2 - Dept. Math, Hokkaido Univ. EPrints Server

where "A" (resp. "E", resp. "D" ) means any (resp. any even, resp.
any odd) integer with 2 (i/5) - 2. ( For instance, if y = 8x - 54, then
x is any odd integer with 2 9 or x = 10 .)
Then there exists a minimal algebraic surface S such that
1) );(CIS) = 2, c?(S) = y,
2) is a birational holomorphic map,
3) S has a jibration oj nonhyperelliptic curves of genus 3.
Remark 3.4 There exists a minimal algebraic surface S of gen-
eral type such that
I ) C; (S) = 4% (S) - 6, q(S) = 0, pg (S) > 4 and pg(S) = 1 (mod 2),
2) QIr, is generically 4 : 1 map,
3) S has a fibration 7 : S i P' of nonhyperelliptic curves such
that Qrc:, is induced by a relativation of the generic projection of the
plane quartic curve which is the general fiber of h.
In fact, in Proposition 1. 7, we set m > 3, n = 0 and k = 2m - 1.
Then the assertioil is verified by the similar argument as in the proof
of Propositioil 3.1.
Reillark 3.5 There exists a degeneration p : S - A = {It1 < 1)
of surfaces of general type such that the main compone~id s(O) of the
degenerate fiber So = ,x'(O) satisfies