Proceedings of Symposia in Pure Mathematics

Volume 40 (1983), Part 2

WEAK SIMULTANEOUS RESOLUTION

FOR DEFORMATIONS

OF GORENSTEIN SURFACE SINGULARITIES

HENRY B.

LAUFER1

I. Introduction. Let X: T ^ T be a (flat) deformation of the two-dimensional

isolated hypersurface singularity (F, p). We assume that Tis reduced. This paper

contains a partial answer to the two well-known questions [49, p. 115]:

(a) Does \it : = ^3\Vt) constant imply weak simultaneous resolution?

(b) Does }i*(Vt) constant imply strong, or at least weak, simultaneous resolu-

tion?

The converses to (a) and (b) are known to be true [31, 49, 7]. In this paper we

shall prove

(a) (Theorem 6.4) If each Vt has a singularity/?, such that (Vn pt) is homeomor-

phic to (K, /?), then A: T-» T has a weak simultaneous resolution. It is known

[35], except for surface singularities, that /x, constant implies that (Vn pt) has

constant topological type.

(b) If ii*(Vt) is constant, then A: T-* T has a weak simultaneous resolution.

This follows immediately from Theorem 6.4, the fact that constant ii*(Vt) implies

the Whitney conditions [48] and the Thorn-Mather Theorem (see also [50]). We

shall discuss strong simultaneous resolution in another paper.

Let us start by only requiring that (V, p) be a purely two-dimensional singular-

ity. Let 77: M - Fb e a resolution of V. Let K be the canonical divisor on M. Let

Sm = 7rjQ(mK), a coherent sheaf of modules on X. Then [41, 19] we may blow-up

V at Sm, f: X -+ V. Then (Theorem 3.3), for m 3, $: X ^ V is the RDP

resolution of V9 i.e., Jf is obtained from the minimal resolution of V by blowing

down the rational -2 curves. In case (V, p) is Gorenstein, Sm is a sheaf of ideals

1980 Mathematics Subject Classification. Primary 32G11; Secondary 14B07.

1

Research partially supported by NSF Grant MCS-8102621.

©1983 American Mathematical Society

0082-0717/81 /0000-0644/$08.50

1

http://dx.doi.org/10.1090/pspum/040.2/713236