Title The flat coordinate system of the rational double point of E8type

Author(s) Kato, Mitsuo; Watanabe, Satoshi

Citation 琉球大学理学部紀要 = Bulletin of the College of Science.University of the Ryukyus(32): 1-3

Issue Date 1981-09

URL http://hdl.handle.net/20.500.12000/8327

Rights

Bull. College of Science, U1liv. of the Ryukyus, o. 32, 1981.

The flat coordinate system of the :rational double point

of E8 type

Mitsuo KATO.* Satoshi WATA ABE**

Introduction

The flat coordinate system of a versal deformation of an isolated singu­

~arity is introduced !by Saito [1] .. Its existence· is :proved for rational double

points by Saito [2] and the explicit calculations for A1 " D1 , E6 types are done

by Saito, Yano and Sekiguchi [3] . In Yano [4] ,varions free deformations

and their flat coordinate systems are considered. A linear subspace by the flat

coordinate syst,em of a ersal deformation sometimes gives an example of free

deformations.

Here we give the calculation for £8 t}rpe singularity perfornlliedi by a

computer (DEC 2020 system at RIMS, Kyoto Universwty, and the language

REDUCE 2)1.

The authors are grateful to Dr.T. Yano for many valuable advices.

§ I

The rational duobJe point of Es type is defined by the equation

!(x, y)' = O/5)x 5+ O/3)y3=O.

A versal deformation of its singularity ]s given by

F{x, y) = f(x, y)- Ilx3y-14x2y- tilx3- t,xy- 4,x2- tloy-lI2X-llS=0.

If t's are generic, Fx = 0, Fy = 0 define 8 simple zeros:

{ Fx = O. Fy =O} = {(ai, bJ. i = 1, ~ •• , 8}

L t g(x, y) b holomorphic alld put

w = g(.t, y) dxdy I f~,· i'y.

Then we define :

,"Js(ai, h,) W =g(a;, b;)!Hess,,(a, , h,),

Res cv = !.,8 I reYa/, bdw,wh r Hess,.. = 1'xx1;' y- Fxy .

It is easily eh eked that R ~(d is a holomorphie fun't.ion of t. If g is a

polynomial, w can be ext.ended on P2= {[x: y: z]} and ,is holomorphic on P ll ­

Ha.: bi : 1], i=l, ""', 8} U {[O: 1: OJ} ... Since the lolal sum of ps~dues oJ

w, on )2 i.s zero, we hav1e

R s w = - res [0 : 1 : 0] w.

§ 2 Reidue pair J(al • a,l'.FoHowing Yano [2] , we deHne the residue pairs.

J(OI, aJ) =Nes [aF / aI, • aF / di,dxdy I F}! • FyJ--------------------

* Department of Mathematics. iUniversi.ty of th Ryukyus.

* * Department of Mathematics. Yamagata Uni\rersily.

2'KATO and WATA ABE: The flat coordinate system of the rational

doub~eJ}oint of E8 type

=-res(o:l:o) [aF/at, • aF/atjdxdy/,f'(·· Fy ]

IfaF/oli • aF/atj=xQ1y b,

then at [0 : 1 : 0]xaybdxdylf""'x. Fy=z(Jdzdxjza+b~lj,z4lFx(xlz. liz). zJf'..y(xlz, liz)'

Ttl r sidue at [0: 1 : OJ is calculated in the foHowing way.

z3F'I(xjz;, l/z) = -Z+t.X2Z+trxZ2+IIOZ3+iIX:il

r=z - t"x2, - t,:a 2

- IIOZ3

z =~,(1+ A(xJ·r))'

1= 1· .A - t.x2:(J +A) - hXr(I+Aj2 - tHJ r 2 (l +Aj3

A.(x, 0) =l~x21 (J - t.x''JA(-(x, 0) =t7 xO +A (x, 0)]2/ (1- t.x2

)

A~;dx, 0)=(4"'x(1 A{-t, O))Adx, O)+2tlQt(l+A(x, OJj3JjO-t4X2

)

A (x, \)=A(x, 0)- ~Adx, 0)+,,2 A,~~·tx, 0)/2+······

resx 0, z.1J xDdzd:t/znlb fx{x/ Z, liz) • z 3 Fy (x/ z, 1/z)

=re'.o,r..o xll(l-' A rArJdrdx/zHbFx(x/z, l/z}· (-'s-+tl.xJ)

.= r. Sx-o x (I -I- A -I- s-Adrb/ Zll+b F,dx/z,. 1/z) I ~.. t,xl

o rdinat transf rmatio:n

SI -./. ~ :I'D "'1/1 -J . J,D.k,ql ~• .t ~

und}· th ndition

{1, if ,: I j - 16

J( ., I) 0, if i+j*16..

Such a coordinate system {s I} exists uniquely (Saito [2J ), and is calffied the

flat coordinate system of the versal deformation. {t I} is also expressed by

I s Il "homogeneously/!

§ 4 Resultsj(z: j) stands for .f(o" 'OJ).

J(15,1) =J(12,4)=J(10,6)=J(9, 7)= 1

}(9,6)=3~1

JOO,4)=J(7, 7) = 3t. 2

J02, 1) =J(.9~ 4) =](7, 6) =9t13

}(6,6) = 2t~+27t] 4

J(l0,1) =J(7, 4) =St, t4 +27tl5

J(9,1)=1(6,4)=3~+2lt12l.t -I-8U16

J(7,1)=J(4,4) = 4tl t, + 1Bt12 4J +2tlj 2 +108t14"t4 +243tl8

J(6,1)= 21-'9 +15t12 t, +81113 tfj + 16fl t4

2 +378tlS~ + 72.9t1

9

](41) =3t1 tID +12/12 /g +3t4 t1 +811.1 4 tq +30t1~ is +324tl 5 f() + 17lt13 t4

2

+1539t11 t4 +2187tl ]I

Bull.. College of ciel'tce Univ. of til Ryuk)'u. o. .32, 1981.

}(1',1) = 6/12 t12 +214,['0 +54,[1 4 t,0 +201'\ t. 19 +216tu15 {g,+ t1

2 +24t1t6 t1

+252tl 3 ~17 + 1134tl 7 t-, + 1t1 2 Ii; 2+ 12t-ol 2 4-, + 1296t\ ~ t4 t6+ 437111

8/6 + 134t,2t4 3+4239t1

6 t42 + 1 )954tllllt~ +.196 t l

14

s\ = tl

S4 = ("'1+ 2 1,14

St; = lil +2/'1 2 t.. + (14/5)tl '6

S7 = Iq +211 t6+ (14/3)/, J ~ + (17 /3)11"l

Sg = {g + (3/2)/1 2,(7 + 411

3 1,;, + (3/2)tl t/ + (52/ 5)lu51" + (....99/30)tt 9

SIO = 110 + 114 +2/\3/" + l-4 t1l + (l1/2)tI4 it, + 3t1

2 /'4 2+ (44/3)1. 6/ 4

+ (572/45)tl 10

Si2 =112 + 112

/ 10 + (7/3)iI3 /g + 2/I'4l; + (2' /5)t1 5 11 + If, 2 + 7t1

2 t4 16

+ (23) / 15)t1 6 it, + (l13)/,.3 + I1tI4'~,2 + (1309/30)tI8 i4 + (476/15)1\12

S,15 = tiS + II J 112 + 1.4 tlO + (11/5)t15 I\()+ 1~19+ 3t. 'l l." /9 + (8/15)t,6l9

+ 0/2)t\ t-,2+ 3/12

/6 /" + (1/2)1-4 2 t17 + .lll,4/4 /7 + {77/ 5)t.1 !It?

+ (11/2)tl 34;2 +3tl / 42/ij + (176/5)l, 5 /./f, + (2002/ 45)t'1 9 Ii;

+ (22/3)tl 3 14 3 + (308/5)/. 7 142 + (5642'/45)/1 11/4

+ (16523/225)tl \5

3

il = I

t4 =S4--2\4

to = S(, - 2512 S4 + (6 / 5J~ 16

til =Sr - 2sI Se - (2/3)S1 3S4 + (19/15)s1 7

t'9 =Sg - (3/2)S12S7- S.3s, - (3/2)s~S42 + (23/5)sI5s4 - (28/ 15)sl 9

tl<O= SlO -.S9 - 0/2) '13

S7 - $4S6 + (3/2)51456 + 0/2)512S/ - (2/15)sl,6s"- (U/45)s.10

t.12 = 12 - a 2 510 - (4/3)SI3Sg - 25.5457 +02/5)s.5 s7 - .'>0 2 + 2512S4St.!

- (7 j30)sl G., - O/:J)s. n ~ (7/3) I ~S4 2 - (107/30).')'1 11:;4 -1 (82/7f)s,12

11/I::;"515-SI3SI2-SI5.510+(4/5)sl'810- .'."it! -(7/1'5)'<;'1 So (l/:"~SIS72

- (1/2}~12~Sl- (J /' 1S/l S1 -I (5/3)81 • .'14 $1 - (~J'/ '.JO).",ll:;,

(1/2)51'3 .2+(l/~j.'iIS.2., -(43/30)"'/'8.: (141 5)SIU,

(l/3) ,. 1I s.3 - (43/4. ,). 1'15.,2 + (121/ 45()).'l',".o;~ - (JO,'J /4fjO)~1 In

nEFEI~I~S

[ 11K. 'aHo: I n ilh pdf I)rimitiv inl I(rals (in D'f IYlntiol'l),

[ 2] I. Saito: n a lin tU" tructur of a quoti nl variety by a fitnit, reflexiOl!

group, to .appear.

[ 3] I. aito. T. Yano, J. S kigu "hi: On a c rtain ~ n ral.or systen, of lh ring

of invariants of a finit . r f1··xiOilI'l gr )Up, to app ar.

[ ,t] T. Vallo: Fre d formati, 118 of isolat d singuladli s" S i, n, I~ ports of

th "'aitama niv rsity. S r~es A Vol. IX. N0,3. 1980, pp.6 J.70.