GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 3
global sections and such that the map
cp :
S ...
P
c
associated to
f((w8
®
L)N)
has connected fibres and a normal image. The map
cp :
S
+
S'
can be factorized
cp
=
cp' ocpc,
where
«4'c :
S ...
Sc
is the map in the
reduction theorem above,
(Sc,Lc)
is cminimal and
cp' : Sc ...
S' is the
map associated to
f((wsc
®
Lc)N).
One of the following holds:
i)
If
dim cp(S)
=
0
then
(Sc,Lc)
is a Gorenstein Del Pezzo surface.
ii)
If
dim cp(S)
=
1
then
(S c•Lc)
is a conic bundle and
cp :
S ...
P c
is
the projection in the definition of conic bundle.
iii)
If
dim cp(S)
=
2,
then the image
S'
=
cp(S)
is a normal Gorenstein
sur face. L8_cpl(F) extends to a nef and big line bundle L' on S'
such that
cp*(w8
,
®
L ') ;;;;
w8
®
L and
w8
,
®
L' is ample on
S'.
The first author would like to thank Consiglio Nazionale delle
Ricerche, and the second author would like to thank the National
Science Foundation (DMS 8420 315) for their support. We would
like to thank the University of Notre Dame for its support.
§0. BACKGROUND MATERIAL
We consider complex analytic normal surfaces with at most
Gorenstein singularities.
We recall that for a normal surface
S,
the
Gorenstein condition is equivalent to the dualizing sheaf, that we
denote by w8, being invertible. For the most part we follow the
notation of [Sol].
Let
n : S
+
S
denote the desingularization of S, that is minimal
in the sense that the fibres of
n
contain no smooth rational curves
E
satisfying
E ·E
=
1.
If
D
is a (Weil) divisor we will denote by
D
the proper transform of
D. If L
is a line bundle we will denote by
L
the inverse image,
n*L
except when
L
=
w8 since this would lead
to a confusion of bars. Note that if C
e
ILl it is not necessarily true


that
C
e
ILl. We use without further comment the equality
L ·D
=
L ·D
where
D
is a Weil divisor on
S.
We have the following
fundamental facts (see [Sa 1 ]).