GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 5
PROOF. We follow the usual proof with the minor modifica-
tions needed in our case. Replacing
L
by Ln for a sufficiently
large
n,
we can assume by (0.4.1), that h0(L) 0. Thus choosing a
C
e ILl
we have for all
k
0 the sequence
Since
L
·C
1
0 for every irreducible component of
C
it follows
that
L
0
is ample and that
h1(L
0
k+l)
=
0 for
k
0. Therefore
h1(Lk)
~
h1(Lk+l) for all
k
0 implying that
r(Lk+l) ... r(L
0
k+l) ...
0
for all
k
0. Since
L
0
is ample we conclude from this that
Lk
is
spanned for all large
k.
From this and the hypothesis we conclude
that the map associated to
Lk
for large k is finite to one, and
therefore that
L
is ample.
0
(0.7.2) COROLLARY.
Let
S
be a normal projective surface and
let
p: S ... S
1
be a holomorphic generically one to one surjection of
S
onto a normal compact complex surface
S
1

If the images of the
positive dimensional fibres of
p
are rational double points, then
S
1
is
projective.
PROOF. Let L be a very ample line bundle on
S.
Let C be a
generic element of
ILl.
Since
C
is generic element of
ILl.
it can be
assumed
that
C
is irreducible and doesn't meet the singular set of
S.
By the rational double point hypothesis it follows that
N ·
p(C) is
Cartier for some
N
0. By a direct application of the Nakai's
criterion (0.8.1) we conclude that [N·p(C)] is ample and S
1
is
projective. 0
(0.8) A pair
(S,L)
consisting of an ample (nef and big) line
bundle
L
on a compact normal surface S is called a
(generically)
polarized sur face.
The following classes of special polarized varieties play a key
role in this paper.
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