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

+

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http://dx.doi.org/10.1090/conm/090/1000592