26
H. B. LAUFER
resolution, II: 91t- T-* S. We may ignore base changes. By Neumann's theo-
rem, for w E W, Aw has the same intersection matrix as A0 A. Then, as in the
proof of [34, Theorem 2.11, p. 189], each irreducible component is homologous in
911 to an irreducible component Awi olAw. Let 5" be the maximal subvariety of S
above which all of the Ai weakly lift [32, Proposition 2.7, p. 6], Then W C S'. Let
a: S" - S' be a resolution of S", a proper map. It suffices to prove the lemma
near each point of S" for the induced deformation over S". Since S" is smooth,
each Ai locally lifts to above S"'. Let fi(As) be the sum of the Milnor numbers for
the (plane curve) singularities of As, s E S". Then fi(Aw) = fi(A) for w E W.
Then Wis the subvariety of 5"' where \i(As) is constant.
LEMMA
6.2. Le/ A: T-* T be as in Theorem 4.3. Suppose additionally that each Vt
has a singularity pt such that (Vn pt) is homeomorphic to (V, p). Then X has a weak
simultaneous resolution such that each fiber is a minimal good resolution.
PROOF.
Let ^ : 91 - T b e the RDP resolution of Theorem 4.3. By Neumann's
theorem, each fiber Nt has an isomorphic set of rational double points. Since JU, is
upper semicontinuous, the deformation which ^ induces on the rational double
points is trivial. So there is a minimal weak simultaneous resolution of the
singularities of 91. This gives a very weak simultaneous resolution of A, II: 911 -*
V. The fibers Mt are all homeomorphic near the exceptional sets A
r
As in [34, p.
189], II induces an equisingular deformation of A = A0. Then additional simulta-
neous blow-ups of the Mt along suitable singularities of the At give a weak
simultaneous resolution. The fibers can all be chosen to be minimal good
resolutions.
Let (V, p) be as above. Let 77': M' - V be the minimal good resolution of V,
the MGR in the resolution of [52]. Let
cor:
911' - (?' be the versal deformation of
M' [33]. Let
y4r
be the (reduced) exceptional set in AT. As in [52, p. 333], let S be
the sheaf of vector fields on M' which map Id A' to Id
Af.
Let 0' be the tangent
sheaf on M'. Let 91: = 0'/S. Then 91 « © % , where 9ly is the normal sheaf to
^4-, an irreducible component of A'. As in [32, (2.2), p. 7],
0 - #
!
( AT, S) -^ H\M\ 0) - H\M\ 91) -^ 0
is an exact sequence. By the construction in [33, p. 313], there is a submanifold P
of Q\ 0 E P, with tangent space TP0 « H\M', S) such that all of the A\ lift to
above P. By [34, Theorem 3.5, p. 194], P is the maximal subspace of Q' above
which all of the A\ lift weakly.
Let Tw = {w E P I Aw = /z0} be the simultaneous blow-down-subspace of P.
Let X^: % P^ be the induced deformation. Since the tangent space to Tw is
contained in H\M\ §), the induced map from Tw to ES of Wahl [52, especially
p. 335] is an embedding. Recall [52, Theorem 4.6(b), p. 341], the induced map
ES U' is injective. So any g: Tw - t/which induces Xw from the reduced versal
deformation T: % ^ U is an embedding (see [22, pp. 205 —18] for a treatment in
the analytic category).
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