4 MARCO ANDREATTA and ANDREW SOMMESE
(0.1)
n*w8
;;;;
wg- +
!J.
where
!J.
is an effective divisor on
S.
(0.2)
w8
·C
=
(wg
+
!J.) · C
~
Wg·C
-
(0.3)
!J. ·
C
=
0 if and only if C meets at worst rational
singularities.
(0.4) DEFINITION. A line bundle L on S is said to be big if
L ·L 0 and it is said to be numerically effective, or nef for short,
if L · C
~
0
for each irreducible curve C on
S.
The proof of the following lemma is straightforward and left
to the reader.
(0.4.1) LEMMA. Let :t be a nef and big line bundle on a normal
compact complex sur face, then h0(:tn) c ·n
2
for some c 0 and all
n 0. In particular S is Moishezon and has a projective desingu-
larization.
(0.5) Since S is Cohen-Macaulay and
w
8
is invertible, Serre
duality for a locally free sheaf E on S takes the usual form hi(S,E)
= h2-i(S,w8
@
E*)
or i = 0,1,2.
The Kodaira-Ramanujam vanishing theorem takes its usual
form (see §0 of [Sol] for discussion and references).
(0.6) THEOREM. Let L be a numerically effective and big line
bundle on S. Then hi(S,w8
@
L)
=
0
=
h2-i(S,L-l) for i
~
I. In
particular
if
C
e ILl,
then C is connected.
(0.7) We will need some projectivity criteria for surfaces
obtained by contracting sets of curves. First we have Nakai's
criterion for projectivity of a normal compact complex surface.
(0.7.1) NAKAI CRITERION. Let L be a line bundle on a normal
compact complex surface S. If L ·L 0 and L ·C 0 for all
irreducible curves on S then L is ample, and in particular S is
projective.
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