12

H. B. LAUFER

of the ideal S for m ^ 2 [51, p. 420]. See also (4.3) below. In case V has

multiple singularities, we define KK for V as K- K := 2(A^ A") , with the sum

over all singular points of V.

Let (F, p) be the germ of V at p. Let A: T-* T be the germ of a (flat)

deformation of (F, p). Let 0 denote the initial point in T. Arbitrarily small

representatives for A, Tan d T will also be denoted by A, T and T. We assume

that T is reduced. Vt :=

A_1(f).

Flatness implies that Vt is two-dimensional. We

allow Vt to have multiple singularities. We can see as follows that each singularity

of Vt is normal and Gorenstein: Replacing T by a resolution of T [18], we may

assume without loss of generality that T is smooth. The special fiber V — V0 of T

is normal, so T is normal. Then Vt is normal. So F, is also Cohen-Macauley since

it is of dimension two. By [17, 1.40(a), p. 13], T i s Gorenstein. So also Vt is

Gorenstein and Kt • Kt is defined as K • K for Vt.

Let II: 911 - °\f be the germ of a holomorphic map. M, := (A o U)~\t). Let II,

denote the restriction of TL to Mr Recall [49, Definition 2, pp. 72-73]

DEFINITION

4.1. The map germ II: 9H - Ti s a very weaA; simultaneous resolu-

tion of the germ of the (flat) deformation A: T ^

71, 71

reduced, if for all

sufficiently small representatives of A, the germ n has a representative such that:

(0) ^ is a proper modification map,

(i) A o It: 91L -* Tis a flat map,

(ii) %: Mt - Vt is a resolution of Vv all /.

Recall [15, Satz 2.3, p. 244] in Definition 4.1 that with M0 assumed to be

smooth in (ii), flatness in (i) just means that A ° II is a locally trivial deformation

of M0.

DEFINITION

4.2. The map germ ^ : 91 -* V is a simultaneous RDP resolution of

A: T-* r, r reduced, if for all sufficiently small representatives of A, the germ ^

has a representative such that:

(0) ^ is a proper modification map,

(i) A o V\ 91 - 7 is a flat map,

(ii) ¥,:#,- Fr is the RDP resolution of F

r

Given a very weak simultaneous resolution II: 9IL - F we may simultaneously

blow down all the exceptional curves of the first kind in the Mt [25, Theorem 3, p.

85; 43, Satz 2, p. 548; 4, Lemma 2.1, p. 334]. After finitely many such simulta-

neous blow-downs, we may assume that II: 91L - V is a very weak simultaneous

resolution with each II,: M, - Vt the minimal resolution. Let Ati be an irreduci-

ble component of An the exceptional set in Mn such that Ati • A', = 0, i.e., ylM is a

nonsingular rational curve with Ati-Ati — -2. By [32, Proposition 2.3, p. 4], Ati

is homologous in 9Hto a (unique) cycle Di on A0 • = ^4 such that Dt-K— 0. Since

n

o

: M0 - F0 is the minimal resolution, all components ^4 • of ^4 with supp(^4y) C

supp(Z)i) satisfy ^^; • K = 0. So blowing down a neighborhood in 91L of all the Aj

in A such that ^ , • K = 0 [52, 42] yields a simultaneous RDP resolution "&:%-+ V