WEAK SIMULTANEOUS RESOLUTION 17
We now show that blowing up a suitable % at essentially the ideal sheaf $w,
m 3, yields the desired simultaneous RDP resolution. We wish to apply [18,
Definition 8, p. 226], where we need a smooth center for the monoidal transform.
^*(^m/^m) ls a l°cally free sheaf on T. Let /
l 9
...,/
;
be elements of$mwhose
images in K(im/$*) generate K($m/i*) over 0
r
. Then the map / = (/
l 9
... ,/„ r):
T- C
/ + 1
is proper. Let % := / ( T ) . /: T ^ ¥ is the normalization map. Let
(JC,,...,*/ ,
0 be local coordinates for
C/+1.
Let % be the ideal sheaf on %
generated by (x{9... ,.*,). Then fjjfm) $. Let (z
1?
... ,z„) be local coordinates
for an ambient neighborhood of V. Then since all products ztfj, 1 / w,
K j / , are in 4W, / ^ ( z ^ ) is holomorphic on ¥. Then /: T - l o c ^ J -
% loc(^) is a biholomorphic map.
By construction, we may identify fh := $/(/ t0)fy with Sw ^. Then
dim Yt/%vt+ x d i m S ^ / S ^ 1 , v^\, and we have shown that this value is
independent of t. dim f®/$t •= d i m © ^ , / ^ equals 1 for all t. So % is normally
flat along loc(^). By [18, Definition 8, p. 226 and Lemma 6, p. 216] and Theorem
3.3, the monoidal transformation $: 91 - % of % at f gives a simultaneous RDP
resolution of %. By Proposition 4.7 and [43, Satz 2, p. 548], we may simulta-
neously blow down the exceptional sets in ^ , f: 91 ^ Since 91 is normal, W is
normal. Then the induced map co: % -* % is the normalization map. But/: T- W
is also the normalization map. Hence % « T.
This concludes the proof of Theorem 4.3 except for the proofs of Lemmas 4.4
and 4.5.
PROOF OF LEMMA
4.4. Suppose that a GJW)0 is not an element of Sm. Then on
the minimal resolution M of V, o has a pole of order r 0 on some component At
of .4. T := a ® ®a, with * factors, has a pole of order rv on ^4Z. Since all
holomorphic functions on V extend to holomorphic functions on T, multiplying T
by a holomorphic function / with a zero on Ai of order less than rv gives an
element of 4
m j
,
0
- S
m j
,
We see, as follows, that there are many such functions /. Let D be a cycle on ^4
such that for all AJ9 D -AJ 0 and (D + At) -Aj
r
-K-Aj. Then, Proposition
2.1, H\M, G(-SD A^) = 0 for all positive integers s. Then
T(M, e(-sD))/T(M, e(sD - A,)) « r(M,
O ^ - J Z ) ) ) .
Let g be the genus of At. Then, by the Riemann-Roch Theorem,
dim T(M, 0, .(-$/)) 5^ JZ) - ^ + 1 - g.
Let ^ be the coefficient of At in Z). Then
d™ 3m,,o/ (S
m
, n 3m,0) 2(-*Z -,4, + 1 - g),
for sd, r *. Take 0 c ((-Ai
D)/2)(r/di)2
to complete the proof.
PROOF OF LEMMA 4.5. Recall Ur:
G)tr
%, the above projective algebraic
resolution of % such that pr •= Ar ° II
r
: 91Lr -^
P1
is a holomorphic map. 9H is
the germ of 9Hr at £
0
: = U;\p). II: 9 1 - Tis the germ of n
r
. Fr : = A-/(0). M;
is the proper transform of Vr under II,.. We may assume that £,., the restriction of
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