14
MARCO ANDREATTA and ANDREW SOMMESE
a)
(w8
®
L 2) is ample,
b) {S,L) is
(P2,
Op2{1)) or a quadric,
c)
(S,L) is geometrically ruled and
w
8
®
L
2 =
p*'t. for some ample
line bundle 't. on R where p: S
-+
R gives the ruling.
PROOF. This follows immediately from (2.1) and (2.2), using
Nakai criterion (0.8.1). D
(2.4) COROLLARY. Let (S,L) be a generically polarized
Gorenstein sur face and suppose it is c-minimal. Then
S
is projective.
PROOF. In case a) of Corollary (2.3),
w8
®
L 2 ample and S is
therefore projective by the Nakai criterion (0.8.3). In case b) or c),
S is trivially projective. D
(2.5) THEOREM. Let (S,L) be a generically polarized Gorenstein
sur face and suppose it is a-minimal. The following are equivalent:
a) (S,L) is neither geometrically ruled nor a quadric nor the minimal
resolution of a quadric, nor equal to
(P2,
Op2(e)) for e
=
1 or 2,
b) (w8
®
L) is numerically effective.
c)
h0((w8
®
L)N) '1-
0 for some
N
0.
PROOF. To prove the equivalence a)
#
b) note that we can
suppose (S,L) to be c-minimal. In fact, let
fie :
S
-+
Sc be the map
given in (1.7) where (S ,L ) is c-minimal. Then
w
8
®
L is nef if and
c c
only if
w8c
®
L8c is nef and (Sc,Lc) is as in a) if and only if (S,L)
is.
By the above observation we can apply Corollary (2.3); with
this the proof is the same as that of Theorem ( 1.3) in [So I). D
(2.6) DEFINITION. Let (S,L) be a generically polarized
surface and A denotes the set of positive integer
n
such that h0((w8
®
L)n) 0. For each
n
in
A
let pn : S
-+
P
c
denote the meromorphic
map associated to
f((w5
®
L)n). The logarithmic Kodaira dimension
of the pair (S,L) denoted by ln(S,L), is defined as follows
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