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
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