GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 15
ln(S ,L)
=
-co
ln(S,L)
=
maxnEA dim pn(S)
if
A
is empty and
otherwise.
The generically polarized Gorenstein surfaces with ln(S,L)
=
-co
were classified in Theorem (2.5).
(2.7) MAIN THEOREM.
Let (S,L) be an a-minimal generically
polarized Gorenstein sur face. Assume that h0(w5n
®
L
n)
"1-
0
for some
n
0.
Then there is an
N 0
such that
(w5
®
L)N is spanned by
global sections and such that the map
p :
S .... P
0
associated to
r((w8
®
L)N)
has connected fibres and a normal image. The map
p :
S
-+
S'
can be factorized
p
=
~p' o~pc,
where
pc : S .... Sc
is the map in the re-
duction theorem above,
(Sc,Lc)
is c-minimal and
~p':
Sc .... S'
is the map
associated to
r((w5c
®
Lc)N).
One of the following holds:
i)
If
dim
~p(S)
=
0
then
(Sc,Lc)
is a Gorenstein Del Pezza surface.
ii)
If
dim
~p(S)
= 1
then
(Sc,Lc)
is a conic bundle and
p :
S
-+
Pc
is
the projection in the definition of conic bundle.
iii)
If
dim
~p(S)
=
2,
then the image
S'
=
~p(S)
is a normal Gorenstein
surface. L5_p-l(F) extends to a nef and big line bundle L' on S'
such that
~p*(w 5 , ®
L ') ;;; w5
®
L and w5
,
®
L' is ample on S '.
PROOF. Except for iii), the proof is exactly as the one of the
main theorem in [Sol], the nefness of w8
®
L being insured by our
hypothesis of a-minimality and our Theorem (2.3), or by Theorem
(7.4) of [Sa3].
To prove that point we first observe that each fibre of the map
pc : S
-+
Sc is contained in a fibre of the map
p :
S .... S' and that
~p*(w 8 c
®
Lc') ;;; w5
®
L. This is clear since, as we have seen in
Chapter 1, every irreducible component of a fibre of the map
~p:
c
S .... S c is a curve C either in the set
U
2
or in the set
U
3
(see
Lemmas (1.4), (1.5) and in both cases it is easy to compute that
(w5
®
L) · C
=
0.
We can thus suppose that S is c-minimal. We will prove that
each positive dimensional fibre of
~p:
S .... S' is an irreducible curve
C biholomorphic to P 1 such that L · C
=
1 and such that
~p(
C) is a
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