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