2 - Dept. Math, Hokkaido Univ. EPrints Server

e is ail elliptic curve with self-intersection number -3. The graphs
(a), (b), (c), (d) correspond to 6 E 0,1,2,3 (mod 4), respectively. The
nuinber of (- 2)-curves of the head part o - - o - - - o in (a) N (d) is
[6/4] - 1. Moreover we have
where the summation moves dl the points of type P(I112 fl11~~).
We set T = T,+4 0 - - - o : S* = SrS4 S, and call it the canonical
resolutionr of S.
Let 4; be a component of E' of type (4). Let El, . - - , El(;) be the
components of E which intersect to $i. Assume that Ej has Z-weighting
(aj, bj) for 1 [(i) '(4
- < j - < 1 ) . Then we have 3 nj = 2 Xi,, Oj. Let A,. be
the strict transform of the discriminant divisor A by 7. If A,. intersect
4; at a, point such that A,. E mg, there is a possibility that S* is
singular at (T,+~ o - - - o
inner double point.
o ~,.)-l (P). We call it an. infinitely near
Proposition 1.7 The surface S* has only inner double and in-
finitely near inner double points as its singularities (if they exists).
&loreover we have
2 Examples.
Let (V, P) he a 2-dimensional hypersurface singularity of inultiplic-
ity 3. By the Weierstrass preparation theorem and the Tschiriihauss-
transformation, the equation of (V, P) is given by t3fg(x, y)[+h(x, y) =