GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 7
theorems are true. In this section we will define these sets of
curves.
As in §0, let
n :
S .... S
be the minimal desingularization of
S
and let
L
=
n*L.
By hypothesis
S
is a smooth projective surface
and L is nef and big.
-
(1.1)
DEFINITIONS. A smooth rational curve C on S is said to
be a -1 curve if C·C
=
-1; it is said to be a -2 curve if C·C
=
-2.
-
-
We denote the set of irreducible curves C
c
S, such that
L ·
C
=
0 by U. By a standard argument
( 1.2)
The sets U and
U
are finite.
(1.3)
LEMMA. Consider the set
U1
=
{C
e
U
1
Cis a
-1
curve and C
c S-A}.
The curves in (\ are finite, disjoint, and can be contracted to give
smooth points on a normal Gorenstein projective sur face S ',
cp:
S ....
S'.
Moreover the line bundle L descends to a unique nef and big line
bundle L' on S' such that cp*L'
=
L.
PROOF. This is fairly classical. The first statement follows
from a standard fact, Theorem
(0.1)
of [Li+So], about surface sin-
gularities that the second author first learned from H. Laufer.
Note that the contractibility hypothesis of that result was only
needed to insure negative definiteness of the set of curves U
1,
which in our case is an immediate consequence of the Hodge index
theorem and the fact that
L
·E
=
0 for all E
e ur
Since the map
p :
S .... S' resolves only rational singularities,
the second statement follows immediately from the fact that L · E
=
0 for all E
e
Ur
0
(1.4)
LEMMA. Consider the set
U
2

{C
e
U
1
C
c
S-A
and
C
is a -2 curve}.
Previous Page Next Page