2 - Dept. Math, Hokkaido Univ. EPrints Server

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Whei (b) is the case, set P = (I, x, y) = (I0,
so, yo) for to # O where
(x, y) is a local coordinate at P. Then by putting 7 = - to, we have
S = d + 310v2 + (g + 3Ii)77 + (I; + gI0 + h),
where ~ (xo, YO) + 3Ei = O ,~Z(~O, ~0)to + hs(xO, YO) = 0 and g,(xo, yo) +
h,(xo, yo) = 0. Hence F is a double point. We call it an inner double
point . When (a) is the case, is on the 0-section < = 0. Wc call it
a target singularity. From now oil, we consider to reduce the target
singulal-itics ol multiplici1.y 3 on S to some isolated double points and
si~nple
co cliinension one double points .
1.2 Let P be a target triple point on S. Assume F = ((I, z, y) =
(0,0,0)) and the equatioil at P is of the form (1). Set rnl = multpG > -
2 and nl = rnultpH 2 3 where multpG is the multiplicity of G at P.
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Let 3-1 : I< i Y be the blow up at P, and set El = r-'(P). Let
G1, HI be the proper transform of G and H by r1 respectively. Then
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we have r;G = GI + mIEl and r;H = + nlE1. Now we set
where [m1/2] is the greatest integer not exceeding m1 12. Put Ll =
7;L - lIFI. Let al : X1 = P(Oli(Ll) @ Oh) + be the bundle,
(1). Put G1 = r;G - 2l1El = c; + (ml - 211)Fi ,--., 2L1
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and H1 = 7;H - 311E1 = HI + (nI - 3l1)E1 ,--., 3L1. We have either
0 5 Inl - 211 5 1 or 0 5 nl - 31' 5 2. Now we set
and set TI = Ox,
wlicre El is the inhomogeneous fiber coordinate of al, (gl) = G1 and
( 2 ) = H The surface S1 on X1 defined by ( fl) is linearly ecluivalenl, to
3Tl. 1,ct S1 - X be the lllorphism associated with the composition
of slleaf homomorphisil~s
where the last map is obtained by tensoring Oy, (-Il El). Let 6 : Sl 4
S be the restriction of this morphism to S1. We complete the first step
of the following commutative diagram ;