2 MARCO ANDREATTA and ANDREW SOMMESE

curves

EonS-

6. with£.£= -1 and

p*L·E

= 0. We say that

(S,L)

is

c-minimal

if it is a-minimal and if there are no smooth rational

curves

Eon

S

with

p*L

.£ =

0

and either

a)

6.-E

=I and £.£ = -1,

or

b)

6.-E

=

0,

E.£=

-2 and

p(E)

is not a point.

The following result summarizes the key points about the concepts

of a-minimality and c-minimality. This result is a straightforward

consequence of results of Sakai [Sa3).

REDUCTION THEOREM.

Let (S,L) be a pair consisting of a

nef and big line bundle, L, on a normal irreducible Gorenstein sur face

S.

For i

=

a or c there exists a nef and big line bundle,

Li,

on a

normal Gorenstein i-minimal sur face

Si,

and a bimeromorphic

holomorphic map

cp.:

S .... S. with L

~

cp.*L' such that the positive

I I I

dimensional fibres of

cpi

consist of rational curves. Further

a)

if

h0(w8n

®

L

n) '1- 0

for some n

0

then

w8a

®

La

is nef,

w8c

®

L/

is ample for k

~

2,

and

(Si,Li)

is uniquely determined by (S,L),

b)

if

h0(w8n

®

Ln)

=

0

for al/n

0,

then

(Sa,La)

is either

(P0 2,0p(e))

where e

=I

or

2,

or

(Sa,La)

is

(Q, Op2(l)lr}•

a quadric surface in

P3,

or the minimal desingularization

7l :

Q .... Q

of the singular

quadric in

P 3

with

La

=

n* 0

p2( I)

1

Q•

or

(S a'La)

is geometrically

ruled.

One notable corollary of this is that given a nef and big line

bundle

L

on a normal irreducible c-minimal Gorenstein surface,

S,

it

follows that

S

is projective. The classification of pairs

(S,L)

when

S

is a-minimal proceeds in §2, exactly as in [Sol]. Here is

the main result.

(2.7) MAIN THEOREM.

Let (S,L) be an a-minimal generically

polarized Gorenstein sur face. Assume that h0(w8n

®

Ln)

'1-

0 for some

n

0.

Then there is an N

0

such that

(w8

®

L)N

is spanned by