2 - Dept. Math, Hokkaido Univ. EPrints Server

P(C n 11), P(N~ n II), P(N n 111), P(I n II),
P(I n 1111), P(I1 n 111), P(II n II), P(II n III~),
( I I ) P I I )
P(III n 111), P(+ n any type),
where P(C n 11) is the point of intersection of the components of types
(C) and (11), P(N2 n 11) is that of types (N) with ,!? = 2 and (11),
etc.
1.6 We proceed our resolution process.
1) Let 6, (1 - < i < - n(C)) be the component of E' of type (6).
Set = T;' (C,). Tlle singularity of S, is t3. + " t + x2 = 0 at a
general point of c, and is t3 + xaY6t + x2 = 0 at the point of type
T~(P(C I)). We call it a compound cusp.
n(C) Let rTtl : XT+l Gi
+ X, be the blow up with tlie center Ci,l
Let be the strict transform of S, by r,+1. Set T,.+l = .r,+llsr+, be
t11.e restriction of T,+~ to The surface ST+l is nonsingular along
1 (ci), 1 c.
- < i < n(C). Let CI be the reduced part of (T,.+~)- ( z).
C: is naturally isoinorphic to PI. And we have
2) Let Ni(l < i < n(N)) be the coniponent of E' of type (N).
Assume that Ni has Z-weighting (1,2) (resp. (1, ,!?) with /3 2 3) for
1 - < i < - lz(N2) (resp. n(N2) + 1 5 i < n(N)). Set r:+, = T,+l o ?r, -
and
N = ( +)(N). The singularity along is the following ;
t3 + xc + xp = 0 at a general point of x,
t3 + Z[ + xliyt = 0 at (r,'+,)-'(P(Ni n Ill.)) for 1 < i < n(N),
t3 + xy6t + x2 = 0 at (T:+~)-'(P(N~ n II)) for 1 < i < n(N2).
We call it a compound node.
n(N)
Let r,.+~ : X',.+2 X,+l be the blow up with the center Xi-, , -
Let be the strict transform of by 7,+2. - We set T,+, = r,+21s+,
-
and r:+2 = 7,+2 o T:+~. The pull. hack (7,+2)*(Ni) is a union with two