WEAK SIMULTANEOUS RESOLUTION 11

the only other At having nonzero coefficients in Z' satisfy At-K— 0. For m still

larger, (mK - Z,)-Ai ^ 2K-Ai9 all i. So by Theorem 3.1, for x2 £ Ax, mK has a

section vanishing at xx and not at JC2. The remaining subcase is that xx and x2 are

not in distinct At, say x„ x2 E ^ H ^42 n • • • n^4y. Let Ys from Proposition 2.1

be the first Yk in which an Ai9 1 / y, appears. Then m# - Ys_x has a large

Chern class on ,4,. So sections of 6(mK — Ys_x)/&(mK — Ys) separate x, and x2.

By Proposition 2.1, these sections lift to sections of mK.

T: M - TV, with JV the RDP resolution of K So we have shown, so far, that

X: N ^ Xis one-to-one (and onto).

Let us now show normality at y(x), x E A". y(x) = y(A"). Let Zx be the

fundamental cycle of A". For m large, (mK - 2ZX)-Ai #^4, , all i. Since mK is

trivial near ,4", Q(mK - Zx)/e(mK - 2ZX) « 0(-Z

x

)/0(-2ZJ . Let m be the

maximal ideal at

T(X)

=

T(A").

m/m

2

is isomorphic to T(A'\ 6(-Zx)/6(-2Zx))

[1, Theorem 4, pp. 132-133].

Finally for x £ ^4', we must show that y is biholomorphic near x. Let x E Ax.

Let Z' as above be the least cycle on A such that Z' ^ Ax and (mK — Z') -Al ^ 0

for all i. Let Z" = Z' - Av Then 6(mK - Z")/Q(mK- Z') is the sheaf of

germs of sections of a line bundle L over the curve ^,. By making m large, L

can be chosen to have arbitrarily large Chern class. Then, by (2.1), for m large,

L has no base points and generators over C of T(A{,Q(L)) embed Ax in

projective space. For m large, by Proposition 2.1,

r(/w, 6(mK - Z")) - T(AX, ©(miT - Z , r ) / 0 ( w ^ - Z'))

is surjective. Let \\,... ,A', E T(M, 6(mK — Z/r)) project onto a basis of

r(v41? 0(L)). Let A'^x) ^ 0. There are two cases. If x is a singular point of Ax, the

embedding of Ax in projective space via the X] necessarily extends to show that y

is biholomorphic near x. For x a regular point of A{, choose A'2 so that in

r(^41? 0(L)), X2 maps to a section of L with a first order zero at x. By Theorem

3.1 applied to mK — Z', mK has a section A'3 which vanishes on Ax to first order

near x. Then (A^: A'2: A'3) embeds M in P

2

near x. Then y is again biholomorphic

near x.

This concludes the proof of Theorem 3.3.

IV. Very weak simultaneous resolution—special case. We now specialize to the

case where V has only normal Gorenstein singularities. Let m\ M - V be the

minimal resolution of V. Look at Fnear a singularity/?. Since/? is assumed to be

Gorenstein, near p there is a meromorphic two-form o which is holomorphic and

nonzero on V-p [5, Proposition 5.1, p. 16 and 16, Satz 3.1, p. 278]. The divisor

of 77*(co) is then supported on A = w'\p) and is a representative for the

canonical divisor K. We also let K denote the divisor of w*(w). Then KK is a

well-defined integer, which is an invariant of the singularity/?. Observe again that

the definition of KK depends on choosing the minimal resolution for V.

Nonminimal resolutions will give more negative values for the self-intersection of

the canonical divisor. Observe that by Theorem 3.1, -m2K- K is the multiplicity