WEAK SIMULTANEOUS RESOLUTION 27
THEOREM 6.3. Let g: Tw U be as above. Let W {u E U | Xu has a singularity
pu such that (Xu9 pu) is homeomorphic to (V, p)}. Then g(Tw) W.
PROOF.
g(Tw) C W be Definition 5.1. The rest of the proof of Theorem 6.3 is
the same as the proof of Theorem 5.6 via the changes:
"W" replaces " S " and the condition {Xu has a singularity pu such that
(Xu9 pu) is homeomorphic to (V, p)} replaces the condition {Ku Ku = K0 K0}.
"Tw" replaces "Ta".
Lemma 6.1 replaces Theorem 5.2.
Lemma 6.2 replaces Theorem 4.3 and [8]. The base change a: Tx - T is not
needed.
co: DR/ - 0 ' replaces co: 9H g.
THEOREM
6.4. Lef X'.'Y-* T be the germ of a (flat) deformation of the normal
Gorenstein two-dimensional singularity (V, p) with T a reduced analytic space. Then
X has a weak simultaneous resolution if and only if each Vt has a singularity pt such
that (Vv pt) is homeomorphic to (V, p).
PROOF.
This theorem follows immediately from Definition 5.1, Theorem 6.3,
the versality of r: % - U and the fact [52, Theorem 4.6(b), p. 341] that g: Tw - W
is biholomorphic.
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