6 MARCO ANDREATTA and ANDREW SOMMESE
a)
(S,L)
is called a
quadric
if
S
is biholomorphic to a possibly
singular quadric Q
c
P3 and
L
is isomorphic to the restriction
of the hyperplane bundle,
Ops(I),
to Q.
b)
(S,L)
is called a
geometrically ruled surface
if
S
is a holomor-
phic P
1
bundle, p:
S
-+
R,
over a nonsingular curve
R
and the
restriction Lc of L to a fibre
f
of p is
0
p).
c)
(S,L)
is called a
Gorenstein-Del Pezzo sur face
if
L
is ample and
w8-
1
=
L
(see [Br] for a classification of these surfaces).
d)
(S,L)
is called a
conic bundle
if there is a holomorphic surjec-
tion, p:
S
-+
R,
with connected fibres onto a smooth curve
R
with the properties that
L
is relatively ample and that
(w8
®
L)k
=
p*'t. for some
k
0 and some very ample line bundle 't. on
R.
The reader should note that the general fibre
f
of the map
p :
S
-+
R
is a smooth rational curve with
L ·
f
=
2.
§I. SOME TECHNICAL RESULTS
We assume from now on that
L
is a
nef
and
big
line bundle on
a normal Gorenstein surface S, i.e., that
(S,L)
is a generically
polarized Gorenstein surface. As pointed out in
(0.4.1),
S is
Moishezon.
The following simple example shows how the set
U
=
{irreducible curves C
c
S
such that
L ·
C
=
0}
being nonempty causes the theorems of [So I] on the nefness and
ampleness of the line bundles
w
8
®
L,
w
8
®
L2, and
w
8
®
L 3
to fail.
Let
S
be the smooth surface obtained by blowing up a point x
in P2,
T :
S
-+
P2, and
L
=
r* Op(I). L
is nef and big and
(S,L)
'I-
(P2,
0(1)), but w8
®
Li is not nef for every
i
as can be seen by
restriction to the exceptional divisor
r-
1(x).
Fortunately the general situation is not much worse than this
example. By contracting naturally given finite sets of curves
c
U,
we obtain a new Gorenstein surface
S
1
for which the hoped for
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