GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 15
maxnEA dim pn(S)
is empty and
The generically polarized Gorenstein surfaces with ln(S,L)
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
Then there is an
L)N is spanned by
global sections and such that the map
S .... P
has connected fibres and a normal image. The map
can be factorized
pc : S .... Sc
is the map in the re-
duction theorem above,
is c-minimal and
Sc .... S'
is the map
One of the following holds:
is a Gorenstein Del Pezza surface.
is a conic bundle and
the projection in the definition of conic bundle.
then the image
is a normal Gorenstein
surface. L5_p-l(F) extends to a nef and big line bundle L' on S'
~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
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
S .... S c is a curve C either in the set
or in the set
Lemmas (1.4), (1.5) and in both cases it is easy to compute that
L) · C
We can thus suppose that S is c-minimal. We will prove that
each positive dimensional fibre of
S .... S' is an irreducible curve
C biholomorphic to P 1 such that L · C
1 and such that
C) is a