Suppose that we have already shown that T:¥(&(mK)) locally free
implies that Sm ° \p r^(&(mK)), for all T. Letting T: M -* N be the identity map
then implies that Sm ° 7 7 = 6(mK). So it suffices to prove the last statement of the
Observe that ^#(TJk(0(mA'))) = ^(©(raAT)) = Sm. Also, since the construc-
tions of the various sheaves are local with respect to V, we may assume that V is
Let x E N. Let A1?... ,A generate Sm^(x). Each A, is the direct image of a A^
defined in a neighborhood of
in M. There is an exact sequence near
(3.5) e ^ 0 ^ - S
- O
with Xi the image of (0,..., 0,1,0,..., 0). Tensoring (3.5) over 0
with 0^ gives
which is exact at each point near x E N.
There is a map of coherent sheaves a: 0^ - 7T^(6(mK)) given by
a(0,...,0,1,0,...,0) = irJ|c(X-)- a induces a map 0: ^*SW - T*(G(mK)). ima =
im /?. /? is an isomorphism off a nowhere dense subvariety TV' of TV. ker/?, since it
is supported on TV', is a torsion sheaf. rHe(0(mAT)) is assumed locally free
(necessarily of rank 1), so ker ft = T(^*Sm) and im /? = Sm ° 1//. Then also im a =
Let a': - 0(miO be given by a'(09... ,0,1,0,... ,0) = A;, By Theorem 3.1,
a' is onto. By hypothesis, r^(&(mK)) is locally free, necessarily of rank one. So
r(r - 1 (*), 6(mK)) has a single generator (over QN
). By Theorem 3.1, the single
generator for
6(mK)) is nowhere zero on r~\x). Then mK restricts to
the trivial line bundle on r~\x). By Theorem 3.1, there is a A; which is nowhere
zero on r'\x). So &(mK) « 6M near r~\x). Since T V is normal, r^ is an
isomorphism on holomorphic functions. So a is onto. This concludes the proof of
Proposition 3.4.
We shall now complete the proof of Theorem 3.3. We do the m ^ 2 part first.
Let \p: T V - F be the RDP resolution of V. Recall that T V is obtained from the
minimal resolution 77: M -* V by blowing down the curves ^4/ onM such that
At- K 0. mK restricts to the trivial line bundle near At such that Ai K 0. Let
T: M-^ TV. Then T^(6(mK)) is locally free. By Proposition 3.4, Sm ° ^ =
5 | C
is locally free. So by the universal property of j: X ^ V, the mon-
oidal transformation with respect to Sm, there is a proper modification A: T V X.
Recall that X « F, the normalization of F, wherever Kis a manifold. T V is also the
RDP resolution of V. So A is an isomorphism except possibly on the
At is an exceptional curve onM, which blows down via m. Each point of V is the
image of only finitely many At. So each point of X is also the image of only
finitely many
T ( ^
) .
If Ar K = 0, then r(At) is a point. If ArK^ 0,
cannot be a point, for then r^(6(mK)) would be the trivial sheaf in a neighbor-
hood of T(At), which it is not. So A maps
to a 1-dimensional set, i.e. A is a
Previous Page Next Page