GENERICALLY AMPLE DIVISORS ON NORMAL GORENSTEIN SURFACES 9

an arbitrary singular point of

S

1

such that

n

doesn't give a biholo-

morphism of a neighborhood of y and a neighborhood of n·1(y). Since

Sand S

1

are normal, it follows from Zariski's main theorem that Ill=

n·1(y) is positive dimensional. By the definition of

n,

we know that

there is an irrational singularity, xeS, with n(x) =

y.

By Theorem (3.1)

of [Sol], it follows that there are global holmorphic sections sk of

w

5

®

L

k fork= 1 and 2 with sk(x)

;f

0 fork= 1,2.

CLAIM. For

k

= 1 and 2, sk trivializes

w5

®

Lk in a

neighborhood of Ill.

PROOF OF THE CLAIM. Since sk(x)

;f

0, it suffices to show

(Wg

@

Lk).

c

= 0 for all of the

c

e

u3

such that

X

e

c.

This will

follow if we show that

[n*(w5

®

Lk)]

·C

= 0 where

C

is the proper

transform of

C.

Since L ·

C

= L ·

C

=

0,

it suffices to show that

n*w5

·

C

= 0. Since

n*w5

=

WS_

+

ll,

and since

fl.

C

= 1 by hypothesis,

it suffices to show that

ws ·

C

= -1. But this is an immediate con-

sequence of the hypothesis that

C

is a smooth rational curve with

self intersection -1.

0

From the claim it follows that L = (w5

®

L2)

®

(w5

®

Lt1 and

hence

w5

=

(w5

®

L)

®

C 1 are trivial in a neighborhood

U

of Ill.

From this it follows that for some open neighborhood

V

of

y,

Wreg(V)

is trivial. From the fact that

Wy

is the direct image of

wreg(V)

under the inclusion of reg(V) in V it will follow that

Wv

is

free and hence that y is a Gorenstein point of

S

1

•

0

(1.5.1) LEMMA. Let n: S ... S

1

be the contraction of (1.5). There is

a nef and big bundle L

1

on S

1

such that L;;;; n*L

1•

Further

w

5

=

n*w5,.

PROOF. Note the proof of {1.5) showed that

w5

and L are

trivial in some neighborhood of any fibre Ill of

n.

From this it is

straightforward to conclude that L ;;;; n*L

1

for a nef and big line

bundle

L

1

on

S

1

and

w5

;;;;

n*n

for some line bundle

n

on

S

1•

Since

n

agrees with

w5

1

on the Zariski open set Z

c

S

1

such that

n

is a

biholomorphism from n·1(Z) to Z and since

w

5 1

is the direct image