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 ®
§^ - Sm+W. Theorem 3.2 gives immediately
3.5. The natural map §w ® Sw - Sm+W w surjective for m 2 and
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 ^
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 :=
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
M A ^ V X
given by
(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
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