GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 13

fibre of the map p:

S .....

R giving the ruling and

w

8

®

L2

=

p*'t. for

some ample line bundle 't. on R.

PROOF. By the hypothesis w_e

~ave

that

n*(w8

+

2L) ·C

=

0

and therefore by (0.2)

tha~ ~

+

2L)·C '0. _

Suppose first L ·C

=

L ·C

=

0 ap.d thus

w

8

·C

=

0, wg·C ' 0 and

C

·C 0. By the adjunction formula the two following cases are

possible.

- -

i) w8

-~

=

'5~

=

Ll·C:.

=

0 and

C

·C

=

-2.

ii) wg·C

=

C

·C

=

-Ll·C

=

1.

Since by hypothesis S does not contain such curves, we can

thus assume L ·C

=

L ·C ;.

1.

Using the proof of Theorem (1.2) of [Sol], with our lemma (1.1)

instead of his Lemma (0.7.1), we conclude that either (S,L) is a

quadric or that

L

·C

=

I and there is a map

p:

S .....

R

from S to a

curve

R

such that

p

is a P1 bundle near

C

with

C

a smooth fibre

and

(w8

®

L2)N

=

p*'t.

for some

N

0 and some ample line bundle

't.

on

R.

Suppose there exists a singular fibre

f

=

rjEJCX/j·

By the

equalities

we get that F. ·F. 0 for each

i

and by I

=

L ·C

=

L

·f

=

L ·E a.F.

I I

J J

that

L·Fk;.

1 for at least one k

e

J.

Moreover by

(w8

®

L2)N

=

p*'t.

we have 0

(w8

+ 2L) ·

f

=

(w8

+

2L) ·I:

a/r

Thus, by the nefness of

(w8

+ 2L),

(w8

+ 2L) ·

Fj

=

0 for

each

i.

This leads to the absurdity

This proves that

S

is geometrically ruled.

0

(2.3) COROLLARY.

Let

(S,L)

be a generically polarized

Gorenstein sur face and suppose it is c-minimal. Then either: