4 MARCO ANDREATTA and ANDREW SOMMESE
is an effective divisor on
!J.) · C
0 if and only if C meets at worst rational
(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
for each irreducible curve C on
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
for some c 0 and all
n 0. In particular S is Moishezon and has a projective desingu-
(0.5) Since S is Cohen-Macaulay and
is invertible, Serre
duality for a locally free sheaf E on S takes the usual form hi(S,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
h2-i(S,L-l) for i
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