GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 3
global sections and such that the map
has connected fibres and a normal image. The map
can be factorized
is the map in the
reduction theorem above,
is c-minimal and
cp' : Sc ...
S' is the
map associated to
One of the following holds:
is a Gorenstein Del Pezzo surface.
is a conic bundle and
the projection in the definition of conic bundle.
then the image
is a normal Gorenstein
sur face. L8_cp-l(F) extends to a nef and big line bundle L' on S'
L ') ;;;;
L' is ample on
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
We recall that for a normal surface
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].
n : S
denote the desingularization of S, that is minimal
in the sense that the fibres of
contain no smooth rational curves
is a (Weil) divisor we will denote by
the proper transform of
D. If L
is a line bundle we will denote by
the inverse image,
w8 since this would lead
to a confusion of bars. Note that if C
ILl it is not necessarily true
ILl. We use without further comment the equality
is a Weil divisor on
We have the following
fundamental facts (see [Sa 1 ]).