Chapter I
Preliminaries
1. Equisingularit y Clas s o f a Branch . Modul i Spac e
DEFINITION
1.1 . Le t / b e a n elemen t o f C[[X , Y]], th e rin g o f forma l powe r
series i n tw o variables wit h comple x coefficients . I t will always b e assume d tha t /
satifies th e followin g tw o properties :
(a) / define s a convergent powe r serie s in a neighborhoo d o f the origi n i n C 2;
(b) / i s irreducible i n C[[X, Y]].
The equatio n f(X, Y ) = 0 the n define s a comple x analytic branch ( a ger m o f a n
irreducible analyti c curve ) C.
These branche s wil l play th e rol e of global curve s a s considered b y Riemann .
We shoul d no w defin e th e topological type o f a branch . On e ca n immediatel y
remark tha t an y branc h C i s homeomorphi c t o (C , 0) (ger m a t th e origi n o f th e
complex line) , i f on e endow s C i n a neighborhoo d o f th e origi n wit h th e induce d
topology fro m C 2. Thi s homeomorphis m ca n b e obtaine d b y parametrizin g th e
branch; fo r example , fo r th e cus p (y
2
= x
3)
on e ha s a homeomorphis m give n b y
t-(*V
3
).
By regarding such homeomorphisms, however , one does not obtain a reasonable
definition o f topologica l type . (On e want s th e topologica l typ e o f th e cus p t o b e
different fro m tha t o f a smoot h curve) . I t i s therefor e necessar y t o introduc e a
stricter equivalenc e relation .
DEFINITION
1.2. Tw o branches C an d D hav e the same topological type, (on e
will write C = D)
1
i f and onl y if C an d D ar e topologically equivalen t a s embedde d
surfaces i n C
2
(o r R
4),
tha t is , if there exist s a homeomorphis m
T : U - U',
U an d U
f
ope n neighborhood s o f the origi n i n C
2,
suc h that :
(a) C i s defined i n U, D i s defined i n U' (thi s has a meaning because a branc h
is defined b y a series tha t converge s i n a neighborhood o f the origin) ;
(b) T(CnU) = Dnu f.
One denotes by L(C) th e se t o f branches D suc h that D = C. I f D an d C hav e
the sam e topologica l type , on e als o say s tha t th e tw o branche s ar e equisingular :
L(C) is called the equisingularity class of C, an d i t i s in thi s se t tha t w e will no w
place ourselves .
As wit h th e Rieman n problem , w e introduc e i n L(C) a n equivalenc e relatio n
(finer tha n th e first), whic h i s the relatio n o f analytic isomorphism .
l
http://dx.doi.org/10.1090/ulect/039/01
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