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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)