2 - Dept. Math, Hokkaido Univ. EPrints Server

We denote by 0' the dimension of the colternel of the map r. Then by
(5), we have
(6)
ho (W2 D2 + ~ ( g- i hi))
= hO(W, Co + 6*(1(6 + Dm + D,)) - k + Of,
h1 (W2, D2 + ~ ( g- i hi))
= hl(W, Co + 6*(Iic + Dm + D,)) + 2k + 0'.
Now we consider the map r.
Any element $' t H0(W2, D2) is written as $' = T*$ for some
$I E H0 ( W, Co + 6' (Kc + Dm + D,)) . Then we have
where $ ' I hi E Ho (hi, D2 1 hi) T C and $ (Pi) is the value of $ at Pi.
We set M = HO(W, Co + 6*(Iic + D, + D,)) and denote by Mhhe
vector subspace of M consisting of the elements which pass through
PI, - - . , Pi. We have a descending filtration
We have dim &fi = dim Mi--' or dim'Mi = dim (MC1) - 1 according
as Pi is a base point of M"-' or not. Than the number 0' is equal to
the cardinality of i such that dim Mi = dim M"'.
Let; (20 : ZJ be a fiber coordinate of n : W C such that
Co = (Zo = 0). Then any $ E HO(W, Co + b*(Icc + Dm + D,)) is
written by
here $I~+~D,,+D, E H" (C, I~c + j Dm + Dn) for j = 1,2. A Since (I < -
i 5 k) is on Co, $(P,) = O is equivalent to $ICc+2D7n+D, (Pi) = O. We
set N = Ho(C, I& + 20, + D,) and denote by Ni the subspace of N
consisting of the elements which pass through E, - - . , E. We have a
filtration
(8)
N= No> N1> -.-3 N!