10 H. B. LAUFER

finite map on r(At). So X~\x) is a finite set for all x G X. Since X is a proper

modification and N is normal, A is precisely the normalization map and N ^ X.

This concludes the m ^ 2 part of Theorem 3.3.

Recall that S denotes the singular locus of V. Let y G V. Every element of Sm

is determined by its restriction (as a germ) to V — S. On V — S9 tensor product

(of tensors) gives a natural isomorphism Sm ® Sn « §m+w- For all j G F , there is

an induced natural map Sm ®

c

§^ - Sm+W. Theorem 3.2 gives immediately

THEOREM

3.5. The natural map §w ® Sw - Sm+W w surjective for m 2 and

n^3.

Let us observe that for the m 3 part of Theorem 3.3, we can in fact choose m

to be arbitrarily large. Let jm: Xm -* F temporarily denote the monoidal transfor-

mation with respect to Sm. Let \ , , . . . ,Xp be generators of § j G K ^ i s the

closure in V X P ^

1

of im /xw, with jnm: F - S - F X P^

- 1

' given by nm(z) =

(z,(Aj(z): ••• :Xp(z))). Let r be a positive integer. By Theorem 3.5, Srmy is

generated by tensor products of the Xi having r factors. There are in fact

n :=

(r+ppSxx)

such products. Xrm is the closure in V X P^

_ 1

of imjurm, defined

analogously. Embed P^

_ 1

in P "

- 1

via the Veronese embedding. This induces the

desired isomorphism between Xm and Xrm.

We may reorder the Xt as is convenient.

With m large, mKAi can be made as large as desired off of the Ai with

K-Aj = 0. So all of the line bundles to follow can be chosen to satisfy the

hypotheses of Proposition 2.1 and Theorem 3.1.

As observed above, we may assume that V is Stein and irreducible at y.

77: M -* V is, again, the minimal resolution. Upon restricting to a smaller

neighborhood of y, we may assume that M is strictly pseudoconvex with con-

nected exceptional set A = UA( — Tr~\y). As in the paragraph following the

proof of Proposition 3.4, there is an induced map y: M -* X. Let X] E

T(A, G(mK)) give Xt as its direct image. Then y may be realized by extending

to A the map

yM-A:

M — A ^ V X

JP~X

given by

yM-A(z)

~

(TT(Z), (X\(Z):

• • • : \'p(z))). So y(z) =

(TT(Z), (X\(Z):

• • • : A;(z))).

A' : = UAj with AJ•- K — 0. Let A" be a connected component of A'. mK is a

trivial line bundle in some neighborhood of A". So, as above, y{A") is a point. Let

us verify that otherwise y separates points. Consider xu x2 G A9 xx ¥= x2 and x}

and x2 not in the same A". By Theorem 3.1, mK has a selection which is nonzero

at both xx and x2. So for y to separate J^ and x2, it suffices to find a section of

mAT which vanishes at one of xx and JC2, but not at both. There are two cases. The

first is that J^ (or x2) is in A". Then let Zx be the fundamental cycle of A". Then

for all /' (and large m), {mK — Zx)-Ai • 2KAt. So by Theorem 3.1 applied to

mK — Zx, mK has a section which vanishes at xx but not at x2.

The other case is that neither xx nor x2 is in A". Suppose that xx G Ax. Let Z'

be the least cycle on A such that Z' ^ Ax and (mK — Z')-At ^ 0 for all /. By

choosing m large, we can insure that Ax appears with coefficient 1 in Z' and that