Contemporary Mathematics
Volume 90, 1989
GENERICALLY AMPLE DIVISORS ON NORMAL
GORENSTEIN SURF ACES
Marco Andreatta and Andrew John Sommese
Let
(S,L)
be a pair consisting of a nef and big line bundle,
L,
on an irreducible normal Gorenstein surface S. In this article we
will carry over the structure theory developed by the second author
in [Sol] to such pairs. For us, the most interesting application of
the results is to study surfaces S which are the minimal desingu-
larizations,
p :
S ....
p(S)
c
PC' of an irreducible surface in projec-
tive space, with
L
being the pullback of Op(l) to S. Nevertheless no
real simplification would occur by considering only such smooth S,
since even if we start with such a smooth S, Gorenstein surfaces
will occur in both the statements and proofs of the results.
The results in [So 1] were for ample line bundles L. If
L
is
merely nef, it is easy to see that the curves
C
on
S
that satisfy
L ·C
=
0 cause the theorems in [Sol] to fail. In §I we consider the
contraction of various classes of smooth rational curves on S that
satisfy L · C
=
0. Let
p :
S .... S denote the desingulariza tion of S
which is minimal in the sense that there are no smooth rational
curves
E
on
S
with
E.£
=
-1, and
p(E)
a point. If
S
is Gorenstein,
then
p*w8
;;;;
ws
+
ll
where
wx
denotes the dualizing sheaf of a nor-
mal space
X,
and
ll
is a divisor whose support equals exactly the
fibres over the nonrational points of S. We say that the pair
(S,L)
with L nef and big is a-minimal if there are no smooth rational
©
1989 American Mathematical Society
0271-4132/89 $1.00
+
$.25 per page
http://dx.doi.org/10.1090/conm/090/1000592
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