12
MARCO ANDREATTA and ANDREW SOMMESE
§2.
NUMERICAL EFFECTIVITY AND AMPLENESS OF w8
~
L
In this section, (S,L) will be a generically polarized normal
Gorenstein surface in the sense of definition (0.9).
We will recover, under the milder hypothesis we have on L, the
same kind of results as in [So 1 ). For the most part, the proofs pro-
ceed as in [Sol]. For this reason we will give only the new parts of
these proofs, referring to [Sol] where they can be read without any
change.
(2.1) PROPOSITION. Let (S,L) be a generically polarized
Gorenstein sur face. Suppose that (S,L) is a-minimal in the sense of
( 1.6) and not
~
(P2,
0
P2(1 )). Then w8
®
L
2
is numerically effective
and
h0(w8
®
L
2)
~
0.
In particular, if (S,L) is a-minimal and not ;;;;
(P2, Op2(1)), then w8
®
L 3 is nef and big.
PROOF. Suppose there exists an irreducible curve C on S such
that
(w8
+ 2L)·C 0; then by (0.2)
(ws
+ 2L)·C 0. We consider
the two following separate cases:
a)
L · C
=
L · C
=
0
b) L·C=L·CO.
- -
In case a), by the Hodge Index theorem we have C · C 0 and
-
by our assumption
ws-·C
0 and
w
8
·C 0. By the adjunction for-
mula C is a -1 curve and by (0.2)
-
0 w8
·
C lll
wg ·
C
=
-I.
- - -
Thus
w8
.c
=
ws-·C
and
t,..C
=
0, i.e.,
-1
curve contained inS-/::,..
By our hypothesis this can not happen.
In the case b), L. C
=
L · C 0, the proof goes as in the Lemma
(1.1) of [Sol], using Lemma (1.8) instead of Lemma (0.7.1) of [Sol].
(2.2) PROPOSITION. Let (S,L) be a generically polarized
Gorenstein sur face and suppose that it is c-minimal. If there exists an
irreducible curve C on S such that (w8
+
2L) · C
=
0 then either (S,L) is
a quadric or it is geometrically ruled. In the latter case C is a smooth
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