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