WEAK SIMULTANEOUS RESOLUTION
25
Since Bx is irreducible, gx(Bx) is contained in an irreducible component Sj of S.
Since gx(Bx) D y °
OL(TX)
by construction, in fact gx(Bx) C Sx. Then gx(R X O)
C Sj n /(Ta). But the deformation X,: Tj - 7^ may be induced, by construction,
from
T : 9 C - £ /
restricted to
T_1(g(/{))
=
T_1(gi(fi
X O)). This contradicts
Lemma 5.5 and our selection above, in the first paragraph of the proof, of
y (E (E n S[) - [E D (Sx n f(Ta))'].
THEOREM
5.7. Let \\°H'-» 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 very weak simultaneous resolution, possibly after a finite base change, if and
only if Kt Kt is constant.
PROOF.
If X has a very weak simultaneous resolution, then Kt Kt is constant by
Proposition 5.4.
Conversely, assume that Kt Kt is constant. X: °V^ Tis induced from T: % - U
via a holomorphic map g: T - U with g(T) C S of Theorem 5.6. f:Ta-+ U has
f(Ta) = S by Theorem 5.6. Let 7" = T X
v
Ta be the fiber product [12, Corollary
0.32, p. 29]. Then the map a: T' -+ T is finite and proper since / is finite and
proper, a is surjective since g(T) C f(Ta). Then a is a finite base change. Via the
map /?: V Ta we see that the induced deformation X': T ' - r
r
has a very weak
simultaneous resolution.
VI. Weak simultaneous resolution. Recall Neumann's theorem [38]. Let (V, p)
be the germ of a normal two-dimensional singularity. Let TT'\ M' V be the
minimal good resolution. Let A' be the exceptional set in M''. Then the oriented
homotopy type of V p determines the topology of the pair (M\ A'). Hence also
if 7r: M - V is the minimal resolution, the oriented homotopy type of V p
determines the topology of the pair (M, A). In most cases, (M, A) is already
determined by the fundamental group of V p. Observe that Neumann's theo-
rem implies that the oriented homotopy type of V p determines the topology of
(V, p).
Let X: T-* T be the germ of a deformation of the normal Gorenstein
two-dimensional singularity (V, p) with Treduced. If one of the singularities/?, of
Vt is topologically equivalent top in V, then in fact/?, is the only singularity in Vr
Indeed, since Kt Kt is the sum of K- K over all singularities in Vt, Theorem 5.2
implies that t E S. By Theorem 5.7, we may resolve X over S, after a finite base
change. Choose the minimal very weak simultaneous resolution. Then by [32,
Proposition 2.5, p. 5] and Neumann's theorem, pt is the only a singularity in Vr
LEMMA
6.1. Let X: T-* T be the germ of a (flat) deformation of the normal
Gorenstein two-dimensional singularity (V, p) with T a reduced analytic space. Then
W {t E T\ Vt has a singularitypt such that (Vt, pt) is homeomorphic to (V, /?)} is
a subvariety of T.
PROOF. AS
observed above, W C S, S from Theorem 5.2. As above, after a
finite base change, resolve X above S with the minimal very weak simultaneous
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