WEAK SIMULTANEOUS RESOLUTION 21

on supp(£0) - M'. S o a ^ O in 0((m + l)K0) ® 0 ( - # J on supp(£0) - M\

near AT. So a^ may be further extended by 0 (regardless of Hm on the rest of E0)

to ak G T(M0, 0((m + 1)AT0) ® 6(-Hm)). That is, ^r E r(M0,r, 0((m + l)tf0)

® 6(F*+] — Hm)). This completes the proof of the lemma.

V. Very weak simultaneous resolution-general case. Weak simultaneous resolu-

tions will appear in our proofs, so let us recall the definition [49, Definition 3.1.1,

p. 105]:

DEFINITION

5.1. The map germ II: 9H - Fis a weak simultaneous resolution of

the germ of the (flat) deformation A: T- T along the image Tx of a section

a: T -* V, T reduced, if for all sufficiently small representatives of A, the germ II

has a representative such that:

(0)-(ii) of Definition 4.1 are satisfied and^_^^,

(iii) the map induced by restriction p : = A ° II:

(Jl~l(Tl))Ted

- Tis simple, i.e.

a locally trivial deformation.

We are interested, of course, in the case where A: T- T is the germ of a

deformation of the normal two-dimensional singularity (V9 p). By Definition

5.1 (iii) and [32, Proposition 2.5, p. 5], if II: ?Jlt- T is a weak simultaneous

resolution along Tl9 then I I " 1 ^ ) is the full exceptional set in 9H and each Vt has

just one singularity. Conversely, if II: 9H - °\f is a very weak simultaneous

resolution from Definition 4.1, let S be the exceptional set in 9H and p: Sred - T

be the induced map. If jo is simple, then each Vt has just one singularity, pr

o: T -* Tgiven by o(t) = /, is a section, i.e. is holomorphic, seen as follows. Let

a,: r- 91L be a section with a^T) C Sred. a! exists since p is assumed to be

simple. Let o = H ° ov So we may define weak simultaneous resolution without

introducing a section a.

For each t, Sred has only plane curve singularities. Plane curve singularities

have a well-understood equisingularity theory [55, 49]. So A: T-* T has a weak

simultaneous resolution if and only if A has a very weak simultaneous resolution

II: 9IL - Tsuch that p: Sred -» Tis an equisingular deformation of A : = Sred0.

(F, /?) is our given germ of a normal Gorenstein singularity, by [2, Theorem

3.8, p. 32], we may assume that V has a projective algebraic representative. Let

T': 9G' - t/' be the versal deformation of (K, /?). T' is often called the miniversal

or semiuniversal deformation of (K, /?). T' is the germ of a deformation. By [10],

we may assume that T' has a projective algebraic representative r/: 9C^ -» £//. Let

r:%^U and v 9Cr - Ur be the reductions of T': %' - £/' and r/: %; - ££

respectively.

THEOREM

5.2. Let A: 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

-K0 • K0 -Kt • Kt for all t G T sufficiently near to 0. Also, S '.= {t E T\ Kr Kt

— K0- K0} is a subvariety of T.

PROOF. TO

prove this theorem, it suffices to prove it for the special case that

A: T-* T is T: % - [/, the reduction of the versal deformation of (V, p). Let