GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 9
an arbitrary singular point of
S
1
such that
n
doesn't give a biholo-
morphism of a neighborhood of y and a neighborhood of n·1(y). Since
Sand S
1
are normal, it follows from Zariski's main theorem that Ill=
n·1(y) is positive dimensional. By the definition of
n,
we know that
there is an irrational singularity, xeS, with n(x) =
y.
By Theorem (3.1)
of [Sol], it follows that there are global holmorphic sections sk of
w
5
®
L
k fork= 1 and 2 with sk(x)
;f
0 fork= 1,2.
CLAIM. For
k
= 1 and 2, sk trivializes
w5
®
Lk in a
neighborhood of Ill.
PROOF OF THE CLAIM. Since sk(x)
;f
0, it suffices to show
(Wg
@
Lk).
c
= 0 for all of the
c
e
u3
such that
X
e
c.
This will
follow if we show that
[n*(w5
®
Lk)]
·C
= 0 where
C
is the proper
transform of
C.
Since L ·
C
= L ·
C
=
0,
it suffices to show that
n*w5
·
C
= 0. Since
n*w5
=
WS_
+
ll,
and since
fl.
C
= 1 by hypothesis,
it suffices to show that
ws ·
C
= -1. But this is an immediate con-
sequence of the hypothesis that
C
is a smooth rational curve with
self intersection -1.
0
From the claim it follows that L = (w5
®
L2)
®
(w5
®
Lt1 and
hence
w5
=
(w5
®
L)
®
C 1 are trivial in a neighborhood
U
of Ill.
From this it follows that for some open neighborhood
V
of
y,
Wreg(V)
is trivial. From the fact that
Wy
is the direct image of
wreg(V)
under the inclusion of reg(V) in V it will follow that
Wv
is
free and hence that y is a Gorenstein point of
S
1

0
(1.5.1) LEMMA. Let n: S ... S
1
be the contraction of (1.5). There is
a nef and big bundle L
1
on S
1
such that L;;;; n*L
1•
Further
w
5
=
n*w5,.
PROOF. Note the proof of {1.5) showed that
w5
and L are
trivial in some neighborhood of any fibre Ill of
n.
From this it is
straightforward to conclude that L ;;;; n*L
1
for a nef and big line
bundle
L
1
on
S
1
and
w5
;;;;
n*n
for some line bundle
n
on
S
1•
Since
n
agrees with
w5
1
on the Zariski open set Z
c
S
1
such that
n
is a
biholomorphism from n·1(Z) to Z and since
w
5 1
is the direct image
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