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).