
extension
of
degree
k,
and over
contained the
v(z).
natural
k. in
15
To p r o v e V_(Z),
use
that the r a m i f i c a t i o n
a formula
as
__
k((z))-module
over
k((z)]
with
•
T
of
rank
k((Z,T)) in
over
Corollary
] .3.6.-
maximal
Yi so
= k((z))(t) prove
that
and
[]
t
integra
is
is
a
Let with
extension
[]
= k((Yl~')
properties
(3),
k((Yl))
the
S(Z,T)
W.P.T.
, there
= Z exist
P((Z),T)
of
s(t)
is
a unit
degree
v(z)
that:
= U(Z,T).P((Z),T).
deduce
of
The
and
series
polynomial
(t),
is
(iii). F o r
over
is
that
P((z),t)
= O,
and
thus
t
is
.
Y]
(i)
variable
such
may
k((z))
deal,
i
prove
we
F = k((z))
over
Proof:
[]
We m u s t
a monic
k((Z))
k((z))
polynomial
i.e.,
ii),
v_(z) . B y
and
U( z , t ) ~ 0 ,
iv)
ts
in
two
S(Z,T)
integral
k((z•)
t
v(z).
the
order
coefficients
Since
over
1 .3.4.:
z ))
formula
z = s(t),
in
U(Z,T)
k[[
I =O
of
[]
i
.@
the
of
.
if regular
=
by
free
in
v(z)-1
[]
iii)
(1)
index
degree
[]
be
Yl
Z 0.
formally
(ii)
each
t
is
a zero
[]
of
an is
v(z).
The
proof
a curve
and
{ y . }I
Then,
the
independent /
YN )"
(4),
(5)
and
from
2 ~ i 4 N,
is d i s t i n g u i s h e d
k((Y1 'Yi)~] /(g)
the
is
in
~
it has
basis
statements
of
hold:
integral.
corol
ary
proposition.
the i r r e d u c i b l e and
evident.
k.
the
above
irreducible
] -.< i < N a n y
following over
k((Yl~ )
(Y2 .....
follow
i,
and
].1.8.
hoId.)
We s h a l l
polynomial
its coefficients
k((Y1) ) ['Yi ~J /(g)
g in
of
k((Yl)~ ,
16 (see
Zariski-Samuel Now,
induction
, r29),
we g e t
on
p.
k((Yl
.....
.....
Yi )) = k ( ( Y l = k((Yl))
Proposition
is
F
I .3.7.-
In
a complete for
It
[]
from
-
is
(Y2 .....
L ..m F
NL
I--]L
then
any
finite
Yi-1 )) ((Yi)']
(Y2 .....
Yi )
by u s i n g
the
= k((Yl
'Yi )) (Y2 ....
Yi-I)
Yi ) "
be a f i n i t e
valuation
by the
similar
[-1L
extension
integral
ring
of the
closure
of L.
If
u
r-] L
is
of
a series
quotient
of
[7
in L
a uniformizing
= k['['u~) , L = k ( ( u ) ) , a n d
extension
finite
to the
and
last
field
-uV (t) = [ L k((t))
is
: F].
of t y p e
xI
proof
by
t.
of theorem
The
equality
irreducible
curve
t.3.1 -uV ( t )
= [ F : L ]
I--I U
extension
be of
an
its
quotient
field
F.
over If
k and
of
the
maximal
ideal
m
of
D
, each
y.
--
series
in
u
of
positive
(1)
Definition representation say
Remark
t .3.9.-
that
{ai}
(2)
order
A set
of the
we s h a l l
by
F by
follows
(1)
Let
as
curve are
{yi}
l~ i <.NC k
= k((u
)
denotes
a a
1~i~
is
a formal
power
I
Yi = Yi (u)
1 .3.8.-
L
{y. } l
basis
. , replacing
corollary.
Let
denote
,yi.]~ = k ( ~ . y ] ) ) ( y i ) .
.
Proof: L,
k((Yl
yi) ) = k((yl~)
.....
1--~. T h e n ,
discrete
particular,
k((u))
Let
of a curve
parameter
Then
i:
k((Yl
field
146 ) .
;
i.e.,
'
1 ~ j ..< N ,
(1)
will
[]
in the
local
have
be c a l l e d
Yi = a i + x . ( u ) l
~ ( y i ) > O.
a t0cal
considered
, I < i ..< N ,
m.
Moreover,
for
of elements
modulo
parametric
basis.
equations
be a s e t
residues
:
with
parametric
1 ~< i ~< N its
we
of
the
['7,
We h a v e :
curve.
and
17
with
x (u)
E: m .
Let
us
assume
that
{x. }
--
set
The
and
( 2 )
the
~
is
point
also
(al
called
local N , a N ) £: k
~...
is 1
..
said
to
of
of
be
m. --
parametrization is
a basis
N
the
the
curve
centre
of
E] ~
the
parametrization. (1) 0 = (0,... unless
,O)C
k
a N
otherwise
Definition (1)
is
is
local .
We
primitive
1 .3.
shall
assume
We
when
1 1 .-
say
representation m .
obtained
Then
t
The
new
Remark
[]
other
= cht
h
always
the
a curve
the
+
...
any
equalities
= y.(u)
centre
is
parametric
-
be
basis
parametric
}
of
1~
that
,
Ch
its
maximal
basis
E: k ( ( u ) )
~
h > 0,
if
and
c i -C k
only
,
[]
which with
has
cenl;re
if
U(y.) --
: k((_X ]] X.
~, t
0
h=l
>
0
,
.
1 ~;i,< N ,
I
those in
k((u))
~,
equalities .
tn
fact,
statement
holds
f
is for
,
y.(u)
t
of
as if
a local
we
consider
X = { X i }1
<:i~< N '
,
1
a prime the
curve
is
,Z O .
ring h o m o m o r p h i s m
kernel
0
representation
a primitive
{y
I
representation
f
the
the
representation
primitive
I
is
in
type
h+l
is
Given
1 -< i
in
of
+ c_+~h/ t
1.3.12.-
local
parametric
a substitution
y.
there
centre
1
a curve
any
a
y. = y.(t),
representation
parametric
that
with
L = F.
Let
of
by
henceforth
that
I
ideal
representation
stated.
1 .3.10.-
Remark
parametric
ideal
p_ c k ( ( , X ) ) , []
= k((X))
and /E
•
trivially
our
18 Definitions
1 .3.13.-
parameter Since non
of
the
is
a domain
[] zero
An
curve
element Let
when with
of
x
m
be
element
z C
the
, z ~ O,
principal
only
is
[]
two
prime
a parameter
of
I--1.
inseparability
index)
of
x
degree
(resp.
inseparability
index)
of
the
these
ideal
ml z
is
ideals
(0)
and
We d e f i n e
(resp.
denote
called
a
m-primary. m,
each
a parameter.
degree
We s h a l l
is
numbers
by
n
to
be
the
the
separability
extension
(x)
and
separability
F
i(x)
/
k((x)).
respectively.
S
If
the
characteristic
n
If
the
instead
characteristic i(x) of p The
inseparable
is
zero,
parameter
x
k
=
[ F
the
is
p > 0,
we
: k((x))]
above
to
: v(x)
equality,
is
said
be
k
be
a field
, and
t
always
have:
.
holds
separable
ff
with
1
i(x)=
O,
and
otherwise,
Proposition x
i(x)
(x).p
S
of
1.3.14.-
a parameter
of
Let
a curve
D
of
characteristic
a uniformizing
p > 0,
parameter
D.
of
Then: i) fOP
which
The
x c El p ii)
is
index
x
The
Proof: t
ins.eparability
is
x
is
separable
of
element the
of
and
only
x
is
the
maximum
integer
if
the of
extension this
if
(dx/dt)
t
F /
k((x))
is
extension;
inseparability
index
then
of
the
t over
0.
n = v(x), inseparabi
k((x)).
i
Put X -
S(T)
polynomial
x = s(tP
, there P(X,T)
exists in
i
k([tP))
degree
a primitive of
:
index
).
By
the
a unit k((X]]
U(X,T)
(X
W.P.To U(X,T)
(T)
-
of
S(T)
applied
to
in
k[(~X,T))
degree
n/
) =
P(X,T).
P
i
the and sucht
series a mon that
and ity
19
i
P obviously
is
this
irreducible
over
must
min
On t h e
be the
other
hence
This
i
is
from
the
=
the
proof
p ) = O,
of
t
over
then k((x)).
-
i dS p ) ~---f-(t
U(x,t
inseparability
of
(i).
The
index
" pl)
of
=~ O,
t
over
part
(it)
follows
a curve
with
quotient
k((x)).
trivially
( i ).
Proposition
1.3.15.-
Assume
{Yi } 1 ~i
that
Then,
there
obtained in
P(x,t
polynomial
pi
exactly
completes
real
and
hand,
UP --(x,t ~T
and
k((x)),
exists
from
k,
Let
such
is
another
the
that
[]
first each
be
a basis basis
by
an
y ~.
of
the
ideal
{ Yi ' } 1 g i --
inversible
,
maximal
2
..
two
cases,
linear is
field m,
Yl'
change
Yl~0.
= Yl '
with
a primitive
F.
coefficients
element
of
the
I
extension
Proof:
F /
k((y])).
We s h a l l
a separable
distinguish
parameter First,
have
F'
on
= L(y 2, ...
linear such of
let
F =: L ( y 2 ,
induction
change that
F'/L
the
.... N.
be
yN )
and
of
By
elements
whether
Yl
is
Yl.
is
trivial.
is
induction Y2' """
write
L = k((y])).
separable
over
Assume
N > 2
hypothesis 'YN-1
with
Y ' ' ' "" " ' Y N'' -I
L.
there
and is
an
coefficients
are
primitive
We We u s e set inversible in
k,
elements
.
F
into
F = L(y~',y
an
N)
by
fl .....
algebraic
there
exists
fn
( n = [ F
closure a ~
of k
t
11
YN = a Y 2
+ YN'
for
a such
F.
such
fi ( a Y2'' + Y N ) ~
Thus
and
each it
the
variables
new
separable,
N=2,
).
to
not.
YI
For
,YN_I
Denote from
or
according
a
: L ]
)
the
Since
k
is
infinite
and
that
fj ( a Y2''+ yN )
,
L-isomorphisms
is
a primitive
for
i /: j .
element
of
F / L.
20 yll
Now, 2
in
the
same
~
way,
we
choosing
an
may
I
substitute
appropriate
i
a.
C
k
to
be
y:
order
inseparable
and
= n
(y ). s 1 e. x t e n s i o n of we
YN
i
'
a primitive 1
. In
n'
i
Yi = Y i + a
I
element
Iv
by
to
prove
set
the
second
p = charact,
Denote
by
k((Yl))
k.
case,
k > 0,
= k((
1: ) )
contained
n = E F
the
in
assume
F/k
i
s
Yl
: k((Yl))
maximum
F.
that
] ,
purely thus
is i = i(Yt),
inseparable
separable
and
have:
[ki Then
for
the
curve
in
with
conditions [] of ~
Therefore
means
an inversible
of
: k((y~))]
the
local
above
case
y2,
p
ring
I'-I X
k d
holds
since
1: i s
. . . ,y N
linear
i
may
be
change
T'Y2'
subst
with
• " 'YN )7,
a separable tuted
by
coefficients
the
parameter ' , y2,- • • ,YN
in
k,
F/k
.
, by
such
t
that
each
y.
,
2 - .< i
-.
is
a primitive
element
of
I
I
Moreover, since is
the
a parametric
primitive, k
i /
y ' + b y ' i j
j,
is
may
t
be
representation
by
since
y '
1.3.14.
infinite,
it
there
is
with
must exist
separable
assumed
be
separable:
a uniformizing
exist
a
values
and
to
y~' J of b
which in
a primitive
is k
as
parameter
so.
Hence,
such
element
In f a c t ,
that,
of
F/k.
for ,
i
Now, F/k((Yt)).
The
k((Yl)) (yl) between y.' p
we
shall
separability
contains k((Yl))
h
prove
-C k
necessary
= k
F.
Let
Since
y.'
s
a primitive over
Y"f
h is
be
the
element
k((Yl))
extension
minimum
a separable
is
n'
ks
integer
parameter,
of
of for it
then
k((Yl)) which
is
I
that
h
k((Yl))(yl)t
over
a primitive
element
proposition
.
of
is
the maximum s e p a r a b l e
and
(( t p ))
Y i
degree
i
I
that
.
>~ i , k((yt)) of
and
actually is
n' .p i
F/k((Yt)).
h = i . = tr F This
Then, : k((Yl))
completes
the
degree
] ,
and
the
proof
of y.i'
hence of
the
is
21
4.
TRANSVERSAL
Henceforth bases of
of
the
1.1.8.
seen
that
to
conditions
we
maximal
These
bases
In
characterizations
transversal
parameters.
by
using
say
that
We
.-
nilpotent
in
gr
it
shall
these
to
9ire
We
impose
several
the
have
already
some algebro-
which
compute
with
properties
bases.
parameters,
also
out,
normalization
suffices
we
shall
[]
x C
( []
pointed
will
be
multiplicity
called of
the
parameters.
Let
a parameter
have
normalized
section
transversal
1 .4.1
Definition
called
of
we
9 the
properties
this
MULTIPLICITY.
as
m__ h a v i n
those
x 1
work,
were
geometric
curve
shall
ideal
obtain
to
PARAMETERS.
be
m
an
is
irreducible
transversal
We
algebroid curve. 2 x + m is not
if
).
m
Remark
1,4.2.-
transversal
For
a geometric
parameter
we
shall
interpretation use
an
of
embedding
the
in
k
concept N
of
by
means
of the
be
minimal.
isomorphism
gr defined
by
a basis
(B)
m
{x.}
14:i
I
If
xlg
2
m-m
k(X 1 .....
,
of
~N
the
m
,
which
x+
m
2= ~
- -
has
some
solutions Suppose
the
linear
x
needs
not
+ m I
lg
2)
- -
i ~
is the
linear
variety
in
k
N
given
foPms:
~I XI
where
that
Xi ( x .
i=1
~.. ~: k ,
a
equation
N
(I)
XN) J
(
+
...
~.1 , , . . , ~ N )
+
;kN X N
ranges
= 0
over
the
set
of
solutions
of
(1).
by
22
]he
dimension
of
~
r-I
is
,
being
r = Emb(r-]).
When
N = r,
i.e.,
X
when
the
basis
is
minimal,
is
hyperplane.
a
x
Theorem
1 .4,3.-
statements
are
Let
x
a parameter"
of
the
curve
El.
The
following
equivalent:
a)
x
is
b)
x~
a transversal
m
2
and
parameter.
for
any
embedding,
the
linear
variety X
does
not
contain c)
the
Every
tangent basis
cone. { x.}
]
of
m
14
i ~.N
--
~ with
x = x 1 ,
is
normalized. d)
There
exists
a basis
which e)
x
is
(a)
<:
)
an
is
{ x.} i
of
m --'
element
(b)
x = x 1,
normalized. of
m
having
the
minimum
N Proof:
with
Suppose
x +
--m2= ( 7 -
~i
x.
value
in
v.
and
let
_a
is
not
_m2
) +
i=1 the
ideal
which
defines
the
tangent
cone.
The
element
x + m
2
N nilpotent
in
words,
x
gr
is
(D)
--m
if
and
only
if
~
~.
i=]
a transversal
parameter
if
and
X.
I
~: V ~ a .
i
In
other
- -
only
if
x ~ m
2
and
- -
does
not
contain
the
(b) By we
] .3.6. must
> (c) thethree
prove
the
x = x 1 = 0 Now,
polynomial series a Xi must
is
of of
we x.i
type
+ b X 1 = 0 have
{x.
latter
(0),
is
take
for
over ( a X.
a ~ 0.
Thus;
}
1 ~
be
N
a basis
of
properties
m
--
are
with
x = x 1
true.
However
"
two,
which
contains
I
normalization
former
X
cone.
Let
EF~k((Xl)) ¢
If Hence
tangent
then
E: p
since
i ~ N,
the
p_
is
prime.
a contradiction. each
i,
k((x])). i
X]
+ b X the
Its 1
)s
tangent
2 4
leading
form
; therefore, cone.
But
irreducible as
the since
two
variable
h~)perplane (b)
is
true~
we
23
u (g((0)
x )) = u ( g ( ( X I)
--
(c)
::~
(d)
:===~ ( e )
~
(d)
It
is
i
--
suffices
toprove
Take
The
a basis
=
{x.}
normalized
--
(see
x=x 1
i
~N.
of
xi
over
1.2.4.).
k((Xl))
Since
g
verifies
is
monic
in
I
g((xl),X
implies
i)
= 0
_V(Xl) ~ _v(x-) .i (e) for
v
~
ideal
p_
a ~ 0 ; or
plane
(b)
in
change
linear
the
2 <
with
1 , ~ i ~ N
equality
I
value
,
polynomial
u(g((X1),X.)
I
X. , the
.< _v(x-)l
irreducible
u(g((0),X.)) --
_v(xl)
i
evident.
t
tt
x )). '
m
of
First
. Therefore~
bases
in
defining
the
same,
X 1 = O~
x ~
the the
which
is
m__
for
because any
to h a v e
curve
tangent
one
2
m
of
is t h e
embedding x-
cone
xI C
we
can
minimum make
hyperplanes
_m 2 . T h e n
no
defining
~
a
in m h a v e a l e a d i n g f o r m of t y p e a X I' is not c o n t a i n e d in the h y p e r -
can
the
v_(x)
series
. x
Remark
1 .4.4.-
condition by
in
the
a stronger
in
algebroid
definition
of
plane
curves
transversal
(E. m b ( [ ]
parameter
)
can
~< 2 ) ,
the
be substituted
condition:
" x divisor
For
is t r a n s v e r s a l gr
if a n d
only
if
x + m
2
is not
a zero
( rl ) ,, m
The a
verifying
equivalence gr
z e r o
( El ) ~
k [_X 1
because
,X2"]_
/
m
divisor
without
being by
the
in this
case
is p r i m a r y ;
the
ideal
then ~ any
a
in t h i s
However
given
is o b v i o u s ,
graded
, for
nilpotent parametric
ring
twisted in
gr
is n i l p o t e n t .
curves,
(El). m representation:
x + m For
2
instance
may
be
, for
a zero the
divisor
curve
24
4 X
=
t
y
= t
Z
=
5 10
over
the
since the
complex
x
has
following
with
2
z + m
Definitions to
be
minimum
2
+ m
t
0
integer
for" A
element
value
)
(z
2
+ m
5.-
The of
which
large
Proposition is
a
We
thus
it
that
,
x ~- y
but
3
in it
st
Any
prove
the
zero
=
0
g
of
exists
some the
be
+ q
order
order
it
a
c
.
,
m
of
[]
for
n
=
]
an
x + m
ring
divisor
since
r~ (L_!),
gr
for
,
2
Take
,
i.e.,
large
is
defined
it
is
the
enough.
curve
c
[29)
[]
> 0
,
parameter
]
for
E]
in
the
above
other.
;
the
integer
e(E[])
for
is
a
which
m n-1
The
p.
x
divisor
285-294.)
of
a
curve
E~
.
definition
containement We
zero
curve
ideal = en
formula
prove
x + m 2
the
Zariski-Samuel
of
fop
to
maximal
transversal
element
suffices
case:
graded
a
shall in
of
::~
is
distinguish ( El )
gr
--
1
the
is
3
) + m
of
x ) F~ m c
I .4.6.-
as
2
= ( y
there
(See
element
according
nilpotent v
dimk(r-]/mn)
enough.
must
superficial
in
element
such
superficial
Proof:
)
the
superficial
x E: m
n
not
multiplicity
( m n : []
for
is
holds
multiplicity e
2
.
1 .4.
the
t
x+m
equality
( x
+
field,
the
11
t
always two
or
true,
cases,
not.
m
is
not
c = 1 .
a
zero
Let
divisor
z E: m
be
. an
element
such
that
xz
£: m
n
n - 1 but
z @
m
(s+l
~< n - l ) .
s . Denote
by
s
the
minimun
s+l
Then
z + m
~ 0,
and
integer
for
which
z C
m
s+l -m
25
s+l + m_
(z
Since
s+2
~ n
m2)
) (x+_
the
right
s+2
= zx+m
hand
side
member
is
0~
which
is
a contradiction. 2
nd
case:
x + m Let
2
us
is
a zero
consider
x. = x. + m 2 ~ 1 4
Set
t
which
I
divisor.
a basis
i
<-<.N. L e t
a
the
tangent
cone.
1 .2.5.),
But in
k/~a
associated
( [] )~
( X2
+
(X 1 ....
(X]
, . . .,X
N)
are
with
x = x 1•
(x I ,
ideal
of
XN
0¢NX 1)
homogeneous Thus~
of
to
XN
is
+
an
unique
ideal,
isolated a
an
is
an
k
divisor
. . . ,
and
as
XN )
therofore
its
Then,
divisor
As
,X N)
component.
all
is z e r o
component.
component
a.
component.
x1
~N X1 , X1 ) = ( X 1 . . . .
embedded
of
embedded
embedded
embedded
and
+
prime
is t r a n s v e r s a l ,
belong
,X N) the
X 1 .....
divisor
0~2X 1 . . . . .
ideal
~2
unique
x1
must
the
a homogeneous
the
since
XI
is
is prime
X 1 ~: V-T'a_ grm
the
m
Since
= ( X2
another
be
of
1 --
--
defines
(see
{x. }
Furthermore, of
a
because
associated
prime
a divisors
. any
primary
decomposition
of
a
is
of
type
v
a_
where primary
where
..............1" = V~a_ decomposition
and
=
ql
~
~/q2 of
the
.....
q2
(X1 ideal
IN
'
, . . . , X N) (0)
;1 7
in
.
It gr
and
m
follows (r7)
that is
of
any type
.....
;.7
is
26
Take z E: m c b e
as a b o v e ,
an
an
integer
element
the
c
such
integer
> 0
for
that
such
xz
that
which
(x 1 .....
n
(~ m
but
z @ m
s+]
z ({ m S - m
xN)cC
n-1
(c+]
q2-_ . L e t
Denote
~; s+] ~
by
s,
n-I).
We
have
(X
+ m
2)
.( z + m
1
Since
-
Xl
~:
~/ -q l
hand,
z = h(x]
,
....
this
relation
,x N)
can
tion
conclude
since
that s+1
z ~ m
Theorem
t .4.?.-
,
z + m
If
) = XZ + m
x
s+2
h
z + m s+l
is
a series
is
Ig --ql •
= (0)
~2
a transversal
,
~ 1 .
•
of
= h(x 1 , ' ' ' , x N ) (~ ( X l s+l
= 0.
--
implies
where
s+1
z+m
We
s+l
--
On
order
.....
other
s >/ c .
Then
XN )c ~
is
which
parameter
the
of
q-2
a contradic-
a curve
I--I,
then:
e(17
Proof: e(m)
Since = e(E]
x) On
thus
we
x
is
the
have
this
elements y tg m n =
[
of
other
,
hand
is
remains maximum
in the
module
is
]
for
I~ • for
elements k((xj).
a finite
to c o m p u t e
[O
there
,y
have
independent
exists
F /k((x)). }
and
: k((x))] . Recall
k[(x)J-lineary
extension
we
k([x))-module,
300):
{ 1 ,y .... Then
El,
294). type
I .2. I 5.
the
fop
page
of
By
1
page
(29),
us
number
primitive
, the
over
,
= v(x).
order
(29),
[]
(Zariski-Samuel
which
independent
superficial
is the
F : k((x))
: k((x))]
(Zariski-Samuel
It only that
) = [F
in
an If
[]
are
element we
set lineary
27
n
Corollary
are
I .4.8.-
For
: k(~x))]
a curve,
(a)
e (F])
= 1.
(b)
[]
integrally
(c)
There
(d)
Emb(El)
(e)
[]
is
exists
~< [ - ~ - :
the
~< n .
conditions:
closed x C m
k((x))]
(and
such
thus
that
normal).
v(x)
=1 .
= l.
is r e g u l a r .
equivalent.
5.
RESOLUTION
Let over
k.
maximal
OF
be the
SINGULAR
local
denote
by
ring F
of
its
TIES.
an
rreducible
quotient
algebroid
field,
and
by
curve
m
its
ideal.
I . 5. ] .-
quotients
and
[]
We s h a l l
Notations of
~< [ 0
[]
= []
For
each
belonging
to
( x-1 m )
the
x
x •
F
which
have
let
the
[]-subalgebra
x
form
of
F
-1
m
be
z/x
the
with
generated
by
set z ~: m ,
x -I m .
--
Let
T'
LJ
=
Spec([-]
XC m--[O ]r elements
m , x ~ O,
of
Spec(D
)
,
and
design
by
f2
X
,
X
f~'
X
). X
Notice
which
we
consider In
T'
that
since
can we
be
give
~ x E; S p e c ( [ - ]
~x
F
is
a field,
assumed the
x)
"J~ ~ y
to
following
'
<
be
all
the
subrings e:quivalence
quotients
of
F. relation:
~ y E; S p e c ( l ~
Y)
( E l x)
y
~
:(El x
,
y
rings
,etc..,
28
Denote
The they
aim
to
finally
results
study
which
the
First,
T
by
witi
set
for
We
set
shall
formed
y,z
by
denote
equalities
by
are
,
Proof:
0.
{0}
the
Let
Then
First,
x
f(X,y)
The monic
are
technical
and
,
let
N
of
be
y,z
(z/y)
type
quotient
ring
N
the r
• []
y~z
muttiplicatively , y
where .
The
r
>/0.
following
z,y
x
x/y
is
is
be
over
=
X
k
same,
+ A
y
in
[]
over
k((y))
k((y))
with
(y)
k-I
is
a transversal
a unit
integral
polynomial,irreducible
new
next
[] y (y/z)
] . 5.2.y {
proved
evident:
y,z
y E; -m -
T'/~
_
elements
yTz
[] y , z
Proposition
be
~: m -
the []
set
T.
_
closed
the
X
k-l
integral
+
, thus
it
is
E] ,
a zero
coefficients
....
+ A
o
k ((x)],
over
for
and
•
y
over
polynomialjirreducible
parameter
in
a monic
k((y)~
,
(y).
thus
k((x))
of
with
it
is
a
zero
coefficients
of a in
k((×)) , 1-1- 1
h
these
g(x,Y)
=
We
going
irreducible
irreducible differ
only
in
m
two
to
Y
find
a
look
contains
and
at
this Since x
and
+...
relation
are
variable
a unit,
(a X ~ b y)r. which
+ Bh_l(X)
polynomials
as
To type
are
Y
in
there y,
o
way,
always and
f they
Moreover O(A
another
between
distinguished,
series. hence
+ B o (x).
this
(y)) the
by
g.
are
Since
atso
W.P.T.
they
= h. leading
exists basis
the
and
a basis is
form of t h e
normalized
of
f
maximal because
is
of ideal
29
x is
transversal,
~(Ao(Y))
we
= h = r Set
f~(x')
= y
Since
y
hence,
the
have
not
x/y.
o
have
A~(y)
0 (see f
is
in
a unit
= f(x'y,y)
and
[]
it
Y k((y))
in
Therefore,
h.
+...+A~(y) o
f~(x')=0.
1.2.3.).
0 = f(x,y)
divisor is
-I
of
We h a v e
a zero A~(y)
b~
order
k+ A ~ (y)x,k-1 k-1
x'
is
and x'=
since
we
h),
k-h
must
= y
that
(because
the
in
the
,),
f~(x
A~.(y) = A.(y)/ j j
follows
Collecting
h
y
h-j
where
C: k ( ( y ) ) .
f~(x')=0, order
same
side
and
of the
A
o
(y)is
terms
of
o
the
last
equality
element
f'(x')
containing
x',
= -A~(y)-I(
we
obtain
x'f'(x')=
yk-hx'k-l+A'~ l(Y)x'k-2K-
1,
+...+A
where
1~ ( y ) )
the
belongs
to
O
.
hen
is
x'
a unit
in
as
[]
Y
Corol we
desired.
Y 1 .5.3.-
ary
have
[]
y
For
x,y
E; m -
{0}
,
where
is
transversal,
x,y
Proposition
1 .5.4.-
If
x
is
transversal,
I~x : S p e c ( F l
x
the
"~
)
map
T
×
is
x
X
bijective.
Proof: Q
It =
is
because
£
x
~
~' x
ze==~ ( E l
x
)
~,
= ([[] ×
X
)
~
' -" "*" " x
Q'
X
X
In y ~ O,
injective
order
y E: -m . -
Then
to
prove
the
that
it
contracted
is
surjective,
ideal
of
£Z y
take by
the
meet
N
~
E; S p e c ( O Y canonical
homomorphism
[] is fore
a prime
ideal
~
x
;
x
of
[]
x
FI
x,y
which
:El
y
does
not
verifies
x
~
X
• y
y
x
x,y
and
there-
), Y
30
Lemma
0,
I .5.5.-
Let
0
and
a local
D'
i
is
are
m
0'
i) T h e
two
ring
such
that
k[[x))
only
one
a
k[[x))-module
the
ideals
such
that
l =m 2 ' w h e n c e
submodule
local
of
integral
We m u s t has
prove k
as
respectively, homomorphism = mt
m
i
O
, then
C El.
finite
of
a curve
Then:
Particulary,
type.
is a local
and
k([x))
, we
that
= dim
0'
is
is i n t e g r a l . two
maximal
Since
[]
m~,
If
--2 m'
ideals is
local
m
l'
,
a coefficient m'
and
and
denote
thus q-adic
is
ring,
be by
the q'
13'
m'-adic Consider
a complete
of finite type.
Furthermore,
since
for
its
Krull
topology,
and
that
maximal
ideals
the
extended
Since
Spec(
of
ideal
13'
0'
and
of
q
by
) = { (O),m'}
k([x)) the
, we
have
m'-primary.
topology
and
domain.
El' is a k{[x])-
and
field.
q
~ q'
of finite type
k[[x).] = ] .
complete
over
as
is
algebroid
have
topololy
its
irreducible
[] / k [ Ix])
noetherian
filtration
also
an
--i
by the filtration q 0' 2 n q ' _~ q ' ~ . . . D q ' D
given
of
O t is itself a k [ [ x i ) - m o d u l e
k((x)]
The
k['[x])
ideal
ideal. of
ring
i.s a k [ [ x ) ) - m o d u l e
over
Let
is
maximal c 0'
[] ' , t h e r e e x i s t of [] ' = m' , i = 1 , 2 . ('~
dim 0'
is
the
ml'=rn ~
iii) 0 '
it
maximal
ring e x t e n s i o n
maximal
[]
in
is
of
is
ii) []
it
element
k.
over
Proof:
-m 2
is
be
noetherian. iii)
curve
an
has
El'
ii 0'
x
since
q'
El'
~ q20' . . . is
,considered
as
~ . . .'~q nO'
This
is
~..
actually
m'-primary
and
k[(x]}-module, ,;
its 0'
i.e.
by
q'-adic
noetherian,
it
topology. the
q-adic
submodule
topology and
O'
over is
free
the
k[[x)]-module of
finite
the
type
0' over
31
13'
k((.x)) , then It field.
The
which
is
is
only
complete.
remains
canonical
actually
Proposition
curve
only
of
ring
map
an
isomorphism
Let x
[]
to
~
[] a
El'
that
FI'
,
/m'
has
induces
k as
a coefficient
a homomorphism
r - ] ' / m,
since
be
the
/m'
local
transversal
(y ~ m-{O})
y
see
k[~x~)
k[[x)]/q
and
type
us
k=
I .5.6.-
algebroid
for
is
ring
algebraic
of an
parameter.
such
that
the
over
k,
irreducible
Then
[]
(:::El
(see
is the
×
map
r-~
: Spec(U
)
Y
)
T
Y Y
Y is
bijective.
Proof:
Since
is t r a n s v e r s a l
x
we
have
0
C. []
1 .4.3.).
x
By
the
above
lemma
Spat([]
)
consists
of o n l y
two
ideals,
(0)
and
x
the
maximal
If
y ~
ideal
m-
{0}
mx.
is
an
Hence, element
by
1 .5.5.
such
that
T t~
--
consists
is
these
two
Since
clear
ideals, ([])
that
t~ ( ( 0 ) ) y
[]
is
atgebroid
defined
quadratic
the
=
Let
to
above
curve of
= []
and
[]
[]
over
the
transform
is
x
be for
results
Then
(El)
~ ((0)) x
be the
transversal
m
-- y "
and
y
parameter
The
choice
(0)
asserts
I .5.?.-
x a transversal of
y -m- y
equalities
Definition
is
exactly
two
bijective,
elements.
Spec(E]
)
y
of
local.
has
is
= F
maximal
and
t~, i s y
t~ (m ) = t~ (m ) . y --y x --x
and
[]
is
y
bijective, The
it
last
of
= (El) = (I--]) = [] x m y m y --x ~y an D
irreducible . The
curve show
same
m
__y
y (0)
and
Y
field,
parameter. intrinsecalty
curve
(strict)...quadratic
9ivan that
algebroid
by []
the 1
is
ring an
which
does
Note,
in
[]
transform 1
= E]
x
irreducible not
depend
particular,
constructed
and
from
the
on
the
that
the
ring
D.
32
Theorem
1 .5.8.-
transform
.
The
i)
0 1
ii)
particular with
O
__
Let
O
following
be a c u r v e
and
properties
hold:
is the local ]
__
is
ring of s o m e
integral
over
0
the
= [] and the 1 associated valuation iii)
transversal
Let
B =
. There
{x.}
exists
O
0,
to
hJ
E1C
valuation
algebroid
[]
C
1
to
O .
[]
curve.
t n
coincides
]
a basis
1~< i ..
a set
irreducible
i.e.,
associated
its q u a d r a t i c
I
of
{a. }
m , where
C k
i
l~
x
--
such
is
1
that
--
B1
= { xl , _~ X
is
a basis
of
the
iv)
basis
B
maximal
I f we
given
x
a parametric
rn]
01
aN }
]
.
representation
,
I < i
for
0
in
the
=
x
With
0 1
=
vi)
The
0,
x
I
01
k((Xl) ) (x 2' ,
t I
is
1
given
we
have
by
(t)
representation
v)
for
1 t x'. = x . ( t ) /
of
of
a parametric
I
the
ideal
= x.(t) i
i
x
centre
N X
l
representation
(2)
The
,
by
(l
then,
have
_ a2 ....
1
(2) , B,
needs
B1,
.... X N' )
contracted
,
(t)
ideal
24
not
x i and
,
the
be X'.l
Emb(rl
of
i<~ N ,
(0, .... as
0).
above,
I) ~< E m b ( O )
ideal
Pl
,
defining
e ( O I)~< I-'1
I I
1
homomorphism T : k ((X 1 .....
X N) ) X
I
X.,
>
k ( ( X '1 . . . . .
I
~-
X'
;
>
X'1.(X'i + ai ).
e(O).
X ~))
I i >~2
,
by
33 is
the
ideal
Proof: in
p
i)
m,
defining
If
then
O.
x E: m
Vz
is
E: rn ,
ii)
OcO
iii)
Denote
I
transversal,
v(z/x)
>/0,
cO
~
by
a.i
it
is
minimally
whence
z/x
valuated
£. F 1 .
Thus
for
v
1-11cF1.
[] I :: O.
the
residue
module
m1
of
xi/x
1
- aN)
,
,
i >/ 2.
Now,
[]
x2
1
0 X
and
xN
(~--1 ' ....
x2
--xI ) = []
xN
(W-
a2 . . . . .
--Xl
.
I
since
x.i = ( - - x
ai ) . x 1 + a.x l i
'
B1
is
a basis
of
rnl .
1 iv)
It
is
trivial
from
iii).
x2
xN
= the
basis
Finally
B , as
x 1 E: E l
is
to is
, trivially
defined
completes
the
I .5.9.-
minimal, minimum
For []
b)
0 = 0
((x'l
the
(1),
= (w.T}
proof
a)
k
by
parametrization
Lemma
are
the
by
p
This
be
)
= k((Xl))
(x 2 .....
we
F~mb(F1 1 ) ~< E m b ( O ) .
of
get
values
of
v
X'N) .
on
If
m_l
and
is
a
of
,...
,X N '))
~
k ( ( )t)
the
canonic
parametrization and
-1
vi)
curve
(2)
of
iv).
Then
(pl).
.
hence
(0)
and
= T
(w
-I
therefore
the
[]
-1
(0))
of
= T
the
-I
theorem.
conditions
regular,
1,
equivalent.
Proof:
El <El
a) ==> b) 1
cO,
in
iii)
e ( m ] 1 ) ~< e ( I - l ) .
Denote
hornomorphism the
taken
e(mt l)
vi)
is
, ....
[]
actoa,,y
is
regular
13 1 = 13.
is
equivalent
to
[]
= O-
Since
w.T
34
b) If
[]
of
m ,and
y/x
is
~
~
a)
not
regular,
[]
then
and
[[] =~ [ ]
13c[3
chain
Proof:
is
[]
module
at
c
I
of integer
[]
is
The
for
...
completes
Definition be
the
1 .5.11
are
both
Remark of
the
, is
the
OF
y/x
a basis
C
SINGULARITIES).
chain
I
it
c[3.
I+I
of
c
finishes
E] 1 and
Let
successive
...
in
[--~-module, is
therefore
ElM = FTM+ 1
:•
M
[]
be
quadratic
c
[]
and
[]
.
each
E].
stationary, , the
tf
above
is
I
M
lemma
a
r-]-sub-
denotes states
the
that
:rq
M
curves
when
the
rings
in
The
defined sequences
their
over
the
formed
same
field
are
respectively
desingutarization
later.
In
to In
will
embedded fact,
idea
of
equiresolution,
transformation,defined
essential
clear
for
and
.-Two
1 .5. 1 2.-
transformations and
Now,
not
said
by
chains
the
1 .5.10.
identical.
quadratic
I . 5.?. made
of
x.
is
proof.
equiresoluble
m~Jttiplicities
[]
{ x }
El.
thus
[]
to
cFl.
chain
and
set
for
O:
which
regular
y ~C r n -
ascending
stationary
[3".
the
x
1
s a noetherian
first
This
exists
the
starting
parameter
> 1 , then
(RESOLUTION
consider
transforms
M
there
1.5.10.-
a curve
a transversal
Emb([])
therefore
Theorem
This
Ta-ke
classify
order be
singularities
to b e s t
studied
in
illustrate
now
from
which an
is
intrinsic of these
based manner
curves,
as
aspects,
a geometrical
on
point
that
in
it
will
quadratic of
view
curves. a quadratic
transformation
T
will
be considered
be
35 as a c e r t a i n
transformation
of a c u r v e be t h e
[]
in
image
schemes,
an N - s p a c e ,
of
This
U
by
idea
Since
A
ideal
is
M,
the
local,
which
be f o r m a l i z e d
a precise
Consider
Thus,
the curve
as we have
has
N-spaces. []
for
any embedding
defined
1
in
1.5.7.
will
T.
can
because,
"embedding"
of
pointed
meaning
N-space
Spec(A)
sometimes
by using
has
out
in
the
language
of
I .1.3.
, the
notion
of
there.
Spec(A)
where
a unique
closed
point,
reasons
will
by geometrical
A = k[[Xl
.....
its
XN] ) •
maximal
be d e n o t e d
by
O. The hence is
Pr°J(@oMn)n==
denoted
by
TI[: B I M ( A ) with
oo Mn @ is n=o is a finite
ring
BIM(A) )
centre
Spec(A)
type
A-algebra
projective
BIo(Spec(A))
some
(5),
or
1)
BIM(A)
isomorphic
is
called
the
well
known
basic
Romo
Spec(A)-scheme The
91obal
morphism
blowing
properties
= O D + (X.)~ , w h e r e i=1
of
up o f
which k-schemes
Spec(A)
of
BIM(A),
D + (X.)l
is
an a f f i n e
open
set
to
X.'''''X. I
is
biD + ( x )i
the
morphism
i
induced
A
A
by the
IX1 A.-E--. . . . .
D
+
(X) I
2)
glueing
17-I(o)
actually
a divisor)
Proj(k[~
1 .....
~N])
togeti~er
is a c l o s e d
isomorphic
'
I
in the
obvious
k-subscheme
to the
, ~ . = , X.+t M 2 ,
inclusion
XN] X-
I
the
type,
( 19 ) ) :
Spec( A[
and
).
of f i n i t e
O. We n e e d
(Bennet,
(or
a graded
projective
and the
manner. of
BIM(A)
space
canonical
(and oo
ProJ(n=@o M=
embedding
n
/Mn+l) =
36
-1
.fl
(M)
,,
BIM(A)
II5
II
~
n
ProJ(nO__oMJMn+l)
is
induced
by
the
natural co
~ Proj(
graded n
e0
@ M n=o Thus, onto
to
such
a point
(al
the
,...
the
closed
directions O'
,aN).
be
n
@ M /.n+l n=o M
points
of J7-1(0)
the
origin
represented
scheme
homomorphism
>
through
may
The
ring
~ M n) n::o
7"1-1(O)
correspond
in
by is
.
an a f f i n e
its
one-one N-space,
homogeneous
called
the
and and
coordinates divisor
exceptional
of/l. 3) only
if
A point
a.~l 0 .
O'=(a I .....
Moreover,
{ xl X is
a regular
Thus,
system
I~ Bl
(A)~O' of M the h o m o m o r p h i s m
is
-1
(O)
is
point
of
x '
XN X.
i'''''
parameters a --J
~
a regular
i
of
for
the
, j ~ i , and
Z
in
D+(X.)I
BIM(A)
aN a.
if
and
and
}
i
ocal
ring
= X
ai
t~
, the
BI
(A),O'' M complection
l
(A),O' is i s o m o r p h i c M of local rings
k[(X 1 .....
g ven
-
~
L~BI
:
by ~
is
at ... a '
i
X. Z = ---J j Xi
setting
induced
O'
aN) ¢
XN))
to
>
k((Z I ..... Z N ) ) and
k[(Z 1 .....
ZN,]) ,
by a
X X
Definitions theoretical is
called
j i
t .5. 13.morphism a formal
1
>
I
>
The T~:
quadratic
Z Z + i j Z.
J
.... a
,
Z
j~i
,
i
i
homomorphism
T (or
Spec(k[[Z
ZN]))
I ....
transformation
in
its
analogous >
the
Spec(k([X
direction
scheme 1.... (al,..
XN))) ) ~aN).
37
Thus,
a formal
a transforrhation
of
On t h e that
open
other
defined
to
be
hand, ,
D (X.), +
transformation
may
be
viewed
as
N-spaces.
Z i C I~'BIM(A),O
in the
quadratic
is
and
it
follows
a local
from
equation
so the
the
for
exceptional
above
the
construction
exceptional
divisor
of
T
divisor
may
be
I
the
hyperplane
Z.=O
(i . e . ,
the
subscheme
1
Spec(k((Z
1 .....
ZN'J)/(Z.))
of
Spec(k((Z
by
T.
1 .....
ZN'J ) )
C
an
).
Intuitively,
I
0
is
blowed
up
into
Z.=O I
Remarks
and
algebroid
definitions
curve
I .5.14.-
in t h e
x.
=
N-space
x.(t)
I
be p a r a m e t r i c i.e.,
x
point
C is by
O'
Spec(A),
,
for
C.
is a t r a n s v e r s a l
1 .5.8.
vi)
T :
2 .... the
and
i ~< N
Assume
embedded
irreducible
let
,
by
the
XI=0 the
is
transversal
to C ,
curve.
it is e a s y
to s e e
that
direction
of the
tangent
a. = x.(t)/ J J
quadratic
A=k(FLX 1 . . . . .
for
I .2.5.
~a N) , w h e r e formal
that
parameter
From I .2.4. and -I E: 71 (O) determined
O' = (l,a
1 ~<
be
i
equations
I
Let
x i (t)
the
closed line to
(mod.(t)).Moreover,
transformation
XN] )
m
k((X~
....
X~4]] ~ - - ~ B I
(A),O' M
!
( I )
verifies schemes
T as
-I
(EL) = Pl
Xl
~
>
X1
Xj
1
)
X'j X I + a j X 1
' and
so
it i n d u c e s
t
a commutative
,
j >~ 2 ,
diagram
follows i
Spec(E]
(2)
I)
I )
Spec(k((X, I ..... X,N)))
°I Spec(E])
1T" i
Spec(k((X
I ..... XN)))
of
38
where
i
and
p] ~ and
Q
is
i1
are
the
the
morphism
Therefore, by
T
by
Pl ' so
by
into
T,
will
be
(compare
this
definition
that
by
Reducible
curves
two
explained
in
transform
and
an
to
give
lines
each to
Thus
ideal
q
has
total
C'
and
transformed
of
C
quadratic
the
exceptional
r~i,
same
way,
exceptional of
by
X1 =0
T(j)
is
be
of
C curve" which
mentioned
as
that
denote
the
total
the
defined
to
iv))
and
r~.l
it
is
r. we
1
~ the
J
may
Then
by
be
p.
X1 = 0
of
and
strict
of C).
~'~ m
is distinct Each
a formal
quadratic
is noted
J
are
--t
set
cone
C'. U E
~trict
,
the
tj
depthPi=l.
C =lrlu...u
divisor
union
the
that
i .e.
the
with
since
write
C,
tangent
the
clear
tangent
of
with
and
defined
to
union
C <.
XN}I
curve
primary
prime,
exceptional
The
by T(j).
=
-tP" i s
(i . e .
algebroid
1 .....
= {t I , . . . , t s }
1
all
of
Tst(C)
C,
general)
irredundant
curves
A(C)
in
of
an
transversal
T(j).
divisor C
is
r.
O'(j)
we
such
q (:: k [ [ X
1. 1 .2.
of
curves
a point
transform
will
in
where
algebroid
[-~""1 L e t
the
they
ideas
an
= 1 (see
assume
transforms
transform
is
transform
quadratic
(reducible
components
transformation
the
certain
irreducible
to
total
I I 1,3. I . , but
embedded
XN))/q)
irreducible
determines
the
E
I , . . . , X'N) }) g i v e n '
The
of
} E] 1 .
X' = 0 ~ i.e. ~ it is a " r e d u c i b l e I considered in t h i s w o r k ~ b y r e a s o n s
q = 1.
quadratic
In
union
the
q = P l N " " " f~-Pm
transversal
J
by
and
curve.
means
depth
1 .....
tangent
the
[]
quadratic
I .5.?.).
p_
and
clarify
give
Now,
t
to plane
embedded
called
C'
not
a
decomposition
The
are
N-sp.ace
dim(k[['X
given
Spec(k[(X
strict
by
incl.usion
C
(in
with
N=2,
curves,
order
To
= q
when
in C h a p t e r
of
an
the
to be
the
curve
C'
called
respectively
X'] = 0 .
is f o r m e d
in
curve
C is d e f i n e d
Note
seldomly
by
embedded
C'
hyperplane
C
the
embedded
of
given
induced
the
transform
are
embeddings
J
(rep.
where total)
by
C'.j E . J
is
qMadratic
be
U t E A(C) J
C-~ J
(rasp.
T
(C)= tot
~ C ~ ) t EA(C) J ' J
39
and
C'
(resp.
j
C .~) j
T
(C)). (In tot disjoint union connected
is
(C) st N-spaces,
14.-
p_ d e f i n i n g
formal
T
of
1 .5.
ideal
called
components
Remark
p
fact
are
of
In an
quadratic
and
1. 5.8.
vi)
it
= f
is
g yen
irreducible
n
the
that
f(X,Y)
of
(X,Y)
+ f
n+l
C is p
(X,Y)
plane
and
We the T
tangent given
Then
(X,Y)
consider line
Y-
a X = 0,
have
X
=
Y
= Y'
f(X,Y)
The
may
be
Notations
of
the
C
under
and
hence
+ ....
c ( Y - a X )n
c,a
f
,~ 0 . S i n c e n thus, if X=0
E: k
and
c ~ 0.
0'= (I ,a) • ~I.-I(0) d e t e r m i n e d
and
the
formal
prime
quadratic
by
transformation
by
we
and
=
the point
X v
it
is
series
,
and
the
Y')
f'(X~,Y
constructed
called
X'
+ a X'
= X 'n f'(X',Y
f'(X'
C'
(resp.
set
is a homogeneous polynomial of degree j and j f is irreducible, f is a power of a linear form and n is transversal to C ~
n
t(C)
s
curve
f.
f
T
behaviour
a!gebroid
Assume
a generator
f(X,Y)
of
(C) are subsets of a topological tot and C'. a n d C are respectively the J J these subsets.
embedded
Take
components
T
transformation.
principal.
where
connected
strict
definitions
~)
= y,n
')
, where
+ X'
is
directly
fn+t(1
,Y'+a)
a generator from
of
the
ideal
f(X,Y)
and
T,
quadratic
transform
of
f
1 .5.'15.-
In
1 .5.10.
~ Y '+a)+..
+ X'2fn+2(1
is
so by
showed
defining this
series
T.
that
a
a
40
sequence
of
quadratic
desingularize are
a curve
studied
will
we be
shall
infinite!y
and
O
near point
Spec(A)
with
M1
the
infinitely
use
point
of
01
the
in
O.
maximal
if
of
is
defined
M.
an
infinitely
near
point
an
infinitely
near
point
of
the
from
O
in
and
of
its
first
such of
A(O1).
O in
its
denotes
N-space
the
A(O
t
in
I
its
defined
blowing ) =
ideal
,
of
to
up
BIM(A), if
O.i
be
of O1
is if
A(O.),
I
Spec(A(O.))
for
An
neighbourhood,
maximal
to
points.
definitions
is
recurrence
order
near
Spec(A).
of
set
1 ' By
i-th
the
basic
neiqhbourhood
O
order
point.
divisor a
infinitely
the
view
in
transformations
of
so
an
exceptional For
considered
quadratic
a modern
point
be
language
language
ideal
point
and
this
a closed
centre
neap
using
now
be
may
Sequences
by
introduced Let
a closed
[].
classically
Sometimes it
transformations
an A(O.)
I
we
define
I
first
A(O.
a+l
of
0
in
O.
of
i+1
order
) =
its
( i+l)-order
the
closed
neighdourhood point
neighbourhood,
O"
B
M. t
O. ( = M . ) i
and
(A(O)),
0
set
i+1 /
02 /
"-. "\
\
~
\
l '
I
II
I/
I/
~
.,,i t l
/
//
r
o1,¢/.
It
x
i//
I l l/ Ii1
I, yl
be
of
L
we
to
./.. .11
////"
./
ff~
~ x x j / ~ , .¢/I 0
FoP sequence
Oo,01
neighbourhood Now, Spec(A)
a sequence
.
of
~ " " " ' O " " ' l" " of let
~.
I
for
C be
all an
infinitely such
near that
O.j+l
points
we
means
a
is
a point
in
the
first
algebroid
curve
in
the
N-space
i >-0. irreducible
41
Proposition-Definition sequence the O
Oo,O1
sequence
is the
' " " " 'O1
of
by
point
the
of
Spec(D)
embedding
given
completely
the
Proof:
O =O.
embedding
The
o
corresponds induces
to
an
points
by
i
of
M
C).
of
graded
n/Mn+I
the
M (resp.
functor
m)
Proj co
(J
is
the
the
maxima
to
the
sequence ~ )
map),
we
oo
Proj(j) Proj(
the
~)
is
diagram
inclusion has
a point (2)
)
in
only
of
Mn/Mn+ 1
i
the
O 1 , infinitely 1.5.14.
takes
closed
origin
point
of
this s e q u e n c e
i
A
,14
~,
[]
, which
A (resp.
On
applying
oo mn / n+l @ n=o -- m
Proj(~ ~)
n
@ M /..n+l n=o M II1 -1 TI (0)
in
BIM(A)
(see
)
form
of
O,
oo
)
P r o j ( @ Mn) n=o II BI
(1 . 5 . 1 2 . )
] .2.5.),
near the
El).
sequence
eo
TI-I(O)
a point
(the
% mn/ n+1 n=o -- m
"~ P r o j (
of
the
of C
rings
Proj(j)
Q mn/ n+l n= o - - m
~ mn/ n+l ) n=o -- m
Proj(-[ the
is
rings
ideal
obtain
O
is c a l l e d
C,
n=o
natural
Pro j(
with
which
C.
by
ocal
,
is
Mn
@ n=o
origin
uniquely
Spec(A)
n=o
where
points
Conversely
curve
)
determines
near
of the
g yen
epimorphism
a homomorphism
C
identified
embedded
Spec(D)
curve
infinitely
near
determines
Let
The
of
infinitely
closed
Spec(A)
1 .5.16.-
then as
M
(A)
and its
desired.
image
by
Furthermore
a
42
i
S p e c (I--]i )
(3)
I )
Q ~
i
~
Spec(D)
and
the
rest
of
the
,
is
let
O
constructed
,
,O.
O
respectively i.
the
We c l a i m
the
claim
and
if
that
for
C=C'
C and
Xt=0
because
sequences
is
for
:
x 1 =
x 1
x.
a
= j
induction.
and
O'
C' , and
by dia9ram
assume
(3)
be
I
O =O'. 1
we o n l y
for
1
need
case,
if A=k([X 1.....
X1 =0
is
In this
we h a v e
O~ O
t o C~ t h e n
O1 = 0 I . T h e r e f o r e ,
C
by
,
C and
regular.
transversal
T~
I
In fact,
C'
)
Spec(A)
)
sequence
Conversely
Spec(A(O1)
to prove
×N))
transversal
parametric
to C',
equations x 1 = x 1
x
+ a
jl
1
2
x
+.
C':
. .
×. j
1
j2
=
a l, jl
× 1
2 +a I x +. j2 1 1~<
Now~ and
since
O.=O'. I
hence
i
,
one
may
prove
easily
a..=a'., jI jI
that
for
j
. .
4N.
all
i,j,
C=C'.
Definition
1.5.17.-
The
multiplicity
e(O.)
of
the
point
O.
I--
sequence is
atl
of
defined
infinitely
to
be
near
points
of
multiplicity
the
the
of the
in
the
I
origin
ring
of
D.,
an
irreducible
i-th
curve
quadratic
1
transform
of
U .
Sometimes, on
the
at
O..
points
0
,0
o
I
simplicity,
, . . . , 0 . ,
l
. ..
we and
,
shall that
it
say
that
the
has
multiplicity
curve
lies e(O.)
I
Remark
I .5.18.-
For
components
and
them,
obtains
from
one C.
adding
To the
the
each
a reducible
sequences a tree
point
for the o r d e r
are
C , taking
of infinitely
of infinitely
of the
different,
of the
curve
of this t r e e
multiplicities
the c o m p o n e n t s 1
for
one
near
near
attach
irreducible by
I .5.16.,
neighbourhood
points
points,
may
its i r r e d u c i b l e
uniquely
large
in it.
multiplicities
enough.
by
constructed
a multiplicity,
components these
determined
by
Since became
43
If then the
it
follows
i-th
strict
embedded
to
formal
is
the
is
l
a point
from
the
of
C may
be
quadratic
the
i-th
the
tree
that
in
for
transform
N-space
embedded
of
1.5.16.
quadratic
in
transform
are
O
each
of
Moreover,
of
the
the
q
lying
on
observation
enough, two
O.
is
I
regular
proximate
end
this
points
double
i-th
strict
is
actually
q
O.~l
quadratic
making
reference
component
at
O!
components
O..t above,
for
transform. in
point~
on
this
i.e.
i large
(If
way,
C is
and
, a normal
enough,
plane,
for
i
large
crossing
of
chapter
for
the
we
case
shall
N=2
introduce
which
will
satellite
be o f t e n
and
used
in
the
. Let
an
ordinary
of
lying
curves). To
sequel
an
q
irreducible
made
O. i s a r e g u l a r point for the i-th strict t the i-th total transform may be defined
neighbourhood,
ring
connected
whose
I
order
without
its
Spec(A(O.))
by
the
intrinsically,
in
transform
local
Thus
transformations:
curve
i-th
component
the
Spec(A(O.)). i defined
the
O
infinitely
be
near
the
closed
point
0
point of
0
of in
a 2-space
its
i-th
(=plane).
order
Take
neighbourhood.
i
Let
E(O
i
)
the
exceptional at
O,
of
total
exceptional
divisor the
of
~
(i-l)-th
at
curve
at
0 1 , and
total
©., t
E ( O )t
quadratic
i.e, is
E(O 1 )
the
transform
is
the
connected of
component
E(O1),
for
i>~2.
I
It
is
evident
Definition free that
1
respect O.
that
is
•
O.
is
I
5.19.0
a regular
We s a y
if
O.
is
a satellite
that
or
ordinary
the
infinitely
a non singular
of
0
if
it
is
double
near
point
of
a singular
point
point
E(O.); point
and
( and
of
O.
E(O
of
I
we
O is
say
actuatly
i
ordinary
double)
of
E(O.). t
Definition satellite called
1.5.20.-The of
O
if
a leading
Definition
1 .5.21
satellite
O. I+1
is
free
point
.-
Given
free of
point respect
O O.
is
I
The
catted point
a terminal O. I+1
is
then
O.
a point
O
in i
the
sequence
of
I
infinitely
).
44
near we
points say
that
of 0 i
the +1
origin '
Oi+
2'
of ...
an '
0
i+r
embedded ape
irreducible proximate
plane points
e(O.), : e(Oi+1)+ e(Oi+2)+... +e(Oi+r)
of
curve 0
i
when
C,
CHAPTER
HAMBURGER-NOETHER
material section
The
purpose
which
will
we
expansions
shall in
expansions, the
remaining
is
is
in
the
there
is
is
replace
ALGEBROID
to
give
following no
the
the
CURVES.
technical
chapters,
possibility The
of
In
the
obtaining
first Puiseux
Hamburger-Noether
Puiseux
EXPANSIONS.
well
known
series. at
all
us
that
field
ones,
are
developed
in
of
true,
and
consider algebroid
DISCUSSION,
every
algebroid
characteristic
We s h a l l the
see
curve
zero
how
in
classical
a primitive []
curve may
positive
methods
parametric over
over
an
be
an exhibited
characteristic are
not
sufficient.
representation
algebracally
of
closed
k:
x.
= x.(t)
I
Assume
chapter
characteristic.
closed
irreducible
field
that
will
It
not
this
used
positive
PUISEUX
Let an
see
I.
a Puiseux
this
of
OF
sections.
algebraically by
EXPANSIONS
be
which
II
that
N = Emb
The sentations
of
U(x --
)
>
0
,
1 ~ i 4 N.
primitive
parametric
I
([[:Z1).
procedure []
,
I
is as
to
get
follows.
all
the
First,
if w e
keep
the
basis
repre-
46
{xi
} ] ..
parametric
of
the
maximal
representation
parameter
by
Now, another
one
we
{x'}
j
= c 1 t'
may
fix
xj
is
J
a
series
Jacobian
matrix
maximum
rank
Remark
2 . 1 .1 . -
in
k((t~)
for
of
which
of
the
order
x
-
t'
and M)
of
the
(it
is
uniformizing
can
,
basis
t
of
zero, is
form
4 J ~< M
>
the
0,
and
series
for
any
f
the has
i
series
always
a substitution
to
use
indeterminate
curve
a
be
}1 ~< i ~
{xi the
order
]
CliO.
the
N.)
sufficient
representation
k
takes
'
degree
each
the
with
actually
for
primitive
the
g:
relation
XN)
there
c.i
changing
characteristic >0,
other
changing
,
variables
is
Consequently, parametric
by
any
+ ...
I .....
forms
n n
~
N
rank
In
,
(N
in
.(]-his
+ c2.t'2
f (xj
fixed,
type:
t
~<j~<M
I
m
obtained of
t
(2)
f
is
a substitution
(1)
where
ideal
Puiseux
obtained
in
any
x of
=
x(t)
(1)
type
coefficients).
type
of
primitive
basis
of
m:
n x I
=
t'
x.
=
x.(t')
I
If
charact,
of
order
n,
In
fact,
for
for
which
Example
show
that
k we
..
2
~
= p >
can
instance,
0
assume
for
and
if
the
x last
= t p + t p+I
x
=
x t)
~
k((t.])
property ,
there
only is
no
is
a series
when
(n,p)=l
substitution
. (1)
x = t ~p
2.1 . 2 . -
privileged relative
then
,
I
basis to
A of
priori, its
a transversal
this
question
nevertheless,
maximal
ideal
parameter, has
a negative
every
having
curve
a Duiseux
VVe g i v e answer.
below
an
could
have
expansion example
to
a
47
Let
us
consider
x
= t p
y
over
a
meter
field
of
the
=
curve
+ t p+] m
t
,
characteristic
p
> 0.
,
with
we
x'
= a x + by
x'
= at p
+ f(x,y)
+ atP+l
conclude
uniformizing
that
+
it
parameter the
g(t)
is
,
not
y
such
that
no
Any separable
cyclic,
if
tn
2.1
_m
+
a
t p
it
para-
which
and
and
if is
-
m
>
p+l
,
x'
write
>/
2,
i.e.
a ~ 0.
= t ~p
for
any
2
always
us
can
/0 a
parameter
transversal
There the
except
infinite
we
for can
parameters.
,
x
=
x(t)
n
for
,
any
O__ ( x ) > /
1 .3.14.).
extension
and
not
true.
We
[]
and
so
it
shall
{ x.}
some
representation
in
a
set
basis
Unfortunately
is give
a basis
Therefore
14i~ < N
x.i function
Xl + c~ x l
of v a l u e s
of
is
~ in k.
containing these
] ,
In
exists
a finite take
t '
0.
always
is p r i m i t i v e .
separable~ is
is
a curve
because
as
(proposition
Galois
transversal.
expressed
/k((x))
this
consider
separable,
be n >
k((t))
(dx/dt)
Let
+P+I
integer
type
sequel.
is
k
to
o(f)
2
any
the
is
and
+t
characteristic
x 1
0
_~(g)
possible
+P + t p
m
of
zero
.3.-
Since separable
for
only
uniformizing
transversal
tp
in
and
in
where
1 ~ i ~< N ,
and
positive
examples
Remark
=
extension
characteristic
of
a transversal
2
= t p
element
t'
uniformizing
a ~ with
2
of
is
curve 2
x
some
x'
,
t ~
Moreover,
is
p+l
~ then:
Hence
is
If
m >
bases
48
are in
not the
stable
under
quadratic
transformations,
2.1
teristic
p > 0
.4.-
Let and
k
let
be
[]
an be
x
the
unique
maximal
ideal
not
in
the
the
over
k
field
given
of
charac-
by
+l 2
tp +p+ tp +p+l basis
transversal of
curve
closed
{x,y}
of
m_
element
in
the
quadratic
is
transversal
basis
transform
and
{x,y/x
of
[]
}
is
y/x
not
be
of
separable the
,
and
it
is
separable.
Example fact,
2.1 let
k
.5.be
An
extension
a field
of
x = t
We
evident
2 + tp 2
Y :
element
algebraically
the
2 tp
x :
but
is m a d e
following:
Example
The
as i t
cl.aim
since
that
the
Remark
k((t))/
identity
2. } .6.-
is
If
k
thus
k((t))/
3
+ t
3
+ t
Moreover,
+ a6t
not
has
6
+ aGt
normal.
and
7
normal.
+
. • •
Indeed,
this
characteristic
of
3,
for
is
trivial
k((t)).
the
curve
4
x'
is
not
normal.
if
we
consider
y =
3
k((x))-automorphism
is
x=t
may
9
parameter k((x'))
4
again
y = t
transversal
characteristic
is k((x)) the unique
X = t
any
k((t))/k((x))
9 t
+t 12
I0
like
the
in
the
curve
preceding
example;
In
49
then,
the
mal"
holds
Remark shown
for-
in
the
whose
k((t'))
k((t)). it be
be
normal
may
be
a
which
is
k((t))/k((y))
not
is
if
true
is
a primitive
are
allow
not
nor-
not
normal
are
Galoisian
commutative, and
not
extension
of
separable
and
the
by
x = x ( t ~)
examples
not
normal
otherwise
is
which
a separable
k((t'))/k((x))
which
x
minimal
Abelian;
Obtain
Those
k((t))/k((x))
Now~ we
examples. group
be
2
m-m
k((t))/k((x))
Galois
Trivially
cannot
y g
Extensions
above
Let
any
it.
2.1 .?.-
extensions
Let
"for
property
normal
k((x))
containing
thus
subextension
extension.
Gatoisian.
But
k((t))/k((x))
would
hypothesis.
is
the
expansion
parametric
of
x
as
representation
a series
for
in
a curve
t'~
in
appears:
x = x(t') y = t'
For
a mope
explicit
X = y
in
characteristic
versal it
is
not
by
3
in
the t3
the
curve
8
9
has
the
peculiarity
extension
k((t))/
that
for is
k((x'))
any
trans-
Galoisian
but
notice above +t
4
that
the
examples by
terms
assumption is
of
that
not e s s e n t i
type
t p
+
al . i+]
t p
k If
has
characte-
charact.k
one
may
=p
> 0,
obtain
ones.
Now, Puiseux
the
t
consider
Abelian.
replacing
identical
t
example,
+
which
x~
Finally, ristic
6
t
-
3,
parameter
x(t').
series
we
try exist.
to
give Also
a characterization we
shall
give
of
curves
requirements
for in
which
order
to
50
know
when
a
shall
assume
that
Proposition there is
Puiseux
1
charact.k
2.1
exists
.8.-
Necessity k
primitive.
I n the
sequel
> 0.
Then,
x E: k ( ( t ] )
O (x)
with
satisfying
x
n
= t'
if
=
n
and
only
if
(n
= k((T
is
))
be
obvious
the
from
greatest
1 .3.14.
Conversely,
because
in
k((t)).
(n',p)
On
separable
extension
of
=
the
Then
we
can
hand,
the
choose
"[ s u c h
T =
so is
k((x))
a
other for
some
.9.-
Let
I--} . T h e n
t'
basis
Corollary
if
{xi}
I~
x.
a
which
= x
inseparability
index
of
s
T
i = i(x)
E; k ( ( t ] ) .
Then
x
= t'
and
thus
In
expansion
Proof:
From
separable
parameter
a
a basis
Puiseux
,
if
(n
order
and
2.1 x.
s
(x
1
for
relative
necessary
remark
has
be
of
the
expansion
maximal
ideal
(relative
to
24
i
~
),p)
=
1.
a curve
to
a
there
Since
n
that
always (x)
multiplicity
transversal
sufficient
.3.
of
= n,
parameter
(n,p)=1
exists the
n
to in
.
a transversal
result
follows
and from
{ x,y
corollary.
2 . 1 . 11 .}
by
an
Given
irreducible
an
equation
distinguished
of
a
plane
polynomial
have
every
S
above
Xl):
I
2.1 . I0.-
, it is
1 4 i ~.N
x.(t')
I
only
Puiseux
Remark
the
= t ,n
and
basis
the
,
n
t 'p
2.1
curve
that
(x)
1 .
x 1
in
S
complete.
Corollary of
n'=n
that
i
proof
(x),p)
nl
contained
and
S
let
s
is
we
1.3.13.),
Proof: and
is
= p > 0.
Let
(()) E; k t
t'
(see
representation
curve over
in
a
basis
k((x)):
51
n-1
n
Y
+ A
(X)
n-I
Y
+
...
+ A
(X)
o
= 0
A.(X)
~
E: k
I
0-~
then
the
degree if
integers and
one
the
the
are curve
[]
under
given
a
The
i(x)
are
index
be
in
made
which
the
polynomial. this
expansion
may
quadratic
that
curve~
Puiseux
bases
respectively
of
a plane
remark
.12.-
stable
and
of
has
similar
2.1
not
equation
curve
A
Remark
(x) s inseparability
the
knows
whether
n
i
for
we
,
,
separablity Consequently, determines
not.
twisted
have
~n-1
equation
or
(()) X
curves.
Puiseux
transformations.
For
expansions
instance,
the
by p2+I x = t p
y = t
has
a
all
Puiseux
the
has
of
parametric
no
any basis
Puiseux of
proposition of
expansion
bases
whose
2
its
m_
+p+1
+ t
relative
(2.1.10.)~
p
x
=
t
y
=
t p
any
its
transver.sal
quadratic
parameter
transform
rT]
2
+I +
t p+I
relative
to
a transversal
parameter
Let
us
consider
a
parametric
representation
type
n ×
=
t
1
r
X. =
a
M
with
I
~
<
r._iz <
,
ideal.
2.1.13.-
(1)
in
ape
expansion maximal
+p+2
to
but
equations
2
p
t
r
ii
/
+ a
ril
" " "
and
t
i2
+
....
24
ri2
a
i r
.
ij
0,
I
~<j
<
~.
i~<
N,
in
52
Then 9.c.d.
of
the
the
representation
{n,r21
one.(Note
Proof: If
that
>
is
1
separability
such
characteristic
of
k((t'))/k((Xl) find
t'
t'
x.
as
I
Thus
q
of
closed
field
k,
algebraic
the
k((xl/n)) may
of
,
be
n =
of
the
der case
every k((t,)),
any
polynomial by
t'
using
q-th
the
that
exists
t'
4:N.
the
p.
By
~< N ) .
sufficient
1 ,< i
with n/q
)
condition.
El k ( ( t ) )
By
2.1 is
with
.8.
the
prime
with
separability using
we
may of
closure
degree
again
2.1.8.
we
1,2 .... as
the
a formal
is (20),
in
a linear
and
fact
pag.
is
power for
by
t'
of
In
the
type
the
w.t q
expansion
and
set
get
(2)
53).
This
x the
like
is
union
known
is
(1).
which
is
classical an
algebraically
indeterminate of
the
over
fields
element over k((x)) ]/n x with a finite
in
(see
Walker,
Ostrowski
theorem
k((x))
the
k an
n > 0
Ancochea, in
in
when
series
a well
coefficients change
k.
algebraic
some
by
in
points
that
every
characteristic with
fact
k((x))
Hence,
and
in
fundamental
exponents
result
t'
t
proof.
zero of
unity
element
the
the is
of
replace
each
completes
of
between
root
characteristic
Waerden, of
if
the
k((t))/k((Xl)
a relation
curves
negative This
(Van
only
}
2 ~ i
there
follows
prime
a
One
expressed
number
is
This
algebroid
It
,
prove
primitive,
k.
is
.14.-
n <~ r i ]
extension
a divisor
contradictory.
study
and
x 1 = t'
in
is
2.]
the
also
there
series >1
Remark
is
w
have
x i El k ( ( t ' ) ) ,
satisfying
= w.t q , where
of
if
PN] ....
We s h a l l
not
p of
Now,
not
that of
)
r31 .....
obvious. is
degree
the
.....
needs
representation
q = _U(t')
may
it
Necessity
the
primitive
set
(2)
is
is
(2),
was
Theorem generalized
who
factorizes
(22)).
proved lineary
in that over
53
m x = t
Indeed,
this
m+l + c 1 t
generalization
is
s +...+Cst
our
,
m
proposition
> 0.
I .3.1.
(see
remark
1.3.7.). But,
as
we
have
seen
in
this
section,
in
positive
charac-
teristic k((xl/n)
K ~
)
n=l is
not
algebraically
expansions. theorem
closed.
G.Rodeja,
is
~.12),
equivalent
expansions
to
In
in
the
positive
work. For
of
to
study
expansions,
sake
Recall
any
Definition {x,y}
the
curves. of
in of
Puiseux
this
case
Ostrowski's
mamburger-Noether
EXPANSIONS
we
characteristic.
the
present
section
in o r d e r
Noether
field
how
availability
NOETHER
preceding
Hamburger-
plane
use
OF
PLANE
CURVES.
appropriate
curves
cannot
has p r o v e d
the
HAMBURGER-
ALGEBROID
not
we
.
2.
are
Hence
of that
the
will
saw
which
are
we
shall
still
how
Puiseux
singularities
We s h a l l
simplicity k
just
develop
denote
the
of in
algebroid
the
sequel
thecnical
begin an
series
with
base
the
in
case
algebraically
of
closed
characteristic.
2. two
2.t
.-
formal
A parametrization power
x = x(u) y = y(u)
series
system
is
defined
to
be
a set
54
in
k((u))
,
Algorithm and
such
2.2.2.-
suppose
Noether by
out
that
Let
expansion using
using
provided
that
Yl
Then
and
only
(A)
consider
and
be
There
of
of
always
(B)
a02 x
There
a0i
£
carried
the
the
denominator)
Stop
k,
from
Continue
in
the
statements
aoi
h
other
case.
holds:
1 ~i.< h,
such
that
h
+...+aQhX
exist
n.
is
term
order.
being >/
and
2 y = a01x+
independent
following
h > 0
which
follows:
positive
order
two
as
{x,y}
Hamburger-
al9orithm
the
of
y (x
the
exist
out
Yl
of
an
system
the
k([u))
pick
a series
one
of
in
a series
instead
construct
means
divisions x
>0.
a parametrization
by
by
Yl
U_(y)
{x,y}
obtaining
division
and
We s h a l l
of
y
> 0
U_(y).
successive
quotient,
one
us
n=_U ( x ) ~ <
Divide the
U_(x)
4x
E: k
z 1 ,
,
with
1~ U_(z 1 ) < U _ ( x ) .
1 -.< i < oo ~ s u c h
that
2 y = a01 x + a02x
In and as
again Ion 9
the
either a s
(A)
Note number
of
steps
z. are J obtained
decreasin
case
(A),
(A) is that we
+ . . .
or
(B)
we
continue
with
the
is
obtained.
The
system
process
continues
obtained. when
n = 1
are
9 .
in
This
(B)
(B),
holds.
Hence,
because
means
that
the
an
after
orders
expression
:
2 y = a01x+
,
.
.
a02x
h
.
.
.
,
,
.
.
.
.
.
.
,
2 Z
=
r-I
Z
a
r2
+
r
h + x z 1 h Zl lz2
+...+a0hx
a l 2 z 12 + . . . + a l h
x =
(D)
{z I ,x }
. . . .
.
1zlhl+ .
.
.
.
.
°
.
.
.
.
.
.
a finite
of
the
series
as
follows
is
,
55
where
a.,
jI
(ajl=O
since
, z.
j
u(zj_
Definition { x,y
E: k
1)
2.2.3.-
}
is
Remark
a set
A of
1 ~ o(z
>
I -g j 4
--
u(z.),
) <
r
of
of
--
expansion
type
(D)
A Hamburger-Noether
representation
< U ( z 1 ) < O(x)..< U ( y ) .
. ..
for
which
are
expansion
a curve,
--
--
r ).
Hamburger-Noether
expresions
2.2.4.-
parametric
@ k((u)),
with
the
system
t~erified
provides
parameter
z
by
it.
a :
r
X
X(Z
=
r
y = y(z r
Sometimes, expansion
by
an
we
shal
expression
denote
simply
a
H a m b u r g e r - N o e t h e r
like h
(D)
together
with
coefficients with
0
Proposition { x,y
}
the
a... JJ ~< r , z
~j
For o
1)
=x,
z
2)
z
ji
j
+
Z
restrictions
instance =y
-1
The
(D)
(or
Z
.
j
on w i II
simply
j+l
the
be
is
uniquely
indices
written
0 ~< j ~ r
Hamburger-Noether
U ( x ) ~< U ( y ) a.. jl
a
i
necessary
2.2..5.with
=
z J-1
if
j
as
(_D)
there
is
expansion
determined
by
or
the
for
the together no c o n f u s i o n ) .
a system
conditions:
Ig k .
k((u)).
c J
3)
1 ~< u ( z - -
Particutary, (and depend
the
fixed on
the
element uniformizing
)
< ...
< o(z l)
r
<
- -
the in
expansion it,
if u.
u(x). - -
g (x)
depends =
U (Y)
only ),
on but
the it
system
does
not
56
Proof: For"
For
each
every
system
L({
We L(
shall {x,y}
expansion {x,y
x,y}
}
) :
prove
the
) = 0,
then
of we
min
type
1 D
proposition
is
using
is
some
U/
would
be
another
an
= h + h
expansion
induction
2
for
on
expansion
y = a~, x + a,.~ x
FoP
L(D)
1
+...+
h
r-1
set
{L(D)
there
let
(D)
(O)
{x,y}
L({
x,y}
for
}.
).
{x,y}
If of
type
+ ....
UZ
expansion
(D')
one
of
the
following
statements
true
(A)
y
(B)
y
2 I
X
+
a
h
X
T
a01
+ . . . + a
02
X
I
h +
X
xh
Z
I
Z
Oh
'
1 ~ U(z
1)
< O_(x).
series
in
2
in
the
case
(A) h )--
X
.... a 0 1
,
(a0i
the
--
+
~
~
X
a02
becomes
a~) i ) x i
( ~-~
=
In a0q
on,
different
~ a Oq ~
the
case
, then
because
(B),
again
if
the
to
be
Now, and
suppose
L( {x',y'} L(D)
= N
they
must
and
)
--
agree
both
sides
in
it
are
u
q
denotes
the
first
integer
for
which
i
-a'
0i
) xi
= 0
a contradiction. let
that )
xi
equality
(a 0
out
.
orders.
i >~q turns
a
i >h
a contradict have
....
equality
i=]
which
+
{ x,y} the
Let
be
a system
assertion us
D'
be
with
the
consider another
is
such
true an
for
that every
expansion
term
system (D)
expansion.
independent
L({x,y})
Then in
the
for a0] series
-
N>
0,
with {x,y} = a0~ 1 y/x.
with ,
since Hence
57
Yl = (y - a 0 1 X ) / x
The
system
are
{ x,y 1 }
respectively
term.
As
has
actually
obtained
expansions
(D)
and
( D 1)
(D')
by
and
(LD'1)
removing
the
L(D
Corollary....
2.2.6.-
representation expansion
The
of
we
some
have
system curve
U(z
Proof: and
Conversely,
if
U(z
the
quotient
is
only
if
for
its
parametric Hamburger-Noether
the c o n d i t i o n s
k((u)) ~ k((Zr) ) , and
) = 1
then
k((u))
= k((z
r
field
actually
and
a primitive
on is p r i m i t i v e ,
imply
--
if
is
we
r
If the r e p r e s e n t a t
y(u) E: k((Zr] ]
{x,y}
hypothesis
which first
= I .
--
tion
two
from
= y'l
) = L(D) - I = N-t, by the induction 1 (D1) = ( D ' 1) a n d c o n s e q u e n t l y (D)= (O') .
have
to
= (y-a(~l X)/x
thus
)),
x(u) C k('~Zr))
--U(Zr) = 1 .
and
since
z
belongs
r
of
the
curve
[]
r
= k(~x(u),y(u)))
the
representa-
primitive.
Remark
2.2.?.-
in o r d e r
to d e t e r m i n e
The
preceding
when
corollary
a given
is a practical
parametric
criterion
representation
is
primitive.
Definition over
k
2.2.8.-
Let
{ x,y
} a basis
and
Noether
expansion
for
Noether
expansion
of
When chosen,
the
If
is
there
expansion which
[] the
of
the
in
this
no of
confusion, []
in
only
the
multiplicities
will
an
essential
2.2.9.-
{ x,y
local
Let
to
(D)_
call }.
ring the
plane
it
the
Now,
we They
curve
1[[] . A H a m b u r g e r be
a Hamburger-
}.
x,
in
that
basis
in i t i s c o m p l e t e l y
singularity
classify
of to
{ x,y
say
[].
m
defined
system
shall
for tool
is
expansion
basis
of
basis
algebroid
ideal
parameter,
we
on the
maximal
parametric
a transversal
sequence
Proposition
be an i r r e d u c i b l e
Hamburger-Noether
depend
give
[]
is determined.
Hamburger-Noether shall
find
wilt
make
of
elements evident
a curve;
so
that
in
it
the they
singularities.
, 0 ~ j ~. r ,
Zo=X,
Z_l=y
, be
a Hamburger-
58
Noether be
expansion
the
quadratic
expansion is
for
curve
transform
for
given
the
D 1
in
[]
of
the
in
L~.
the
Then
transformed
basis
{x,y}
. Let
[]
1
a Hamburger-Noether
basis
{ x,Yl}
( Yl
=(Y-a01
x)/x
by )
If
h > 1, h-1 Yl
= a02x
h-1 +
+ " " " + a0h x
X
Z
]
h z
i)
f
= ~
j-1
=~-a j-I
induction are
on
needed
proof the
to
Proposition
j
z
+ z
j
Jz
j
,
j+!
t
~<,j~<
r.
h = 1,
z The
a i
J
i
proceeds
minimum
from
i
j
h
+ Z
j
.
Z
j
,
j+l
proposition
number
desingularize
2.2.10.-
Z
of
1 ~ j
2.2.5.
quadratic
~r.
by
using
transformations
which
D.
Let h z
the {x,y}
, x=
nj
=
0 4j
Z 0 , y =Z_l
i)
different
The
sequence
+ z
fop
the
~
v
is
the
z
j
0 ~< j
,
j+l
curve
[]
valuation
in
~
the
basis
associated
to
~],
Then: multipticites
I .5.10.
for
which
[]
are
occur
exactly
in
the
the
desingu-
integers
nj
,
~< r .
4
h0
r,
Each =
iii)
Proof: and
If
0 -< j
ii) <j
i
a.. z J, j
expansion
_v ( z j . ) ,
larization
0
j-1
Hamburger-Noether
and
=
iii) ii)
h,
hr
The
is
n
is J = oo
repeated
integers
h
a consequence
simultaneously
by
J
and
from
indution
in
that
n
i) on
sequence
depend
J
and the
exactly
only
ii). minimum
on
We s h a l l number
h. times, J
F-I.
prove of
i)
)
59
quadratic
transformations For
M(E])
which
= 0,
the
are
needed
curve
is
to
desingularize
regular
and
its
M([--~)
>0
El.
expansion
of
type 2
y = ao]
The
result
is
that
+
. . .
evident.
Let assume
x + a02 x
[] the
tion
h~pothesis
for
[]
be
a curve
result to
such
holds
[]
1
that
for
M-
M(I--~')
and
using
[]
and
< M.
By
proposition
and
let
us
the
induc-
also
holds
applying
2.2.10.
it
.
Corollary
2.2.tl
.-
Let
Hamburger-Noether
expansions
t~ ~
be curves
h
Z
=
j-I
whose
respective
are
a
..Z
Jt
i
+
j
Z
•
z
j
0
j+l
{
j~
r,
h~ Z
Then,
[]
and
Remark that of
F"]~
j-t
are
2.2.12.-
=
curve
C defined
by
equation
f(X,Y)=0
an
equation
f (X
Moreover,
follows
the
origin
of
multiplicity
Proposition
of
j
iff
from
choice for
,Y.)=0 I
+
.
the
expansion
the
for
i-th
~Y Z
J
,
j+t
proofs
basis by
j
0~
r = r ~4, h . = h . ~ , a n d J J of
2.2.9.
determines
the
then, the
,
which
of
C~
Z
and
uniquely
the
{ x,y
if
}.
Hence
f
and
quadratic
2.2.10.
a sequence
desingularize
1.5.14.,
strict
n . = n . ,0~< j ~ < r . J J
(D)
embedded we
take
determine
transform
of
C.
I
according
of
J~
Z
transformations
an
I
..
equiresoluble
It
quadratic
a
i
a Hamburger-Noether
formal
L.--
C is
2.2.10., formed
the by
sequence
h points
of
of
infinitely
multiplicity
near n,
hI
points points
n 1 ,etc...
2.2.13.-
defines
a curve
Proof:
Consider
Any
expansion
[].
the
expansion
(O)
of
Hamburger-Noether
type
60
h (D)
"
z j_ 1 =
ajiz
j
i
Jz
+ zj
j+l
0 (j~< r,
'
x =z0'Y=Z-l"
i
Let
v
be
the
associated
valuation
to
kl,l,z'"
--
to
)).
Since
each
z
in
the
r
that
ring~
we
have
parametric
x = x(z
belongs
J
equations
) r
y = y(z
) r
defining { x,y
a curve,
}
is
actually
3.
irreducible
us
quotient
and
consider
[]
MULTIPLICITY
OF
embedded
curve
the
f(X,V)
let
field.
expansion
basis
(D__).
series
ring
Hamburger-Noether
INTERSECTION
Let
local
whose
C k~(X,Y))
= k~t))
We h a v e
the
plane . Let
[]
integral
a parametric
PLANE
C
CURVES.
defined
= k(~X,V))
closure
/(f)
[]
of
in
by
the
be
its
its
representation
x = x(t) y = y(t)
of
the
curve. Now,
general,
defined
Definition defined
let by
2.3.1.to
be
D
the
be
the
series
The
v
is
the
embedded g(X,Y)
curve,
intersection
associated
D ) =
multiplicity
~(g(x(t),y(t)
valuation
not
irreducible
in
C k((X,Y}).
number. ( C,
where
another
to
I--] .
of
C
and
D
is
61
Remark
2 .3.2.-
multiplicity
The
are i)
( C,D If
iii)
and
D
=
=
D
If
...D
1
>~
not
Now,
and
expansion
{ X+(g),
the
intersection
of
their
component
irreducible
for
each
of
D,
i,
then
) .
in
D
equality
strict
).
holds
quadratic in
.e(D)
the
if
and
only
in
maximal
transform
of
plane,
then
ambient
+ ( C 1 ,D1).
rreducible
D
) = ( D,C
common.
is
and
( C,D
The
D ) is the 1 transformation
that C
Y+(g)}
D.
( C,D
) = e(C
for
irreducible
then
a tangent
assume
an
e(C).e(D).
C (resp. 1 a quadratic
by
m ~-i=1
irreducible
( C,D
Noether
of
I
is )
is
with
m
) =
D
have
D)
D
C
~==~
~
( C,D
( C,D
v) (resp.
)
If
iv) C
properties
trivial:
ii)
if
following
the
and
the
respective
ideals
are
Hamburgerbasis
given
{X+(f),Y+(f)}
by
h z j_ I
=
---
aji
zj
i
+ zj
Jz
j+l
0 .< j . <
'
r,
i z' j-1
Let h
s-1 least
=h'
s-1 index
s and
'
such
be a
the
h; a ~.. z 'i , + z' J z ' Jl j j j+1
= £ i
greatest
integer
= a'
jk
that
for jk .t a' .
a SI
for
j < s (i
~
Sl
and
0
which k
~
+1 , i ~; h ' + l ) . S
~<j-~
r'
'
h= h', j
. Let
i
hl
=hl
be
Finally
i
,..
• ,
the set
S
S
S
The infinitely
near
integer points
=
N
7j=O =
that
h
.n J
h
.n' J
J
+h]+...+hs_1
both
curves
+ have
i -I
is t h e
in c o m m o n ,
number (see
of I .5.16,).
62
2.3.3.-
Proposition i)
If
Keeping
i 4 h
and
s
( C,D
ii)
)
If
i = h
s
proof
2.3.2.,
using
Remark
2.3.4.-
cities
of
commQn~
proceeds
induction
By
the
infinitely
one
gets
4,
S
+
above,
we
have:
,
i .n
s
.n'
s
) = S
+ h
) =
+
S
h'
trivially on
.n
s
.n'
+ n'
.n
.n
+. n
.n'
S
s+]
S
s
.n'
s
from
s
s+l
property
s
v)
in r e m a r k
N.
adding
the
near
their
S
+t ,
( C,D
The
=
h' s
as
s
( C,D
Proof:
i~
notations
i= h'+l,
If
iii)
the
products
points
of
that
intersection
two
the
respective
given
curves
multiplihave
in
multiplicity.
HAMBURGER-NOETHER
EXPANSIONS
FOR
TWISTED
CURVES.
We
shall
expansions
for
of
of
the
case
over
Definition to b e
a set
twisted
k
be
in
curves.
plane
Let minate
generalize
this We
section shall
the
find
Hamburger-Noether
results
analogous
to
those
curves. an
algebraically
closed
field
and
u
an
indeter-
k.
2.4,1 { x,} I
.I~i
An
N-dimensional
~N
of f o r m a l
parametric power
series
system in
is d e f i n e d
k([ u]]:
63
x.
= x.(u)
I
such
that
u (x.) --
Notations by
> 0,
,
1 .
L. e t
MN(A)
A-module
1 ..
I
2./4.2.-
denote
,
I
(or"
A
be
a Pin 9 and
simply
of
(N-])xl-matrices
If
A = k((u)),
M(A) with
there
v
:
if
is
N
a positive
there
is
no confusion)
coefficients
a map
V
M(k((u)))
in
defined
~,
v
is
the
Lernma
2.4.3.i)
=
assume
that
V
o~
2)
Let
as
,
dividin provided
use 9 all that
successive the all
x. the
x
k((u)),
~ 2
obtained
k((u))
¥cM(k((u))). Y1 , Y 2
) , 2 ~ i 4 N.
divisions
by
z e
c
as
~
in
,v(YN)}
Y E; M ( k ( ( u ) ) ) .
x2]
~< v ( x
2) . . . .
a parametrization
Y
We s h a l l
,
follows:
proper'ties:
min(V(Y]),V(Y2)),
consider
n = v ( x 1)
with
fotlowin9
Y = 0
+ v_(v) >~
us
the
the
z
associated
has
4:===>
= z{z)
V(YI+Y
2.4.4.-
valuation
map
v(z.7)
iii)
and
The _V ( Y )
~)
Algorithm
natural
We
A.
min {v(y
where
inte9er.
E: M ( k ( ( u ) ~ ) .
system
{xi}
Let
M(k((u))).
the and
case continue
with
N=2. the
independent
We s t a r t division tePms
l-.~i-~N
64
removed
Z 1 ~
be
a series
One
of
(A)
There
with
= A0I
1 ~< V ( Z
y
1)
element
considering z13,... may
be
exi
case
2 1 +---
st
A
E: M ( k ) ,
Oi
true: 1 -.< i x< h ;
and
(A),
the
E: M ( k )
0i
since
in
matrix
the
parametrization the
h 1 +
{ A0hX
h Z 1 x 1
V(Z
Z 1
=
1 (
i < ®
= nl
~ such
< n = v ( x 1)
that
system
{Zl
elements
precisely,
Z0
1)
such
remaining
More
,
that
2 x 1 + • • •
z 1
repeated.
A
be
~< V ( Y ) .
x 1 + A02
are
0;
will
that
There
,ZiN
n.
situations h>
< v(x 1 )
the
>i
exist
x 1 + A02x
= A01
tn
order
following
such
(B)
an
two
M(k((u)))
y
of
v(z 1)
= n 1
, x,z13,
..
in
Zl,
the
,
there Then,
, ~Z1N above
by }
of
two
situations
if
[zi3] •
(A) g
as
There
M(k(Cu))),
above
exist
h
1
will
be
>0,
A
again
I~<
i x
I
; and
such that h
2 ~0
true:
E: M ( k ) ,
li
= A11
Zl + A12
+...+
Zl
Alh
z1
h
I
1
+ Z2z 1
1 with
1.<< v ( z 2) < v ( z l ) = v _ ( z i ) . (B)
There
exist
A
li
E: M ( k )
,
1 ~< i
< oo ,
such
where
process
I, Zl.J one
is
that
65
2
Z0
All
The as
in
the
process
case
obtained,
+
Zl
the
his
.
.
continues
N-2,
and
+
A12z1
that
.
when
after
algorithm
.
the
a finite
situation
number
(A)
of
hol;ds.
steps
(13)
Note,
will
be
finishes.
algorithm
yields
a set
Y
= A01 x 1 + ...
Z0
.
of
expresiens:
h h + A0h x 1 + Z 1 x 1 h
. . . AllZ1+
h
+ AlhlZl I +
Z2z
1
D) .
.
.
.
.
.
.
=
r-1 such
.
.
.
.
.
.
.
.
.
.
,
A rI
z r + .....
g
M(k((u)))
,
°
that
A element
of
g
ji the
M(k);
Z.,~. j
matrices
j
Z
and
Z.
J
] ~< v(z_ r)
< "'" in
<
short,
we
;
V(Z.)
J
k((u:l)
is
z
an
J
=v(z.) J
and
-
; and j
_V(Xl) ~< _V(Y) . shall
write:
"Zj-I = ~--
(D__)
; z. g j
Aji zj i +
Z
j+1
zj j
'
0 4
j~< r.
the
element
i Remark exactly
2.4.5.in
the
The place
Definition
2.4.6.-
the
system
{xi}__l~
are
verified
Remark not
in
in
z
We d e f i n e i ~N
When
determined it.
which
to
Aj1,
be
l is
j-]
~< j ~< r , placed
a Hamburwera set
of
have
in
Z"
Noether
expresions
of
0
j-1 expansion type
(D)
for which
it.
2.4.7.-
uniquely
x 1 fixed
by
matrices
In
fact
N > 2
the
by
system
the
the
expansion
Hamburger-Noether {x. } I
depends
1 ~
on
the
expansion and
the
is
element
arrangement
of
66
the
elements
element
in
z'.
in
I
the Z
matrices. ,
i
z'/z. I
Even, such
i
V_(Z.)
row
in
more
the
row
Definition
If
does
2.4.8.field
(D)
k,
is
system
} L
expansion
and
The
The
any
[]
the
say
arrangements.
uniformizing
a twisted
over
a basis
of
for (D)
next
Further-
u.
curve
that
the
its
the is
the
algebraically
maximal
ideal.
parametrization
a Hamburger-Noether
in the b a s i s
B.
explained
in
the
sequel
works
as
in
system
f
I~
,
~N,
I
parametric
representat
on
expansion
of
for
a curve
that
if
and
system
only
U_ ( z
if
) = 1.
r
Let
--
ZJ-1
(D)
a Hamburger-Noether }
\
= x.tuj
Hamburger-Noether
2.4.9.2.-
of
I -.
parameter. transform given
another
N = 2.
a primitive
i
exist
completely
to
expansion
shall
I
{x.
up
1 ~ i ~
results
x.
for
be {xi}
for the c u r v e
2.4.9.1.-
is
on
[]
we
determines
i
expansion
B =
,
may
1 --
2.4.9.-
case
z.
a Hamburger-Noether
{ x.
Remark
of
depend
Let
it
I
choice
not
step,
that
--
Hambuger-Noether
that
closed
the
some
= v(z!).
I
However,
at
by:
1
T--
/___.A i ]i
expansion its
Then []
=
maximal
z
for
ideal,
i
j
h + Z
the where
a Hamburger-Noether of
[]
in
the
transformed
j+l
z.
j
,
j
curve Xl=
.
[]
in
the
z0
is
a transversal
expansion basis
0 ~ j
for the (relative
basis
quadratic x
1
) is
the
67
i)
If
h
> 1,
h-t Y1
= A02Xl
+ Z
+ " " " + A0hXl h
=
j-1
ii)
i
z ji
+ Z
.
z
j
j+l
,
j
.
1 4
j
I~<
j ~
h = 1
If
7~
2.4.9.3.-
A
h-1 1
X
1
=
jit
A
z
ji
i
i j
h + Z
j+l
z
j
,
j
Let
h Zj-1
=
Aji
zj
+ Zj+l
zj
,
04
j
~
i be
a Hamburger-Noether
its
maximal
ideal. i)
If
The
J
,
0 ~< j
0
= h,
h
r
Each
A of
h. J
Hamburger-Noether formal
quadratic
the
multiplicity
of (see
some
of
the
(Xl=Z0)~ rings
basis
of
then
which
1 .5.10.)
occur
are
the
in
the
integers
exactly
h
times,
J
0.,<
j
..
embedded
and
by
depend
only
on
D.
J expansion
determines
transformations curve
the
n
C
sequence
of
defined of
by
uniquely
N-spaces the
infinitely
h points
of m u l t i p l i c i t y
and
r] •
are
two
respective
which
choice
of
near
n,
a
points
h
I
the of
points
n I ,etc...
Two
integers
curves h.
and J
expansions
[]
repeated
integers
} . Moreover, i l~
. 5 .-
for
is
J
{x
the
..<j ~< r ,
multiplicities
n.
The
desingularize
if
0
in
= oo) .
2.4.9.4.-
2 . 4.9
,
[]
..
iii)
sequence
for
= -v ( z j )
sequence
ii) (h
nj
different
desingularization n
expansion
(D)
[] n
of J
and
(D ~ )
agree
.
equiresotuble
if
Hamburger-Noethe.r
and
only
basis the
of
68
2.4.9.6.A of
jl
Any
stated some
set
in
the
curve
in
of
type
remark some
(D)
which
2.4.5, basis
of
are its
verifies
the
requirement
a Hamburger-Noether
maximal
ideal,
on
expansi~
CHAPTER
CHARACTERISTIC
This irreducible
of
the
computed
in t h e
way
in
(n,p)=l,
they
Since begin
it
Zariski's the
case
of
of
in
means
of
the
the
germs
of
complex
all
this
ON
EQU
be
chapter,
case
are
In
positive
compute
considered
plane
they
them
by
polygons,proving,
a short
analytic
called
be
characteristic
section, fop
will
shall
Newton
classical will
first
system
expansions.
we
field.
a co,mplete
zero
sections, and
closed
invariants
Puiseux
of
exponents, in
this
account
of
curves
and
its
chapter,
meaning
curves. the
word
"curve"
will
stand
curve.
1.
REPORT
1n this
section
definitions
in
the
( 28 ~ , and
we
consider,
plane
by
equisingularity
of
characteristic
equisingularity
Throughout plane
the
CURVES
algebraically
These
In
with
an
of
construction
equisingulaPity
giving
theory
the
ALGEBROID
study
over
expansions
agree
the
by
is
successive
Hamburger-Noether"
when
for
idea
PLANE
the
equiresotution.
usual and
to
curves
exponents.
characteristic
in
the
OF
devoted
plane main
for
characteristic
we
is
atgebroid
invariants
using
EXPONENTS
chapter
Here
I I I
curves.
we
SINGULARITY
g ve,
equisingularity in
THEORY.
without theory
particular,
proofs,
the
initially the
case
basic
developed of
complex
facts by
and
Zariski
analytic
70
We C-over
an
begin
algebraically
reducible
and
denote
of
C and
components of
C.
The
(Pesp. 1
•
5
.
strict
Trot(C)
t8.
that
by by
(resp.
>
a pairing
71
Remarks
3.1 .2.-
pairing.
If
not
(C)
)
Definition
C is
two
C
are
(C'.) J
Tt, j
(C J~)•
:
D
are
tangentially
pairing
the f o l l o w i n g
e(r.) as
iii) (a)-equivalences
definition
in The
be
then
)
(D') J
).
(D
in
general.
curves. D.
A bijection
We s h a l l
that
two
are
~(R)
induces
,
say
irreducible and
I
TI.(P.)J
a tangentially
rl
j
and
stable
pairings
I
~<j~;
s
,
~) j
irreducible,
the
~
(C)
>
for
all
(D) C
and
•
(C
I-'
i
chapter
trivial
pairing
s said
to be an
D
are
(a)-
regular,
or
stands
for
. ( e(-)
I
I .)
pairings all
4.
stable.
= e(Tl(r.))
for
= ;~-1J . C'
~. ) . J
C and
so
tangents
conditions:
i
multiplicity
(D)
distinCts
1 .5.1 C
plane
iff
iff e i t h e r
i) T~ is t a n g e n t i a l l y ii)
in g e n e r a l ,
stable.
(of s i n g u l a r i t i e s )
~!~ v e r i f i e s
)
I ,
provided
tangent
stable
of
Tst(C)
(resp.
between
ordered
Tt,'. : J
A
C' . J
embedded
"~ : ( C )
set
chapter-
stable
suitably
and
3.1 . 3 . -
equivalence
of
1) L e t is
to be,
transform in
a pairing
tangentially If
(D)
C,D
tangentially
A(D)
2)
,ts }
quadratic
is
j
C
the
called
P
assume
curves
~(C) = { tl,...
is
I
We
components
Let
F'. a n d
are
k.
algebroid
the set of i r r e d u c i b l e
s connected
(D)
components
which
field
plane
(C) = {r I ,... ,l-'m}
total)
s
3.1.1.-
(C)
closed
embedded
= ~_~lj_,.= C ~'j ) , i n t r o d u c e d
has
,
Definition ]1:
by c o n s i d e r i n g
11'. : ( C ' . ) J J
)
(D'.) J
induced
used
by
TI
are
j.
The
coherence
of
the
inductive
method
is
guaranteed
by
the
fact
by
that
a finite
in
the
number
above of
71
successive
quadratic
of
has
C
which
Remarks they
3.1
regular
.4.-
then
by If
the
of
the
definition
of
iff
either
ii)
]
that
are
and
total
may
be
said
a strict
transform
components,
to
be
transforms
(see
1 .5.18).
(a)-equisingular
are
characterized
is
an
used
when
Then
equivalent
TI : ( C ) ordinary
simultaneously
without
components.
a pairing
D have
)
, by
the
~
(D)
double
reference one
to
the
is
a
point
may
to
prove
the that
following
one:
(b)-equivalence
or
Tt,
verifies
the
does
means
of
formally
of
that
double A third not
the
TI: : (C~) .... ~ (D') J J J (b)-equivatences for all j
are
fact
stable.
pairings
components
ordinary
which
induced
coherer~ce
connected
3)
tangentially
( D ~) J
J
1 .5.]8.
is
The
The
be
connected
(a)-equivalence
~
( C ~)
only
obtain
conditions: i)
:
curves
irreducible
C and
following
as
we
(a)-equivalence.
strict
We s a y
the
an
(a)-equivalence
multiplicity
~.
curves
1)Two
correspond 2)
transformations
the
definition
by
a finite
of
the
formal
guaranteed,
number
successive
of
according
quadratic
total
to
transformations
transform
of
C
have
points. definition,
make
is
and
equivalent
reference
to
if
and
the
a concrete
equivalence.
equivalent
to
Two only
if
pairing
curves one
two
of
C the
precedent, may
and
D
be are
following
given said
by to
conditions
holds: a)
C
and
D
are
b)
C
and
D
have
c) and and
an
There
ordering
C ~
and
on
simultaneously
a tangentially
A(D)
D ~• , a r e
;
such
formally
that
an
stable V
ordinary
pairing
t. 6 J
A(C),
double T~ : ( C )
point. ;, ( D )
C[ J
and
D' , J
they
are
(a)-
equivalent.
J 4)
equisingular
exists
regular.
If
C
iff t h e y
and have
D
are the
irreducible, same
sequence
clearly of
multiplicities
in t h e
72
desingularization
respective
equisingularity
agrees
In
the
equivalence
if a)
with
general and
V
P
V
e: ( C ) ,
and
P,
i-'. e: ( c )
i
stands
for
curves
is
the
Reduced
irreducible
so
identified. not
relation two
on
than
study
they
its
be
local
to
shall
are
:
is
an
'(a)-
equiresotubte.
(T'I ( F ' )
j
, ~ (I-'.))
I
that
of
consider
of
,
where
(-,-)
j
for
the
plane
algebroid
equiresolution
and
therefore
equivalence are
only
for
said
be
a change
of
with
Jacobian
9enerat,
a number
is
of
will
formally
is
Thus,
by
Recall
the
0 ,
under
k
iff
series,
and
hence
multiplicities
invariant
does
that
over
formal
from
it
a weaker"
equivalent given
be
i.e,
curves.
different
are
irreducible
intrinsic,
embedded
variables
k-isomorphic.
case
equiresolution
fop to
the
equiresolution
equiresolution
transforms
an
invariant formal
ring
of
formal
iff
the changes
mean
consider closed
for
equivalence,
a complete
a family
sequence
Remark
when
associated each
i.e.,
to
curve when
embedded
t depends it
can
be
curves only
on
is it
constructed
said
class
from
the
. By
the
by
(D)
variables.
modulo
we
are
(a)-equisingularLity
quadratic
In to
we
curves
rings
successive of
formal
>
curves.
y'=9(x,y),
local
(F' , P.)
essentially
concept
correspond
x~=f(x)y))
of
embeddings)
embedded
,
(a)-equisingularity
The
depend
~(~)
I
aigebroid
and
TI : ( C )
multiplicity.
Henceforth, curves,
(a)-
1
j
intersection Thus~
a pairing
i
r-'.,
,
if
i
b)
Therefore
eguiresolution.
case,
only
1 . 5.10.
sequences
of
3. 1 . 5.only field
of
sy.ste m of
invariants
multiplicities
which of
the
(Characteristic
irreducible of
invariants
characteristic
determines
zero.
the
equiresolution
and
is
desingulariZation
exponents).
algebroid
for
curves
I n this over
an
determined chain.
remark algebraically
we
by
73
For ideal
such
metric
such
that
a curve
x
is
D
,
if
{ x,y
transversal,
we
}
is
have
a basis
of
a primitive
its
maximal
Puiseux
para-
representation
n
X =
t
y
/~ i =n
ai t
If
6
(1)
Write
6
=
= n.
0
6
We
then
we
set
proceed
the
(see
2.1
= min
greatest then
( 13 0 , • . • , 6 g )
= ] .
the be
shown
nor
on
the
be
In
that
for
in
/
0
exists
an
of
is
the
(2)
i=n y
=
t, m
the
b.
t li I
of
of
is
<
t~v )
( 60 . . . .
a . ~Z 0 } such
111),
of and
are
t,
)} •
1
that are
(1). the
basis
hence
it ix,y}
they
E ] X. a complete
a consequence
in
> 1
~
is
of
system the
1: y
}.
130, . • • ~ B g
choice
exponents
order
0
6 0
parametrization
exponents
representation
x = ~
g>
I
<
(B 0 .....
6~) , i )
{i
~ 2 8 .~,
1
and
set
the
This
60 )
integers
on
(Zariski,
[
(i,
....
integer
depend
Abhyankar, m
the
the
equiresolution. of
and
(60
of
exponents not
define
s defined
situation,
do
we
{- 0
and
characteristic
(1)
a
By
characteristic
formula If
I
divisor
this
they
the
invariants
parametric
a.i
parametrization
called
inversion
I
there
Moreover, of
i
characteristic
can
may
{
i
If
commun
.13),
called
{
inductively.
6V+1
Since
= rnin
1
> 1 ,
0
we
have
another
74
for be
a suitable
t'.
defined
for
Characteristic
(2)
a)
If
in
the
n < m <
exponents
same
way.
61 , t h e n
~
~ - n , ~ l , . .~.
,
~!g ,
can
Then:
g'=g+l
,
=n
' =6 V +n-m 6~+1
and
,
X
1 ~< v
b)
If
n=m
or
Remark
3. 1 .6.-(Characteris..tic
of
of
pairs
integers,
1 , then
m=6
pairs)..
called
and
g'=g
In
6~ =6.~
the
characteristic
case
pairs,
+n-m,
k=C play
I<
V~< g .
a system
an
important
role. They exponents the
are
in t h e
conditions
been
defined
following 61/n
then
verifies
way:
ml/n
I
,n,)
is
I
_
recurrence
The
first
and
by
n
((m.,n.))
set
I
. .n
1
n
i-I
pairs
pa
for
3.1
of
of
the
set
of
and
determines
rs
with
. If
is
given
i-I
by
pairs
have
conditions
g.c.d.(m.
and
i
the
m
n i-I
conditions
.?.-(Toroidal positive
integers
knots).
integers
,n.)=l
.
i
characteristic
< m., i
exponents
2 ,
Conversely,
i
is t h e
The
image
set
of
characteristic
= m
knot
I
g.c.d.(m
I
1 ~
,n )=1, I i by
inductively
-
i
be
-~g
and
cf
mi_
the
1
n
i
+ p
i-1
n
i-1
n
i
~
i
>~
,
.
I
)
of
type
S
2
"
map
1 x
S 1
( z ml , znl
(S )
(m 1 ,nl).
I
= unit
a set
(p.) I I ~
1
S 1
toroidal
F =((m.,n.))
= ml
Tml n
z
Let
with
constructed
Pi
called
I ,ni)=I
completely
)=1
these
Pl
is
characteristic
a curve.
Remark pairs
the
( m I ,n I )
i
g.c.d.(m.,n
of
pair
the
I
any
from
g.c.d.(m
given
i
n
and
by
m
i
set
=
(m
J~
The
constructed
circle)
75
If
the
toroidal
knot
of
type
m 1 • • .mi_ 1 T
is
defined,
we
define
T
n1 •..ni_ 1
( m i _ 1 ~ni_ 1)
. . . . .
as
follows:
Take
a tubular
n 1 ...n.i
neighbourhood S 1
( m l , n 1)
m1 . . . m l
Ki_ 1
m1 • . . m i _ 1 Tnl...ni_ 1
of
ml""m" T I n I ...n i
x S 1 . Then
is
the
whose
image
bond p. T t n.i
of
is
homeomorphic
by
that
to
homeomorphism.
m I ...m
A loroidal 2
~i
~< g ,
F is
the
Pi
> Pi-ln'i-1
set
of
3.t
.?.-(Toroidal
Remark
singularity). the
¢
of
of
IR 4 .
and
The
there
with
point
Brauner
characteristic
Remark
¢
, and
is
analytic
pairs
in
the
determine the
of
above the
following
Theorem:
C
the
power
irreducible
the variety
of
an
of
isolated
There
differentiable
the
origin
singularity
type
remark,
toroidal
result exists
by g
manifolds
C(Sg~C
B
) is the real o the closed ball with
is s topological morphic
, if
curve at
algebroid
to of
f(x,y)=0
a surface
C
is C
o (17 ])
(Milnor,
C.
of
a sin9u!arity).
that
the
associated
Milnor
[ 17 } :
> 0
such
The
characteristic
knot
(S
that
, S g
if
i,
series
I122 of e q u a t i o n
V
with
E
, 0
<
.
spaces
(B£
~C g
projecting centre ,Bgt~C
o
)
)
are
fact,
pairs it
o of
m.i , i . e .
plane
irreducible
a neighbourhood
O
all
a curve.
a complex
let
<
mi_lni
for
f(x,y)=0.
3.1 .8.-(Topological
explained
with
for
a convergent 2
if
if
o g > 0 such that ~ g , 0 < g-< g the sphere S o o £ O and radius g , meet C transversaly and the o C f-I S is a nonsingular real curve, which by a theorem o g ( ? .~ i s a c t u a l l y the toroidal knot defined by the
centre
curve
of
algebraic
exist
intersection by
equivalently
knot
of
in
is c a l l e d
pairs
¢2 i l R 4 ,
locally
g
n1 ...ng
or
equation
Taking identified
n''l
f(x,y)
O=(0,0)
over
T
characteristic
Let
origin
curve
knot
, is
~:~ ~o
of
a
complemented
the
pairs
diffeomorphic.
Moreover,
o
cone
of
O and
radius
and
(Bg
S N C E: o E, the ,C(SgI'~,C
o
from
O
pairs
of
))
are
and
homeo-
if
76
This
theorem
of (B ,B c ~C - - g --0 determined by teristic
)
and
the
establishes the
of
C,
Conversely, ,(27
]
result,
case
The
curve
determined
by
components.
Remark
3.1 .9.-
the
problem
analytic
of
curves,
may
Zariski
in
of
is
be
studied
to
2.
be
formal
algebraically
an
closed
Definition
3.2.1
recurrence
on
the
.-
the
8
]
and
analytically of
the singularity.
following
a complex of
algebraic
The
easily
in
may
terms
complex
type.
This
for
global
techniques
to
the
"invariant" of this
,(see
problem
plane
of
consider
problem
algebroid
word
pairs
all
topological
equivalent all
any
isomorphism,
purely
analytic
analytically
of
One
moduli
is
each
multiplicity
Riemann
of
of
curves
with
introduced
moduli
problem.
EXPONENTS.
irreducible
plane
alg,ebroid
curve
[]
over
k.
We s h a l l number
an
the
of
embedding
class.
field
have
type
analytic
equivalence
CHARACTERISTIC
of
singularities)___:.
, it
interpreted
Consider
we
topological
b.y m e a n s
particular
characteristic:
topology
a singularity
a given
(a)-equisingularity may
the
analogous
modulo
3.1 .4.
of
charac-
, [ 27 ) ) :
modulo
with
In
curves,
local
classifying,
( 26 ) ) .
classifying a given
(Moduli
which
the
type
uniquely
? ] , Bur'au,[ pairs
on
topological
adjetive [
and the i n t e r s e c t i o n
singularities
problem,
and
only
the
, S s F ] C o) are . . . . hence by the
g
9 the
Zarisk'i
type
distinct
(S
Br'auner,
reducible
1 5 ] , and
component
that
characteristic
topological
is
irreducible
by
depend
of
of
justifyin
the
curve
(Lejeune,(
Theorem: plane
the
thus
that
plane In
type pairs,
results
show
irreducible
isotopy
characteristic
exponents
Zariski
therefore
of
define quadratic
the
genus
of
a curve
transformations
[] needed
by to
an
77
desingularize
[] LeL
~ in
I-'1 1
be
(i
If
[]
(ii
tf
e(r'])
the
the
following
quadratic
transformed
> e(l~]
1
If
e([-])
)
,and
> e(E]
g([:])
I I
e(r-I
Remark
and
curve
using be
g(r-])
=
) , and
: g(F1
t~
above
:
Consider for
[]
same
not
divide
e(E]):
e(D)
:
g(l-]l).
definition
g(r-])
= 0
iff
expansion,
by
n1 n2
n]
does
I
g([-I)
computed
n
)
e(r-I ) divides
Hamburger-Noether
( {
the
1
I.
) +
e(t-I I ) :
trivially
model having
:
the
the
3.2.3.-
complex
field
From
can
Definition a
e(P"l)
3.2.2.-
regular, the
If
D,
).
I
(iv)
of
g([]) = O.
iS regular:
g(Fl) : g ( • (i i)
way:
the
n r-1 n
r-]
the
is
genus
of
formula:
} I~ Z
)
r
a plane
algebroid
to be a p l a n e c u r v e singularity
curve I"]
reduction
[]
over process
. the
We d e f i n e complex
(1 . 5 .
!0)
and
[]
as
D. More
precisely~
D
the
cD
condition
t
requires
c_ . . . .
cD
D c c (E]~:) 1 c .... are
the
respective
desingular
zation
M
that
=12
C(E]C)M,
sequences
if
: "G~: for
C '
then:
Proposition
expansion
(i)
M = M'
(ii)
e([--I )1
3.2.4.-
given
by:
TM
e((ElC)
Let
r-I
)
,
be
0
~< i ~< M.
a curve
with
a
Hamburger-Noether
7B
(-D- )
z j-i
=
aji
h
i
zj
+ zj
j
zj+ 1
,
0 ~< j ,< r` ,
i
and
denote
by
the
complex
Then
F
: k
¢
curve
any
[] ¢
map
which
F(x)
~ 0
.~
,
0 ~< j
)
x ;~ 0 .
has
---
hj
,i
z j_ I =
(D¢)
verifying
F ( a j t ) . . z.i
+ z'j
,
z.1+1
~
i as
Hamburger-Noether
Proof:
First,
(see
expansion
recall
that
such
check
a curve
that
order
the
to
prove
values
h
the
model
with
proposition
and
n
J
agree
for
expansion
[7.
(__DC )
exists,
in
notice
both
that
expansions
it
suffices
(D)
and
J
to (D¢)
--
_
(2 . 2 .10.) . By
have
follows
n
r"
construction -n'
assume
=1 ~
r
and
that
that
0¢ []
complex
cur`yes
Proof:
Since and
Corollary
)
the
Iool
= n'.
,
J
J
Let
[]
(resp. I-7 ~
are
17
and
[7
0
is
3.2.6.-
the
All
~)
a curve
over
k , and
model
for
[]
if
and
only
(resp. if
r-I~).
the
(a)-equisingular.
sequences is
n ~ = v(z'.) J j on F , it
requirement
a complex
are
If
~< r .
(a)-equisingular
proof
the
coincide.
J the
0
desingularization agree
h at
-{j
(resp.
r7¢)
and
the
[]~¢
and
by
n
3.2.5.-
curves
E] ~
r
trivially
Proposition
The
a complex
2.2.13.). tn
we
is
for
[]
and
E]¢
(resp.
evident.
complex
models
for"
a curve
[]
aPe
(a)-
equisingular.
Proposition g(E1)
Proof: in
the
=
3.2.7.-
g(13
¢ ).
Let
n
if
121 ¢:
is
, 0 4 j -.< r , b e t h e ] desingularization sequence
a complex
different for'
[]
m,odel
for
multiplicities (actually
for"
0,
then
which I--I¢).
occur
Then
79
we
have:
g(r-1)
:
&
({n
n nr-1 } r
n ' "o " ' ' 1
Definition closed
3.2.8.field
k,
We d e f i n e
the
exponents
of
Remark all
of
do
not
Consider and
By
ape
of
on
above
integers
also
e(E])
(ii)
60
....
exponents.
In
>(
other
Lemma
3.2,11.-
the
Two genus
.
Let
following (a)
of
[]
statements If
g = g' ,
is
[]
be
(resp.
the may
,
3.2.?.
the
. and
the
charac-
El. exponents
are
non-
>
g
( 60 .....
are
6g)
(a)-
same
set
state
equisingular of
the
char
of
denote
If
61 = h n
6'.~ = 1 3 -
n,
and
"The
set
invariants
by
9) 0.<<'g~
(resp.
if
only
tcteristic
following:
system
and
= 1.
[] be
+ n1 ,
hold:
6' 0
3.2.5.
that:
a curve
Ell).
By
exponents
by
h >1,
60 =
r].
characteristic
models.
coherent,
ring
a complete
( 6V ) 0 ~< v~
Let
exponents
we
g(t~)+l
• . • <6
curves and
the
for
characteristic
characteristic
61 ) > . - -
exponents
si ngul ar'i t y"
<
model
be
complex
therefore
such 1
to
the
is
the
6
words,
(a)-equi
transform
<
0
60 ,
same
characteristic
ristic
6
3.2.10.the
that
algebraically
Furthermore,
on t h e
(6.,)) 0 -.
(i)
if they have
only
[]
exist
exponents
depend
the
a complex
of
model.
is
g(E]C).
ovcr
is
Then
definition
:
C"
there
complex
We r e m a r k
Proposition
[]
3.2.4.
the
exponents
negative
I--] C
curve
2 )
[]
exponents
characteristic The
teristic
that
(a)-equisingular.
depends
number
assume
complex
3.2.9.them
a curve
characteristic the
n
1 4 9 ~< g -
1
for"
its
the where
of the
quadratic
chaeacte0
80
(b)
f
h = I ,
and
n 1
does
not
divide
n,
i
g = g'
'
B0 = n 1
(c)
f
h = 1 ~ and
t--I
If
plex
1._I C
model
result
for
only
model
for
in
(see
Remark
3.2.
any
the
this and
the
so
the
This
Pr'o#osition
~< "g ~
then
(I
curve
may
be
IC) 1
is
com-
a
the
field.
taken
derives
t--I
g-1.
to p r o v e
is the c o m p l e x
proposition
to
formula,
find
as
a complex
from
quadratic
may
relations,
exponents
be
as
given
tran~sform
the
"inversion
.
by
in
used the
when
above
Puiseux
This
too,
lemma,
series
leads
,
in
k
for
particular,
result:
3.2.1
3.-
Let
k
characteristic
exponents
of
any
characteristic
exponents
of
any
c u r v e
I 1,
1
it is sufficient
inversion
zero,
following
,
l
5.).
1 2.-
its
n,
for
k
any
1 ~< ,~ ~
6~ +1 - n
model
in w h i c h
case,
3.1.
and
B~=
*
,
divides
1
. Consequently,
I
characteristic
curve
n
a complex
characteristic
among
to
[]
itself,
formula"
has
is
in the c a s e
But,
B,0' = 13 - n 1
B0 = n l
g = g +1,
Proof:
'
be
a field
curve
of
characteristic
over
Puiseux
k
agree
series
O.
with
The
the
representing
the
.
Proof:
By
induction
transformations
which
for-
each
(I~)0~<'# quadratic
M = M(r-])
needed
For for
on
M = 0,
M = M(E-]) El' ~
with the
to the
proof
is
> 0,
and
assume
< M.
1).
evident.
( B'~)O 4,~ ~
represent
E]
(resp.
Let
that
[]
). I
of
quadratic
same
a curve
over
proposition
is
proved
( ~',~) 0 ~< V.
(resp.
the
by
exponents The
number
[]).
Denote
characteristic []
minimum
desingularize
M([]')
transform
(the
for denote
exponents
[]
[] by
be
(resp. (B~)
by
the (resp.
0 ~
k,
81
By On
the
the
ether
induction
hand,
using
the
above
and
6.g = IB,~
3.
it
over
the
maximal
and
the
-~ ,
an
g' = g'
that
60=
remark
and
171V;'- J3v
a nd
60
3. 2. t 2. ,
, 0 4V
61 = 6 ~: 1
we
'
deduce
~
hence
g = g
'
~4
0,.< ,,) < g .
CHARACTERISTIC
EXPONENTS
AND
HAMBURGER-
EXPANSIONS.
3.3.1
[]
evident
each
NOETHER
Notations
is
lemma
for
hypothesis
.-
Consider
algebraically ideal
m
the
irreducible
closed
field
is c h o s e n
and
k.
plane
algebroid
Suppose
denote
that
a
curve
basis
of
by h
(D)
z :l_a
--
the
j
Harnburger-Noether
= v(z -
are
j
),
v 0..<
invariants
is j
0
shal
ji
expansion the
of
Let
j,
a. i
If n
=
the
s.
j
z
j
Ell
h
(see
< s_ < .... < s g Z ~<j ~< r , for which n I n j j-1 also put s = O. N o t i c e that
j
,
n.,
to 0
j
be
the
For g
is
0
~< j .< r ,
basis
[7,
2.2.
(h
o
o
we
set
=h,
h
r
, y=z
). -1
= oo ) ,
10.).
ordered
set
convenience
of
exact)y
(x=z
and
(. j ~< r ,
proposition
/
,
j+l
in t h a t
valuation
integens
curve
+ z
for
associated
the
z
the
of
the
notations
genus
of
indices we L~.
o
Proposition teristic
3.3.2.exponents
130 =
6,~+ I =
Proof:
We
use
Let are
again
an
algebroid
6,~ )0.< "0 ~
no = n
~ j=O
be
[]
curve,
whose
charac-
Then:
,
h
n
+ n
J
induction
÷n s,~
--
n
sv+ I
on
M(E])
= M.
For
M = 0
the
result
82
is
evident. Let
( a
6"0)0
4"0 ~
certain
( 6.~ )0
[] and
basis.
We
the
we
obtain
induction
4g.
(D)
n1 ~
h = 1
and
It
in
is
For
,
and
#
~3 a s '0
desired.
n.
We
in
s
that
130 = to
as
case
trivial
expansion
transform
as
applied
in
of
17,
lemma I
,
I'-]1 ,
6. = 6 . 0 - n , and
g'
3.2.11
the
= g,
.:
1.~ v 4 g "
above
relations
6 ~) = n t '
6 v' = 6`0-n I '
have
g' = g-1
,
6 ~) = n 1 ,
nl
+
6.~ = n
the
:
1
~'-
+
and
6 1 = n + n]
induction
hypothesis
2.-j=l
h
hjnj+
J
n
J
+ n
3.3.3.-
expansion
given
Let
[]
be
+ n
so
+n
n
to
E] 1
sv+l
-
n 1 =
-n.
sv
Proposition
s`0+1
a curve
with
Hamburger-Noether
by h z j_ 1 =
aji
zj
i
Jz
+ zj
j+l
0 ..< j ..< r .
'
i Keeping
the
statements
notations
in
the
above
proposition,
t he
following
hold: (a)
Furthermore,
as
If
j /
in t h i s
sv
,
case
0~
6~ = 6 v + l - n l ,
= 1.
60=n
applying
60
have
We 1
in
get :
6v+ 1 =
and
(a).
n 1 I n.
this
`0 >1 ,
cases
g'=g,
h = 1
(c)
quadratic
three
for
follows
exponents
exponents
expresions
But
the
Actually
proof
characteristic
a Hamburger-Noether
be
consider
The
1 ~< `0 -.
be
[]
hypothesis
the
with
1 characteristic
h > 1.
(b) 1 4V
let
shall
(a)
a curve,
Let
its
-.<"0( g '
By
be
j-1
then
a.ji
= h. n + j j
n
= 0 ,
j+1
1..<
i ~< h J
~ we
83
(b) such
that
If
aji
(
(a)
indices
1 <
60 ,
and
61
i
,
1~< *g -.
<%
,and
....
'
(b)
By
there ajkv/:'
) = (n
proceed
exists
s
0 . , n
_1
directly
an
tn
this
s.~_l
from
integer case
) = n
+1
the
k v<~ h s M nj_ 1 = %nj. ,
sv
1~< V ..< g .
properties
of
the
s,~ . We
(a)
= s.~
= 0,
(c)
Proof:
j
and
shall
(b)
prove
n
is
(c) the
by
last
using
induction
remainder
in
o n .,~ .
the
For
Euclid's
".~= I , b y
algorithm
s 1 for
n and
nl
,
so
it
is
its
( B 0'
Now, By
using
if
a similar
3.3.2.
we
(c)
greater
6 1 ) = (n,n
holds
for
argument
as
,6.~
{x~y}
3.3.4.the
in
Thus~
( 6 0 .....
6~)
s 1
then
the
Keeping
1
the
Hamburger-Noether
may
V,
= n
case
) = (n
s,~
tion~
1)
divisor.
"~ =1
and
= (nsv,
according
6V+l). to
obtain
(n
Remark
common
be
written
in
the
,
n
s v
) = n ; sV+ 1
%+1
notations
as
in
expansion
of
a curve
following
the
x + ...
+ aOh x h
x =
k =
as 1 k I
+.,
g
-1
z
s
g
k
+
z g
s
g
g
s
Sl+l z
st+1
+
....
hs I
z
a
Sl,h
k s
.
s I
=
a
z 1
1
Z
=
h + x
h
z
a basis
z 2
(D')
zsl
in
l
z 1
zsl-I
[]
proposi-
way:
h y = ao1
preceding
s 1 +2
S
I
hs +
1
Z
I Z
s I
Sl+l
84
with
a
{
0
,
1
~
s,ok.o Notice 1 < I) ~< 9 ,
that
the
determine
trivially
Thereafter, notations
which
expansion
(D)
expansion
is
Remark expansion points
of
should actually
h
=
in
reduced complete
the
origin
O
(A)
Free
points.
1.
The
h
of
the
points
of
we
k.,~
,
we
shall
of
17.
keep
the
a Hamburger-Noether
shall
assume
reduced
form
and class
specified, When
the
, g
(D')
form
of
that
this
(D').
the
Hamburger-Noether
information
about
the
infinitely
curve
1.5.1
5.-1
.5.20.):
(see
multiplicity
n
and
the
first
near
one
of
and
the
n1 2.
one
here.
considered,
written
The
~ 0 ~< j ~< r : s J (a)-equisingularity
otherwise
used
be
provides
multiplicity
first
been
h
the
unless
are
3.3.5.-
integers
of
The
h
-k s.g
last
'2
multiplicity
n
points ,
s~+ 1
of
(14
v
multiplicity
x<9).
Here
n we
,
sv
may
suppose
that
co
s 9 Note free
infinitely
not
necessarily
that near
free
exists a o n e - o n e
points
onto
the
correspondence
coefficients
in
from
(D')
which
Satellite
are
the
points.
points
which
are
not
included
in
the
above
points.
(C) They
Leading are
free
the
O '•4
(D)
the
zero.
(B) All
there
Terminal
points.
following
=
9
Oh+ h t
satellite
points:
+...+h
points.
s
+ k ",}-1
,
1,< "a ~< g .
list
of
85
They
are
the
following
On
=
g
points:
lj h+hl+...+h
(E)
Proximate
points
of
s 'J-1
+k
-t,
14
"~< g .
O. I
poi,nt
of
If
e(O.)
2.
If
n
(i)
If
j
= e(O
f
)
i+1
,
0
is
the
only
proximate
l+l
O. I
the
h
points J n j+ 1 ,(here hs
= e(O.) i
j-1 ~ s
,
"g
>
1.< ~ 9 ,
multiplicity
of
=co
).
If
j
e(O
i+1
the
n
,
then:
proximate
and
J
) = n. j
the
points
first
one
of
of
O.
are
i
multiplicity
g (ii) O. ,
are
the
k,)
We proximity I 1,
5)
curve
Lemma
first
remark
(which , for can
= s.,~
that may
of
the
be
'#,1-.< ,.) 4 g ,
multiplicity
classical
found
near
obtained
3. 3.6.-
some
points
infinitely be
for
for
trivially
Consider
n, j results
instance
points
the
of from
the thi,s
a rational
= n
on in
proximate
% satellitisme
Zariski
origin
points
of
,
an
and
~27),
irreducible
analysis.
number
m/n
,
m >n
> 0.
that
m
:
c
n
+
r
r
+
o n
=
1
c I
r
denotes
the
Euclid's
s-1
::
C
We
shall
use
.
J
induction
2
r
s
algorithm
j=O Proof:
C
r
I
s
for"
r
+
J
the
r
-
s
on
s.
integers
n
GTS,
=
m
m
~
and
(
n.
=
r
0
Then
n
).
Assume
86
For holds. we
s = 1,
Applying
the
it
is
evident.
induction
Suppose
hypothesis
to
the
for
s-1
rational
the
formula
number
n/r
1
obtain s
s
~'-- C j r j
+ rs - n = c0 n + r1 +
j=0
3.3.?.-
Proposition
(6,)
) 0 ~< V<~ g
60
,
the
= n
Proof:
For
suppose
For
I)> ] .
0 Using
6V+ 1
6j=
I~
- ns
+
By
the
preceding
completes
the
Remark also
s~
lemma
-k
the
infinitely
The
then
near"
is
12
)n
exactly
the
bracket
a free
-9
point
sV - I +
V<~
9-1
are
evident.
Now,
l +"""
+(%-l)n
sV
]
+
s,)+l
has
the
value
=
denotes anterior
the
as
and
L
set to
exponents
n
and
s~
this
e(O.)
formed O V
can
be
expressed
follows: the
near
ic b I
1 - l )n
+n
s.g
denotes
genus
By
where
I~<
have
sV - 1 +
points
(O ~) v 0 < v ..
,
sv+l
equalities
characteristic
If
the
n
proof.
3.3.8.using
[(h
exponents
hold:
+
we
+
(h
characteristic
corresponding
3.3.2.
•
;
kv) ns. ~
the
1
with
= h n + n1
( hs~
n = m
j=l
equalities
61
By+
~--- c j r j + r s - r l -
a curve,
following
;
6v+ 1
in
that
set
points we
by
terminal
of
the
l(
-~ < g ,
satellite
origin
of
the
points curve,
have
,
i
of
the
indices
for
which
O.
i
is
87
Remark
3.3.9.-
expresions
Proposition
of
the
characteristic
Hamburger-Noether
of
( I~v ) 0 ~ ' ~
the
and
3.3.?.
exponents
in
give
function
us of
directly
the
expansion.
Conversely, curve
3.3.2.
if
~
we
we
know
may
Hamburger-Noether
the
characteristic
calculate
the
expansion
exponents
integers
in
the
13,) )
,
h,j ~ n j
following
of
a
and
k
way:
First,
n
= (
%
60 . . . . .
0
~ v ~< g .
(1) h
Now,
we
may
get
n
s
s~
- k.,~ =
from
ns
the
,
equality
"g+ 1 ns~
= 6.~+1
-
6V-
+1 Finally,
by
I ~< V -,< g .
n
expressing
s
/n s
( h sv
-
k
)
as
a continued
k,)
are
n
s
fraction,
one
finds
~+1 the
rest
of
values
But h
s.g
may
be
unless
actually
independent. gularity hj
~ 0.~
integers parameters
that The
class) j
the
computed
Notice
k,~ , are
h
the
from
the
the
values
second
determined subset
1 ~: "~ ..< g , certainly
s.)
numbers
characteristic
are
< r, b y
the
formula
in
known, (1)
then
the
.
h., n. , and k are not of course J J exponents (hence the (a)-equisinby
of
already
the
indices
satisfying independent.
the
number
r,
Sl ,... ,Sg condition
by
the
integers
,
and
by
k
,,)
~
s
the These
88
4.
CHARACTERISTIC
EXPONENTS
AND
THE
NEWTON
POLYGON.
We to
compute
from
give
the
any
of
its
by
origin
of
embedded
successive
of
[]
as
{x,y}
a basis
By
1 .5.1
of
the
the
the
4. ,2.2.8
us
directly
in
series
that
m
f~
points.
of
( 6
and
defining
g
these
exponent
exponents
of the
, f f~
'
at
x and
points
.
1'"
[]
with
basis
2.2.12,
-~ o f g
characteristic
characteristic
of
near
and
. . , D
r7 1 ,.
first
curve
infinitely
uniquely
transforms ,
of
satellite
x determine
3.3.?.
must
~)
solve
order
if
obtain
the
f~ "0-1"
Within
get
plane
algebroid
the
the
first
polygon
of
and
1) n
h~ ~ v
) 0 ~< ,,# ~
BI(I
exponents
of
diagram
6~)
first
its
of
one
th;
s two
3.4.1.-
curve
, and
[]
6v
are
+1
-
13V+
known,,
n
s,g
then
successively. consists
6,m 1
induced
k((X,Y))/P__
equations, from questions
Let
a basis
us
cons
{x,y}
be
der of
giving
that
we
a method
a curve
second
Newton will
isomorphism
[]
of
the the
For
in
exponent
f ~%)
notations
is an
IV)
characteristic any
=
sv+l
formula.
the
section
there
+ n
sv
and
above
questions:
this
tions
-kv+
g,~
the
Newton
Defini
1.1.3.
by
find
Newton
sv
60 . . . . .
shall
two to
= (h
V
computed We
By
curve
enables
, by
be
the
which
a plane
and
f(X,Y)=0
curve.
function
Thus
in
of
curve
terminal
quadratic
BI([[]
may
the
parameter
given,
an a l g o r i t h m
exponents
equation
0 ~g
Using is
section
an irreducible an
0 T ,...,
transversal the
be
Take
the
present
equations.
[]
transversal.
the
characteristic
Let
denote
in
one
diagram
from to of
solved.
an
irreducible
its m a x i m a l
ideal.
the
89
where be
the
prime
ideal
a generator
k((X,Y))
of
p
p .
is
The
(resp.
Z
real)
(resp.
+
numbers,
IR the
D(f)
be
called
A
C c~ weighted
f
is
k
is
of
the
f(X,Y)
L:0
:
determined
upon
BX
a unit
in
convex
to
of
set
Z 2 / +
for
f.
of
A
if
( c~, 6 ) 6:: D ( f ) of
definition
hull
the
nonnegative
,
the
we
;{
c~,13
0
}
value
shall
(or
say
"mass"
that
it
is
)
the
f.
the
Newton
polygon
of
f
is
segments and half lines in the 2 + IR which are not contained
D(f)
integer
set
diagram
straight
denotes
dia,gram
attached
Newton
)
+
.... { ( 0 ; , B ) g
Newton
By ting
series
Let
. If
will
principal.
the
set
consis-
boundary in
+
the
of
the
coordinate
axes.
Remark
3.4.2.-
generator
given
by
The
Newton
of
the
ideal
In
the
particular
6=1
p_
,0~ >~0.
polygon
does
not
depend
on
the
chosen. case
The
p = (Y)
same,
if
p_ -
the
Newton
(X)
,
it
is
polygon given
is
by
c~=_l ,
6 >,0. Excluding we
may
lying
assert
these
that
respectively
and
n
are
there on
chosen
n = co ,
transversal form
of
Lemma v(x) to
1:3).
if m=
1.
points
A-(m,0)
coordinate
,
shall
for
axes.
to
p ~= ( Y ) We
and since
[-] ,
we also
be
f ,
We
is
B
irreducible
=(0,n)
shall
in
assume
, D(f)
that
m
minimal.
shall
set
assume
(i.e.
X does
n
1 ,
that
x
not
m=
0o
; and
= X + (f)
divide
the
if
is
and
If v(y)
A = (m,0) = m.
(v
and being
B = (0,n) the
natural
are
as
valuation
above,
a
leading
).
3.4.3.= n
the
parameter f
are
cases
respectively
Moreover, p_ = ( X ) ,
trivial
then
associated
90
Proof:
Let
C'
parametric for
be
the
curve
representation,
C'
If
C
v (y)
is
C'
therefore
our
= (C',
y = 0.
curve,
C)
we
= (C,
has x
x = x
is
,
y = 0
,
as
a uniformizing
a
parameter
have
C')
= U
(f(x,0))
=
m .
--X
An
:identical
argument
Lemma
3.4.4.-
straight
line
E ~ (X)
since
Proof:
We s h a l
holds
If
is
x=
E ~z ( Y )
, the
which
joins
segment x
for
0.
Newton the
polygon
points
A
prove
it
by
induction
on
x
[]
M(E])
in t diagram
this is
case: {x, basis
subject
homologous straight must
be
by line
this
> 0,
the
curve
curve
=(m,0)
M([~)
the
m y/x
> n.
}
is
We r e f e r
of
its
maximal
obtained
to
the
the
from
plane
:
that
AB
the
a curve
that
with is
for
M(FI
true.
')
We
cases:
quadratic
[]
transform
equation the
with
assume
El'
two
when
evident
fl
weighted
=0
of
Newton
transfromation
>
polygons
transformation. is
f:=0
is
is
B = (0,1).
lemma
ideal.An
( cz,6 )
Newton
it
the
result
and
every
segment,
necessarily
= 0,
be
x
T
Moreover
( Note
actually
consider
basis
B.
The
Let
x
B=(O,n ~ A
the
the
and
M([7)
because
x
to
is
M(EI).
regular.
st
f
transversal).
If
1
for
Since same
for
( 0~+ 6 - n ,
of the that
f
6 ).
and polygon of
f
fl of This
are fl
is
a
segment
<
91
2 is
nd
a power
(0~,
in
polygon
is
the
of
Since
order
Newton
n.
the It
diagram
multiplicity
follows
with
[]
of
that
there
0~ + 6 <
n.
is
are
Hence
n,
no
the
f
points
Newton
AB
3. 4. 5.-
Newton
m=n.
series
6)
Lemma
case:
Let
polygon
f
of
,
f ,
A
,
B
~ m
L(X,Y,f)
=
,
n
and
as
above.
Let
P
be
the
and
~
A
XC~Y 6
(c~ , 6 ) C P
a,6
Then: (a) a unit
in
I f
k([X,Y)) (b)
exist
a,
m = oo
L(X,Y,f)
= Y
U(X,Y),
where
U(X,Y)
is
.
If
~ C: k
,
m
/ o~
, a.~ 0
and , ~k ~
0
with
= m'/n ,
m/n
, such
( m' ,n')
= 1 ~ there
that n/n' n ~
L(X,Y,f)
Proof: on
(a)
M([--I).
A = (m,0)
is e v i d e n t . For
and
M(D)
We = 0,
B = (0,t)
shall the
,
M([[]')
< M(E]) 1
st
preceding f
1
take (b) case:
lemma
holds. > n.
=
[] As
maps
(b)
Newton
f)
a curve
m
prove
-?t X m
by
)
using
again
joins
the
polygon
induction
points
then
L(X,Y
Now,
= a (Y
I
with
above
The
the
a ( Y
plane
Newton
-I. X m )
M([7)> we
,
a ~ 0
0,
consider
and two
of
~/
assume
O.
that
cases:
transformation polygon
,
T
f
onto
by
the
in t h e that
of
. If
hypothesis:
(re,n)
= r
,
then
(m-n,n)
= r
; thus
induction
for
92
- ~Y
L(X,
) = (c'--
'fl
yn ~
-
n'
a
xm,
, r
n )
,
a ~ 0
c'¢
0.
,
then
X Hence
Y
L(X,Y,f)
nd
2 it
is
we
of
type
must
also
Proposition
case:
m = n.
a(Y-x
X) n
have
v_(y) .... m = n d ,
with
a ~ O,
),/
L(X,Y,f)
,
a ~ O,
since
With with
O.
notations
d
Then
>0,
is
the
f
is
if
,
n
Y'
leading
- a X 'm)
form
irreducible.
of
f
As
m = n,
by
Proof:
The
change
between
the
respective
the
and ,
above
lemmas,
L(X,Y,f) we
have
= a(Y-
assume n ),X d ) ,
v(y)
>d n
>
{x,y}
the
as
in
~ x+dy =c~+d B
Particulary, is
in
y ...... y - ~ x d
A=(m,0)
polygon
as
d ~g Z ,
B=(0,n)
Newton
) = (c'
X ~ O.
3.4.6.-
that
= X n L(X,-~-,fl
the
change
displaced
{x,y
}
A=(m,0)
to
:>
Newton
the
{x,y} right
{x,y}
side
induces
diagrams
•
the
picture.
a multivaluated If
(
functiG~
0~6 ) E: D ( f ) ,
6
XeyB=
then
the
0
~: 6 '
~i
point All
X G ( ~ + ~xd)B= X G ( ~ (B) i i=O ( 0~,6 )
lie
on
the
is
transformed straight
line
into
)i xdi~6-i)
the
x + d y =
points
(o~ + d i , 6 - i )
C~+ d 6 ,
i.e.,
, on
93
the
parallel
line As
plane
points
to
complete
x+dy
the
proof
by
the
of
f
not
is
the
successively
Let
If
the
be
an
is
.
in
which
the
regular
(b) we
get
If
y
is
not
[]
a change
of
.
in
P
Since
the
joins
v(y)=
m
,
m >d n . the
change
the
get d
= 0
x
as
curve
the
Newton
A = (m,0)
~ we
i
plane that
~ with
whenever
)~
has
that
remains
polygon
m >/dn
Assume
3.4.6.
= y -
curve
the
after
and
i=l for
with
prove
~_ y
Thus
irreducible
k((X,Y))
as
[]
,
diagram
point
diagram.
B = (0,n)
changes
>/ 0 .
to
( G ,13 ) .
Newton
because
[]
points
(a)
y
A = (m,0)
to
C
through
new
x >~ O ,
evident,
f(X,Y)
joins
the
suffices
be Ion9
3.4.?.series
,
it
passes
,
and
this
does
Corollary
which
>~ d n
B = (0,n)
But (d n,0)
P
a consequence
region
the
to
n
i
(d
'
a new
regular,
of
~
<
polygon
n ~
divides
a change
defined
m. m.
Apply Then:
type
d
i+1
)
,
equation.
after
a f
,
(d
nite
number
of
steps
type s y
d
= y-
)ti
x
< d
)
i = t
such
that
if
first
characteristic
Proof:
v(y
(a)
It
(b) By
tookin9
{ x,y} m after
changes It
is
one
evident
has of
(m []
since
be the first 1 Hamburger-Noether that as
follows
m = v(y)..< in that
3.4.6. there
,n)
z. n .
Actually
i s the
m
.
n=l.
13
the
sees
m
exponent
Let
at
one
=
t3 1 are is
characteristic
exponent
expansion Thus bounded a change
all
of the
by
[]
possible 131
in
of the values
~-I . basis of
94
s
14
d
)---
y
= y -
i
hi x
(di
< di+l
)
i=0
such
that
n
does
not
Hamburger-Noether
divide
m
expansion
14
14
= v(y
of
[]
) . By
(now
looking
in
the
again
at
the
basis
{ x,y14})
3.4.6.)
may
one
t4
can
see
that
Remark
= 131"
3.4.8.-
computed
The
dir"ectly
Indeed, ape
m
for
the
points
The diagram
fr"om
ever"y
only
value the the
which
may
change
m
must
is
given
y,
0)
In point
(
be the
the
(~1 '
way,
131) E:: Ly
is
be
on
the
(¥,
str"aight
( y ,0)
substitution.
line
into
in
0)
any
be
L
: c~+d6- Y
Y
(T,0). in
the
weight, ed
by
(c~,13) C L
T y for"
which
associated
giving
by
x6
A((y,13)
integer"
=
without
tr`ansformed A
the
~( ~' 13t
diagr"am
~
=
least
same
proposition
points
mass
~( Thus,
(in
Newton
y>dn
associated
after
m
the
mass
A(y,0)~
0.
~,-(0~.1,131) i n
any
formula
x
(~,13) c L
6 - 131 (~1) A(
~' 13)
Y 6/>13 1
We h a v e beginning of
any
of
this
curve
Notations section,
c;~0
expansion. by
the
3.4.9.we
to Newton
With
shall
study
has and
solved
section:
from
f 14 1=fl ~(u,w) where
just
order n
s1 We s h a l l
For
the
find
first the
first
outlined
at
characteristic
the
exponent
diagram.
assumptions
as
now
the
between
relation and
, a
its
are given slk 1 use in the sequel
sake
question
the
el=nSl
f I (u ' uV)/ue
fl (u'v)=
the
leading by the the
in
the
form
beginning f and is
of
f~ . The /
the series e
c(W-aslkl
u)
Hamburger'-Noether"
auxiliar
series
fl
defined
1 of
simplicity
we
shall
find
a relation
between
1
95
the fl
Newton " Notice
the
series
v(x)
= n,
of
diagrams that
of
the
now
I:
chosen
v(y)
and
relation
Assume is
f
= m =
instead
between
that
the
curve
such
61
fl
that
(where
of
fl '
and
[]
is
if
those
f ~1 not
is
first
f
and
trivial.
regular,
x = X +(f)~
61 i s t h e
of
and
y = Y + (f)
characteristic
that , then
exponent
El). The
]oining
the
Newton
points
polygon
of
A = (m,0)
f
and
is
thus
a straight
line
segment
B = (0,n).
n : v_(x): 8 0 . m= V ( y ) = 6 B=(0, n)
(n,m)
(hn,0)
The of
[]
in
that
S l +1
top
rows
basis
(using
in the
1"
< n.
A=(m,0)
the
Hamburger-Noether
notations
as
in
expansion
section
2)
are
h y
=
X
Z 1
h x = Zll
z 2
hs1-1 z
sI
-2:
Zs I
I
z
-
s I
k z
By mined the
by rational
m
3.3.3. and
n.
fraction
Sl
-
]
the
=
as t k
z
Sl
1 + ....
integers
Namely~ m/n
1
the has
h'hl
,
0. l
k
1
' " " " 'hs
continued
these
as
values
-I ,k I are deterI fraction which represents as
partial
quotients:
96
h,hl,...
m n
--
=
h +
,kl
]
=:
h
1
+
t
h I ,.-
. ,kl]
h2 + +
1
-
-
k1
where i.e.
the
brackets
~ the
[
]
polynomials
respectively
the
this
is
fraction
in
are
the
Finally,
the
welt
partial
numerator
and
expressed
the
as
known
quotients
denominator
Euler
polynomials~
whose of
the
values
give
fraction,
when
irred, ucible.
sequence
of
transformations
h
y = X
Z I l
h I X
(t)
=
Z
Z
1
2
........ k
zSl_l leads
us
to
the
= zsl f l' (u~v) ~ w h e r e
series
u=z
s1 fl' (u,v)
Intuitively, with
centre
transform
0 I
of t h e
Lemma new
may
(O,a
) (see
1 .3.9.)
[-'1 '1 o f
at
first
origin
3.4.10.-
of t h e
The
transformation
the
X
U
1 .....
.. ' "
Proof: composed. we
have
By
induction For
N=I,
on
as
successive free
an
equation
quadratic
infinitely
near
point
f=O.
the
transformations
kl]
[h,h
1 .....
v
[h] =
of
therefore
(1)
is
the
by: [h,h
u
the
curve
composition
y =
for
leading
embedded
given
be thought
kl]
[h I
'
.. ' "
h
s I -I
h '
]
] s 1 -I
V
the
the
number
result
N
of
is e v i d e n t .
transformations If it is t r u e
which for
N=
are i,
97
h,...
~
,hi] z
i
..... ×
=
Z
. . .
h
]
'
y = 2:.
I-I
i+I
hi]
[h 1 . . . . . Z
i
hi_l]
i+l
h.
As Euler
z.
I+1 z i+l i+2
= z
polynomials,
we
, by
[h .....
hi+l]
Hence
it
is
also
Proposition ments
[h,h
If
1 .....
the
hi]
[h 1 ..... Z,
N = i+1 , and
With
notations
e 1 = (m,n) kl]
hi]
i+2
this
completes
as
above
)
, m' = m/e
( O',T )
is
the
(i)
(; >tO
~:m'
-
(% n '
2 T..< n '
evident.
I
the
the
proof.
following
state-
=
of
and the
.
,
d
2
shall
~
T = [h t .....
diophantine
I
m'.
prove
(b).
h,h I , .... hs1_1] hl ' .... 'hs I -I]
1 , n' = n/e
t
,
= n'.
conditions:
"~ >1 0
We
kl]
hs 1-1]
solution
two ,
[h 1 .....
.....
unique
following
is
and
d = p'hl
the
(a)
(e 1 = nsl
= m'
If
(ii)
Proof:
of
i+2
hi+l]
i+l
for
I
verifying
Dreperties
hold:
(b) then
Z.
3.4.11.-
(a) then
true
of
Eh . . . . . Z
+t [h 1 .....
=
use
obtain:
y = z.
X
making
Since
equation
hsl_1
] ,
98
it
follows
(Wall,
(1/ T
that (23),
continued
page
is
the
15).
fractions
(s
Thus,
we
-1)-th
1
aproximant
using
the
equivalently
((;
known
to
m'/n'.
properties
of
have s
or
well
fraction
m I
a
n ~
T
,1-)
is
1
-I)
-I
n ~
a so
ution
of
the
diophantine
equation
s I -1
"rm'
(2)
If
(
O" ,
)
T ~
is
( T-
and
hence
as
(m',n')
d-
~
- cr n '
another
P)
m'
solution
= ( ~-
T-- T
and
= 1 , there
(-1)
=
exists
Z
d
=
O"
+
q
mY
T
=
T
4-
q
n v
(2)
, then
~*)n'
have
q •
of
the
such
same
si gn.
Furthermore,
that
(3)
i.e.,
with
the
( a
according
product
,
as
T
q
)
>/0
Relation
(3)
(~)
reca
hs
or
usual
shows
" It
order
or
q <
that
-1-1
the
-<< ( a , T )
0,0)
Now,
T = _Fhl . . . . .
of
( o
Z x Z,
on
,
)
>
we
(a,
have
~)
O. that
(2)
~< ( ( ; , T )
our
real
verifies
has
only
one
solution
verifying
..<(m',n').
solution
the
i's
following
(:r = F h , . . . L.
,h
s 1
-1]'
inequalities:
1 m t
[h,
h 1 ,.
..
,kl]
= k t [h,
...
,hs
-1] 1
+ [h,
....
hs
-2] 1
>t
2d
.
99
n' = [h l ,. . . ,kl]
: k I[h
I ....
,hs
-I ] + Ehl ' ....
hs
1 Finally, of
0"n'- Tm'
= t
Therefore, That
will
be
T
onto
and
:
TO +
"[I = n~ "
of t h e m
can
The
following
(a)
The vertical
and
plane
(
2 ,[ .
~)
this
c(n'
(resp. (4),
( 0"
completes
)) I 'T
I
then
( 0 , 0 ) ~< 2(0" , "[) < (m'
linear
>
To )
satisfying
verify
solution,
B)
the
the
, n
')
.
proof.
transformation
+ ~ m',
oT£+ B0. )
properties: Newton
polygon
straight
AB
line
segment
Newton
diagram
of
f
is
A'B'
transformed joining
by
A'=(mn'
, m'c)
B' = ( m ' n , n o ) . The
in
(~,
=-1)
0"1 = mt
2.-
(0" o ,
a solution
0"0 +
right
3.4.1
the
take
qn'-Tm'
one
the
T
we
(resp.
only
Proposition
satisfies
if
-2] I
the
image
external (b)
of
region
the
bounded
A
(m, 6)
straight
is
therefore
lines
OA'
contained , OB'
((~, 6)
X
Y
,
,
f'1
=
(~= m . n / e (c)
I
' U (~ -
A (m',6')
where
the
f
tf
f:F then
by
of
and
T-1(m',6
6 V
13'
-Y
')
y = m i n ( . [ m , 0. n).
If
L(u,v,f'])
=
A
( ~ , , B')CA'B"
I(
1~ ~',B')
u
v
and
~.'B'.
100
there
exist
a,
)t E: k ,
a { 0,
), /
0,
such e
L(u,v,f
Proof:
I
Since
n' T
1
I ) = a (v-)t)
m' I
z
-+
1
,
T
is
that
bijective
d
y
/
V'
/ /
x
R
/
/
/
/
/ /
× /
B
/
X /
n
m
R '
X
f
A' /
X
>A
/
/
/
×
/ X X
//
T
mT
/
/I
/
/
iU
/"
/.,/
/
/
/ t
j/
j7
U>
B'
/
/ i "/
t/
/
ii
/
ff
/
I/
¢/
,×'>
0 ~-- ~.~/et-->
(a) e 1 = I Tm
It
is
- o'nl (b)
trivial.
follows
f
that
{ x,y
}
But,
since
is
to
y = u
the
{ u,v { x,y}
that
the
length
of
A'B'
is
mr
}
0
v
n ~ x
u
=
u
"1: v
of
induces
transformation
the
{u,v
then
v
composition
.... >
~
quadratic
transformations
} transform
f
T
on
the
into
f~
,
from plane. the
result
trivial. (c)
and
ice
).
xC~y 6 =
It
Not
)t ~ 0 .
By
Hence,
3.4.5.
L(x,y,f)
= a
(yn
~
-Xx
m t
e
)
]
,
with
a ~ 0,
101
L(x,y,f)
= a u
6
n'o"
( v
-
~,
v m'
"[)
e1
e
If
y = n(/
, then
L(u,v,f'
1)
= a (1-Xv)
If
y = mT
, then
L(u,v,f'
1)
= a (v-~)
f'l
in
e
In
both
cases
Remark 7t C
If
m
3.4. k
as
exactly
series
(c),
in
maximal The
fl)
holds.
13.-The
is
the
i ,
(c)
of
the
v
D(f'l)
p.(v) J
is
that
3.4.14.-
ble
f(X,Y)
series
beginning
of
this
Let successive By
(see
the
one
tions
m0 0)
only
in
{u,
v }
k((U,V)). is
a basis
of
acts
on
:
v
E; k ( ( X , Y ) )
, and
keep
has
a sense
,
each
curve
of
thus
Uj
for"
the
v = v -?t
and
p.(v) J
diagram
D(f).
substitution
£ j:0 in
(Newton
j.
D
defined
the
notations
by
the
irreduci-
used
from
the
section. D(f)
be
the
Newton
of
type
diagram 3.4.6.
(A)
The
process
is
infinite,
(B)
The
process
in
finite,
obtains and
of
f,
and
whenever
make
n
divides
m.
a diagram n=e
0
on
with the
or i.e.
a
~ after
Newton
a finite
polygon
coordinate
axes
algorithm
finishes
verify
number
whose e0<
of
projecm0
and
e0 -
<
n the and
a unit
, either:
steps
(e0,m
D(f'])
on
picture)~
Consider
substitutions
3.4.7.
acts
the
a polynomial
Algorithm
~ then
3.4.6.
f'l (u,v)
where
is
v = v-~
= v -~.
note
~ R'
(b)
__1 rq
change
Moreover', because
setting
ideal
change as
1
if
n=l).
case
(A)
the
(this
case
holds
if
102
In and
solve
tions
the
the
case
3.4,11.
vertical
The
being
v
= v-X
(see
(A
O)
,
m'=
mo/e
t ~:m' -(~ n ' I =
)
1
one
m1
The
in
1
1
with
3.4.12.
,
n'=eo/e
the
leads
1
9
condito
a
on
the
> e
).
1
in
substitutions
3.4.12.
(c)
infinite,
is
3.4.6.,
, the
first
of
these
1
or
finite, with
axes
i.e. a
verify
In
the
case
( A 1)
in
the
case
( B 1)
,setting
the
e 1 >
...
,
after
Newton ( e 1 ,rn 1 )
algorithm
a finite
polygon <
e1.
finishes
e2=
number
whose (Note
projec-
that
it i s n o t
(e]~l).
( e 1 ,rn 1)
,
it
continues
in
way.
As straight
line
e0> Thus
the
a result
of
segments)
projections e
on
and
'# (i)
9
"0+I
3.4.15.-
irreducible
exponents,
above
algorithm. (a)
this
< e
=
(e
Then:
0
, m
)
[]
be
(P
e •~
)
~
g'
finitely
many
polygons
are
(and
obtained.
of
these
,~ ~ < g ' - I
0
The
polygons
for steps.
actually lengths are
of
respec-
.
~< ~
g'-1
algebroid
k((X,Y)),
0,~
integer
that:
,
the
an after
g'
'J~g axes
recall ,
f(X,Y)
g.
finishes
) "~ 0 4 coordinate
)
exists
algorithm
Finally
and
g'=
there
algorithm
Let
series
ristic
.
, m
e
,
(P
the
m
(e
(ii)
Theorem
is
a diagram
m
e g' = 1.
tively
as
of
3.4.13.).
process
e]
Since which
is
process
The
that
same
~.
cycle
either
obtains
and
necessary
a new
remark
again
(B 1 ) steps
the
e 1 =(eo,m
transformation
~ where
Then9
the
linear
afterwards
substitutions
tions
, set equation
polygon. Make
of
(B)
diophantine
~) ~ < g ' - I
.
curve
defined
( B'g )0~< ",~ ~
polygons
its
given
by
the
characteby
the
103
(b)
~ 0 = e0
'
{3") +1
= m0 + m l
+ "" .+m
0 ~v4g-l.
,
e0
m0
m1
Pg~
'-1
,=1 mg,_1
Proof:
It
Assume
that
{x,y}
is
is
evident the
written
ml
= (hs 1
of
F] ~
-k
1
is
1
that
60
in
) ns
the
the
+ ns
1
1
+1
exponent
hand,
is
get
by
also
Now,
in
m v g=
=
(h
s ',)
V
) n
s
form
Indeed
3.3.4. the
have
13 I = m 0 .
[]
in
We c l a i m
first
= (hsl-kl)n
construction
our
- k 2) n s2
-k
we for
the
basis
that
charact,eristic
exponent
+ n
Sl
of
tlne
Sl
+1 + n s
polygon
1
P1
'
this
m 1 + nSl if
m 2 .... ( h
3.4.7.
by
given
other
By
expansion
reduced
l-'In 6 1 ( L J 1)
On
= e 0.
Hamburger-Noether
assumptions + n
s2 +
n
v
s ÷1 v
[]
and s2+1
, for
by
is an
replaced
by
inductive
I" - ] ~I
'
we
method
' all
v
,
0~<
v
~ g'-1
.
In
particular
g'.
Finally
the
formulaein
proposition
3.3.6.
lead
us
to
the
104
expressions:
6,~+ 1 = m 0 +
Corollary
3.4.16.-
exponents
(
V
has
Let
flY)0
genus
g-
be
Then,
gv<4 g
and
'J
a curve for
its
It
is
,
0 ~
with
each
'J
O
characteristic ,
1 ~V
characteristic
~ ) = 6v
-
1~v +
O
Proof:
+m
~
O
the
exponents
curve
ape
0
0
By(
[]
m 1 +...
evident,
ev
0-<
o
since
*a<~ g - v
.
o
6,~ +,~
-6 v o
= o
z~
mi •
~o~i < '#+~) O
Remark
3.4.17.-
Consider
C k((X,Y))
f(X,Y) the
algorithm
the
Newton
steps
without
at
each
f
in
the
step
(m,n)
a
The
straight
segment
to
a straight
line
e g'
fraction
expansion
of
type
y
m
and
projections
row
representation determines
= 1 .
on
D(f)
precisely,
the
successive
(Hence,
the
Newton
segment.)
of
this
= y-a01 n,
may
be
associated
of the
x-...-a0h
with
m>
Hamburger-Noether
h
the
that
More and
a series
way=
y = a01x
Now,
of
Assume
applied.
line
Hamburger-Noether
with top
D(f)
irreducible.
modif.ications
is
diagram
completely
a transformation
a polygon < n.
be
a
following
By obtain
is any
Then, to
can
polygon
weighted
necessarily
3.4.]4.
lead
polygon
not
the
+ ...
the
following
+ a0h x
rational
n,
h
we
and
expansion
is
h + x
z I
number
rows.
x
Namely,
m/n i f
by h,h
a continued l , ....
h Sl_ 1 '
k I
are
the
partial
quotients,
these
rows
are
h zj_ 1 = zj
J
zj+ 1
,
1 ~< j
~<Sl-1
.
105
he
second
step
of
the
algorithm
provides
I
which
yie
v
= v - asl
ds
the
klU
-...-asl,kl+Sl
polygon
P
.
1
We
with
h
as
= k 1 + I
Note
that
rest
of
in
the
since
1
Zs
add
+...+a
the
the
Theorem
s
g
3.4.
=
-I
h
Sl
expansion
last
step k
Z
1
row hs]
1
type
s 1
z
hs 1 + z
Sl
zSl+l
Sl
1
s 1 The
k
1
of
u
k I Sl-I
a change
a
I 8.-
s
k
g
Z
g
e
g
constructed
= 1,
the
in
bottom
the
row
same
is
of
way.
type
g +
s
is
....
g
Keeping
the
assumptions
as i n
the
above
remark,
if
h
(_D)
zj_
1 =
aji
z]
i
Jz
+ zj
j+l
,
0
.<j
.< r
to
f,
then:
expansion
of
'
i is
the
Hamburger-Noether (a)
f
(b) []
(__D)
= k((X,Y))
ideal
is
expansion irreducible.
is
the
Hamburger-Noether
in
the
basis
/(f)
(D)
defines
irreducible
a primitive
V + (f)}
of
its
curve
maximal
x
representation
of
an
x(z
y = y(z
multiplicity
of
f
irreducible
it that
=
) r
The
implies
parametric
curve:
(i)
this
{ X + (f),
the
.
Proof:
is
associated
this
is
)
.
curve
suffices
f=O
r
an
is to
n =
check
equation
u(f). that
for
Hence f(x(z
the
r
, ),y(z
irreducible
to r
prove ))=0, curve
that because (I)
.
106
Furthermore, from
this
the
part
(b)
of
the
theorem
prove
f(x(z
),y(z
)) = 0
r
If
n = ] , the
transforms no
and
so,
proceeds
trivially
equality. We s h a l l
has
also
f(X,Y)
point
on
formal
into the
f(X,Y)
a series
n.
~ aoi X i in k((X,Y)) i=O whose Newton polygon
follows
a0i
on
q/ = Y -
f(X,Y)
It
= ( Y-
induction c o
change
x-axis.
by
r
that
F(X,~)
X ~') h ' ( X , Y )
= ~ h(X,~),
.
i=0 As
the
parametric
equations
X
=
(1
are
actually
X co
y
~--- a o i
=
x
,
i=O
then
f(x,y(x))
= O.
Now, hypothesis any
9
let
of
with
the
into
fi(U,V)
the
we
cut
the
parametric
and
and
into
order
n > 1
that
the
9 also
those
satisfying
result
is
the
true
for
hypothesis.
h xi Um' ~ n' 1" , and Y = V , X = U V , Y = Y - £ a0i i=0 "[- a r e d e f i n e d by the algorithm, transform (see
fl(U,~)
series
of
assume
satisfyin
U6 V ' Y ' f l ' ( U , V )
turns The
if
q
a seres
theorem,
changes
m' , n ' ,
f(X,Y)
be
U (9) < n
The where
f
By
E: k ( ( U , V ) )
fl(U,~-)
expansion
3.4.]2.).
(D)
allows out
the
~-:
v-a
, being
U(fl)
algorithm
to
through
the
Slk 1
'
= e 1 =(m,n) continue,
corresponding
thus term,
equations
u = u(z
) r
q = ~(z
) r
vet
fy
fl (u(zr)'v(zr))=O
(by
the
induction
hypothesis).
It
follows
107
that
f(x(z
),y(z
r
Corollary power s,
2.4.19.-
series,
0
)) = 0
r
Let
where
< s < oo .
For
f
each
h
Then, such
f
is
that
the
is
m
The
only
on
Remark
gives
Newton
diagram
a practical
set
of
method
to
find
5.
m>.. m
the
o
the
can
Newton
degree
o
theorem
applied
to
since f"
depends
D(f).
points
this
if
that
m
an i n t e g e r
above
be
out
algorithm
curve
CHARACTERISTIC
the
algorith,m
from
can
be
the
viewed
as
equations.
f(X,Y)=
diagram
how
expansion
0
Hamburger-Noether of
of
.
from
parametric that
polynomial
there-exists
Hamburger-Noether
remark
the
if
3./4.18.
particular
from
only
in
In
be a f o r m a l
.
m
3.4.14.
.
,
f
directly
the
C k((X,Y))
set
for
points
components
fashion,
and
us
curve
irreducible
if
proof.
homogeneous
+ f2 +...+
Theorem
We a l s o reducible
1,
1
the
+f2 +''" a
algorithm
3.4.20.-
3.4.14.
is
derives
"the
a finite
s m~
irreducible
result
property
completes
f=fl
= f
m
irreducible h
Proof:
which
the
equation
expansions can
for
is
be
obtained
of ,
of
a
the
in
a similar
f.
EXPONENTS
AND
PUISEUX
SERIES.
We h a v e not In
always this
series
possible
fifth exist
seen
section and
its
to
in
chapter
find
a
we
Puisedx
consider
characteristic
I I
the
that
for
series curves
exponents
a given which
curve
it
represents
for
which
can
be
Puiseux
computed
is it.
108
using
these
series.
Since is
obvious
,
in
we
algebraically will
be
the
characteristic
shall
assume
closed
kept
Theorem
as
in
the
3.5.1.-
in
of
plane
algebroid
n of
[]
prime
this
preceding
p,
section p
over
and
the
above
that
k
>0.
discussion is
The
an
notations
sections.
THEOREM).-
curve to
case
characteristic
(PUISEUX'S
cible
is
field
zero
k .
Assume
Let
[]
be
an
that
the
mult
irreduplicity
that
n
x = t
£
y =
ai t
i=n is
a (primitive)
of
its
parametric
maximal
curve
ideal.
agree
with
sentation.
(These
Then,
the
We
shall
trivial
since
[]
is
[]
be
which
in
the
a curve
{ x,y
}
the n
and
Denote teristic sets
B I=B'I
integers = m.
For
of
the
above
2.1
.13.
and
n = 1
the
{ x,y
}
the repre-
3.1
result
.5.)
is
multiplicity
n >
a parametric
I ,
(n,p)
= 1 ,
representation
n
)
theorem
a i
by
are
to (
of
t
holds
prime
exponents of
of
has
i=0
than
to
of
s
co y =
less
n.
a bas
exponents
according
on
in
regular.
(1)
that
[]
exponents
induction
x = t
Assume
of
characteristic
a sense
use
basis
the
characteristic
have
Proof:
Let
representation
the
for
any
curve
whose
multiplicity
is
p.
B.~ ) 0 ( v []
i
~< g ( r e s p .
(Yesp. same.
of I t
is
(1)
( 6 " J ' ) 0 - -< "~< g' ) ).
evident
We that
must
prove 6 0 =6~
the
characthat
= n
both ,
and
109
We for
i < m,
those
may
assume,
since
of
without
these
terms
loss
do
not
of
generality,
affect
the
that
values
a
of
= 0
i
13~
nor
13 The
proof
will
be
long,
so
we
shall
divide
it
in
several
steps.
1
st
step.
Outline. Let
satellite r-~ 1
is
n
s 1
v
y
be
= u
an
applied
to
n'
appropriate
V V
the
< n
1
to
3.3.5.). (e 1 ,p)
'
= t.
the
first
terminal
The
multiplicity
Thus
the
is
{u,u.v
of
induction
1"
ideal
of
[~
= -+ I
;
1
}
,
where
by
o' ;
"["
Tm'
-o" n'
uniformizing
U
=
t ~ oo
v
=
~-i=O
(2)
According its
(see
maximal
e
and
associated
point e
of
m'
curve
being
given
X = U
For
,
1
basis are
the
near"
= e
can
A and
be
infinitely
hypothesis
u
El T
t'
characteristic
we
m/e
0~<
2d -~ m '
I
I
0 ~ ; 2 T~< n ' .
,
have
t
b.
t 'i
with
,
b 0 / O.
I
3.4.
to
G []
~ n' = n / e
m'=
t6 • the
exponents
curve
r- } ~ 1
( 67)0~<~)~ " g-]
has are
genus
g-1
given
by
13 ~ = ( 6 u . 1 - g 1 ) + e 1 By of
the the
induction
hypothesis
these
are
the
representation e U
=
t ~
1 co
i+e U,V
=
7--
i=O
b.
t ~ I
1
characteristic
exponents
110
nd
step.
Auxiliar Since
t/t'
then
t
and
and
t'/t
t'
are
Consider tively
¢]
curves
are
units
the
0
and both in
uniformizing
[]
auxiliar
'
parameters
curves
[~
and
1~'
defined
(3)
e1
These
[~ :
e
will
representations be
proved
t
equisingular. suffices
are
later).
to
prove
its
But
this
(a)-equisingular.
el E] ] = k((t
Hence tions
agree,
respectively
the
curves
[]
step.
(6 0 .....
by
each
13.,) ) ,
and
(e .0,p)
[[],
~(0-)'/
by
same
and
r5 '
are
it (a)-
multiplicity,
it
transforms
are
because
0' ~'[0)(0) ) )
e
= k((t'
] '
exponents
the
not trivial and
quadratic
]]
O(0)
is [^]
that the
evident,
t
fact
1
t'
induction
t't
of
the
two
hypothesis
exponents
of
= 0
representa-
they
the
two
i
are
(a)-equisingular
and
= t).
= w
r the
between -~ , let
f1)
be be
A r-I,
and
1.< ~) ..< g - I wv
Let
[ [ 3 ~1-
,
let
el) = ( 6 0 ' " " " ' 6 " g - 1 )
a primitive the
e,a-th root
k-automorphism
of
of
unity
=
(note
k ((t'))
defined
t' v T= x /
f
(4)
with
have
respective
charact,eristic
For
I" = T p
now
characteristic
Relation
f (t')
(this
see
they
is
e O' = ( t / t ' )
t'
sinoe
Since
If
'
the
(3)
shall
since
that
El':
1
primitive
We
Indeed,
= t t
~'
0 = (t'/t)
that
respec-
by e1
rt
r-I,
.
= t
3
for
with value
t)
u
n'G
( v ) "[" -
('~ , p) =1 , in
v
of
,
we
have
vT = f (t/t') 1)
and the
v_y. i s left
v 1: = ( t / t ' )
n-
the
hand
(t/t')
Then:
n.
natural side
n.
valuation
member
of
associated (4)
is
111 -
p r v (f ~
(This
equality
is
( v ) "[- - v
verified
f
where
>
v
The the
same
,j i
v
i+j=
= pr
) = pr
v(f (v)-v ~
( 13 ~ - e ) ~) l "
because
T
(v)-
-
v
=
]
(f(v))
(f
,
,J
(v)
-
v
)
. v
v~(0) =¥ v(0) / 0).
with
"~-1
v-value
of
the
right
hand
side
member
of
(4)
is,
by
reason,
v(
Now,
the
imply
that
of
¥)
pr
equalities the
f,j(t/t')-
(t/t'))
( 8~ -e
v(fv 1) = _
representation
characteristic
(3)
(t/t')
for"
-(t/t'))
I~]'
has
, the
I -.
following
,
system
exponents:
6,~
{ pr(
_ el ) + el } 0 <~) ~ g - 1
tn
particular,
and
since
that
the
4 th
step,
g (t)
by Ell=
proof
= w
using
DI of
'
2.1.13. so
these
Relation
between
g,) be the
t
1 ~< ~ < g - I
As
y = u
m'
the
points
Let ,
is
v
this
representation
other
representation
had
[]
been
where
g =+
1
13
and
in
(3) the
Recall
second
step,
of
k((t))
defined
by
.
0"
,
we
have
m)
according
y
gx)(y/t
v
n'O"
= t'
mT
n
(t/t')
ge]
as
m )
rn'l:
= u
= (t'/t)
Tm-
,
o.n
"[ (5)
out
primitive,
.
k-automorphism
T (y/t
left
is
=- + e 1
It
"t-
(y/t
m)
follows
that
ce 1 = g (t'/t)
ge 1 -
(t'/t)
Then,
112
The
v-value
pr
The
v(gj(Y/tm)
v-value
since
t'/t
of
the
left
_ (Y/tm))
of
the
right
and
g
(t'/t) ,~
hand
As
above
the
imply
that
system
of
1 _ m)
member
units
in
g'a ( t ' / t )
th
step.
we
2).
have
is
1
1
depend
as
(e
1
,p)
B '1 ) "
on
c
,
= 1,
it
is
v
p
,
the
same
1 < V -.49- 1 ,
exponents
of
[]
is
_
( Bv+l
B'1 ) + e 1}
~g'-I
Conclusion. The
(step
not
and
- (t'/t))
0 4~ v 5
,
(5)
= pr(B~+
does
r-I
characteristic
r
{
(Bj+
of
equalities
' 1 - B' 1 ) = _v ( ( g,O+
the
= pr
member
- (t'/t)).
r
p
side
side
are
v(9v(t'/t)
hand
characteristic
Thus,
by
g' = 9
and
P
r
(
using
~ By-el)
exponents the
+ el
of
formulae
= P
r
(
in
B' v+l
I~ the
B'I
and third
) + e
]
El' and
'
are fourth
steps
1 ,.< ,# ~< 9 - 1
whence
~4 i i] Bv = B,a+l - 13 + e 1
But,
in
the
step
1
we
B'~~
have
=
B~+
pointed
1 -61
out
+ el
,
1
that
< g-1
61 = 6 ' 1
.
and
that
'
1
then
[3V+1 = BI,,+ 1 '
] <~ ~ ) ~ < 9 - t .
This
completes
the
proof
of
the
theorem.
113
Corolla of
ry
its
3.5.2.-
maximal
Let
ideal
[]
be
a parametr
x = £ i :n y = t
where
n < m
of
this
and
(m,p)
representation
c
b.
which
has
representation
,
t
b
in
of
the
basis{
x,y}
type
/0, n
I
m
= 1.
( B v' ) 0 < .g~< g
I f
a curve,
(
6 0' = m)
and
(
are
6v)0
the
characteristic (6 0 =n)
~'~ ~<9
exponents
the
ones
of
0
,
I.<: v ..< g - I
,
then: (a)
tf
n
divides
g = g -1 (b)
Proof:
60 = 6' 1
If
n
g = g'
Let
does
,
["]'
m,
60
be
=
the
not
are
6~0
proof
the
= m
follows
Remark series, case
the
The
= I
Thus, inversion
formula" I he
expansions
exist,
for
the
characteristic
I ~< v
shows
exponents (n,p)
corollary
following
,
corollary
although this
the
-<
Now,
exponents since
of
O' 1 =0
O ~
, the
3.2.11.
characteristic
( 61 , p )
by
i+m
theorem,
lemma
]g v 4 g.
m
~ V = 6 V' + m
from
3.5.3.-
t
above
and
defined
- n
I
i=n
to
+ m -n
6.~-- 6 v! + m
curve
t
,~+~
m,
and
b.
y =
6~=
divide
6'I
x =
According
and
how may
,
by
be
using
Puiseux
computed
also
in
the
> 1. may
Puiseux examples characteristic
be
considered
as
a
"generalized
series. show
that
if
exponents
(n,p) of
> these
1
and
Puiseux
expansions
114
have
no
relation
Example
with
3.5.4.-
the
Let
characteristic
p >
characteristic
k
0,
be
and
an
exponents
algebraically
consider
the
of
closed
curve
[]
the
curve.
field
over
of
k
given
by
3 x = tp 3
(1) y = tp
First, (see
2.1
since
.13.),
expansion
the
this
3 + tp
+p
parametric
multiplicity
of
[]
2
+p+l
representation 3 is p Its
is
primitive
Hamburger-Noether
is
y
X Z1
=
p p+l x = z 1 + z 1
z 2
p p+l z t = -z 2 + z 2 z 2 =
Thus,
2 +P
the
z3P
curve
(v(x)
z 3
=p3,v(zl)=P2,V(Z2)=p,
....
+
has
genus
3 and
its
characteristic
exponents
3 60=P 3 6 6 B
while
the
1 2 3
= P = p = p
2 +P
3 3
+2p +
2
+ p
2p2+2 p + 1
characteristic
representation
(1)
B0=p P
6 ~ =
p
3 2 +P 3
+p
,
exponents are
3 6 i =
v(z3)=l
2
+p+l
.
of
the
Puiseux
parametric
are:
).
115
Now,
we
(i) two
It
summarize has
characteristic
3,
while
of
the
the
E~),
B2
and
For
no
neither
expression
62
of
z
Puiseux
series
B3
do
not
occur
as
effective
parameter
nor
as
63
series
z C m ( m
has
only
exponents
occur
in
as
being
effective
the
in
follows
where
a,b
from
E: k
and
Let
R
maximal
exponents
the
in
ideal the
t.
3 This +g(t)
F-I:
curve
series. (iii)
of
genus
anomalies
exponents.
(ii) Puiseux
the
the
fact
that
z
U_(g) > 133
=
P >/ 3 ,
if
3 2 3 2 + b t p +p + b t p +p + P + ] +
at p
(If
p=2
(iii)
is
evident).
Example of
3.5.5.-
Puiseux
type
.
and
Between
R'
the
be
two
parametric
following
representations
statements
there
is
no
relation: (a) (as
Puiseux
R
are
R'
have
the
same
characteristic
exponents
series). (b)
R'
and
The
algebroid
curves
defined
respectively
by
R
and
(a)-equisingular. In
fact,
(a)
~
let (b)
k
be
a field
Consider
the
of
characteristic
following
p >..3.
parametric
representa-
tions: 3 x = tp (R)
3 y = tp
2 +P
3 2 3 2 + t P +P +p + t p +P + p + ]
3 x = tp
3
(R') Y = tp
They they
evidently are
not
have
the
2 +p
same
(a)-equisingular.
3
2
+t p
+p
set
of
3 +p,
tp
2 +p
+p+]
characteristic
indeed,
the
two
exponents, top
rows
in
but the
116
respective
Hamburger-Noether
y
=
expansions
p
the
in
the
first
(b)
~=#~=~ ( a )
p+l
case
p+l
+z I
v(z
2)
Consider
representation
R''
Hamburger-Noether
both
x z 1
x = zI - zI
however,
are
in
expansions
y
=
x zl
x
=
z
2
= p+l
the
used
z
'
while
in
the
second
representation
3.5.4.
and
R' again
as
p >/ 3 .
v(z2)=p. above, The
are
1:)
p+l -
z 1
p+l + z 1
z 2
(R')
z 1 = (I/2)z# z
= -32
2
y
=
x
x
=
p Z 1
(R")
R' but
they
define
and
R"
+l + Z # + 1
Z
P + .... 3
z]
+
p 2
Zl=-z z2=
z
+ (3/8)z~
p+l z 1 p+l + z 2
z 2 z 3
P z 3 + ....
do
not
(a)-equisingular
have
the plane
same
characteristic
atgebroid
curves.
exponents
and
CHAPTER
OTHER
SYSTEMS
OF
INVARIANTS
PLANE
In riants
for
this
the
relationship The formed
by
second
section
using the
the
of
them
Newton we
semigroup
of
the
fourth
study
the
main
to
section
1.
As Newton
(Lejeune,
of
generalize
of
the
have
and
systems curves
of
and
inva-
their
the
by
degree
the
of
any
section
is
zero
case
the
of
it
of
it
In
is
the
genus, devoted
and
case.
positive
conductor
and
(15).
of
We c o m p u t e
characteristic to
in
contact third
curve.
section
Lejeune
maximal The
first
to
obtain Finally
in
characteristic any
curve.
COEFFICIENTS.
pointet
coefficients (15)),
the
the
complete
in
given
a plane of
OF
exponents.
explained
briefly
those
NEWTON
we
is
EQUISINGULARITY
algebroid
expansions.
we
properties
plane
coefficients
values
analogous
of
new
characteristic
Hamburger-Noether
results
of
first
THE
CURVES.
we consider
equiresolution the
FOR
ALGEBROID
chapter
with
tV
of their
out, an
this
section
irreducible
expression
is
devoted
plane in
terms
to
algebroid of
the
the
study
curve
characteristic
exponents. Let defined k((X,Y))
over
C the
. Denote
be
an e m b e d d e d
algebraically by
E1
its
irreducible closed
local
ring
field and
plane k set
by
algebroid the
x=X+(f),
series
curve f(X~Y)
y=Y+(f).
E:
For intersection we
any
embedded
multiplicity
curve
of
F'
P and
we C.
shall By
denote
varying
by r
(r to
,c) be
the
regular,
define
B'I ( C )
Definition contact
contact
4.t
.1 . -
A
regular
with
C
iff
If
C
is
regular,
with
C
is
itself,
regular,
= sup P reg.
(C,P)
requirements
curve
=
i-1
is
said
to
have
the
maximal
-61(C). the
for
(C , P )
only
curve
and
actually
the
maximal
which
"6
(C) I contact
=
has
the
~o
If
are
maximal C
is
in
the
that
C
given
not
following:
Lemma not
are
4.l
regular.
Let
The
be
a regu
following
(i)
The
(ii)
r"m
By
may
ar
curve
and
assume
is
statements:
multiplicity has
the
of
C
maximal
proof
u s i n g an a p p r o p r i a t e
suppose,
parameter
for
follows
Remark chapter
Definition the
without []
4.1 .3.show
and
from
does
not
contact
divide
with
( C , I '~ ) ,
C,
that
results
that
if
C
= h n + nI
4.1 .4.rational
loss
formal
change
of g e n e r a l i t y , y=0
of v a r i a b l e s that
is an e q u a t i o n
x
for
in
k(~'X,Y)~
is a t r a n s v e r s a l P
. Now,
the
3.4.7.
The
also
~'1(C)
be
r
equivalent.
Proof: we
.2.-
The
number
]J1 ( C )
the
section
is
not
regular,
= 6 1
first (and
of
,
Newton
n = ~_ ( f )
not
-
in
integer)
hnn+ n]
the
preceding
then:
coefficient
actually "BI(C)n
4
= e (C).
for given
C
is by:
defined
to
119
We m a y local
ring
then
with A=
r
C
)
since
is
1.1 ( C )
a regular curve (c,r ~ ) = - . Moreover,
if
is
1
diagram,
pl([]
this
value
is
intrinsic
of
the
[]. If
C,
write
if
x and
(m,0)
by
n
transversal
only
and
having
if
the
B=(0,n)
then
has
polygon
(m,n)
maximal
turning
y=0
Newton
with
the
contact
back
the
of
to
the
maximal f
with
Newton
contact
joins
the
points
and
let
B:(0,n)
A=(m, 0) Notations
4. 1 .5.-
Let
us
consider
[]cD be
its
desingularization
transformation
0
a 1
=
<
a 2
<
(i)
above, the
first
a,.j
)
g([~j)
4.1 .6.the
,
"~-th
Newton
c~
[]
~E]
1..5.10.
definition
such
M
curve
of
genus
and
as
by at m o s t a u n i t , t h e r e
under exist
a quadraintegers
that:
g(E]).
g(r'J
(iii)
the
decreses
< ag
g =
(ii)
Definition
to it
. , ,
c ....
1
sequence
According tic
a plane
=
g+l-V
,
1 ~< v
= g+l-'~
, if
[]
a plane
Let
Newton coefficient
be
a v ~<j <
coefficient
of
of
curve
the
~ g.
curve. []
# []
a +1
Keeping ]Jv(~), ak)
is
notations defined
as to
be
120
Proposition ideal
m
given
by
4. 1 .7.of
0,
Assume
the
that
curve
in
[]
the
has
basis
a
z j_ I
z
aji
}
of
the
maximal
Hamburger-Noether
h
(D')
{ x,y
.
+ z. j
j
expansion
z
,
j+1
0
-.<j ~< r .
i Assume
that
(D')
is
written
in
the
reduced
n
+ n
h
p
(0)
=
s.~
~)+1
Proof:
It
is
evident,
s~ n
form
s.9+1
3.3.4.
Then,
1 -< '3 -< g - 1 .
,
s)
since h
z j_ 1 =
is
a
Hamburger-Noether
aji
i
expansion
zj
i
Jz
+ zj
for
the
j+l
'
curve
s
~< j
~
[] a~+1
Corollary
4. I .8.-
characteristic
[]
Let
exponents
be
a curve
and
of
genus
(111)) 1 ~<~)~< g
g, its
(
By
)04.~,< g
Newton
its
coefficients.
Then,
131
130 • !41
,
13~ = 6.,0_1 + e v _ l (
Proof:
It
Theorem only
if
4.1 they
Proof: sely, teristic
proceeds
trivially
.9.-
Two
have
the
Necessity we
claim
that
given
same
is
form
obvious the
3.3.7.
curves
genus
and
from
Newton
Hv- kv_l)-
and
are the
the
the
above
proposition.
(a)-equisingular same
Newton
proposition
coefficients
if
coefficients.
4.1.?.
determine
the
exponents. t
I n fact n
I
= e~)-I / e ~
so
,
if
]Jv
m.~ n~0
with
( m 'o '
, n'~ ) = 1 '
and
we
have
Convercharac-
121
6 0 = n =
Now, formulae (see
the
in
4.1
3.3.9.).
)
6~
.8.
4.1
. 10.-
formulae
for
characteristic
C
local
ring.
MAXIMAL
Let
us
over
If
g
0 ~ "a ~< y ,
we
in
,6.~
4.1.8.
CONTACT
an
the
are
( 6 , ~ ) 0 -<'~; g
6 0 ....
viewed
Newton
[]
and
e~_ 1
and
as
the k'V '
inversion
GENUS.
plane
field
k.
Denote
y
an
integer,
and
characteristic
60
coefficients.
irreducible
closed
of
be
HIGHER
embedded
genus
the
and
from
determine
can
OF
algebraically
be
inductively
exponents
consider the
Let
that
Formulae
2.
Ig
be c o m p u t e d
(Recall
Remark
curve
can
n 1 , n 21 . . . . n
exponents
of
algebroid by
[]
0 4
E] , for"
its
y
-.
each
',~ ,
put
By
Bv =
S~f )
( I3 0 . . . . . Denote genus
Y
system
of
by
embedded!
the
set
is
(i)
6 ~
<
0
x
6 ~0 =
argument
as
in
for
curve
t
0
,
>(60'6 ~ y =
3.2.4. over
non-empty.
< 6~.
....
t
one k,
of
plane
The
"t~y(C)
set
ambient exponents.
curve
some
in
the
characteristic
(ii)
the
'~y(C)
iS
may
which
irreducible
curves
having
In
r
( 6~)
fact,if
of as
k = C,
since
By)
= 1,
7
~1) i
eh,
> .... in
see is
> C).
that
clearly
this in
( 60, If
....
k ;g t£/, curve
_~ y ( C ) .
is
by
using
a complex
an model
I22
Definition genus
4.2.1
Y
to
be
.-
We d ~ e f i n e
the
the
value
of
the
maximal
contact
of
number
6%t
=
sup
( C , F' )
r c ~y(c) Definition
4.2.2.-
contact
of
A curve
genus
Y
with
P C
(c,r)
Remark
4.2.3.-
1)
curves,
the
of
the
number
having the
value defined
the
maximal
maximal
which
by
contact
has
2)
If
the
maximal
Proposition
y=
In with
genus
g,
~0
contact
4.1.1.
of
since
maximal
contact
the C
C)
of
same
are
Let
is
have
the
genus way,
the
0
set is
the
curves
the
max
mal
of
regu
ar
nothing
regular in
but
curves
~0(C)
having
0.
~" ( C ) _ co , a n d g+l contact of genus g.
4.2.4.-
to
By+l(c)
y =0,
the
s said
iff
=
If
C ~y(C)
P
be
a curve
,
0..<
v~<
C
is
itself
of g e n u s
y .
the
The
only
curve
following
statements
are
(a)
6v(P
(b)
p.) (I-')
equivalent.
Proof:
) = =
(pV(--)
Bv
lJv(C)
,
denotes
y.
l,..< V 4 y • the
,~-th
coefficient).
Newton
Let h z]-1
=
. aji
z
+ z
]
]
j
z
]+I
,
0 (
j
~
i
be
the
notations
Hamburger-Noether
of
C.
We s h a l l
use
the
habitual
. Let
is
expansion
C
Y
be
the
curve
whose
Hamburger-Noether
expansion
123
h -
T=
zJ-I
aji
2---
i
i
zj
+
z
jz
j
0 .< j ..< s
,
j+l
k -
Z
=
-
a
s T-I
sy,ky
,
h Y
z
-1 Y
+
-
. • .+a
s
z
Sy
sy ,hs
Sy
Y
Y It
is
trivial
that
Cy C ~ y ( C ) ,
since
by
3. 3.7.
we have
~M
6v(C Y)
On t h e
other
hand,
Now, 3.2.10.
and
ducible the
~
of
ones
(a)-equisingular
C.
to
C
4.2.6.-
Corollary
also
,
1 ~ v ¢ y.
tivial
the
that
because
statements
according
(a),(b)
is
to
equivalent
(a)-equisingular".
may y
0 ~ ~ ~ Y.
inmediatly,
of
are
genus of
pv(C)
=
one
~y(C)
is
it
follows
%
4.2.5.-
"y f i r s t
(Cy)
each
and
curves
4.1 .?.
proof
4.1.9.
"
Remark
the
,
%
using
~
~
. . . .
be also
whose
Even,
it
defined
Newton is
the
to be the
set
coefficients
set
of
curves
of
irre-
agree
with
which
are
Y
Let h
(D')
the
Hamburger-Noether
3. 3.4. expansion
Then is
aji zj i + zj
expansion
I-' E: ~ ( C ) Y of type:
y
(1)
E.
z j_ 1 =
if
and
b01 x +
=
of only
--
if
written its
+boh x h
'
in
0~< j
the
~:r,
reduced
form
Hamburger-Noether
+
-X h - Z ]
h I_ Zl z2
=
............. ~Sl
•
C,
J z j+l
-
1=
h
k1 bslkl
~ s 1 +...+b
h Sl
Slh
sI
s1
S
+~ sI
I~ Sl+l
to:
124
ky
z
with
bs,jk,~d 0 '
Proposition if
0
=
b
z
+
.
.
.
.
I"< "~ "<'Y "
Keeping
4.2.7.-
~< y ~: g ,
the
hypothesis
and
notations
as
above,
and S
Sy
1
-
n
( £
~y
h.n2 ) + n j j Sy+ 1
j=o
'
then:
(a)
If
(b)
(C
Proof:
Taking
follows
from
] -< s
Y
,C)
into the
=
S
(m , c )
account
that
and
the
i ~ h
Y
~ sy
Y that
n.(P ) = nj/n j Sy multiplicity formula
intersection
Notice for
e##c),
r
equality In
holds
particular
if
and
(b)
, 0 ~ j ~< s
Y
, (a)
2.3.3.
only
if
in
(1),
a..=b., ji jJ
follows.
]
C0r'iiollary
4.2.8.-
For
each
y
,
0 ~< y
we
have
Sy :~y+1(c). n particular, the
local
Theorem
ring
the
values
[-'} .
4.2.9.-
expansion
given
of
We m a y
Let
C
the
write
be
maximal
contact
therefore
a curve
are
invariants
of
~y+l([~).
with
Hamburger-Noether
by
h z~_ 1J
=
a..j,
zj
+ zj
Zj+l
,
0 ~< ] <~ r .
i
Then,
another
curve
r
E: ~.,{(c) has
the m a x i m a l
contact
of g e n u s
y
125
with
C
if
and
only
if
its
Hamburger-Noether
expansion
is
of
type
h -
~
= /__ i
z J-1
=
Z s Y -I
-
a.. jJ
I", <
z
i
-
+ z
j
a
h
j-
z
Syi
z
j
j+l
+ g(z
sy
0 ~< j
'
,
)
%
<~s - 1 , y
sy where
g
Proof:
is a s e r i e s
in
It is t r i v i a l
position
/4,2.7.
Remark
4.2.10.-
curve
having
z
by
using
From
the
of o r d e r
sy
the
maximal
the
>
remark
above
h
made
theorem
contact
with
sy
in t h e
we
C ef
may
any
proof
of p r o -
deduce
genus
that
a
exists,
but
it
t is
not,of
course, The
the
unique.
maximal
terminology
of
near the
but
sequence
points
of
Remark contact from
of
this
I
, so
4.2.11.2.
C,
with
is r e f e r e d has
with
C I
in
be
then
if
a
{ x,y}
it w i t h
Let
C
of
main
curve
has it w i t h and
if
y = 0
the
origin
be
some
Y
= n
the
with
C
are
infinitely points
C (i.e.,
of the
maximal
properties:
of
contact
particular the
1.5.16.):
inmediately
contact
maximal In
has
these
in
)".
Sy
obtained
maximal
C.
of
of
of
of
the
y
first
properties
having has
genus
sequence
>~ e
can
sketched and
h+hl+...+hay
( 15 ~
a curve
of
the
icity
We s u m m a r i z e
C ] , it also to
the
be
1.5.15.
contact
points
the
may
(see
which
multip
of
Lejeune
4.2,9.
for
near
Most
by
maximal y
genus
points
contains
infinitely
Let with
genus
sequence
theorem
Y ~< gl
C
of
the
origin
4.2.11.-
4.2.11.1.
C
its
shown
gl "< g
of
higher
near
having
curves
points
of
infinitely
"Curves nothing
contact
maximal
genus of
, for contact
genus Y=O, with
C.
1
be
a curve
having
maximal
contact
with
C of
if
126
genus of
gl~<
genus
g.
Then,
Y < gt' Note
having
the
general
.
so
that
of
in
the
y
,
transforms
.
C
be
plane Then
having
of
of
contact
with
C
gl
not
with
C
P
hold
are
having
since
not
the
curves
unique
in
maximal
n/ey
of
Consider
genus
g,
a formal
denote
by
the
maximal
has
does
a curve
is
a curve
r
maximal
statement
genus
C
and
the
C i .
the
]
with
1 ~ "Y < g .
ambient
wiAh
y =g
y
Let
genus
it
multiplicity
genus
4.2.11.4. of
if
curve
contact
The
contact
has
maximal
/4.2.11.3.
any
and
P
another
quadratic
C 1
and
q
transformation
their
contact
curve
respective
with
C
if
strict
and
only
if:
C
a
1
have = b
01
4.2.1t C.
(i)
Ul(F'
(ii)
PI
) =
Pl(C)
has
To
show
the
the
maximal
the
( or
BI(P)
maximal
sufficient
= B~I
contact
they
"
with
condition
contact,
)
C 1
note that,
have
the
same
since origin
~ ]
and
, and
so
01
.5.
Assume
that
I-'
has
the
maximal
contact
of
genus
y
with
Let
CM(C
"~
CM(C)_I
>
....
[-~ M(P)-I
~
....
>
C
>
1
C
(i) F'
be
respective
sequences
quadratic
transformations
(1 . 5 . 1 0 . )
simultaneosly
we the
have genus
consequence
:~
M(P)
g(r ~ )= and
the of
g(C
I
fact 4.2.9.)
)+
of
strict which
transforms is
for
C
y -
g.(This
that
1
and
by
a sequence P
.
Then
follows
e(r-~i)/e(F..)
a sequence
of
for from
of
formal
desingularization
each the
= e(Ci)/e(C),Which
i,
0 ~< i ~
definition is
M(P of
a trivial
),
127
4.2.11.6.
Let
genus
g,y
sequences. tions
of
the
genus
genus
Assume
sequence 1-I
0)
Y )
with
.7.
for
(1)
C
are
if
near
another
those
condition points
first
for
transforma-
I-'.
has
the
maximal
the
maximal
implies
in
with
desingularization
M ( r TM )
has
one
contact
that
common,
contact
and
F'
(of
and
hence
C have
the
result
the
maximal
4.2.10.)
Let
C
a curve
with
CM(C )
a sequence
>
genus
of d e s i n g u l a r i z a t i o n
with
C
, then
g,
CM(C)_I
there
and
>
for
a T = inf { i I g(C.)i = g - ~#} . If U contact
the
with
I-~
the
C
respective
PM(P)
then
fact,
Y and
that
agree
and
CM(p)
C.(In
genus
assume
infinitely
from
4.2.11
with
that
C ~C)_
with
h + h t +...+hsy follows
a curve
Furthermore,
If (of
r-'
it.
...
For
>
each
is a r e g u l a r
exists
C 1
"( < g ,
curve
a curve
>
I~ of
C
set
having genus
y having
the
ay maximal by
the
contact above
4.2.9.
the
with
sequence,
the
closure valuation
such
that
then
THE
Let
us
algebraically of
[] of
of
SEMIGROUP
consider
an
U
OF
~
quotient
:
F
together that
of
with P
the
.)
VALUES.
field field
plane
k.
>
algebroid
Denote
F,
F:
v
of C
determine
irreducible
closed
in its
if
aT expansion
expansion
3.
is the ay-th transform of ay = U.(To prove t h i s , n o t e that b y
r~
Hamburger-Noether
Harnburger-Noether
over
C,
Z
and
by by
v
curve
[]
the
the
natural
[]
integral
r~
128
If E]
= k((t
t
)),
is F
any
= k((t)) []
Since
v([]-{0})
is
Definition
4.3.1.-
an
values
Remarks
and
notations
by
[]
its
parameter
an
parameter
v(z)
=
integral
set
of
for
domain
and
of
Z
S ( E ] ) = -v ( r - I
for
[~ ,
we
any
series
r-]ci--I
have z = z(t)El'-].
CF
the
set
+
-{0})
will
C Z +
be
called
[~].
4.3.2.-
Let
defined
ring
u(z)
subsemigroup
curve
local of
is
The
p of
algebroid
and
additive
semigrou
plane
uniformizing
and
by
C
be
f(X,Y)
assume
an
~
that
embedded
k((X,Y))
irreducible
. Let
x = X + (f)
is
us
denote
a transversal
r-I .
Let h =
z J-1
be
the
Hamburger-Noether
written
in
the
in
the
on
the
often
preceding
the
the
values
section
put class
following
NO=
having f
~)-1
the
(X,Y),
maximal we
follows
that
for
j
z
f,
60 = n, of
`# ,
,#-I e9
j+l
C
,
,
1 ..~ ~) @ g ,
contact
which
,
0
~<j ~< r
wilt
be
assumed
1 ~ `# 4 g = g ( r - ] ) ,
(these (see
values
defined
depend
4.2.8.)).
I -.< v ~< g ,
with
let C.
P If
~) - 1 l'~`#_l
have:
-6,)c
'
We
only
shall
e`# = n
s`#
I.
~v : ( c , % _ 1 ) It
j
notations: e
each
+z.
6 ~ ) = 6") ( C )
and
NV
For
zj
i
form.
(a)-equisingularity use
aji
i
expar~sion
reduced
Consider
7-
s(E])
,
: z(%_ 1 (x,y)). 1 .< ~) .
be
a curve
of
genus
is d e f i n e d
by
the
"#-I series
129
Moreover,
rv_ 1
, so
x
fV_I(X,Y)
if
+ (f
y = 0
)
is
n
N 1 . N 2 . • .N.)_. 1
for
the
pute
these
Lemma
as
The
c
,c
o
,...
t
a continued
ponding
algorithm
Let
m/n
,c
are
s
aim
to
a rational the
also
get
more
will
be
is
~
.
-I of d e g r e e in
k((X))
useful
in
to
Hence
(4.2.11.3.).
appropriate
If
fraction,
partial
Let r =n, 0
' • " " 'rs
rl
By
for
induction
expressions order
to
com-
s-1
,
by
of
>n
> O,
its
assume
representation
)
(r s = ( m , n )
and
be the c o r r e s -
then
s
s.
2 = m.n j
q..r j
j=0
on
m
quotients
s
A
true
it
for
coefficients
3.4.14.
fraction.
residues.
Proof:
then
a polynomial
with
results
C,
parameter
to be
Y
to
values.
4.3.3.-
that
in
ev_ I following
"~v's.
tangent
a transversal
v -I be c o n s i d e r e d
may
The
is
For
s=l,
considering
the
the
result
rational
is
evident.
fraction
n/r
If
1
,
we
it
is
have:
s A
Thus,
A
s
= c
o
2
.n
+ A
s-1
E
=
c
j=l
= Co.n
s-I
j
2
. r
j
2
=
+ n.r
n ° r
1 = n.m
,
which
completes
the
proof.
Proposition
4.3.4.-
(algorithm
3.4.14.),
-6x)
Proof:
If
=
-
1 %-1
According
.,
are
1 .my_ t)
~
(ev'rn~)04v
as
in
chapter
then
( e0
to
,
m 0 +...+
/4.2.8.,
we
e-
1 4; V ~
have,
s ~ -1
1 6"x;
e '$-1
>
-j=O
h .n 2 j
j
+ ns
+1 '9 -1
'
1 ~< '~ - ~ g
"
•
I I [
130
Hence,
by
using
the
preceding
(])
Now,
from
sition
(1),
may
by
e.o-~v+l-
e)_]~V=
+ (h
n
an
-%)
sv
4.3.5.-
( 13
)0.< "0~
the
values
of
inductive
For
Let
by
.n
s v+ l
the
- n
.n
sv-1
e
sv-1 +l
formulae
in
the
g
and
.m
V
propo-
be
a curve
maximal
v=0
80
6v
Nv-1
=
and
v>-2,
we
V=l have
genus
characteristic
contact)
B'0 =
with
of
[~.
exponents
(resp.
Then,
' ' 6v-1
, it
is
(see
-6v-1
+BY
evident.
above
proposition)
= %_2.¢v_1
+ ev_l-%_
1
3. 4. 1 5. ,
Corollary
4.3.6.contact
(i) (ii) (iii) particular,
%
method,
[]
ev_l.~ v
maximal
+ n
obtain:
s°-I 2 2 "~ hj nj + kv ns ) + j = sv_ 1 +1 v
(
( "~V) 0~<'O ~
the
If
Hence,
2
%
we
be obtained.
Proposition
Proof:
lemma,
Let for
-#re
the
(-6")0
~
curve
s([7),
r~
system
of
values
of
the
Then:
0-< ,~ ~g.
-#0 < ~ < "'" < -#g ( -B-0 . . . . .
~-,0 )
=
~-0 > ( - 6 0 ' -~-1 ) > . . . .
( 60 . . . . .
13) = e?
> ( ~'0 . . . . .
,
0 <
6"g) = ] .
V ~< g"
In
131
Proof: tion
(i) on
using
"g.
the
and For"
Corollary if
of
formulae
the the
to
conversely
(Note
from
6 0 ....
Theorem the
maximal
]
~ x a transversal
associated
to If
d
1
of
maximal
the
for
by
"0-1,
induc-
then
the
8,) = %
(a)-equisingular
same
values
contact
if
of
are
the
and
maximal
a complete
equiresolution).
/-4.3.5.
can
-~0'
• . " ' ~'g
in
NV_ 1
, B.)_])),
The
proof
[~
be
Let
(iii)
s0 . . . . .
are
and
that
may
be
viewed
terms
be
as
of
obtained
is
from
~ and
(~)O~
v
the
where
g
-~0' ....
and
~-1
evident.
with
a basis
inversion
[80, . . . , 13g
therefore
a curve
{x,y}
parameter
the
-
maximal
natural
as
values
ideal
of
valuation
[[]. z C []
curves
in
contact.
holds
:(
genus
the
Let
of
degre
for
4.3.8.-
it
prove
have
plane
values
compute
If
.....
same
formulae
We s h a l l
evident. we
Two
invariants
The
is
=(80
have
(i..e.,
evident.
proposition,
.....
they
Proof:
(or
it
4.3.7.-
contact, system
are
~) = 0 ,
above
(%
only
(ii)
... N
and
z=g(y),
is
a polynomial
in
y
of
, then
Y
Y v_(z)
In
6
~ v=-O
6v Z +
particualar, g
"0=0 Proof: basis
{x,y
regular a
Denote
curve
change
the
}.
maximal
such
by
C
We m a y y=O as
contact
the
embedded
suppose,
has -y = and
the y-
without
maximal a01 x
z
algebroid
can
contact
-...be
loss
aOh
expressed
curve of
generality,
with xh
C;
, the as
defined
a
by
the
that
the
otherwise
curve
y=O
polynomial
by has in
y
132
of
the
same
degree
as
9.
We s h a l l
prove
the
If
, we
have
y=l
theorem
by
induction
on
h'
•
d z =
A (x)
yi
,
d
<
N1 .
i=0
Since it
is
_v( A ( x ) ,
sufficient
to
y
) qg T 0 Z +
see
that
these
if
v(A
x)
Indeed,
+ B1
values
Z+
for
are
each
pairwise
yi ) = v(A.(x)yJ)
for
i,
0..<
i ..~d,
different. indices
i,j
J
0..< i , j and
~
since
we
have
I i-j
I < N 1 , we
Now, z = g(y),
a curve
C.
4.3.2.
of
assume
where
Choose By
degree
(i-j)-6
9
is
Since
f
has be
B
o
(Y) By
9(Y)
and
C
'
1
have
the
the
Hence
(i-j)
=
0 (,mod.
is
Y-1
result in
maximal
is
true
y
of
for
y-1 , and
degree
contact
of
defined
by
a polynomial
monic,
we
may
(Y)
= B
o
(Y)
E: k ( ( X ) ) ( Y )
repeating
above
in
the
same
g(Y)
=
continuing
+ C
1
and
the
let
zE:['~ ,
d < N1 . . . N ~ genus f
y-1
y-1
(X,Y)
with E: k ( C X ) ) ( Y )
obtain
(Y)
f.. f
wit,h
we
Y) '
degree(B
division way,
(X -I
'
o
(Y))
< N
CI(Y)
1""
.N
instead
y-1 of
have
S
(t)
~
Bi(Y)
(f_I(X,Y))
i
,
i=0
with
B.(Y),
more,
we
C k((X))(Y)
have By
the
s
T
and since
induction
I
degree(B.(Y)), d < N 1 ...N
Y
hypothesis,
-9=0
N1),
i=j.
1
g(Y)
with
0(mod.n).
a polynomial
can
N1...NT_
must
that
which it
1 =
+
<
N 1 ...Ny_
1
Further-
133
As different
above
v(B.(x)) -
recalling so
and
this
must
prove
that
these
values
for
indices
i,j
have
(i-j)
are
pairwise
.
If
and
we
that
(rood.
completes
Remark
4.3.9.-
[]
{x,y}
, 0 ~ i,j
~< s ,
j
v_(fy_l(x,y))
(i-j)-~O
Let
= v(B.(x))
i
= By , w e
N
). Since Y proof.
the
Let
C
thebasis
of
be
I i-jt
a plane
its
< N
it
Y
algebroid
maximal
0 follows
curve
ideal,
(mod.
wilh
that
with x
ey_ 1 i=j,
local
ring
a transversal
parameter.
having
the
(X,Y) is a p o l y n o m i a l in Y Y maximal contact with C of genus
every
z 6
[]
If
f
has
an
expression
of
which
defines
a curve
Y ( 0 - .< y ~< g - 1 )
type i
(2)
z = 0~(-[y< Ny
Ai I ..... i g(X)
then
i
f0(x,y ) I " " "f g-1 (x,y)
g
I ~;,y.< g
In appropriale
fact,
it
suffices
induction
to
use
hypothesis
to
induction each
on y , applying
B.(Y)
in
(1)
of
an
the
above
I
theorem. Furthermore~ (2)
is
i0
1.< y ~ g . (2)
for
-B0 + i1 - 6 ]
Since z
the
is
+'''+
each
two
unique.
It
Corollary semigraup
free
4.3.10.of
values
of
~g,
g thi.s
the
general
where
values
follows
{fo
of the
i
of
i
is a basis
v-value
that
i0~
are the
term
0
and
different
the
in
the
sum
iy < Ny
for
expression
set
i
I . . . .
f g - /u} i
Y
k((x)]-module
Let
S(E]),
0
be
an
Y
FT.
irreducible
and h a v i n g
ptane
( ~,o) O.~ ~
g
curve as
with
system
of
134
values
of
S(D) the
the
maximal
contact.
(a)
"~ 0
(b)
For
each
does
not
which minimal
set (c)
S(17)
such
Proof:
(a)
is
of
the
minimum y ,
belong
each
that
have:
s ( F 1 ) - {0 }
of
of
y
we
1 <. y <~g, -~y i s t h e l e a s t integer in Y-1-to ~ _ ~ Bx) Z + , ( i . e . , { T 0 . . . . ,B-g} is
generators
For
,
( -~0 . . . . .
I t is
Then
the
semigroup
1~< y .< g , ~-y-1 ' 6
)
S(O)
~" i s t h e Y < ( -~ 0 . . . . .
(see
least
5.1.1.)).
integer
6
in
~-y-I ) "
evident. Y-I
(b)
If
6 E: S ( l ~ )
136 ~---~v Z+
and
, then
V=O
8
where
av
~ 0
( 4.3.6.(c))
for" the
(c)
It
:: a o
some
t30+at
"g> y - 1
result
are
4.3.
if and
B >~ 6 y a n d
a
since
>/0,
V
Y-t ~--,~'~ Z
-By~
trivially.
2y)
< ( ~o . . . .
plane
curves
only
if t h e i r
2¢y_]).
defined
over
respective
the
same
field
semigroups
of
agree.
Proof:
If
complete
we
suffuci
is
ence
Proposition
of
that
a trivial follows
different if
they
the
invariants
from
fop
the
Two in
have
values
of the
consequence
4.3.12.-
are only
recall
system
necessity
and
fwo
(a)-equisingular
values
which
1 1 .-
,
~'g
(b) , since
from
-F0 . . . . . Theorem
. So
follows
derives
~'1 + " " " + a g
general,have a complex
maximal
contact
(a)-equisingularity, of
above
given
the
4.3.8.
are
a
the
, and
the
corollary.
plane the
model
curves,defined same
in
semigroup
common.
In
over of
fields
values
particular,
if
+
135
the
semigroup
of
values
of
a curve
and
that
of
any
of
the
of
its
models
agree.
Proof:
According
depend
only
on
proposition
to
the
4.3.5.
the
values
desingularization
derives
from
the
process
above
maximal
contact
(1 . 5 . t 0 . ) .
theorem
applied
Then,
to
the
the
complex
field.
4.
THE
DEGREE
In
Chapters
III
(a)-equisingularity the
have
desingularization
coefficients,
replaced 4.3.5., model
fields. integral
model
4.3.12.).
The
of As
the
closure
of
algebraically
closure
of
[] The
Samuel,
~ 2?~)
its
constant
when
a 9iven
method
study
to in
to
be
this
,3. 2.8.
of
a curve
to
extend
case section
of
for
of curve ,4.
the
is
I .8.
~
a complex
certain curves
values.
well over
known arbitrary
conductor
of
the
curve.
k.
quotient of the
the
3.2.3.
of
Newton
remain
field
conductor
exponents~ semigroup
it ~ (see
for
sequence
and
for
[].
invariants
contact,
an i r r e d u c i b l e
in
of
IN
maximal
a plane
closed
[]
multiplicity
characteristic
curves
we
OF
systems
introduced;
substitution
complex
a example,
CONDUCTOR
several
a successful
Consider an
the
systems
sometimes
properties
IV
a complex
and is
and been
of
these
by
THE
process,
values All
OF
Let
algebroid
curve
= k~(t))
the
defined
(see
[]
[]
over
integral
field. []
set
plane
in
[]
is
Zariski-
136
If (which
is
{z 1 .....
Zq}
noetherian)
is
and
a set
if
we
of
generators
put
z
=(x
'
l
~< i g q ,
then On
neously, with
the
Thus,
t
as
in
[]
.
the
integral
the
the
main
4.4.1
c
is
said
of
the
Proof:
tf
t
in
the
.-
c
is
the
c
[]
c'
Finally
we
an
we
t c []
for
degree
E: E l ,
that
and
of
Yi
y.;g 0 , I
follows
degree
have
of
~/ []
0.
simulta-
valuation
some
the of
just
integer
only
c
ring
>0.
conductor
the
of
conductor
t:
then
c,
c
and
the
second
note
that
c
verifies
t
[] of
t c,,
one
>
0
c, D
generalize
which
[]
The c'-1 both
devoted
El.
C
~
trivially
will
for
¢
implies
is
case.
c-1
t
out,
which
zero
that
and
pointed
integer
and
such
tc
>z
~=
,
Fl-module
i
a discrete
characteristic
integer E:
'
[]
is
the
the
ones
is
be
as of
m
L.~,
implies
to
of
[]
the
x
I
tt
as
curve.
t c []
c' > 0
¢
called
section,
(])
ideal
have
properties
corresponding
Proposition
we is
present
an
I
y~O.
~ since
c
closure
and
j~ i s
particular
Sometimes,
obtain
hand
integer
The to
E: ~
a uniformizing, The
in
other
" "Yq
[]
/y.)
i
Y = Yl"
of
and first
< c~
condition so
c = c'.
properties
in
the
proposition.
P. r o p o s i t i o n
4./4.2.-
Let
plane
~.
c
curve
Then
(2)
Proof:
From
Now, It
is
then with
evident v(z) w
~ O
abo~ve let
that
= c-t
--
k
is
i ¢-_. Z
the
for and
be
-.
v(w')
the
the
i
z ~C E l . it
semigroup
integer
>/ c
c
least
We s h a l l
>/1 ,
the
least
,
+
be
proposition
c' c'
S(E])
integer
would
c for
satisfy
(2).
for
which
any
of
a
s(r-]
es
= c.
values
which
i C
verif
Then
for
~
prove
of
Indeed, w E; E l ,
2)
is if
true. c'
w = w
< c, O
+w',
137 Z
W
=
Z
W
+
Z
W I
~--
D~
0
so
Z ~
tion
.
)~
Corollary
4.4.3.-
depends
only
Corollary over
= t c r7 ~ whence
The
on
the
and
respective
degree
Let
[]
of
be
[]
a complex C conductors for
The
>/ c ,
and we
the
proof
of
an
class
this
for
and
it.
get
c of
irreducible
model
[]
would
conductor
(a)-equisingularity
4.4.4.-
k
v(z)
of
in
[]
[]
[].
plane Then
a contradic-
the
algebroid
curve
degrees
of
the
E](E a g r e e .
corollary
is
trivial
from
4.3.11.
and
4.3.12.
Proposition integral
4.4.5.closure (.i)
Let
of If
a plane []
desingularization
c
= []
the
degree
curve C
o
1.5.t0.
of
r7 . The
the
conductor
following
of
the
statements
hold:
. . . C r']
and
= '~ iS t h e s e q u e n c e of M e i = e(F] i) , 0 ~< i ( M , t h e n
if
M
o
~-"
:
~
(~.-1)
i=0
(ii) ~],
If
( 13"0 )0 ~
are
thee c h a r a c t e r i s t i c
exponents
of
then
c =
where
(eg_t-1)
e.o = ( 6 0 ' (iii)
81 . . . . . With
If
(eo
1 ( n - e 1)
-
(n-l),
13-o ) "
notations
c
(iv)
+13 g-1 ( e g - 2 - e g - 1 ) + ' ' ' + 6
=
N
g
~
, m a)o
as
g
-
in
B
section
g
~ `O -.
2,
(n-l).
are
the
values
obtained
from
138
the
algorithm
3.4.14.,
then g-1 c = 7-"0=0
Proof:
To
complex
a
as
well
prove model
as
contact
the
(see
the
and
In
fact,
characteristic the
-
(n-l)
(iii)
it
suffices
since
for
both
exponents
formulae
Zariski,
(iv) and
(ii),
El.
agree,
curves
(i),
mv ( e ~ - l )
are
the
and
same
to
[]
the
semigroup
curves
the
as
replace
values
those
of
for
the
by
maximal
complex
(26")).
proceeds
proposition
trivially
form
(iii),
the
theorem
3.4.15.
4.3.4. i
Proposition [-7, i
then
/4.4.6.for
and
It ,
is
, 0
belongs
is
is
<~i ~ c - 1
to
Proof: we
sufficient
to
4.4.7.-
ring
ring,
the
only
, the
the
conductor
one
of
the
semigroup
complex
considered
Since
any
[-}-submodule
)
of two
S(E])
[]
in
integers is
case
(see
Zariski
,
I--I/~
as we
irreducible
plane
algebroid
)
as
F]-modu of
es).
El/E]
is
also
the
same
way
In
a
k-vector
space,
I (r-I/~)..
).
{1+ I~,t+ have
an
,
..< _ _ d i m k ( [ ' 7 / E ] ) "
~< d i m k ( r ' 7 / ~
of
::
[~/~
I(r-I/~'
[]
i.e.
and
Now, space
of
and
(i.e.
consider
The
a Gorenstein
have__ I(E]/F-I)
and
degree
, one
S(EI).
I(E]/I:])
( El/E]
the
page 2 8 ) .
Proposition curve
i
c
.
Proof" (26)
each
c-1,-i
symmetric)
tf
~ .....
dimk(E]/~
tc-l+ ) = c.
~ } Actually
is
a basis t(FI/~)=
of
the c,
k-vector since
139
c t c-~ ~-/t~ c...
c0)
is
a strict
chain On
[]/~
as
by
the
F']-modutes.
other
k-vector
in
s(r-])
hand,
space
elements
c/2
gers
of
in
+ ~(FI//:)
Theorem an
is
account
as
any
set
v-values
of formed
the
inte-
dim
= I(E]/~)
(I~/~)
= c/2
,
= c,
= o/2,
.< d i m k ( O / ~ )
= o/2
,
trivial.
4.4.8.-
irreducible
Let
C
be
the
embedded
plane
curve
defined
by
series
,(×,Y)=
7-
,(x,Y)
n > 0
and
a
f
c
I
i=n with
choosing
a basis
that
-
.< a i m k ( O / O )
I (Old)
since
c.
= dimk(E]/~)
~(0/0)
proof
obtai.ned
than
into
) =c/2
respectively
less
taking
~(E]/D)
the
be
having
are
dimk(F'}/D)
dimk([-l/~
may
[]
whioh
Finally,
ct-D/~: c~/~:
homogeneous
,
polynomial
of: d e g r e e
i.
If
i
each
m "# n
we
put m
h
(X,Y)
=
~ i=n
m
and
we
design
by
C
the
curve
f (X,Y) i
defined
by
h
m
an
integer
m
> O, 0
fop
m >~ m
o"
such
, then
there
exists
m
that
C
and
C
are m
formally
isomorphic
for
140
Proof: []
By
(resp.
3.4.19. []
)
m
,
for
denote
m
the
large
local
enough, ring
h
of
is
m
irreducible.
(resp.
C
C
Let
) . Assume
that
m
h z j_ 1 =
aji
zj
z
+ zj
i
is
the
Hamburger-Noether
where
x= X+(f)
and
if
basis
m >/ m {x,y
o
}
,
to
the
,
for
theorem
3.4.18.
,
Hamburger-Norther
x=
E]
in
0~<
the
j
~
basis
{ x,y
}
,
y=Y+(f)
According that
expansion
,
j+l
X+(h
)
,
there
exists
expansion
y = Y+(h
)
m
is
m
for
given
[]
such
O
in
m
the
by
m
-
&--
=
zj-I
-
L_
i
z
aji
j
h
-
+ z
j
j
-
z
' 0~< j
j+l
~
'
i --
Z
where the
g
is
a series
conductor
m
C,((Zr)).
and
whose
2
z
r
--
+
order
...
is
+a
pc
z
C
+
r
greater
g(z
than
r
)
,
the
degree
of
= z-
z
,f
,
we
may
suppose
that
[]
Ck
( (z ) )
and
r
r
~s t h e
v
natura,
Z(x-x)
~
v(y-y)
:~ c
va,uation
of
,((z
r
)).
we h a v e
c
,
so ,
where []
ar_Z
c.
Setting
[]
--
=
r-]
= []
g m
and ~_ k [ ( z
g' r
]),
x
= x + g(x,y)
y
= y + g'(x,y),
are
series
which
of
completes
order the
>~ 2 , proof.
(see
4.4.1
.).
Hence
CHAPTER
TWISTED
This
chapter
singularities
of
algebraically
closed
coincide
for The
lution
of
lution
using
the
are
dered
as
with
On t h e
a better
definition
1.
PRELIMINARY
As
in
previous
Z+
A minimal
(i)
go > 0
(ii)
"~'~ +1
the
we
the
Z
by
over
shall
prove
none
of
set
of
Z+
,
wh
means
of
one the
thi~m
given.
equireso-
by
equireso-
third
one
in
general
that may
ch
be
by
consi-
ones.
NOTATIONS.
s meant
consider
an
are
second
AND
+
classifying
equisingularity,
and
other
CONCEPTS
us
of
(a)-equisingularity,
hand,
than
Let
such
of
transformations,
other
problem curves
singularities
chapters
integers.
5 . 1 .1 . 6 g } C::
the
However,
different.
Definition
algebroid
projections,
quadratic values.
the
definitions
one classifies
space of
Three
plane
CURVES.
essentially, twisted
field.
generic
nonnegative
{~'0 . . . . .
treats,
curves
first
semigroup
they
of
the
ALGEBROID
irreducible
plane
V
to
denote
the
a subsemigroup
generators
of
S
semigroup
S C Z
is
+
•
a set
lhat,: .
v
~: ~-i=0
g ~'i
04
~ ~< g - 1 ,
and
S=~-- I~i Z+. i=O
142
(i~i)
( -F 0,
According generators the
values but
of
the
set
and
system
such
of
if
Let
in
values
of
+
-
S
S.
If
S
set
of
maximal
minimal
is
( ~)0~<
minimal the
The
definition
Thus of
the
= 1.
this
unique.
curve,
S
+
(ii) is
.2.Z
-Fg)
generators
of
5.1
only
Proof:
it
a plane
Proposition if
to
exists,
minimal
-F 1 .....
set
if
a minimal
set
"~g
may
be
called
semigroup
of
is
the
generators
contact
of
is
(see
nothing
4.3.10.
generators
of
of
(b)).
exists
S
finite.
finite,
Then
there
exists
an
integer
c > 0
that:
j
Set
6"0
C Z
= min (S-{0})
4-
and
,
j
>~c
define
--'~
j
¢
S.
inductively
~1 . . . . .
6"
by
g
-g +l
i=0
4-
I
-,)
(S-
whenever
"~--~
i Z + ) ('~
i=O
We c l a i m j
< c+
6"0
it
is
{ 1 ,2 .....
c+-~
we
have
a~) C
27+
=~
¢
.
g
that
S = ~-- ~ Z In + i i=0 g that J • Z ~" i 2'
evident
i=O
exis;ts
0 }
such
that
c ~< j -
a~) ~" 0
fact,
let
If
j
j E; S . >/ c + ~ " 0
If , there
+
< c + "~'0"
Since
j - a~'0eS,
:
J - a0 60
a0-F0
+ al
61
+ " " " + ag
-6
g
,
a.
I
E;
Z
+
~
0 ~<
g
Hence
j ~
~i=O
(
~'~)0~<'~
..
is
~.I
z
as
desired,
Moreover,
it
is
+
a minimal
set
of
generators
of
S.
evident
that
i.-.< g .
143
Conversely, for
generators a 0
,
.
.
S.
such
,ag
.
assume
the
elements
•
~0
for
S
implies
that
Now,
~
let
closure
of
[]
associated a nonzero
the
to
the
[].
be
As
exist
set
of
integers
called
(a+l)
degree
of
the
k.
5.1 . 3 . -
=
1 .
completes
by and
by
v
the
~=(t c)
an
= -v ( F I
[]
for
p of - {0 ~
curve
of
of
,of
the
El
valuation
in
in the
is
[]
r-~
has
c > O,
[] of
the
integral
integer
closure
values
over
natural
of
conductor
integral
semigrou
proof.
[] = k((t))
~
the
the
algebroid
the c o n d u c t o r
of
two
S,
This
Of t h e
S([])
C
+
field
degree
The
subsemigroup
Z
Denote
4.4.
conductor
-#g
g
irreducible
thus
the
+...+a
>/ 0 ( r e s p . a < O) we obtain J ] b - a = 1. I n o t h e r words, we have
an
in
and
be
61
is finite.
field
element,
Definition
a minimal
a
in its q u o t i e n t
will
which
+
+
-S
[]
algebraically c l o s e d
a]
that
Z
+
+
which
such
a
which
is
that
terms
a~b
( ~ - ' # ) 0 ~<'0.<9
( B0 . . . . . "6"g) = 1 , t h e r e
Since
a0
Collecting
that
[]
or
curve.
defined
to
Z+ . B
Lemma least
5.1 .4.integer
The in
degree
S(['-I)
such
j C Z
In
particular,
Proof:
It
Proposition
Z
works
+
-
as
5.1 . 5 . -
of
S(t--I)
in
+
the
conductor
of
in
[]
[]
is
the
that
,
j>~c
is
finite.
>
j c s(E])
4./4.2.
A
subsemigroup
S
of
Z +
is the
semigroup
be
144
of
values
of
Proof:
Assume
according it
a curve
and
only
S = s(El)
to
suffices
if
the
to
if
for
preceding
..<9
Now, fl
v(m)-
2'
lemma.
is
+
[].
finite.
Then
Conversely,
~ if
-
+
Z
S -
+
is
S
finite
is
finite
take
(BV)0.~'~
by
-
a curve
[] = k((t where
S
we
is
.....
a minimal
shall
set
denote
< f2
<'"
"< f N
the
v(m 2)
is
finite
since
Let
x i El. m_
t rg)), of
by
m
elements if
generators
the
in
maximal
v(m)
i >/ c +-6
-
S.
ideal
v ( m 2)
and
0
of
of
(Note
v(z)
= i
['7 t
and
that
we
have
z El m 2 ) .
Lemma { x.
}
5. 1.6.-
Proof: tot
is
a minimal
must
prove
l~i~
We
space
m/
--
m
2
such
basis
that
of
that
_v(xi)
m.
In
{ xi +m2}1~<
= fi
,
1 ~<
particular,
i ~
is
i ~< N .
Then
N = Emb([-1).
a basis
of
the
k-vec-
. N
First,
Z
if
a.
El m 2
x. I
I
,
with
a.
--
El k ,
we
have
I
i=I N
v(
E i=1
--
so
a.=O
for
I
i,
all
a.
x.) I
E: v ( m 2 )
,
I
1 ~ i ~N.
Thus
{x.
I
2 + m } --
1~
i~
are
N
lineary
independent. They either
v(z)
z 1 El m v(z {) so
2
are
C
v(z-a.x.)
I
or
that If
I
2)
v(m
such
> v(z).
also
,
v(z)
> v(z)
if
generators. --
(
fact,
;
v(z) ~ v~m2j . If • z = z - z I , we 2
6Z v ( m for
In
a.
)
then
El k .
I
v(z) It
for v(z)
any
El
= v(x.)
for
some
i
)
2 a.z.
+ z' t
i=I
,
with
a
El k i
and
z ~ El m --
-
m
,
exists and 1 ~
that
N z =
m
El v ( m 2 ) , there ~) 2 v_(z El v ( m )
have
follows
2 z
,
145
N ~'--
z + m2 =
whence
--
Proposition is
the
set
generators
of
,f2 .....
= {fl
Emb(r-])
the of
as a b o v e .
S(E])
( 6 " ~ ) 0 (v~< g
-6 g }
~< g + l .
I
be the
i
f
j
greatest
6: S ( [ ~ ] ) ,
i
integer
yM g
v ( m 2)
Pa--: s u c h
which
Corollary
a ~) -6~2
that
av C Z + ,
,
..,
Then
v ( y . ) = 8.~
a contradiction.
5.1 .8.-
0 ~< "~ ~
is
Let
without
v(
Proof:
Necessity
8.)@ _v(m 2)_ polynomial
y,#6
8",~E: f ( [ ~ )
,Y_I)
~ 0= "~
is
and
{ Y~u . . . .
q(Y0'
" " " 'Y
if
linear
P(Y0 ....
P(Y0
~
•
•
by collecting
.
~
Y all
~(P(Y0'
_1 ) its
.
is
. Then Thus
a j>p 2
f
the
" ' . ~ - 1.
i)
C
is
order
is
8,j
no polynomial
that
' Y ' ~ - I ) = ~',2 "
Conversely,
is a basis g without linear
Yg)
polynomial
monomials
whose
there
terms,such
"" "Yi
i
only
if
Yo
a
o
6 .
and
'Y} )
=
j
= v(
be an e l e m e n t
obvious.
g
fj
m
Z( q ( Y 0 . . . . .
If
whioh
we have
a
Take
for
i
= v7 - ,= 0
fJ
If
, we have
f N } C. { - ~ 0 ' -61 . . . . .
f. C f(r~), and let J Assume 6. < f.. Since
j
desired.
notations
Take g f..
as
--
We k e e p
particular
8.
I
5.1 .7.-
minimal
Proof:
+ m2)
(x.
I
i=1
f(r-~)
In
a.
in
)) . = z ( q ( Y 0 '
let
of
-SV~C f ( L - J ) - S i n c e
_m , t h e r e
terms,
such
exists
a
that
) = 8V
obtained Y0'"""
from
'%-1
' ' Y g ))
'
=
gv
q(Y0 ~ . .. ,Y
we have
g
)
,
146
Corollary []
5.1 .9.-
is
plane
Remark
if
5.1
minimal
of
g > 1 ,
S = S([-]) (see
only
if
fl
4.4.6.)
if
-60
S =
=
6"0
'
curve
, there
is
f2 =-61
no
,
plane
Consider
a singular
g
(~j)
f = {fl is
>1).
= {-60'
( where
not
instance
curve
true
the
}C{-~-U . . . . .
S(i~)
symmetric
= S.
PLANE
algebroid
curve
integral
[]
closure
in its q u o t i e n t field~ v the n a t u r a l v a l u a t i o n of [] and g s ( r ~ ) = ~--0_ 13V z + the s e m i g r o u p of v a l u e s of [-] , w h e r e the m i n i m a l
i-'7,
such
F-I v
we
set
of g e n e r a t o r s
Let
p([-l)
that
k
denote C [[]'C
the [-]
v(r-I,-{0})
Since
c
set
formed
and
by
Emb([-]')
the
..< 2 .
sequences
ordered
Z
of
Z
+
+
(-B'~)0 ~)~< g
alqebroid For
such
subsets
in
Z
+
,
s(O)
=
of the
Z
c z
curves a
curve
may
+
be
+
viewed
= { v([]'-{0 --
by
the
order
}) I
as
strictly
set
Z
be
of []
S(FT).
v([-I-{0})
infinite
S(P(l--I))
may
over
have
--
increasing
of
B" } g and
f = f(]~})
is not
GENERIC
the
6"1 } "
that
S
that
to d e n o t e
Moreover,
0 ~<~;4 9
' "" " 'fN
if
such
rreducible
continues
( if f([-])
E.s.1.
I ONS.
[]
1
or
it
For
EQUISINGULARITY
section
= ~
= {B'0}
E] .
PROJECT
In this
f2
g /~ ~R'V2' + "~=0 of S) and
and
some
and
f(E])
generators
for
2.
k.
=
.10.-
set
with
and
ft
El'
--~ i n d u c e d
e
p(I-])
by
the
}c
z
+
+
lexicographic
order
147
Lemma
5.2.1
Proof:
Let
by
(S(P(r-])),
.-
M
induction,
be
a
~
)
nonempty
a sequence
is
a well
subset
of
of
integers
ordered
set.
S(P([-])).
We
m 1 < m2
< . . .
may
construct,
< m.<
. . .
as
I
follows: First, and
we
put
the
m
we
we
= min
i+l
put
M
m
= min { 1 £ M I m 1 = min S'
inductive
constructed,
and
define
M 1 = { S' In
are
we
i+1
Now,
if
I
t = rain
S'
for
some
S' C M}
} .
m I , . . . ,m.i
and
Mi C M,
M i ~: (~ '
define
{
= {
by
step,
t
t
t
S'
the
= min(S'-{m
E; M .
I
i
above
m.
1 .....
= min
i+1
m })1
(S'-{m
construction
the
for
1
set
some
.....
S' C M }1
m.}
)}.
l
S~ =
{ m . I l~:
<~o }
1
is
infimum
of
M in
that
S ~ @ M.
To
prove
such
that
It
the
m
follows
S ~ as
> r
i
and
that
above,
we
Furthermore,
2 Z+ +
for
this, if
S'
the
note ~ M.,
a semigroup
have
m+m'
if
that we
I
S ~< i s
texicographic fop
+
Z
,
r
> O,
S ~ ~ {1 . . . . .
m,m'~
E; S't-1 {1 . . . .
-~ S ~< = v. , £ :u -BY
any
have (if
order.
,r
}
S ~, and
We if
i
must is
see
an index
r } = S'th{1 .....
taken
hence
r > m+m' m+m'~;
and
S~).
where,
=
-#0
min
(S { - {0}
~V ~
= min
)
,
-2 +I
r we
may
be
have
Notation
S~ C
along
Definition
this
A of
[]
r
>
S' ~
minimum
section
5.2.3.-
projection
that
S v, w h e n c e ,
The
S~<-[
6"~ Z i
i=0
such
5.2.2.-
denoted
,plane
chosen
(
by
curve when
69 Sm:,
.
)
+
Thus, and
element
'
since
therefore
for
~
B.g C S~
in
=
S',
S'
S(p(T-}))
0.~-~.,.4; g ,
E: M
will
S~(Ft).
}--I'c s~(r-I)
p(r-I)
will
= v(E~'-{0}).
be
called
a 9ener!c
be
r}.
148 Proposition
5.2.4.-
Proof:
If
On t h e
[]
other
[]
is
is
plane
hand
plane
if
we have
since
S ~ (0)
El
and
only
if
C P(O)
C S(E])
it
s(F-I)
and is
= s~(F']).
so
S~(['-[)~-~S(I--]).
evident
that
s(O) ~ s ~ ( [ ] ) . Now plane
assume
projection
z C El, with
we
z
C
that
of
may 0',
17.
S:~([--I)
We c l a i m
construct such
= S(FI) that
inductively
and
[]'
let
= 0
.
Ft'
In
be a generic
fact,
a sequence
z
for
1 'z2~
any
" " "
,z
,
m
.
.
.
that
i
lim
Hence,
since
0'
which
completes
Lemrna
5.2.5.-
is the
a closed
Let
Proof:
and
see
(i)
and
:
assume
Thus
Now, that
El,
z =~
we have
z
eO', I
plane
projection
E],
of
then:
= 2.
that
Z + • If we consider = "~ 1 ' t h e i n e q u a l i t y
contradiction.
such
be a g e n e r i c
= v(z)
v(y)
of
= "61
61(E]')
(
6 0, 6 1
are
exponents),
To
S~(I--])
subset
=
i=O
El'
Bo (1:]')
characteristic
(z1+...+Zm))
proof.
Emb(E]')
(i)
_v( z -
let
v ( x 1)
Emb([7') {x I , x 2}
does
not
[--]'
= k((z)),
x,y
E: [ ]
5([7')
z ~
such
~
[--].
that
Then v(x)
v ( k ( ( x , y ) ) - {0 })
=B is
a
= 2. be a b a s i s
divide
of
v ( x 2)
the a~d
maximal
ideal
v ( x 1) < v ( x 2 ) .
of
0'
As
above
s(i:],) ~ ~_(k((x,y))-{0}) implies
easily
v(xl)
Lemma
5.2.6.-
2"
parameter
t
for
+ []
= ~0
and
-S~([-])
is
is
also
v-(x2)
finite,
= ~'I "
i.e.,
a uniformizin9
any
uniformizing
parameter
for any
generic
149
plane
projection.
Proof:
Let
a basis
of
expansion
['-I' its
be
a generic
maximal
in
that
ideal.
basis
( D ~)
plane
is
z ~1_J
projection
Assume
given
that
of
the
[--I,
and
{Xl ,x2
}
Hamburger-Noether
by
=
a j,
h
i
zj
j
+ zj
.
z j+l
'
0
4 J
.~ r .
i
According
2.2.6.
we
must
prove
v(z
) = 1.
--
z
= b r
with
b. C k and s > J expansion given by
t
s
I ~ we
z j_
+ b
may
~
=
1
s
t
s+l
s+]
+
consider
-
. aji
Otherwise,
if
r
z.j
i
...
the
h
-
+ z.j
j
Hamburger-Noether
-
z j+l
,
0
<j
< r-1
,
I
(D'')
z r-1 z
c
where
v(z0-z and
0)
so
is
the
~ c E~''
ar2
= b
r
degree and
2
=
of
s
the
v ( z 1 - z 1)
E; P ( E ] ) .
Finally
t
c
Zr
s
+ b
+...+arc t
s+l
s+l
conductor ;~ c ,
we
from
c
Zr
+ Zr
t
+...
of
[]
have
in
['-t''
4.3.4.
and
['1.
Since
= k((z 0,z_l)) 4.3.8.
it
C
follows
that
z(N"-{0}) which
is
a contradiction.
Now, plane
we
projections
projection {x 1 ,x2 }
F'I' a basis
aim
to
defined of of
D. the
If
This
completes
give
a geometric
the
Let
us
m
the
maximal
maximal
ideal
of
proof.
interpretation
above. is
= s'~(I-q),
consider
r-I,,
of
a generic
ideal x 1 +m
of 2
[] and
generic plane and
x2+m
2
[]
150
are
lineary
therefore
independent it
minimal
may
basis
embedded
if
curve
to
we
m.
want)
~ and
of
consider
be the
the
open
(N-2)-planes
planes
H
that
and
(even a 1 -,
I
by
1 ~ i ~< N ,
that in
dense
subset
pass
throught
Hr~
5.2.5.),
{ x. }
Denote
in
2-plane
which
such
a basis
,
I
representation !]
and
, 5.1.9.
that
= x.(t)
I
Let and
of
assume
x.
a parametric
5.1 .6.
be completed
(l)
is
(see
T[ = { O } .
X]
+ )`3
X3
X2
+ la3 X 3
k
basis. N
defined by X3=...=XN=0, N,N-2 G C G ( = grassmannian O the origin) formed by those
Any
(N-2)-plane
+ "'"
+)` N X N
of
G
is
given
by
= 0
(2)
where
( )` ' N) = ( )'3 . . . . .
equations by
of
type
(2)
define
U3 . . . . . an
= 0,
UN ) C k 2 N - 4 .
element
of
G
Conversely,
which
will
be
any denoted
H( )`,I] ) . The
is
the
plane
(3)
parallel
curve
C( )`, ]j):
=
given
over
projection
algebroid
Equations
A
X N'
+ " " " +]a N X N
an
k()`,P).
curve
H( X ,
of
C
on t h e
k
X = x](t)
+
t 3 x3(t)
+...+
X N xN(t)
Y = Xg(t)
+ '4 3 x 3 ( t )
+-..+
P N XN(t)
(3)
may
be v i e w e d
closure has
curve
given
]a)
over
algebraic This
to
F
as of
plane
by
a parametrization
the
quotient
a Hamburger-Noether
for
field expansion
(over
h Z j_ I
=
a.jI
Z J
+ Z.j
Z j+l
, 0
~<j .< r .
i Furthermore,
since
X(t),
Y(t)
C: A ( ( t ) ) ,
the
a
of
by
(D)
ir[
coefficients
F)
151
a.. ji
are
actually
in
k( 1 , ~
Consider maximal
ideal
the
).
localization
m = ( X , ]J).
One
A
has
of
m
the
ring
A
relative
to
ep
morphism
the
(.evaluation
map):
e : h(
A
~__,!J )
h'(_~,
which
has
natural
a
if
localization i
i
0
{q0
of
IS A ,
/
is
e(p0...pg)
~ 0, a..
IS
'qs
(ii)
to
((t))
m
>
we shal
the
k((t))
denote
.
by
multiplicatively
A
z
(iii)
'qs
such
P0 tS A
,
above,
04
j
there
exist
be the
~< r
,
1
..< i ,<
h..
,
j
" " 'Pg
leading
coefficient
(D)
of
Z
written
in
coefficient
the of
~ 0 . Moreover,
is
J
v ( Z j ) = -v ( e ( Z . ) )j -
is
leading
e ( p 0)
p0, . . . ,p9
J
((t)) Po'"
that
have
as
that:
IS A
The
Assume
} .
notations
(hence
in
A P0,,-,Pg
).
reduced X.
a0i
a unit
form
Since
IS A
3. 3.4.
x]
is
and
= Pl
Now,
let
91
that
e(gl)
~ 0.
~(0)1
In for
h n + n 1 = "61 " T h u s
/P0ml fact,
be the
if
leading
e(91)
k(('Xl 'x2))
= 0
= ~]'
coefficient
the would
[--I'
= k((X(
t,U
), Y(
of
value
be
greater
;~,U ) ) ) ,
C(t)). Po
first
setting
trans-
Z I E: A
Po
contact
qo''" set
closed
P0' " " " 'Pg
J
we
A
IS A ,
A
JI
versal
e :
>~ 0
Keeping
(i)
Let
h'(0,0)
relative
i 0 .....
5.2.'7.-
Proof:
h ( O , 0)
)
14)
qo'''" A
k
s
" " " qs
Lemma
I
extension
Finally, the
>
m
Z l . We c l a i m
of
the
than
maximal
152
where
2N-4
(I , ]J) E: k
have
[[]'' g P ( [ [ ] )
verifies
is
((t))
A
for and
any that
j
,
the
I ~< j
..<sl ,
leading
It
follows
that
a
po,Pl I ((t])
.
If
it
is
evident
coefficient
r-I,,
= k((X(~
. C A , s 1 ,i po,Pl
P2
pom2 plm~
g2
of Z 1 ' we must - = ( 0 ) Sl + I~ 2 of the maximal of
have
denotes
e(P2)
contact
#~ 0 .
for
of
Z. J
, 1.1), y ( ~ ,
p))),
is
as
above Thus,
with Z
~. i
~
in
o
.
a unit
in
, and
s]+]
coefficient
the be
E:
Z
second
greater
(_~ ,'P )
value
than
that
that
such
,1.1 ) ~ O.
the
we
may
~
A
leading
construct
PO'"""
~Pg C A
PO' " " " 'Pg
coefficient
of
each
Z
is
a
completes
the
proof,
" ' ~Pg
that
if
exists ( ;k
--O
, p
--O
- - 0
an open
) E: ~
, -P- o )
is a g e n e r i c
plane
projection
[-]( I
with
( •
, ~__o)
would
C
sI leading
, ]J)-P2 (t
aji
[]
--0
is
Zj
j
5.2.8.-There such
(t
that
and This
P0'"
(0_.,O)
the
that
" 'Pg
A
Theorem
1
choosing
induction
such
((t)) PO'
unit
using
=A 0
(~ A J
would
a contradiction. by
e(P0...pg)
k
Otherwise
r-i,
P0 ( ~,iJ ) * P t
This
we
-{o]) 4 z(E]' -{o]) : s'(F]),
P0,Pl
Ap0,p
0,
a contradiction. Now,
A
I , ]J) /
and
z(E]" which
p 0 ( ;k,~ ) . p l (
)~
of.
the
dense
~ Ck
2N-4
containing
curve
= k((X(t
o,~o),Y(to,
['7. In p a r t i c u l a r (a)-equisingular
Zo)))
the c u r v e s
Define
Proof:
~ : {( ;k
, l.lo)
--o
po ~ • . .,pg
where evident
that
E: k 2 N - 4
/
p0(~,
are
(0,0)
the
E: ~
polynomials
. On
the
• , ~1 ) and [7(0,0 ) = 17' "-o --o respective Hamburger-Noether
~'j-1
=
). • .P
the
hand
are
()~
g
--o
in
other
[7(
, IJ
--o
--
, 1.t ) :ag O } ,
--o
--o
previous for
Iemma.
(:k
,
--0
) E; ~
1.1 --0
(a)-equisingular
It
is
,
(since
[heir
expansions,
~--.
~
aji
i
zj
h
~,
+
j
~..
J
,
j+l
Or,
j
.
h Zj_ 1 =
•
e (aji
) zj i + z
j z
j
,
j+l
0
.<j
~
r,
I
v(2".) = v _ ( z . ) , 0 $ j~< r). J J
verify
v(l:::]( X_o ' £ ° ) - { o } )
Remark
5.2.9.-
of v a l u e s
of the c u r v e
Definition and
5.2.10.-
[7 2
s~(l-1
I
The
over
) = s~(l-I
Remark
2
projections. E.s.I.
A complete
of the
curve
s~(rT).
They
actually
are
said
also
given
to be
by
is a c t u a l l y
the
sea[group
(3).
atgebroid
equisingular
to this d e f i n i t i o n
curves
E.s.I
contact
since
be
of i n v a r i a n t s
set of g e n e r a t o r s
maximal
called
a complete
and
equi r e s o l u t i o n
system
, characteristic may
([7)
S ~
irreducible
E.s. I . means
Moreover,
plane
given
According
is the m i n i m a l
of v a l u e s
F
= s~(Fl).
.
[]
iff
).
5.2.11.-
equisingularity
over
are
= z(Fq,-{o})
semigroup
Two
k
Therefore:
for
s~(r-l)
any
of the
S~([-I),
generic
of
exponents invariants
i.e.,
plane
( B~ )0.<'~<, g
characteristic
generic
plane
for e q u i s i n g u l a r i t y
is the s e a [ g r o u p
exponents
system
of
4.3.11.
projection.
of v a l u e s
are of
for
the s y s t e m
of
defined []
E.s.1.
and
a
by
154
We s h a l l
consider
Equisingularity
E.s.1
description,
from
saturation the
of
concept
.
an
the
case
has
a more
algebraic
view
algebroid
of
now
curves.
sat uration
,
caract
precise point,
is
in
which
description. is
Therefore,
which
k = 0
made
we
in
shall
extensively
This terms
of
introduce
studied
in
briefly Zariski
(27). Let
[]
algebraically closure
may
[] of
choose
that
an
closed
of
valuation
be
irreducible
field
of c h a r a c t e r i s t i c
it its q u o t i e n t O.
Take
field,
generated
by
t
y
such
is a G a l o i s
the
and
a parameter
a uniformizin9
k((t))/k((y))
algebroid
curve 0.
over
Let
denote
0-
by
v
an
be
the
the
integral
natural
of the
curve. If m = v(y) we m y= t It is trivial to c h e c k
that
extension
whose
Galois
group
G
is c y c l i c
k-automorphism
f
:
k((t))
>
k((t))
t
wt
,
unity.
Moreover,
m
where is
w
is
a primitive
invariant
by The
ring
all
of
satisfies
1)
root
automorphisms
saturation
which
Y
m-the
in
[]
the
of
LJ = k ( ( t ] ]
G.
with
respect
following
to
y is
defined
to
be
the
conditions:
cr7 c D Y
2)
If
z C []
, z'
e []
and
v(g(z')-z')
>~ v ( g ( z ) - z )
Y
V
9 C G,
then 3)
[]
In respect
to
ring
with
that
[]
z" C []
y
Y
is
Y the
general,
a ring
when
respect
smallest
2) to
r-],~-
holds
for
y which
is itself
an
We s h a l l
need
ring
is
it.
said
]-hus
contains
irreducible
satisfying
U
]) to
and
be
2).
saturated
with
U .
algebroid
is the smallest Y On t h e o t h e r hand, curve
over
k (see
saturated note 1 .1 .5.).
Y
(Zariski,
O=V1 X
[ X t
°
2? ] ) :
If
an
x and
important x'
are
property transversal
of
the
saturation
parameters
for
[7,
then
155
Now, is
an
we
algebraically Let
{ xi
shall
}1
closed
[]
~i ~
discuss
be
any
an
the
field
of
Equisingutarity characteristic
irreducible
basis
of
E.s.].
x1
curve
is
k
zero.
algebroid
m , where
when
over
k,
transversal.
and
Assume
that
n x
=
t
y
=
(4)
i a ij
,
t
2
4j
N,
~<
i>zn
is
a Puiseux
Let
I
(4)
primitive
denote
the
, i.e.,
parametric
set
6 C I
of
representation
integers
z. "2
b
t
which
are
0
for
by
using
in
effective
some
j
or
that
basis.
exponents 6=
of
n.
J6 Define
6~) = n
and,
induction,
1
I
min
6,#+ 1
Since
(4)
is
The
1~i
absolute to
any
curve
..
it
transversal k
65
/
'I)
>
( 6'0
is
[]
that,
if
!
, 6v , 6 )
.
g>
does
not
by the then
"S
.
0
is
F].
In
is
)
=
on
fact,
saturation an
semigroup
y
.
,6,~ ) }
.
.
.
that
6g
depend
[] its
'
.
i
< ( 6 0
such
> (13 0 . . . . .
(i.e.,
parameter), such
.
'
....
determined
of
,
exist
' ) 0 -<'0~: g ( 6~
set , and
I
there
( BO'6'
saturation
over
",{ 6 6
primitive,
60' >
{x.}
set
1.
the if
of
basis []
~
is with
irreducible of
the
values,
respect algebroid we
have
= n I
B ' ~+1
= min {
6 E: "~ /
(6 '0 . . 6'.1 . • .
18X),6 ) < ( 6 O, • .. , ,
,6 v ) } ,
0 ..< "0,< g - t
(see
Zariski,
(27),
and
J.L.
Vicente,
(21)).
.
156
Lemma
5. 2
12 •
the
-
Keeping
of
Proof:
k(
It
as
above
( 6 ' ) '
characteristic
closure
notations
"
of
1~3, . . . , ~ N )
is
Proposition
exponents
closure
of
_ x 2 + !J 3 x 3 + . . . +
according
The the
k( X,14
~
over
g
the
algebraic
x 1
y,
of
defined
0
by
=
5.2.13.exponents
curve
X '
evident,
teristic
the
are
v
to
set
curve
the
t~ N x N
•
definition
( B ')0 ~<~ C
of ( 6~)0
is
g
the
defined
set
over
~
of
the
v<~ g "
characalgebraic
by
X :
x 1 + ~3
x3
+""
"+ ~N
XN
x3
+'"
"+~N
XN
(3) Y
Proof: the
There
curves
C( t., .
exponents --o
= ( ~
exists
and 3,0
basis
{X(
Proposition
1.!) .
.
and
) = x
:t.
) ) ,
N,o
the
proof
~_o),X2 .....
By
teristic
using
Corollary
closed are
1
is
XN}
For field
equivalent:
CK
have
N-2
the
x 3 +...
such same
+ )t
that
X C ~' --o
characteristic xN
N,o
if
(where
'
transversal. from
lemma
5.2.12.
applied
to
the
.
(see
above
~'
~_)
3,0
follows
defined
5.2. 1 5.-
algebraically statements
the
exponents
o + )t
The
( 6 ' ) 0 ~
dense
C(~.
.
X()~
.....
+ I]3
open
5.2.14.-
exponents
Proof:
an
--o
Now,
:= x 2
semigroup 4.3.
5.)
proposition in
two
remark
given
defined is
by
the
characteristic
S~(E]).
(
13 ' ) 0 ~
5.2.11.
algebroid
of c h a r a c t e r i s t i c
"~'~ g
The
proof
curves zero~
are
the
t he is
over'
charac-
tivial.
an
following
157
Remark tion be
(a)
They
are
(b)
They
have
(c)
its
given
works
temma
as
given
by
is
the
x'
(n,p)
8,10,31 = t
8
+t
13
~
5.2.12.
characteristic
are
in
,
I I I ,
4 th
there
does
=
t
y
=
t
Z
=
t
y
= t
if
hold.
8,10,15
saturations
> 0
way
and
many
hold,
and a
are
(n,p) result
isomorphic.
=
1
as
the
satura-
5.2.14.
may
patological
since
if
cases.
charact,
k
For =
2,
instance and
[]
10 13
of
8 10
we
use
the
basis
exponents
X v
=
t
y'
=
t
means
13
+]_it
characteristic
. This
' "g~
section).
8
are
.
8
exponents
= t
(
= p
are
not
x
; but the
k same
x
set
absolute
the
1 ,
E.s.1
same
charact,
chapter
tf a
If
(see
the
respective
5.2.16.-
criterion
equisingular
,
where
of
13 +
10
{x',y,z}
t
+
that
13
!jt
a
saturation
criterion
does
not
158
3.
EQUISINGULARITY
be
the
tions.
This
called
definition
definition
definition
over
k
5.3.1
will
same
was
QUADRATIC
Tv~o
be called of
Recall
that
uses
actually
space
given
irreducible
(E.s.2.)
quadratic
in
will
transforma-
1.5.12.
algebroid
e~uisingular
resolution
Hamburger-Noether
equisingularity
and
it
was
Namely:
.-
process
of
which
equiresolution.
Definition
the
second
classical
SPACE
t ONS.
TRANSFORMAT
Our
E.s.2.
E.s.2.
curves
iff
[]
[]
and
is c h a r a c t e r i z e d
by
and
[]~
~--}~
have
1.5.10.
equiresolution
expansions
(see
2.4.9.5.):
using
If,
h Zj-1
=
Aji
i
i
z j
Z
+
i
Z j-1
are then
respective those
7"-i
A j i~ z j~
Hamburger-Noether
curves
are
J z
J
j+l
0
<j
J Z j ~+ l
0
-.<j ~< r
expansions
for
E.s.2.
iff
[]
r=r
and ,
h J
and
n.=v(Z.)
j
--
j
=
v(Z ~)
--
Remark
5.3.2.-
Noether
expansion
j
= n
The
(i)
n
(ii)
If
must > nl > n. ~
j
J
0
j-1
n
r
=h., J
=
1 ,
h
.I
>~I ,
in
the
0 ~< j ~<
Hamburger-
r.
, then
nj-l.1
hj ~ L ~ J (iii)
Furthermore,
~,
r.
h ~ n ( 0 ,~. j ~< r ) , J J some requirements:
satisfy
n
n
'
values
... >
~ j-<
..< r
h
Z ~J
+
equisingular
'
if
and
n'j+ I ~ n j_ I - h j
nj
159
nj+o,
/f
nj_l
-
hj
nj
-
" " " -
h j + o . -1
nj+o, -1
'
0 ~< ~
s,
then
hj+s
( I
j-1-hj
nj-
. .. -h j+s-1
n j+s-
l
nj+s
and
nj+s+
Conversely, (0
~: j ,.< r )
over
k
verify such
(i),
that
able
in
Emb(r-])
and
to with
Hamburger-Noether following
a plane
(i)
(ii)
state the
are
then
...
if
-
hi+ s nj+ s .
integers
there
actually
h. , n J J a curve
exists
those
of
any
of
its
the values
dimension existence
N
are
of
h. , n. (0 J J Indeed, for
is
given,
we
a curve
[--I,
with
~< j 4; r )
in
N=2
we
its
have
the
result:
Keeping
the
notations
as
above,
h.~-
0
~:j .< r .
if
[]
hj
nj
is
then: n>
If
nl
>...
If
> n
n. A n j j-1
hj (iii)
that
(iii),
integers
5.3.3.-
curve,
and
expansions.
evident
Proposition
(ii),
prove
if enembedding
general
= N
can
h.j n J -
expansions.
However, not
~< n j _ 1 -
one
these
Hamburger-Norther
1
r
,
j
1,
then
=[ "nJ-1]
L nj j
n j I n j-1 h. j
=1
n j-I >.. - n J
, then
and
nj+ I
nj_ 1 -
.
160
Now, between
the
shall
E.s.1.
Examples by
we
give
and
examples
E.s.2.
5.3.4.-
there
Assume
following
in
k
parametric
=
is
~,
(R)
t
y
=
t
z
=
t
+
t
+
t
12
are
On expansions
the
,
r
Thus,
they
=
"
=2
are
;
h=l
;
h'=
not
(R')
are
t
y
=
t
Z
=
t
z
y
curves
given
8
(R)
~ ~
:
t
=
t
=
t
equisingutar and
(R ~)
by
looking
hand
10 12
13
+
t
+
2
E.s.]
t
. ,
15
since
the
charac-
agree. at
, 1
n= 8 ,
n
,
hl=3
"=8
,
the
n1=2
h~ ....5
equisingular with
,
Hamburger-Norther
10
t
k
=
~
11
,
the
(R '~)
15
exponents,
y z
E.s.2. so
but they
h2= ,
oo
h2
,
= co ,
n2=
1
n "=1 2
E.s.2.
x +
,
nl" =2
,
8
equisingular
teristic
y
15
Conversely,
=
the
have
2
x
(R ~)
other
we
r
13
of
that
representations:
trivial
exponents
check
relation.
{
10
to
consider
X
They teristic
no
and
8 X :
order
they are
have not
curves
~
~ ~
=
t
=
t
=
t
not
8 10
+
t
13
15
evidently
equisingular
equal E.s.
] .
charac-
161
4.
EQUISINGLILARITY
SEMtGROUPS
As
we
saw
equiresolution For of
is
twisted
of
equisingularity to
(26)
every
we
this
Definition
shall
ity
.-
Let
E • s . 3.
natural
is
of
[]
values
let
to
be
of
using
the
considered
as
section,
a graded may
be
curves,
semigroups
therefore
this
E.s.3.
[~<
k.
be
of
values.
coincidence a new
and
ring
according
associated
characterized
algebroid
We s h a l l S(I[[])
that is
us
any
We h a v e
consider of
and
by
i >~ 07
the
say
curves
that
[]
with in
over and
= S ( [ - I ~) ~ w h e r e
the
minimal
=
[] m
-i
m
system
in the
its
of
of
a curve
-
we
set
a
S
the
[[]~
are
denotes
[]
generators
over
quotient
maximal
graded
for
k,
and
field,
ideal
of
/
v(z)>,
by []
.
set
r~
= M°..~
ring
of
invariants
[]
= {z
e
[]
filtration
[]
(1)
the
a complete
closure
valuation,
defines
and
in
by
when
Mi
which
out
,
field
evident
integral
For
algebroid
values.
Now, the
OF
fashion.
equisingular
[]
plane
coincidence
(E.s.3.).
equisingular,
closed
semigroup
, for
true,
turns
that
algebraically
tt
not
prove
5.4.1
of
iV to
is
values
algebraic
semigroup
chapter
criterion
curve,
purely
in
COINCIDENCE
VALUES.
equivalent
curves
semigroups
OF
E.s.3.
M1 _~ . . .
D Mi Z) . . .
i}.
the E.s.3.
denote ~
the
by
162 co
e,:oM/M+l
grM(t--1) = Lemma of
5.4.2.-
S(E]).
Let
Then
we
( 6",~)0 ~ < have
an
g
be
isomorphism
-¢ gPM(I--]) Proof: y~ ¢
Take []
~
a uniformizing
such
the
k( t
minimal of
o
t E: E l .
set
graded
of
generators
rings:
-¢ .....
g)
t
Since
~',aIBS(L'])
there
exists
that "~x) Y'V
If
gr(t ) k((t))
is
the
[]
we
have
an
injective
t
graded
this
ring
,
v(y,0
given
by
= mO D m] ~
filtration
of
grM(Fl)
induces
the
,~
the
filtration
of
[]
:
. . . ~) m . ~ . . .
homomorphism
.. since
+ y.~
graded
.
rings,
gr(t)k((t) ) :k(t),
filtration
(1)
over
E].
-¢g As if
~v
= i n M Yv
I n M z E:: g r M ( r ' l )
with
a 0, . . ,a
g
>/ 0 ,
H(InM(z))
with
Im H
z C: [ ] ,
~
. . . ~t
k(t ~0
then
) . Conversely,
v(z)
= a 0 '60 + . . . +
E: k (
t
-
a
g
6" g
so
=
c t
a0-¢0
...
t
a ~ g
g
T0
.....
t
-¢g
),
c E: k .
Proposition E.s.3.
, with
,
5.4.3.if
and
only
if
Two
curves
[]
there
exists
an
g r M( [-t )
~
and
L~ "~
isomorphism
grM(E~)
are of
equisingutar graded
rings
,
163
Proof:
If
S(E~)
= S(Eg ~)
gPM(G)
where The
~
(-6~) ) 0 ~V~ converse
is
k ( t is
g
then
7o
the
evident
by
the
above
-~ t g )
.....
minimal
since
~'~
#et
k(t
lemma,
gPM(L~J ~) ,
of
g .g_ e n e r a t o r s
B0 . . . . .
t
g)
of
S(E])=s(r'7~).
determines
the
semigroup.
Remark Pity
5.4.4.-
There
definitions.
neither"
Recall
E.s.1.
23
. , = v(
second
xz-y
they 2)
we
E.s.3.
(R)
but
relation
E.s.2.
/'i
representations
no
that
implies
E.s.1.
E.s.]
is
and
among
have
already
nor-
E.s.2.
The (R ~)
in
section
over are
not
the
same
semigroup
of
belongs
to
the
first
semigroup
but
that
E.s.1.
~
5.3.4.
3,
with
parametric
equisingular
values, it
since
does
not
so t o t h e
one.
given
~ ,/>
E.s.3.
It
suffices
to
consider
4 =
curves
over
z = t
5 6
,/'~
E . s . 3 .
x=t y = t
same
4
t
y = t
the
the
by
x
neither
in
equisinguta-
implies
have
E.s.2
have
three
seen
curves
as
the
E.s.1
(resp.
t
y'
= t
z'
= t
+ t
9
4 2' E.s.1.
+
+ 6
nor
2"
+
E.s.2.
?
over
{;
they
are
,
4
y~
=
z'
= t
+ 15
5
curves
x~=t
semigroup
equisingular
=
E.s.2.).The
4 6
x t
t
Z as
+
6 15
~ but one
may
easily
check.
REFERENCES
1.
Abhyankar,
S.S.
, "Inversion
pairs". 2.
Aneochea,
Am.
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la
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Real
Academia
G. , "Die
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Atiyah-Mac
5.
Bennet,
B.
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Brauner,
N. , K. ,
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91
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Burau,
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"
Endler,
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Wertehalbgruppe
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89
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13.
Hironaka,
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D~'ng T r a n g ,
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Jorge
"Noeuds Grenoble
]5.
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18.
Moh,
T.T.
,
"On
C. ,
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der
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Wail,
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,
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Zariski
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Math.
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yon
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.
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.
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ria'.
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de c o u r b e s
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117-126.
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broides
20.
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points
(1973), Romo,
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for
19.
de
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near
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Am.
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Memorias
de N e w t o n " .
Paris
J. , " A b s o l u t e
J.,
infinitely
Juan,
(1972),
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of
l'equivalence
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theory
Algebriques".
Coefficients
16.
the
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singular
14.
to
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Springer-Verlang-Berlin.
(1972). to
C. I .M.E.Varenna.
the
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of
equisingularity". (1969).
1
166
26.
Zariski,
O. , "Le
probleme
planes".
27.
Zariski,
O.,
B.
Theory
J.
Math. Zariski,
in
les
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branches de Math.
(1973).
Math.
and
de
Appendice
(11 I )
saturated
93 (1971),
local
5?3-648.
872-964.
(I 1 I)
Am.
(1 I ) J.
415-502.
equisingular'ity".
"Commutative (1958).
saturation
93 ( 1 9 7 1 ) ,
507-535.
972-1006.
ton
J.
97 (1975),
O. , "Studies
Zariski-Samuel,
of
Am. Math.
(1965),
29.
pour
Teissier.
rings".(])
28.
donn~
Polytechnique.
"General
Am.
modules
Cours
I'Ecole de
des
(I I) Am.
(I)
Am. J.
AI9ebra".
J.
Am.
Mat!q.
Math. Van
90
J.
Math.
87
87 ( ] 9 6 5 ) ,
(t968),
Nostrand.
96]-1023. Prince-
INDEX (a)-equisingularity, associated Blowing Centre
70,71
valuation, up,
of
Local
13
Maximal
35
system
of
model,
conductor,
77,134,135
characteristic
twisted
Embedded
equiresolution,
89
-polygon,
89
plane
basis,
10
system,
53
curve,
,62
9~ 69
(loc. par.
proximate
point,
repr.),17
44,84
-E.s.2.
, 158
Puiseux's
theorem,
-E.s.3.
, t61
Quadratic
transformation
-of
divisor, formal
points,
Generic Genus,
q.t.
,37
quadratic
projection,
147
expansion 55
-for Infinitely -of intersection inversion
a basis, twisted near the
57
-total
origin
irreducible
40 of
C,41
multiplicity,62 73
-generalized, curve,
,
, 38 curve~
Satellite,
43,84,88
saturation, of
values,
twisted
superficial
cone,
the
143,161
24
9,38
knot, of
128,
curves,
element,
t oroidal Values
38,69
1 53
-for
Tangent
, 31
,31 ,38
Reducible
semigroup
curves,65
point,
formula,
transform
-strict
Hamburger-Noether
in
(formal), 36
76,77
-
] 08
36
43,84,94
plane
42
, 3
primitive
E.s.l.,t53,156
exceptional
Free
2
34, 59,72,158
equisingular
point,
-N-dimensional
dimension,
24
105
-diagram,
Parametric
3
a curve,
coefficients,
N-space
curves,153
curve,
e mbeddin9
78
118,122.
of
normalized
exponents,
3
an inf.near
Newton
135,143
-for
a curve,
contact,
-at
invariants, 71
complex
of
multiplicity
a parametrization,17
complete
ring
73 maximal
contact,1
113 2
] 8, 122
Weierstrass
Prepation
Theorem,
1
SYMBOLS
, ~ ;~ IR
+ +
;~ +
,
,
integer
non
negative
integers
,
non
negative
reals
,
infinite
I .....
k((t))
XN) )
, power
embedding
Spec
,
spectrum
Proj
,
projective
,
"~0
blowing
,
,
of
,
graded , the
series field
dimension
scheme up
,
ring
root
of
,
intersection
(re,n)
,
greatest
, ,
ideal
multiplicity
(C,D)
dim
an
dimension
length
multiplicity common
complex
non
, order
gr
I
power
series
,
BI
real
sequences
Emb
e
,
,
+
k((X
{:
divisor
negative ring
numbers
integers