Introduction
The moduli problem fo r algebrai c curve s of given genus g is a classical proble m
that goe s bac k t o Riemann . Thi s problem , o f a globa l nature , ha s alread y bee n
solved. On e know s that th e modul i spac e i s an ope n subse t o f an algebrai c variet y
of dimensio n 3g 3 (i f g 1). Th e compactification s o f thi s variet y hav e als o
been studied . Th e mos t recen t articl e o n thi s subjec t i s that o f P . Delign e an d D .
Mumford (Publ . Math , d e TI.H.E.S., no. 36) .
In thi s course , w e stud y th e analogou s loca l proble m o f th e modul i spac e o f
branches o f th e same equisingularity class, wher e analytical equivalence replace s
birational equivalenc e whic h i s use d i n th e globa l problem . Ver y littl e ha s bee n
written o n this problem . W e cite tw o articles :
(1) S . Ebey, The classification of singular points of algebraic curves, Trans , o f
the AM S (1965).
(2) K . Wolfmardt, Variation of complex structure in a point, Amer . J . of Math.
(1968).
The proble m o f th e complet e descriptio n o f th e modul i spac e M o f a give n
equisingularity clas s i s entirel y open , an d th e fe w example s o f Chapte r V sho w
that M ha s a structur e tha t i s too comple x t o hop e fo r a complete solutio n t o th e
problem.
The somewha t mor e restrictive questio n o f the determinatio n o f the dimensio n
of th e "generi c component " o f M i s no t solved . W e giv e howeve r som e partia l
results fo r th e characteristi c sequenc e (n ; ra) where n , ra are relativel y prime. 1
Chapter V I is dedicated entirel y t o thi s particula r problem . W e determine th e
dimension q of the "generi c component " whe n ra = n + 1. We have als o been abl e
to find q in the cas e ra = 1 (mo d n ) bu t w e give the formul a withou t proof .
In Chapte r V we give a complete descriptio n o f the modul i spac e fo r th e char -
acteristic sequenc e (n ; ra) G {(2; ra), (3; ra), (4; 5), (5; 6), (6; 7)}. I n th e cas e (6 ; 7),
the mos t interesting , on e observe s severa l specia l feature s tha t th e modul i spac e
can exhibit .
In Chapte r I V w e sho w tha t M i s quasi-compact ( M i s alway s non-separabl e
unless it is a single point) only if the characteristic sequence is (n; ra) (ra, n relativel y
prime) o r (4 ; 6,2s - h 1).
In Chapters I, II, III, we treat th e structure o f a singular branch and the princi -
pal numerica l invariant s o f an equisingularit y clas s (th e exponen t o f the conducto r
c, th e characteristi c o f th e class , th e semigrou p V o f positiv e integer s associate d
to th e class , etc.) . Th e majorit y o f thes e notion s ar e classi c an d wel l known , bu t
it wa s essentia l fo r us , an d useful—w e believe—fo r th e younge r reade r t o includ e
them togethe r here .
1
On thi s subject , se e th e articl e b y Ch . Delorm e i n th e Group e d'etud e de s singularite s
1974-75. Publ . d u Dept . d e Math. , Universit e d e Paris-Su d Orsay .
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