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
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