GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 13
fibre of the map p:
S .....
R giving the ruling and
w
8
®
L2
=
p*'t. for
some ample line bundle 't. on R.
PROOF. By the hypothesis w_e
~ave
that
n*(w8
+
2L) ·C
=
0
and therefore by (0.2)
tha~ ~
+
2L)·C '0. _
Suppose first L ·C
=
L ·C
=
0 ap.d thus
w
8
·C
=
0, wg·C ' 0 and
C
·C 0. By the adjunction formula the two following cases are
possible.
- -
i) w8
-~
=
'5~
=
Ll·C:.
=
0 and
C
·C
=
-2.
ii) wg·C
=
C
·C
=
-Ll·C
=
1.
Since by hypothesis S does not contain such curves, we can
thus assume L ·C
=
L ·C ;.
1.
Using the proof of Theorem (1.2) of [Sol], with our lemma (1.1)
instead of his Lemma (0.7.1), we conclude that either (S,L) is a
quadric or that
L
·C
=
I and there is a map
p:
S .....
R
from S to a
curve
R
such that
p
is a P1 bundle near
C
with
C
a smooth fibre
and
(w8
®
L2)N
=
p*'t.
for some
N
0 and some ample line bundle
't.
on
R.
Suppose there exists a singular fibre
f
=
rjEJCX/j·
By the
equalities
we get that F. ·F. 0 for each
i
and by I
=
L ·C
=
L
·f
=
L ·E a.F.
I I
J J
that
L·Fk;.
1 for at least one k
e
J.
Moreover by
(w8
®
L2)N
=
p*'t.
we have 0
(w8
+ 2L) ·
f
=
(w8
+
2L) ·I:
a/r
Thus, by the nefness of
(w8
+ 2L),
(w8
+ 2L) ·
Fj
=
0 for
each
i.
This leads to the absurdity
This proves that
S
is geometrically ruled.
0
(2.3) COROLLARY.
Let
(S,L)
be a generically polarized
Gorenstein sur face and suppose it is c-minimal. Then either:
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