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.

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,



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~

~

= 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

~i E m b ( F l )

k ((_X)) /

17 ~

, Xi),Xi+

M-primary.

I .I . 8 . N

~ 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

~

x +m

m i

ring

'

m

Then

given

([[])

--

2

I 4i(N

to

be

10

( ix.}

t-

being

~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 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

-.

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 ~
-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~


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

} ] ..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

~

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

~

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

~

I


1 ,

0

we

have

another

74

for be

a suitable

t'.

defined

for

Characteristic

(2)

a)

If

in

the

n < m
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


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

~(

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.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 ] .

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,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




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

~


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
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
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
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~< ",~ ~

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 -. 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

.... 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..


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 .< ~) . . . . .

( 60 . . . . .

13) = e?

> ( ~'0 . . . . .

,

0
/ 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 ~ 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~
~ 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~
~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 ~
/ 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 ~~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

..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 ~ 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~~ 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

~

( 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 -

{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

~

~ j-
.. - 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.

[~
~ 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

(1977),

267-282.

algebra".

Adison

(1969). a local

ring".

25-87. Hermann.

Paris

Funktionen Math.

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Zweier

Sen.

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Hamburg,

6

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~ "Teoria

funzioni

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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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cuerpos

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characteristic

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

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Ann. 6.

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sobre

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,

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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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1 a Reuni6n

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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-

(1976). in

J.

die

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Springer.

de

curvas

Berlin

Geome(t939).

atgebroides

alabeadas".

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fractions"

Inc.

Van

New

Nostrand.

York New

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.

York

"Contribution

Paris

771-790.

Univ.

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"Algebraic New

25.

un

Madrid

(1967)

Universit@

.

"Singularidades

"Continued

23.

planes.

hypersurfaces",

of

p".

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"Algebraic

algebroides

(1975),

Princeton,

pairs

Verlag

Tesis.

97

singularidades

Tesis.

trie".

Fourier,

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

61 .

sobre

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

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branches de Math.

(1973).

Math.

and

de

Appendice

(11 I )

saturated

93 (1971),

local

5?3-648.

872-964.

(I 1 I)

Am.

(1 I ) J.

415-502.

equisingular'ity".

"Commutative (1958).

saturation

93 ( 1 9 7 1 ) ,

507-535.

972-1006.

ton

J.

97 (1975),

O. , "Studies

Zariski-Samuel,

of

Am. Math.

(1965),

29.

pour

Teissier.

rings".(])

28.

donn~

Polytechnique.

"General

Am.

modules

Cours

I'Ecole de

des

(I I) Am.

(I)

Am. J.

AI9ebra".

J.

Am.

Mat!q.

Math. Van

90

J.

Math.

87

87 ( ] 9 6 5 ) ,

(t968),

Nostrand.

96]-1023. Prince-

INDEX (a)-equisingularity, associated Blowing Centre

70,71

valuation, up,

of

Local

13

Maximal

35

system

of

model,

conductor,

77,134,135

characteristic

twisted

Embedded

equiresolution,

89

-polygon,

89

plane

basis,

10

system,

53

curve,

,62

9~ 69

(loc. par.

proximate

point,

repr.),17

44,84

-E.s.2.

, 158

Puiseux's

theorem,

-E.s.3.

, t61

Quadratic

transformation

-of

divisor, formal

points,

Generic Genus,

q.t.

,37

quadratic

projection,

147

expansion 55

-for Infinitely -of intersection inversion

a basis, twisted near the

57

-total

origin

irreducible

40 of

C,41

multiplicity,62 73

-generalized, curve,

,

, 38 curve~

Satellite,

43,84,88

saturation, of

values,

twisted

superficial

cone,

the

143,161

24

9,38

knot, of

128,

curves,

element,

t oroidal Values

38,69

1 53

-for

Tangent

, 31

,31 ,38

Reducible

semigroup

curves,65

point,

formula,

transform

-strict

Hamburger-Noether

in

(formal), 36

76,77

-

] 08

36

43,84,94

plane

42

, 3

primitive

E.s.l.,t53,156

exceptional

Free

2

34, 59,72,158

equisingular

point,

-N-dimensional

dimension,

24

105

-diagram,

Parametric

3

a curve,

coefficients,

N-space

curves,153

curve,

e mbeddin9

78

118,122.

of

normalized

exponents,

3

an inf.near

Newton

135,143

-for

a curve,

contact,

-at

invariants, 71

complex

of

multiplicity

a parametrization,17

complete

ring

73 maximal

contact,1

113 2

] 8, 122

Weierstrass

Prepation

Theorem,

1

SYMBOLS

, ~ ;~ IR

+ +

;~ +

,

,

integer

non

negative

integers

,

non

negative

reals

,

infinite

I .....

k((t))

XN) )

, power

embedding

Spec

,

spectrum

Proj

,

projective

,

"~0

blowing

,

,

of

,

graded , the

series field

dimension

scheme up

,

ring

root

of

,

intersection

(re,n)

,

greatest

, ,

ideal

multiplicity

(C,D)

dim

an

dimension

length

multiplicity common

complex

non

, order

gr

I

power

series

,

BI

real

sequences

Emb

e

,

,

+

k((X

{:

divisor

negative ring

numbers

integers