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On the weak simultaneous resolution of a negligible truncation of the Newton boundary. (English) Zbl 0682.32011
Singularities, Proc. IMA Participating Inst. Conf., Iowa City/Iowa 1986, Contemp. Math. 90, 199-210 (1989).
[For the entire collection see Zbl 0668.00006.]
Let $$\{f_ t\}$$ be a one-parameter family of germs of holomorphic functions in n variables such that $$V_ t=f_ t^{-1}(0)$$ has an isolated singularity and the Milnor numbers $$\mu (f_ t)$$ are constant $$=\mu (f_ 0)$$. For $$n\neq 3$$ this implies the topological stability of $$V=\{V_ t\}$$, i.e. $$V_ t$$ is topologically equivalent to $$V_ 0$$. For $$n=3$$ there exist only proofs under additional assumptions. One of them requires the existence of a weak simultaneous resolution of V. This motivates the author’s theorem: Let $$f_ t(z)=f_ 1(z)-(1-t)b_ Az^ A$$ (A a triple index) satisfy certain conditions concerning the Newton diagram of $$f_ t$$. Then a weak simultaneous resolution exists. It is obtained by toroidal embeddings. The author gives examples where his criterion works whereas the other known sufficient criteria for topological stability fail.
Reviewer: K.Lamotke

##### MSC:
 32Sxx Complex singularities 32S45 Modifications; resolution of singularities (complex-analytic aspects) 32S30 Deformations of complex singularities; vanishing cycles 14E15 Global theory and resolution of singularities (algebro-geometric aspects)