Algebroid Curves in Positive Characteristics

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.

G. , " C u r v a s

la

J . Math.

de

AngermiJller,

Real

Academia

G. , "Die

Madrid

Atiyah-Mac

5.

Bennet,

B.

M. , " O n

Bourbaki,

7.

Brauner,

N. , K. ,

"AIg~bre " ZuP

Co.

91

Ver~nderlichen'.

8.

Burau,

W. ,

"

Endler,

O. ,

10. E n r i q u e s ,

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F[sicas

153

irreduziblen

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267-282.

algebra".

Adison

(1969). a local

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25-87. Hermann.

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Funktionen Math.

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Zweier

Sen.

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Hamburg,

6

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Theory".

~ "Teoria

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Schlauchknoten".

Abh.

Math.

125-133.

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W. ,

der

Hamburg,

delle

I 1 . Fulton,

ebenen

functions

Abl.

Kennzeichnung

New

Exactas,

1-36.

Z.

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Memoria

1-54.

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einer

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Geometrie

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cuerpos

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characteristic

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to c o m m u t a t i v e

Public. the

Ann. 6.

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Donald,"Introduction Wesley

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de

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

(1967)

Wertehalbgruppe

algebroiden 4.

89

caracter~stica

y Naturales. 3.

invariance

algebraicas

cerrados de

and

geometrica

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delle

vol .2,

equazioni

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e

quarto.

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"Algebraic

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Benjamin

Inc.

New

York

( 1968). 12.

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Rodeja,

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"Desarrollo la

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13.

Hironaka,

H. , "Introduction

Instituto L@

D~'ng T r a n g ,

Lejeune,

Jorge

"Noeuds Grenoble

]5.

Ivl. , " S u r

Milnor,

18.

Moh,

T.T.

,

"On

C. ,

Van

der

Waerden,

75-91 de

21.

Vicente,

J.L.,

22.

Walker,

23.

Wail,

R., H.

S.,

2/4.

Zariski

,

O. ,

Zariski

, O. ,

local

algebroid Inst.

Ann. Press.(1968).

plane

curves

Math.

Sinica

de

Madrid

yon

de

variedades

alge-

caracter[stica

arbitra-

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die

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de

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Berlin

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alabeadas".

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Inc.

Van

New

Nostrand.

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.

York

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771-790.

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25.

un

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Universit@

.

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23.

planes.

hypersurfaces",

of

p".

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Verlag

Tesis.

97

singularidades

Tesis.

trie".

Fourier,

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

61 .

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B.L.,

(197/4).

inst.

Th~se.

complex

characteristic

ria'.

Ann.

de c o u r b e s

of

characteristic

"Resotuci6n

Madrid

117-126.

Math.

Stud.

broides

20.

J.

points

(1973), Romo,

Matem~tica.del

saturation

Math.

for

19.

de

CSlC,

near

(1973).

Am.

"Singular of

Memorias

de N e w t o n " .

Paris

J. , " A b s o l u t e

J.,

infinitely

Juan,

(1972),

rings". 17.

of

l'equivalence

VII, Lipmann,

theory

Algebriques".

Coefficients

16.

the

points".

singular

14.

to

surfaces".

Springer-Verlang-Berlin.

(1972). to

C. I .M.E.Varenna.

the

problem Septembre

of

equisingularity". (1969).

1

166

26.

Zariski,

O. , "Le

probleme

planes".

27.

Zariski,

O.,

B.

Theory

J.

Math. Zariski,

in

les

au C e n t r e Paris

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