16 MARCO ANDREATTA and ANDREW SOMMESE
smooth point of
S
1

Since
L
is algebraic
Ls-c
extends over the
smooth puncture on
S
1
to give a nef and big line bundle; the rest
of the theorem is now trivial.
Let now :._ b_: an
irreducibl~
curve in
S
such
th!t
Jws
®
L) · C
=
0.
If L·C
=
L·C
=
0 then wg·C '
w
8
·C ' 0 and C·C 0 by the
Hodge index theorem.
That implies, by the adjunction formula and (0.1 ), (0.2), that
either
C
is a -2 curve such that
t:.·C
=
0 or
C
is a -1 curve such
that
f:..C
=
I; i.e., Cis either in the set
~
or in the set U3; that is
impossible since
S
is c-minimal.
Therefore we can suppose L ·C = L ·C 0. In this case wg·C '
w8
·C '
1.
The hypothesis dim p(S)
=
2 implies
Since
n*(w8
®
L) · C
=
0, it follows by the index theorem and what is
above that:
- - -
C·C
=
-1
=
wg·C
=
w8 ·C and
L·C
=
L·C
=
1.
-
This in particular implies that C is a
(-1)
curve contained in
-
S -
t:.. The conclusion follows from Lemma (1.3) using the nef and
big line bundle (w8
®
L) instead of
L. 0
(2.8) THEOREM.
Let (S,L) be a generically polarized
Gorenstein sur face. Assume that L is spanned by global sections and
the genus, g(L) of a smooth
C
E
ILl
equals
h1•0(S).
Then
ln(S,L)
=
--~~
and the associated a-minimal pair of
(S,L)
is classified in Theorem
(2.5a).
PROOF. Consider the sequence
Since h1(w8
®
L)
=
0 by the Kodaira-Ramanujam vanishing theorem
(0.6), we conclude that
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