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
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