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Arcs and wedges on sandwiched surface singularities
C. R. Acad. CkomCtrie
Sci. Paris, t. 326, algkbrique/A/gebraic
Abstract.
Version
S&ie I, p. 207-212, Geometry
1998
.A wedge on a surfacc singularit! (.S’. 1’) is ;I l’ornial parametrization series in two variables. locully at I’. A gencricity condition, which enough to guarantee that a wedge lifts to the minimal desingularization is proved to be so if the singularity is wdwichecl.
frangaise
of .S by pww ib expeztccl to be of the wrf’acc.
abrkgke
on entend le voisinage formel 3 d’un point 5ingulier sur une surface obtenue en Cclatant un faisceau d’idkal complet I dont le lieu des zCros e\t un point fermC 0 d’une surface algkbrique non singuliitre dCtinie WI. un corps algkbriquement cloa A.. Si I :- I(, est simple. on dit que la singularit est primiti~v. On appelle courbe (resp. (vin ) un X,-morphismc local de l’anneau 0; dans I‘anneau des skier formelles 2 une variable (resp. deux variables) 5 coefficients dans X,. n’ayant pas I’ideal maximal pour noyau. S’il existe un morphisme local if,, ICI que if 1: ir.,, o 4. on dit que 1~ cwi~ + O.STcrnrr& sur itr courhe It,. On dit que Ir ejt cc,~~I~~~~LI/PI si \a transformCe stricte sur la d~sin~ulariaa~ion minimale .$ de s^ est lisse et transverse 2 la cow-he exceptionnellc de .q en LIII point g6n&al du cellc-ci. Par
LsingularitP
sandwich
Note prksentbe par Jean-Pierre
dr
.YLU~~~~Y~.
DFMAILI.~.
207
M.
Lejeune-jalabert,
Le
rksultat
Pour d’un
A. Reguera
principal
une
singularitk
4Cment
/I I := c.
!J
identitk
, jr,.
I :;
(Ji,,.
Tout
. 2,.
d’abord.
formel
2,,
S,
I’Cventail
dans
sur
fine
.I’,.
,? cst
toute
c6nes
stricte
ZX;_2x1
complkte
de
la
dCsingularisation
DEF[NITION. - On o;,~
=
(;,.
de i2,).
aswcionx
clans
que
(I),
r&uItc
systkmc
une
I <
combinatoire
de ?,I,,.
g.
i
uric
cl
On
oblige
c’or.
tous
apparticnt pas le
le5
commence
de
cylindre
dc
si.
il exi\te
base
C’cst
leurs
vecteur On
208
de
;ILI
des
on
suite
1. no .Y,
; elle =
0 et
et on
pose
plongc
le voisinage
Y(,+,]j
en
25; r donnt’es
et
27.r
X:,, danh
r^, associt-e
envoyanl
respectivement
qui
par
est la subdivision jr,,.
au ctine
On
la
dkmontre
(;,,.
quc
f,,. (j,,, ,)
’ ‘.
r6gulkre
minimale striL?e
plongCe vdrifie
en
pour tore
que
de
fait
leur5
que
I$,
ces
XI
et la
transformke
e\.t une
intersection
de Newton.
vari&&
chacun
de
TX:,
polykdrex
sur
2~~
,. le
et on
rencontrent.
en
d’oii
une
dc
iw,,,(l?,,.
.
+ soul
dan\
grand cvii
qui cas
d‘une
coi’ncidc, singularitti
Jes ;ivt~c
cleux
et
‘I,’
A appartenir
in ff
XI
que
;
Ic : oji
:
eat
la
au
vecteur
deux
tels
que
1 <
r’
I 5 g +
tore
un
soit I. et
squclette.
de E:. II s’agit
(rap.
On
cl’une
au
interprCtation minimale
minimalc
d’un
primitive
: 9 &mt
centr6
minimaux
CJ, de4 fl t 5.
(T E 5 qui
~;, uric
singularitf? g&kWeurs
cntier c ctim*\
tel
U
que
(;,
q ait On
un
vecteur
en d6duit
sur
I,,
caractkristique quc
tous
0 I$
Le premier clans
les
vectours
I.:! mT,
, ,.
j =
1. 2. et on
conclut
que
p a un
fkli
primitive
1,
dCmontre
appartient
:I la dtkinguiarisation
rkgulicre
tlej)
Old, h(.r,)I,<,‘e -. ‘, valuation II-adique.
caractCnstique
se rekve
la subdivision
lex systt‘mcs
1 !. alors
suit
/I!]
son
dimension
pour
le plus
I’u~I
:=~ (A, ~,. ?j.
Sq rencontrant
X.[[o. si
aussi de
lc th~%r?rne 4 / eht
wulcment
7 danh
c611e
cbnes
(T,,]
comme
caractCristiques
c81w
par
sur
“+’ L,:,,, dans
et
caractr*rihtiques
que
coin)
un
(I p ,,) dans
4
des
des
faw-.
d&nontre
un
E form6
irrkductibk
cles
engendk
caractCristique se ramtine
cc faire,
la transform&
du
, caract&istiques
le diviscur
telle
q +
I’axt~
dkfinissant
la rkunion
tresp.
2
161. ()n
montrcr
h montrer de
i ::i
sur
approchi-es
par
I une
, ct l’kventail
AZ
On
orhitc\
~eckwrs
hur
8.3
au semi-groupe
consiste
Ies d’uu
vecteurs
par
I) 5
clt;sin~tllarisation
.? l’dventail
cwrbe 11 c$t
g&&ale
de ,? si. et seulement contienne
. .;i,),
la subdivision
dti~PnCr6ec
WY
clans
i 5
g&A-atcur du
-
minimnle.
-k\;
. ../‘,,+l
Spc(.X,[[,Y,,.
Zz,,
rencontre
non
.s~~ut~/c~r
oil
courbe
minimal
: 23,:,>
es1 une
ct
fl?~
cnsuite
(,,,(,-i.~.,i),~~,~~!,~~~. triangulaire
Ir est
:=
de I’orbite
Z\;,
(J:,,.
Pour
2,)
toriques
7h,,
racines
:
1,
dan\
formel
dc~ I/ donke
unitaire
fnriyw.
?3 T induihant
~~ec’~~rurl.s) c,~irLic.f~;ri.sticlU(I(.I I (I I, (I tsp. (Y;,,, = L’in6galitC
!/ -
essentiel
de [S]
explicitly.
appelle
h I’image
Nous
( ji,
XLIr
pIon&
vecteur
de iwT>f,’ par Rif,$+ .~ des T/ thnctions
\‘~ de
cn
de
i y
varitit&
au voisinage -. tic X,, pi-
_-II::,
- 27~. , qui
~qul(:i,,.
81, le
la problitmatique
la valuation
Soit
dc Pirwton
I’orhitc
transversalitC
adhkent
les
r~guli~re dont
0 2
XL:,, cst S;I dCsinFul~u-isaticln ,F U,i dCfinie per !J lonctions
dans
dCduit
\;J-
pnr
singuliere.
ii011
consickre
strictc
), ( , =
par
un usage de
autcurs).
1111 rrl\,i,n/l/lc/rlc~rll
6lkmentaire
subdi\:ision CT F
morphisme
des
On
j.
fait
gknkratrice les
I/(.I.;
irl,.7,
wrfaue
isomorphe
la transform&
=
,? clr/rls
la
suite
(111 ddsigne
de Yi, et des &entails
la singularit Pour
; if
motivC
combinatoire
d’une
I/,
irzsttrllc
suivant
I 161 (suivant
,q $- I
X,1 subdivision
moins
des
de
i 5
I’Cnonz6
lit preuve
,I,?,
0
est
I 1 I] ou
t 1 ). soit
on
de
la variable
sur
dc
de I’dclatement Its
h, =
Note
primitive.
gtnCral
exceptionnel vkrife
de cette
en considkmt
les
f’acteurs
simplcs
de
1.
wul
1. About By
sandwiched
a .sand~~ic/~etl
a surface the
the
surface
now
minimal
denotes
and
chain the
S
X,,
defined
is said
10 bc
/)~.i/~rti~~.
end
of
wedges
tormal
on sandwiched
p1bintz
from
minimal
.x,,
X,,
anti
I
: = &,
3, of
by
0
for
thl.
I
.I,,
of
.i’.
mot-phiam
to bc
the
1~s
.
.
.
.
.
.
.
.
-I-
such
that
to
labelled
the
is obtained
the
curve
:’ , :=
f/(.1.,).
by
strict
the
;;.(..(I.
(ii,,.
exists
a unique
system
~I,,,~i.
Moreover.
=
where
y =
in
2. A toric Let
2(,
0
in
of S~(Y, 1
k[[,‘il,.
(.,.-,AY,;‘l
R>o(??,,. I (
.I(’
(-yo.....?,).
a proof
I(<‘) and
by
of’
of
CUI~VC
not
-
we
C’,
J’ll
&,r'l,
characteristic
nay
‘I,,
I‘(,
the the
.
-
ii
c
hc
let
where
tt;ttural
of
to
X(C~I
tnorphism
of‘
Vornt
.
>
0 and
/!,,I,.
,',
any
-
E
A-..
the
161 Remark
choose by
other
and
cle~ingularization 01’ the
I’, E II’. .I’,
in
(X,,
I) I is
exceptional
i,,ml ,
.I’, s;;tticfy
I.,,(
b,,
fOl lowing
“‘.l’,‘.
i\ either
f',,..
the
that
1.
fatnil~
: I/( I).
WC
set
/ 5 j < !I+
’ 11 ,.
I 5 j
1. i
1.
itlcnliliea:
II/<
;I unit
.g + transversal The
CL:I-1~'.
:ind
function
j = 0.
I /. I /‘I ,. I ‘2 i 5. g $ I, l.L)t.
c / -,
c
minimal valuation
ot‘ r/i /,’ \ 0).
I,, :=
that
I
the
JJ,,. 0 5 ,j <’ i, such
;ts~~me
1,
of divisorial
ddincd
\yjtent
inlqer\
‘.I’,
graph the
I (‘. We
C ',
intcrkect
gzneritting
nc~nnc~.ati\~e
dual I/ be
I’ “mi ’ in
the
does
tnintmal
the L.~I
ji, ). 0 5 i 5 q i- I 1 /ii, ~= II. :tntl
.I’, ., , =
for
and
I’ If’ I
I l.J
I . :tnti I$!‘~ ‘.
represented
I”
q, is the
. f,‘+
deleting
on by
i 5
f,“.
curve
c’, :=
II,,,
~71 ,.,)
;I
.
VI
by
transform
represented 0 5
I‘(,
there and
assume
II. I. 14).
on of
.
ends
t’rom
defined the
Then,
P 0
-I.
I
K(&)
tnay
ik invurtihlc.
.
II2
field
II’ :=
(‘.
.
I
have
point point
Coral.
graph the
.
we
in
in
dual
I,,
i,,. We
(171.
I OX!,,,
point\
The
=C(c’ j --
l
(S. I’)
ticld
simple that
;I singular
;tt a closed
if’ ncccssary
\uch
blowing-up
-I--
where
singularities
yx t ’
L’O -
of
surface
of
supported closed
Z
on
desin~Ltlari/.alioll
curves
neighborhood
shcuf’
an algchraically
Section
near
obtained
the ide;Ll
over
changing
the
exceptional
mc;1n
S’. by
infinitely
is the
we
;t complete
on
until of
surface
2
irreducible
rirzgrkrr-if>
point
and
singularities
blowing-up
singular singularity
Assume
1’ : X(C1
by
algebraic only
simple, the
.SU/;$KP
S obtained
nonsingular be
surface
Arcs
f/
(I)
In 1: (II’ (1. ke
[ I ] sec.
9.
X. IO
environment = &
AZ+’ is
endowed
. -\-,,+ ” s;,. . Ti,,
with
a nonsingular l]]
of 0
1
i < ~1, .PPP [3])
and
in 2,) rf.
We
Newton their
Its
natural
complete
f’an least
cietined
by
consider ?:\tine
‘I’
x~
/,,*!‘-I1
intersection
surt’ke
the the
(least subdivision
function\ Itn
fine
l;.
action. in F; =
The the .I-,+,
elementary
I‘ormal
f’ormal --- .\-:I’
+ (‘, .\‘i;
subdivision
subdivision
of
\; I (.sw
121. chap.
the
”
(11. 1
Newton I for
neighborhood
I?,
neighborhood
t'an~
the
! : \'
$I ‘Y::‘, :== Rx,:,’ of
the
= ’ + by F’,.
I- I col-respondence
209
M.
Lejeune-jalabert,
3. Arcs
A. Reguera
and wedges
on a primitive
surface
singularity
.,\ Let
<1 be
n : .I We
--
will
the
only
consider
Sketdt of’ptnof: implies w,.
III E (F,,.
210
.a; I <
For-
-
of
that
order
3. --
polyhedron by
completion
U { $- x ) such
Ptavst,rto\
This
I>-adic
rn,<
Z>(,
ctt7)’
that
Ihe
i 5
q.
functions or&t.
Because
I<; in
of
the
is
not
12. By
fki, On
luo
o&t-
-CX j,,<-
inquality. intersect5
strict the
cc)ndilions
transfornt other
hand. ch;traclcri/e
by
by
2-,,
proposition the
it7
supportin a face
I,,
:i.
*t’
111.I’ + !/
tnean
a function
> tliitt(vj.1.).
vi(;)
i y: y -t- I.
lirs
;<, ,,+,
along of
and
. 0 5
the A;
on
,firt7cTiott
) + I/(!/)
1.1 I’, ) f
(I ,, : q = i t/( .I’, j
the* triangular of‘ Ihc
3ti = //(.I
that
/:~71~ttof7.
empty. The5e
~3,
yuch
directton
intersection
-‘,. ?,, .~ t )“.
of
I/’ I,.’ I == 0. vi.!.!/
skeleton
of’ ,?.
117~~ .sk/c~/otr
g hyperplanc i’,
of
dinicn\iort
-
Z,, 5.
the
Newton
at least
\h ith
I (i ). iit,. of‘
of
(I,,}
the
orbit I’
enc. given
0 Cot- any
1.
Arcs
and
wedges
on
sandwiched
surface
singularities
Dtt:tNrrtoN. - An a/~ (resp. a ~vrc/,~:r) on ,5 ilr ;1 X,-local morphism from .i to the lormal powet series ring in one (resp. two) variable(s) with coefficients in 1,. whose herncl ih not tt~, I.. The wedge i\ c~/~feru~/ at the arc Ir. if /i factor\ through 4. we will restrict our attention to arci and wedgea whtch do not tanish on .I’,. 0 1:~ I I. ‘J + I. Given an arc II (rap. a wedge 3) on ,F. wc associate to any dtscrete \~aluation of rank ant: f . tlc~nnegative on X:[[t]] (resp. X:[[U. ,I!]]). t h e order function 11 := r’ o I, (t‘e\p. I’ c $1 : 1 ~- Z,,,, ii {-i- x ). By convention /J in /,.[[I!. f.1’. WC denote by v( ,f) = +x it’ and only if ,/’ pi K(*r /I (rc\p. 9). Fork any irreducible v,, the p-adic order function. PROPOSITION ,qi\~fvf
ig
-- DEFI~II‘ION
II
017 tlic-.
tiot7.sitigdrrr
tt~itiitttul
poitit
flitr7f~t7siot7al
of
f’otif’
/x~/otfg.s
to
;1I’c
ihr
n.it/i
the
ff
hr
oirtsiflr
//
rhf)
t/if>
:
i
.sAelrtoti
Ll/JO\‘f’
\w/i
E.
f/f~.sit7,~7ri~7ri,Ltti,)II
tliot
/Jtu/“P/?ic’s
of
0
E .s/rcV7
t
t/lot
op.,,
if’ftt7tl :=
(I/,
o/ tlir
(trc’
/Jr
fJ/
.hfiifi
(i rw/,cy if’thcw
I., , )j,,.
,
Thr
.?.
/t~frris~~f~t~~~~//~ ( ‘,,
0 -C _
f/w
; c_
IO
tlic
fl
/if,\
i’i(rIafJ
if’ t/rf,w
:=
(1 ml,
Z’l
It
\I 11. lit
fhr
on
E
c.i-ists
fit
ft
II 1 .I’, ) j,,.:,< t/If’
ft
tn~r~ (,/+ I
.sf~ylle/.
f1/7
0115.
(171 *S. 777~~ tJtor/J/ti.stti
,
c~.w~/Jfiotttrl
tk!, i:
of‘
2.
f’.\-t.vfs ,,’
lrft17.~,~ot-tt1
oti/\~
~w~iot~
~CllCtd
hfl
.sttkt
CJ. if’rrtrd
\fwii,y,r~trp
oi Propo\ilion
cur/\.
017
1,iirtrrrr./f,t-i\tif,
(i.c.
rl,ill
~oroll~u-y
f!f’.?.
I JV U/I
,2 ititrt-\f,fT.\
rrf’n
5. - L.CJlr> : .-r -- XJU. f,] /Jf’
PtWrosrrtoh
A.[[/]]
tt~rrti~,f~Jrrt7
.s>.\7f’t77
in an immediate
-01
.cit’ii.I
grt7c~t~trtit7g
f~c/7li\Yrlct7t
tJ7it7itt7ttl Jo/t/e
it7
-
rlr.\it7~7~/ut-i~frtiot7
F,‘,
tt7itfirt7trl
PIW?~. - This
3.
ft fwt7f it7
T . ,/or
SIWC’
T it7
tl7c
fit7.v
it~rrflirf~ihle
k[\l,.
I,]]
ttiitiitt7fd
-’
2
t?qtr/rrt-
lift.\
to
l/70
,s7fhdii~i.sioti
/I.
Proof: -- The morphistn SIHY /,,[[I(. v]] mmm+ .S lifts to tllc minimal dcsingLtlari~atit,n oi‘ ,G il‘and only if is ;t regular suhdt\iaic~n of !Zr it lifts to 2!:. where 1 ;I\ in Proposition 2. Thts is also
Let
--
,fidlo\r~itig
thw
(i) 0, _ t t
fwtittrit7.s
(ii)
rttfd
of’ l/w
T/7(>
oti/y
A,
hf,
f~ottrlitiotis fit
tl7e
.stt7ft//f~.s/
it7
P
ccJt7tfrit7itrg
ft f.ltrtlurr.t~ri.\tif,
IY*(.IcJ~-
c!f’ 9,
Ot7p
o/‘
0,
i ,
holfls: /ccr.st
0170
f~/trct~frf~7rt~i.~tic.
“I,., ,, I 5 I,. 5: i. ,j f./7ftt.f/f,trt.istif,
rwt7f’
~‘fv’tors
:
~~rf~/~~r~ i?/
;
trrlcl
tlir\t
it//
\i)it/7
0 5
Iif>
it7
tl7fj
itt7ioti
I frtt,/
h,.
01
I L!.
orrt~idr
n,+
, z /if,
rJt7 R
,,,:t,
A, <
, t, =
Applying q to ( I ), we derive frotn the existence of o,, E IX”,,:,’ that the minimum ix achieved same side by all o,, f 0. i.e. either thL, cone CT, ! I 01 CT, ,, in \‘*-2’ , contains than all.
0.
on
the
211
M.
Lejeune-jalabert,
A. Reguera
I’
frj, also lies in CT, ‘, Therefore tklr is located with 0 5 i -<. y. Hecause for an! 11. c,, ,’ E (T,. on tl;e’ .Yll-a\,ls of a general XC’ in the union of (T, +, I and irk,,. I :;: X. < /. NOM the coordinate Ir on ,? such that 07~ E (T( .,,. I < I <_ ‘1 + I, oolncides with its multiplicitv ancl is bounded below ii., . with equality if‘ and only if 07, t Iw ,.,,b, i (61 Car. 8.4). So. by ‘77.1 . ‘7ilpI. and above by 71, p having one characteristic vector in CT,_ , .I and in view of Prop. 4. (17, bcIong\ to the mininlal generating system 01‘ (T,+ , , (I:! C: (<-,,, . Y,. 6,). Th e intclaection of this Ia\t Write 07, =: (tI + (~1 with 0 f C\ t iJ,~++l , d cone with the plane generated by ,T, , , is a cone T,.,. , and the pair (CT, ,~, ,7 : T, + , ) is isomorphic to the pair ((( 1.0). ((,,.J,, , - r/,3, I): (0 I 0). II,. - 77,,j, i)). Therefore cl’2 = 0 and thth ~)nly characteristic vector of 9 is (~7,. Suppose now that Condition (ii) 01‘ Prop. 6 holds with 0 5 i < !I. Here for any I), o+-,, E m, and we can write (~7, = n, + cv2 uilh (I, t IT,+I ?. ckl f (I. and o2 t (c,,J.. .;, 8. But (I? lies in the plane generated by h,.-, and c, t ,. Its coordinate on the -iv,,+, -axis may not bc ICI-~ unless 07, = 0 and as above. the only characteristic vector of F i\ it;.. In general. choose X0 and / as it1 171. II. Car. I. 1-l. and let lj be the distinct simple factors 01‘ 1. There exists i (may be not unique) ~LIC~I that p induce\ ;I wedge q, on the forn1aI neighborhood .<, of the image of I’ on the blowit+-up with center I,. centered at a general arc‘ ‘7,. We prove that the minima1 desingulariz~~tions oi‘ .‘i and .5”, art’ I~KxII~ isomorphic at the point\ through which II and /I., lifts respectively.
References
212
Sci. Paris, t. 326, algkbrique/A/gebraic
Abstract.
Version
S&ie I, p. 207-212, Geometry
1998
.A wedge on a surfacc singularit! (.S’. 1’) is ;I l’ornial parametrization series in two variables. locully at I’. A gencricity condition, which enough to guarantee that a wedge lifts to the minimal desingularization is proved to be so if the singularity is wdwichecl.
frangaise
of .S by pww ib expeztccl to be of the wrf’acc.
abrkgke
on entend le voisinage formel 3 d’un point 5ingulier sur une surface obtenue en Cclatant un faisceau d’idkal complet I dont le lieu des zCros e\t un point fermC 0 d’une surface algkbrique non singuliitre dCtinie WI. un corps algkbriquement cloa A.. Si I :- I(, est simple. on dit que la singularit est primiti~v. On appelle courbe (resp. (vin ) un X,-morphismc local de l’anneau 0; dans I‘anneau des skier formelles 2 une variable (resp. deux variables) 5 coefficients dans X,. n’ayant pas I’ideal maximal pour noyau. S’il existe un morphisme local if,, ICI que if 1: ir.,, o 4. on dit que 1~ cwi~ + O.STcrnrr& sur itr courhe It,. On dit que Ir ejt cc,~~I~~~~LI/PI si \a transformCe stricte sur la d~sin~ulariaa~ion minimale .$ de s^ est lisse et transverse 2 la cow-he exceptionnellc de .q en LIII point g6n&al du cellc-ci. Par
LsingularitP
sandwich
Note prksentbe par Jean-Pierre
dr
.YLU~~~~Y~.
DFMAILI.~.
207
M.
Lejeune-jalabert,
Le
rksultat
Pour d’un
A. Reguera
principal
une
singularitk
4Cment
/I I := c.
!J
identitk
, jr,.
I :;
(Ji,,.
Tout
. 2,.
d’abord.
formel
2,,
S,
I’Cventail
dans
sur
fine
.I’,.
,? cst
toute
c6nes
stricte
ZX;_2x1
complkte
de
la
dCsingularisation
DEF[NITION. - On o;,~
=
(;,.
de i2,).
aswcionx
clans
que
(I),
r&uItc
systkmc
une
I <
combinatoire
de ?,I,,.
g.
i
uric
cl
On
oblige
c’or.
tous
apparticnt pas le
le5
commence
de
cylindre
dc
si.
il exi\te
base
C’cst
leurs
vecteur On
208
de
;ILI
des
on
suite
1. no .Y,
; elle =
0 et
et on
pose
plongc
le voisinage
Y(,+,]j
en
25; r donnt’es
et
27.r
X:,, danh
r^, associt-e
envoyanl
respectivement
qui
par
est la subdivision jr,,.
au ctine
On
la
dkmontre
(;,,.
quc
f,,. (j,,, ,)
’ ‘.
r6gulkre
minimale striL?e
plongCe vdrifie
en
pour tore
que
de
fait
leur5
que
I$,
ces
XI
et la
transformke
e\.t une
intersection
de Newton.
vari&&
chacun
de
TX:,
polykdrex
sur
2~~
,. le
et on
rencontrent.
en
d’oii
une
dc
iw,,,(l?,,.
.
+ soul
dan\
grand cvii
qui cas
d‘une
coi’ncidc, singularitti
Jes ;ivt~c
cleux
et
‘I,’
A appartenir
in ff
XI
que
;
Ic : oji
:
eat
la
au
vecteur
deux
tels
que
1 <
r’
I 5 g +
tore
un
soit I. et
squclette.
de E:. II s’agit
(rap.
On
cl’une
au
interprCtation minimale
minimalc
d’un
primitive
: 9 &mt
centr6
minimaux
CJ, de4 fl t 5.
(T E 5 qui
~;, uric
singularitf? g&kWeurs
cntier c ctim*\
tel
U
que
(;,
q ait On
un
vecteur
en d6duit
sur
I,,
caractkristique quc
tous
0 I$
Le premier clans
les
vectours
I.:! mT,
, ,.
j =
1. 2. et on
conclut
que
p a un
fkli
primitive
1,
dCmontre
appartient
:I la dtkinguiarisation
rkgulicre
tlej)
Old, h(.r,)I,<,‘e -. ‘, valuation II-adique.
caractCnstique
se rekve
la subdivision
lex systt‘mcs
1 !. alors
suit
/I!]
son
dimension
pour
le plus
I’u~I
:=~ (A, ~,. ?j.
Sq rencontrant
X.[[o. si
aussi de
lc th~%r?rne 4 / eht
wulcment
7 danh
c611e
cbnes
(T,,]
comme
caractCristiques
c81w
par
sur
“+’ L,:,,, dans
et
caractr*rihtiques
que
coin)
un
(I p ,,) dans
4
des
des
faw-.
d&nontre
un
E form6
irrkductibk
cles
engendk
caractCristique se ramtine
cc faire,
la transform&
du
, caract&istiques
le diviscur
telle
q +
I’axt~
dkfinissant
la rkunion
tresp.
2
161. ()n
montrcr
h montrer de
i ::i
sur
approchi-es
par
I une
, ct l’kventail
AZ
On
orhitc\
~eckwrs
hur
8.3
au semi-groupe
consiste
Ies d’uu
vecteurs
par
I) 5
clt;sin~tllarisation
.? l’dventail
cwrbe 11 c$t
g&&ale
de ,? si. et seulement contienne
. .;i,),
la subdivision
dti~PnCr6ec
WY
clans
i 5
g&A-atcur du
-
minimnle.
-k\;
. ../‘,,+l
Spc(.X,[[,Y,,.
Zz,,
rencontre
non
.s~~ut~/c~r
oil
courbe
minimal
: 23,:,>
es1 une
ct
fl?~
cnsuite
(,,,(,-i.~.,i),~~,~~!,~~~. triangulaire
Ir est
:=
de I’orbite
Z\;,
(J:,,.
Pour
2,)
toriques
7h,,
racines
:
1,
dan\
formel
dc~ I/ donke
unitaire
fnriyw.
?3 T induihant
~~ec’~~rurl.s) c,~irLic.f~;ri.sticlU(I(.I I (I I, (I tsp. (Y;,,, = L’in6galitC
!/ -
essentiel
de [S]
explicitly.
appelle
h I’image
Nous
( ji,
XLIr
pIon&
vecteur
de iwT>f,’ par Rif,$+ .~ des T/ thnctions
\‘~ de
cn
de
i y
varitit&
au voisinage -. tic X,, pi-
_-II::,
- 27~. , qui
~qul(:i,,.
81, le
la problitmatique
la valuation
Soit
dc Pirwton
I’orhitc
transversalitC
adhkent
les
r~guli~re dont
0 2
XL:,, cst S;I dCsinFul~u-isaticln ,F U,i dCfinie per !J lonctions
dans
dCduit
\;J-
pnr
singuliere.
ii011
consickre
strictc
), ( , =
par
un usage de
autcurs).
1111 rrl\,i,n/l/lc/rlc~rll
6lkmentaire
subdi\:ision CT F
morphisme
des
On
j.
fait
gknkratrice les
I/(.I.;
irl,.7,
wrfaue
isomorphe
la transform&
=
,? clr/rls
la
suite
(111 ddsigne
de Yi, et des &entails
la singularit Pour
; if
motivC
combinatoire
d’une
I/,
irzsttrllc
suivant
I 161 (suivant
,q $- I
X,1 subdivision
moins
des
de
i 5
I’Cnonz6
lit preuve
,I,?,
0
est
I 1 I] ou
t 1 ). soit
on
de
la variable
sur
dc
de I’dclatement Its
h, =
Note
primitive.
gtnCral
exceptionnel vkrife
de cette
en considkmt
les
f’acteurs
simplcs
de
1.
wul
1. About By
sandwiched
a .sand~~ic/~etl
a surface the
the
surface
now
minimal
denotes
and
chain the
S
X,,
defined
is said
10 bc
/)~.i/~rti~~.
end
of
wedges
tormal
on sandwiched
p1bintz
from
minimal
.x,,
X,,
anti
I
: = &,
3, of
by
0
for
thl.
I
.I,,
of
.i’.
mot-phiam
to bc
the
1~s
.
.
.
.
.
.
.
.
-I-
such
that
to
labelled
the
is obtained
the
curve
:’ , :=
f/(.1.,).
by
strict
the
;;.(..(I.
(ii,,.
exists
a unique
system
~I,,,~i.
Moreover.
=
where
y =
in
2. A toric Let
2(,
0
in
of S~(Y, 1
k[[,‘il,.
(.,.-,AY,;‘l
R>o(??,,. I (
.I(’
(-yo.....?,).
a proof
I(<‘) and
by
of’
of
CUI~VC
not
-
we
C’,
J’ll
&,r'l,
characteristic
nay
‘I,,
I‘(,
the the
.
-
ii
c
hc
let
where
tt;ttural
of
to
X(C~I
tnorphism
of‘
Vornt
.
>
0 and
/!,,I,.
,',
any
-
E
A-..
the
161 Remark
choose by
other
and
cle~ingularization 01’ the
I’, E II’. .I’,
in
(X,,
I) I is
exceptional
i,,ml ,
.I’, s;;tticfy
I.,,(
b,,
fOl lowing
“‘.l’,‘.
i\ either
f',,..
the
that
1.
fatnil~
: I/( I).
WC
set
/ 5 j < !I+
’ 11 ,.
I 5 j
1. i
1.
itlcnliliea:
II/<
;I unit
.g + transversal The
CL:I-1~'.
:ind
function
j = 0.
I /. I /‘I ,. I ‘2 i 5. g $ I, l.L)t.
c / -,
c
minimal valuation
ot‘ r/i /,’ \ 0).
I,, :=
that
I
the
JJ,,. 0 5 ,j <’ i, such
;ts~~me
1,
of divisorial
ddincd
\yjtent
inlqer\
‘.I’,
graph the
I (‘. We
C ',
intcrkect
gzneritting
nc~nnc~.ati\~e
dual I/ be
I’ “mi ’ in
the
does
tnintmal
the L.~I
ji, ). 0 5 i 5 q i- I 1 /ii, ~= II. :tntl
.I’, ., , =
for
and
I’ If’ I
I l.J
I . :tnti I$!‘~ ‘.
represented
I”
q, is the
. f,‘+
deleting
on by
i 5
f,“.
curve
c’, :=
II,,,
~71 ,.,)
;I
.
VI
by
transform
represented 0 5
I‘(,
there and
assume
II. I. 14).
on of
.
ends
t’rom
defined the
Then,
P 0
-I.
I
K(&)
tnay
ik invurtihlc.
.
II2
field
II’ :=
(‘.
.
I
have
point point
Coral.
graph the
.
we
in
in
dual
I,,
i,,. We
(171.
I OX!,,,
point\
The
=C(c’ j --
l
(S. I’)
ticld
simple that
;I singular
;tt a closed
if’ ncccssary
\uch
blowing-up
-I--
where
singularities
yx t ’
L’O -
of
surface
of
supported closed
Z
on
desin~Ltlari/.alioll
curves
neighborhood
shcuf’
an algchraically
Section
near
obtained
the ide;Ll
over
changing
the
exceptional
mc;1n
S’. by
infinitely
is the
we
;t complete
on
until of
surface
2
irreducible
rirzgrkrr-if>
point
and
singularities
blowing-up
singular singularity
Assume
1’ : X(C1
by
algebraic only
simple, the
.SU/;$KP
S obtained
nonsingular be
surface
Arcs
f/
(I)
In 1: (II’ (1. ke
[ I ] sec.
9.
X. IO
environment = &
AZ+’ is
endowed
. -\-,,+ ” s;,. . Ti,,
with
a nonsingular l]]
of 0
1
i < ~1, .PPP [3])
and
in 2,) rf.
We
Newton their
Its
natural
complete
f’an least
cietined
by
consider ?:\tine
‘I’
x~
/,,*!‘-I1
intersection
surt’ke
the the
(least subdivision
function\ Itn
fine
l;.
action. in F; =
The the .I-,+,
elementary
I‘ormal
f’ormal --- .\-:I’
+ (‘, .\‘i;
subdivision
subdivision
of
\; I (.sw
121. chap.
the
”
(11. 1
Newton I for
neighborhood
I?,
neighborhood
t'an~
the
! : \'
$I ‘Y::‘, :== Rx,:,’ of
the
= ’ + by F’,.
I- I col-respondence
209
M.
Lejeune-jalabert,
3. Arcs
A. Reguera
and wedges
on a primitive
surface
singularity
.,\ Let
<1 be
n : .I We
--
will
the
only
consider
Sketdt of’ptnof: implies w,.
III E (F,,.
210
.a; I <
For-
-
of
that
order
3. --
polyhedron by
completion
U { $- x ) such
Ptavst,rto\
This
I>-adic
rn,<
Z>(,
ctt7)’
that
Ihe
i 5
q.
functions or&t.
Because
I<; in
of
the
is
not
12. By
fki, On
luo
o&t-
-CX j,,<-
inquality. intersect5
strict the
cc)ndilions
transfornt other
hand. ch;traclcri/e
by
by
2-,,
proposition the
it7
supportin a face
I,,
:i.
*t’
111.I’ + !/
tnean
a function
> tliitt(vj.1.).
vi(;)
i y: y -t- I.
lirs
;<, ,,+,
along of
and
. 0 5
the A;
on
,firt7cTiott
) + I/(!/)
1.1 I’, ) f
(I ,, : q = i t/( .I’, j
the* triangular of‘ Ihc
3ti = //(.I
that
/:~71~ttof7.
empty. The5e
~3,
yuch
directton
intersection
-‘,. ?,, .~ t )“.
of
I/’ I,.’ I == 0. vi.!.!/
skeleton
of’ ,?.
117~~ .sk/c~/otr
g hyperplanc i’,
of
dinicn\iort
-
Z,, 5.
the
Newton
at least
\h ith
I (i ). iit,. of‘
of
(I,,}
the
orbit I’
enc. given
0 Cot- any
1.
Arcs
and
wedges
on
sandwiched
surface
singularities
Dtt:tNrrtoN. - An a/~ (resp. a ~vrc/,~:r) on ,5 ilr ;1 X,-local morphism from .i to the lormal powet series ring in one (resp. two) variable(s) with coefficients in 1,. whose herncl ih not tt~, I.. The wedge i\ c~/~feru~/ at the arc Ir. if /i factor\ through 4. we will restrict our attention to arci and wedgea whtch do not tanish on .I’,. 0 1:~ I I. ‘J + I. Given an arc II (rap. a wedge 3) on ,F. wc associate to any dtscrete \~aluation of rank ant: f . tlc~nnegative on X:[[t]] (resp. X:[[U. ,I!]]). t h e order function 11 := r’ o I, (t‘e\p. I’ c $1 : 1 ~- Z,,,, ii {-i- x ). By convention /J in /,.[[I!. f.1’. WC denote by v( ,f) = +x it’ and only if ,/’ pi K(*r /I (rc\p. 9). Fork any irreducible v,, the p-adic order function. PROPOSITION ,qi\~fvf
ig
-- DEFI~II‘ION
II
017 tlic-.
tiot7.sitigdrrr
tt~itiitttul
poitit
flitr7f~t7siot7al
of
f’otif’
/x~/otfg.s
to
;1I’c
ihr
n.it/i
the
ff
hr
oirtsiflr
//
rhf)
t/if>
:
i
.sAelrtoti
Ll/JO\‘f’
\w/i
E.
f/f~.sit7,~7ri~7ri,Ltti,)II
tliot
/Jtu/“P/?ic’s
of
0
E .s/rcV7
t
t/lot
op.,,
if’ftt7tl :=
(I/,
o/ tlir
(trc’
/Jr
fJ/
.hfiifi
(i rw/,cy if’thcw
I., , )j,,.
,
Thr
.?.
/t~frris~~f~t~~~~//~ ( ‘,,
0 -C _
f/w
; c_
IO
tlic
fl
/if,\
i’i(rIafJ
if’ t/rf,w
:=
(1 ml,
Z’l
It
\I 11. lit
fhr
on
E
c.i-ists
fit
ft
II 1 .I’, ) j,,.:,< t/If’
ft
tn~r~ (,/+ I
.sf~ylle/.
f1/7
0115.
(171 *S. 777~~ tJtor/J/ti.stti
,
c~.w~/Jfiotttrl
tk!, i:
of‘
2.
f’.\-t.vfs ,,’
lrft17.~,~ot-tt1
oti/\~
~w~iot~
~CllCtd
hfl
.sttkt
CJ. if’rrtrd
\fwii,y,r~trp
oi Propo\ilion
cur/\.
017
1,iirtrrrr./f,t-i\tif,
(i.c.
rl,ill
~oroll~u-y
f!f’.?.
I JV U/I
,2 ititrt-\f,fT.\
rrf’n
5. - L.CJlr> : .-r -- XJU. f,] /Jf’
PtWrosrrtoh
A.[[/]]
tt~rrti~,f~Jrrt7
.s>.\7f’t77
in an immediate
-01
.cit’ii.I
grt7c~t~trtit7g
f~c/7li\Yrlct7t
tJ7it7itt7ttl Jo/t/e
it7
-
rlr.\it7~7~/ut-i~frtiot7
F,‘,
tt7itfirt7trl
PIW?~. - This
3.
ft fwt7f it7
T . ,/or
SIWC’
T it7
tl7c
fit7.v
it~rrflirf~ihle
k[\l,.
I,]]
ttiitiitt7fd
-’
2
t?qtr/rrt-
lift.\
to
l/70
,s7fhdii~i.sioti
/I.
Proof: -- The morphistn SIHY /,,[[I(. v]] mmm+ .S lifts to tllc minimal dcsingLtlari~atit,n oi‘ ,G il‘and only if is ;t regular suhdt\iaic~n of !Zr it lifts to 2!:. where 1 ;I\ in Proposition 2. Thts is also
Let
--
,fidlo\r~itig
thw
(i) 0, _ t t
fwtittrit7.s
(ii)
rttfd
of’ l/w
T/7(>
oti/y
A,
hf,
f~ottrlitiotis fit
tl7e
.stt7ft//f~.s/
it7
P
ccJt7tfrit7itrg
ft f.ltrtlurr.t~ri.\tif,
IY*(.IcJ~-
c!f’ 9,
Ot7p
o/‘
0,
i ,
holfls: /ccr.st
0170
f~/trct~frf~7rt~i.~tic.
“I,., ,, I 5 I,. 5: i. ,j f./7ftt.f/f,trt.istif,
rwt7f’
~‘fv’tors
:
~~rf~/~~r~ i?/
;
trrlcl
tlir\t
it//
\i)it/7
0 5
Iif>
it7
tl7fj
itt7ioti
I frtt,/
h,.
01
I L!.
orrt~idr
n,+
, z /if,
rJt7 R
,,,:t,
A, <
, t, =
Applying q to ( I ), we derive frotn the existence of o,, E IX”,,:,’ that the minimum ix achieved same side by all o,, f 0. i.e. either thL, cone CT, ! I 01 CT, ,, in \‘*-2’ , contains than all.
0.
on
the
211
M.
Lejeune-jalabert,
A. Reguera
I’
frj, also lies in CT, ‘, Therefore tklr is located with 0 5 i -<. y. Hecause for an! 11. c,, ,’ E (T,. on tl;e’ .Yll-a\,ls of a general XC’ in the union of (T, +, I and irk,,. I :;: X. < /. NOM the coordinate Ir on ,? such that 07~ E (T( .,,. I < I <_ ‘1 + I, oolncides with its multiplicitv ancl is bounded below ii., . with equality if‘ and only if 07, t Iw ,.,,b, i (61 Car. 8.4). So. by ‘77.1 . ‘7ilpI. and above by 71, p having one characteristic vector in CT,_ , .I and in view of Prop. 4. (17, bcIong\ to the mininlal generating system 01‘ (T,+ , , (I:! C: (<-,,, . Y,. 6,). Th e intclaection of this Ia\t Write 07, =: (tI + (~1 with 0 f C\ t iJ,~++l , d cone with the plane generated by ,T, , , is a cone T,.,. , and the pair (CT, ,~, ,7 : T, + , ) is isomorphic to the pair ((( 1.0). ((,,.J,, , - r/,3, I): (0 I 0). II,. - 77,,j, i)). Therefore cl’2 = 0 and thth ~)nly characteristic vector of 9 is (~7,. Suppose now that Condition (ii) 01‘ Prop. 6 holds with 0 5 i < !I. Here for any I), o+-,, E m, and we can write (~7, = n, + cv2 uilh (I, t IT,+I ?. ckl f (I. and o2 t (c,,J.. .;, 8. But (I? lies in the plane generated by h,.-, and c, t ,. Its coordinate on the -iv,,+, -axis may not bc ICI-~ unless 07, = 0 and as above. the only characteristic vector of F i\ it;.. In general. choose X0 and / as it1 171. II. Car. I. 1-l. and let lj be the distinct simple factors 01‘ 1. There exists i (may be not unique) ~LIC~I that p induce\ ;I wedge q, on the forn1aI neighborhood .<, of the image of I’ on the blowit+-up with center I,. centered at a general arc‘ ‘7,. We prove that the minima1 desingulariz~~tions oi‘ .‘i and .5”, art’ I~KxII~ isomorphic at the point\ through which II and /I., lifts respectively.
References
212