WEAK SIMULTANEOUS RESOLUTION
7
We now continue as in [26, Theorem 7.5, p. 137]. Let/
l 9
f2 G T(A, G(LX)) have
no common zeros. Then the sheaf map 0 ( L
2
) © 0 ( L
2
) ^ 0 ( L
1
+ L
2
) given by
(#i £2) ~* (/i ® &i + A ® £2) is surjective near ^4. Let 5Cbe ker y, so that
0 * X  0(L
2
) 0 0(L
2
)  0(L! + L2)  0
is exact. For all « ^ 0, multiplying by the ideal sheaf 6(nZ) gives the exact
sequence
0  DCe(nZ)  0(L2  nZ) 0 G(L2  «Z)  e(L, + L2  nZ)  0.
r(/i,e(L2/iz)) © r(A,e(L2nz))  r(^,e(L,
+ L
2
*Z))

Hl(A,xe(nz))
r(^,e(L2» © r(A6(L2))
T4
r(^e(L,+
L2))

H\A,%)
is commutative with exact rows. By [14, §4 Satz 1, p. 355], a* is the zero map for
n sufficiently large. Choose such an n. Then any h E T(A, 6(LX + L2 — nZ)) is
in the image of y*. This concludes the proof of Theorem 3.2.
Return to F, our given purely twodimensional analytic space. Let £: M'  F
be a resolution of F[20]. Let Sm = ^(6(mK)). Since all resolutions of Fmay be
obtained from each other via quadratic transformations, Sm is, in fact, indepen
dent of the choice of the resolution £. So when convenient, we may take
£: M'  V to be 77: M  F, the minimal resolution.
Recall that AT is also a resolution of the normalization of F Observe also that
in case Fis normal and has only Gorenstein singularities, Sm is a sheaf of ideals.
Let S be a coherent analytic sheaf on F S is necessarily locally free off a
nowhere dense subvariety S of F Suppose that S has constant rank on V — S.
Then [41, pp. 268272] there is a unique proper modification mapping J : X  F
which is the monoidal transformation of Fwith respect to S. In [41], it is assumed
that F is irreducible. But irreducibility is used only to conclude that S has
constant rank on V — S. In case S should be a sheaf of ideals, § coincides with
the usual monoidal transformation at an ideal [41, p. 271]. We are especially
interested in the case that S = Sm. We may choose S to be the singular locus of V.
Then Sm has rank 1. ^
w
is then given locally as follows. Let y E F Let A
l5
... ,\p
be generators of Sm near y. Near y, let JUW: V — S * F X P^
_ 1
be given by
/xm(z) = (z,(\x(z):   • :Xp(z))). Then fim is an embedding. Xm is the closure in
F X P ^
 1
of im jum. jm\ Xm  F is induced by projection onto F in F X P^"
1
.
Then jm: Xm » Fis (locally) the monoidal transformation of Fwith respect to S.
Recall the RDP (rational double point) resolution of F [36], as follows: Let
77: M  F be the minimal resolution of F Blow down in M all nonsingular
irreducible curves Ai such that gt = 0, AiAi — 2 and 77(^4,) is just a point. Call
the blowndown space TV. All of the singularities of N are rational double points.
7 7 induces a proper modification mapping $: N » F, called the RDP resolution of
F
Theorem 3.3 below is similar to [46, Corollary 6, p. 7]. See also [36, Theorem, p.
38].