8 MARCO ANDREATTA and ANDREW SOMMESE
The finite set of curves in
U2
can be contracted to give (Gorenstein)
rational double points in a Gorenstein normal projective sur face S
1
,
rJr. S ...
S
1

The line bundle L descends to a unique nef and big line
bundle L
1
on S
1
such that
IP*
L
1
=
L and
cJJ*w8
1
=
w
8.
PROOF. Let S" denote
S
with
t:.
contracted and let p: S" ...
S
denote the induced map to S. Denote by U2" the proper transforms
of elements of U2 by
p.
Let denote the union of U
2"
and the
curves in the positive dimensional fibres of p. Since p*L ·C
=
0 for
C e Y,
it follows by the Hodge index theorem and Grauert's con-
traction theorem that
Y
can be contracted to give a normal surface
S
1
with induced map
p :
S ... S
1

Since the proper transforms of
the elements of U2 in
S
don't meet
t:.
it follows that all the curves
C
E
Y
are -2 curves; it is well known that these curves in
Y
con-
tract to give rational double points. The projectivity is immediate
from (0.8.3). D
(1.5) LEMMA. The curves in the set
Us
= {
C
E
U
I
C
·ll
=
1,
C
is a -1 curve}
are finite and can be blown down to give a normal Gorenstein sur face
S 1,77:S-S
1

PROOF. Let f denote the set of the irreducible curves C
e
S
that are in the fibres of
Tl
and let Us denote the set of proper trans-
forms of irreducible curves C
e Ug.
Since
L
·C
=
0 for C
e
II
and since
L ·L
0, it follows from the
Hodge index theorem that any subset of curves of
U
has a negative
definite intersection matrix. Since
r
u
us
c
II
it follows by
Grauert's contraction theorem that f
u
Us can be contracted, q:
S ...
S
1
,
to give a normal surface S
1

Since f, the fibres of
Tl,
are
among the curves contracted, it follows that there is a map, 77: S ...
S
1
,
so that
q =
non.
It is immediate that the map
Tl
is precisely the
contraction of the curves in Us·
It only remains to check that
S
1
is Gorenstein. Let
y
e
S'
be
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