10
MARCO ANDREATTA and ANDREW SOMMESE
of
Wz
under the inclusion
Z
c
S ',
it follows that
n ;;; w8'·
D
(1.6) DEFINITION. Let
(S,L)
a generically polarized
Gorenstein surface.
(S,L)
is
-
a-minimal
if there are no -1 curves
C
in
S -
11 such that
-
L·C
=
0.
b-minimal
if it is a-minimal and there are no curves
C
in
S
-
-
that
L · C
=
0 and its strict transform,
C,
in
S
is a -2 curve
with fl.C
=
0.
c-minimal
if it is b-minimal and there are no -1 curves
C
in
S
-
such that L ·C
=
0 and fl·C
=
I.
(1.7) PROPOSITION.
Given any generically polarized Gorenstein
sur face (S,L) there exist a-minimal, b-minimal, c-minimal generically
polarized Gorenstein sur faces
(Si,Li)
and maps
pi:
S
-+
Si
each obtained
by contracting a finite number of curves in U
= {C
c
S
I
L · C
= 0}
such that (Si,L) is i-minimal. There is a commutative diagram of
holomorphic maps.
sa sb
---i··
sc
al a2
PROOF. It follows by applying Lemma (1.3) a finite number
of times, then Lemma (1.4) and finally Lemma (1.5).
D
(1.7.1) REMARK.
It
follows from (0.8.3) that Sa and Sb are
projective if
S
is projective. It will be seen in the next section that
the Sc is projective. From the above results it follows that there
are nef and big line bundles La, Lb,
Lc
on the spaces Sa, Sb, Sc
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