WEAK SIMULTANEOUS RESOLUTION 13
By a finite base change for A: T- T, we mean a germ of a deformation
A': T ' -
71'
which is induced from A via a finite proper surjective map a: T - r,
a_1(0) = 0'. a expresses V as a finite branched cover of T. By [8] if A: T-* T has
a simultaneous RDP resolution, then after a finite base change, A has a very weak
simultaneous resolution.
Recall the definition for X to be a Moisezon space [37, 3, p. 126; 41]. X is then
the reduced analytic space associated to an algebraic space. By [3], for every point
p E X there exist a projective algebraic space 7, and a holomorphic map
o\Y - X such that a has a local holomorphic inverse near p. So in proving
Theorem 4.3 below, we may without loss of generality assume that % is projective
algebraic.
THEOREM
4.3. Let A: T-* T be the germ of a (flat) deformation of the germ
(V, p) of a normal Gorenstein two-dimensional singularity. Suppose that Kt Kt is
constant. Suppose that the map germ A: T- T has a representative \r\ % -
P1
such that % is a Moisezon space and Xr is a meromorphic function on %. Then A has
a simultaneous RDP resolution.
(V, p) may split into more than one singularity. For example, let/? = (0,0,0),
V= {z2 = x3 +yn] and T = (z2 =
JC3
+ (y2 + t)6}. For V = V0, on the
minimal resolution, A = Ax U A2 with ^ and A2 nonsingular and intersecting
transversely. gx = 1, g2 = 0; ^ -^4, = - 1 , A2-A2 -2 ; # -2AX A2\ K-K
-2. For t ¥= 0, Vt has two singularities, at x z = 0,
y2
= ^. Each singularity is
equivalent to z2 = x3 + ^ 6 at (0,0,0) with # K = - 1 . So ^ - K
t
= -2, all /. So
A: T-^ r = {t} has a simultaneous RDP resolution. In fact, following the proof
of Theorem 4.3 below, we blow up Tat the ideal
% =
(z,(y2
+
t)\x{y2
+
t),x2).
We then see that a 2:1 base change is required for a very weak simultaneous
resolution.
An outline of the proof of Theorem 4.3 is as follows. Since V is assumed to be
Gorenstein, there is a nonzero holomorphic 2-form o: on V p. Division by
u®m
exhibits Sm as a sheaf of ideals. Take m 3. We wish to blow up T simulta-
neously along the ideal sheaves Sm
r
, at least in case each Vt has a single
singularity. To do so, we need that all of the Sm
t
have the same Hilbert function.
We also need a coherent ideal sheaf im on Tsuch that §m restricts to Sm , on Vn
for all t E T sufficiently near to 0. By Theorem 3.3, the blow-up of Vt at %mt is the
RDP resolution. A slightly different argument is actually used so as to include the
case of Vt having multiple singularities.
Let 7r: M V be the minimal resolution of V. Let D 0 be a cycle on
A = 7T~\p), A UAf. Recall that Q(-D) is the ideal sheaf of germs of functions
on M which vanish on At to that order given by D, 6D := 0/S(-D). Let
hl(QD)
denote dim
Hl(M,
(9D). Recall the Riemann-Roch Theorem [45, Proposition IV,
4, p. 75]
(4.1) x(D):= h°(8D)-hi(eD) = \(D-D + D-K).
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