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
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