GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES II
respectively, such that
La;; a 1*L2,
Lb;;
a 2*Lc,
and L ;;
19i*Li
for all
i. Further for all k
~
0, the above maps give canonical isomor
phisms between
r(w8
®
Lk), r(wa
®
Lak), r(wb
®
Lbk),
and
r(wc
®
L/)
where
wi
denotes the dualizing sheaf of
si.
The following result is a slight extension of a result given by
[Ko+Oc] in the smooth case and with
L
ample.
(1.8) LEMMA. Let (S,L) a generically polarized Gorenstein sur
face. If
w
8
;;
L3
then (S,L) ;;
(P2,
Op2(1)).
If
w
8
;;
L2 then either
(S,L) ;; (Q,
Op2(1)
1
i·
a quadratic sur face in
P3,
or the minimal de
singularization
n : Q .... Q of the singular quadric in P3 with L
=
n*
Op2(1)
1
Q"
PROOF. If w8
;;
L 3 (or L 2) then S has at most rational sin
gularities by proposition (0.2.2) of [So2]. Suppose then the set U
=
{curve C
c
S such that L · C
=
0} is not empty. By our hypothesis,
using (0.2) and (0.3) and the fact that there are only rational sin
gularities, we have w8
·
C
=
Ws·
C
=
0 for
C
e
U.
Moreover, by the
Hodge Index theorem, we have that
C · C
0 and thus that the C's
are 2 curves.
By Lemma (1.4) we can contract all these 2 curves to a gen
erically polarized Gorenstein surface with only rational points,
19:
(S,L) ....
(S
1
,L
1
),
such that
19*
L
1
=
L. It is easy to see, by the Nakai
criterion, that
L
1
is ample. Using (0.5), (0.6) and (0.7) with the
same argument as works in the case when S
1
is smooth, L
1
is
ample, it is straightforward to prove that if
w
8
;;
L 3
then
w
8
1 ;;
L
1

3
and
(S
1
,L
1
) ;;
(P2,
Op2(
1 )). Since P2 is smooth the set U was
empty and therefore (S,L) ;; (P2,
Op2(1)).
If instead w8
;;
L 2 then w8
1 ;;
L
1

2
and
(S
1
,L
1
) ;;
(Q,
OQ(l)).
This gives the conclusion. D