2 - Dept. Math, Hokkaido Univ. EPrints Server

Prooj When C = P' : By (a), we can choose Dp and D, such
that the
-
conditions (2), (3) and (4) are satisfied.
When g(C) > - 1 : Let C be a hyperelliptic curve of genus g > 1 and
let I, : C P' be the liyperelliptic involution. We choose divisors Dm
such that ;
(c) When m is even, Dm := ~*O~l(m/2) ,
(d) When m is odd, Dm := L *O~ ([m/2]) +& for a ramification point
Q E C.
We choose divisors D,, Dg, D, and + - - - + in a similar way
caref~llly. Theii the conditions (2)) (3) and (4) are satisfied by (b).
q.e.d.
Remark 1.8 In the proof of Proposition 1.7, if C is a general
curve of genus g > I and Dm, D, are general divisors on C, then the
invariants (x, c:) of the resulting surfaces s do not cover wider area
than the above ones in the surface geography.
2 Process of res~alluti~n.
In this section, first we resolve singularities of S explicitely. This
method is analogous to that in [ll, $21 and [2, $31. Second we cal-
culate the invariants of surfaces in each process of the resolution.
2.1 We go back to the situation in 1.6. Our way of resolving
singularities of S consists of the following six steps :
1) Let fl, - - -,fk be the fibers of W i C containing PI,. . ., Pk
respectively. Let q : Wl -i W be the coim.position of blow ups at
IDi(1 < i - < k) and set g: = rcl(Pi). Denote by .f: the strict transform
of fi and by QI the point on intersection of ff and g:.
2) Let 72 : W2 + Wl be the composition of blow ups at Qi, 1 _< i 5
k. Set hy = 7c1(Q;) and denote by f/ and gy the strict transform off(