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The Lê-Ramanujam problem for hypersurfaces with one-dimensional singular sets. (English) Zbl 0657.32005
This paper is a very nice generalization of the excellent result of Lê Dũng Tráng and C. P. Ramanujam [Am. J. Math. 98, 67-78 (1976; Zbl 0351.32009)]. The author investigates the case of a family f: ($${\mathbb{C}}^{n+1}\times D,0\times D)\to ({\mathbb{C}},0)$$ such that the singular set of each $$f_ t$$ is one dimensional at the origin. First some notations: If $$\alpha$$ is an ideal in $${\mathcal O}_ 0^{n+1}$$ and $$f\in {\mathcal O}_ 0^{n+1}$$ such that $$\dim_ 0(V(\alpha)\cap V(f))=0,$$ then $$(V(\alpha)\cdot V(f))_ 0$$ denotes the scheme-theoretic intersection number. If $$f\in {\mathcal O}_ 0^{n+1}$$ and L: $${\mathbb{C}}^{n+1}\to {\mathbb{C}}$$ is a linear form, then f can be assumed to be a polynomial in the coordinates $$(x_ 1,...,x_ n,L)$$. Let $$\nu$$ denote the ideal in $${\mathcal O}^{n+1}$$ generated by $$\frac{\partial f}{\partial x_ 1},...,\frac{\partial f}{\partial x_ n}$$; $$\nu_ f$$ the ideal generated by $$\nu$$ in the ring of fractions obtained from $${\mathcal O}^{n+1}$$ by inverting $$\{f^ n\}_{n\geq 0}$$; $$\gamma_{f,L}$$ the ideal $$\nu_ f\cap {\mathcal O}^{n+1}$$ in $${\mathcal O}^{n+1}$$ and $$\Gamma_{f,L}$$ denote the scheme Spec $${\mathcal O}^{n+1}/\gamma_{f,L}$$. Let $$\sigma_{f,L}$$ denote the ideal which is the intersection of the remaining isolated primary components of $$\nu$$ and $$\Sigma_{f,L}$$ denote the scheme Spec $${\mathcal O}^{n+1}/\sigma_{f,L}.$$
The main tool of the paper is a Iomdine-Lê type formula for families of polynomials. If $$f:(\mathbb{C}^{n+1} \times \overset\circ D$$, $$0\times \overset\circ D)\;to (\mathbb{C},0)$$ is a polynomial such that for all $$t\in \overset\circ D$$, $$f_ t=f_ t(x,s)$$ defines a hypersurface in $${\mathbb{C}}^{n+1}\times \{t\}$$ with $$\dim_ 0\Sigma V(f_ t)=1$$ and the coordinate s has been chosen so that $$f_ 0|_{V(s)}$$ has an isolated singularity at 0 and $$\overset\circ D$$ is chosen so small that $$f_ t|_{V(s)}$$ has an isolated singularity at 0 for all $$t\in \overset\circ D$$, then the author proves the “Uniform Iomdine- Lê Formula”: For all k sufficiently large, there exists $$\tau >0$$ such that for all $$t\in D_{\tau}$$, $$f_ t+s^ k$$ has an isolated singularity at the origin and $\mu (f_ t+s^ k)+\mu (f_ t|_{V(s)})=(\Gamma_{f_ t,s}\cdot V(f_ t))_ 0+k(\Sigma_{f_ t,s}\cdot V(s))_ 0.$ The major application of this formula is the possibility of establishing numerical conditions for $$\Gamma_{f,L}=\emptyset$$. If $$\lambda_ t=(\Sigma_{f_ t,s}\cdot V(s))_ 0$$ and $$\beta_ t=(\Gamma_{f_ t,s}\cdot V(\frac{\partial f_ t}{\partial s}))_ 0$$ are constant for all small t, then the polar curve of f at the origin is empty. In this case the Milnor fibre of f at the origin is diffeomorphic to the cross-product of a disc and the Milnor fiber of a generic hyperplane section of f.
In the special case when $$f_ t$$ is a family of line singularities the author proves the “Non-splitting Theorem”: If $$\lambda_ t$$ and $$\beta_ t$$ are constant for all small t, then the singular set of f does not split. In the case of constancy of $$\lambda_ t$$ and $$\alpha_ t=(\Gamma_{f,s}\cdot V(f_ t))_ 0$$ (which implies the constancy of $$\beta_ t$$ too), and $$n\geq 3$$, using both the h-cobordism theorem and the pseudo-isotopy result of Cerf, the author establishes the constancy of the ambient topological type of $$V(f_ t)$$ at the origin. Moreover, it is proved that the diffeomorphism type of the Milnor fibration of $$f_ t$$ at the origin is constant.
Reviewer: A.Némethi

MSC:
 32S05 Local complex singularities
Keywords:
local singularities
Full Text:
References:
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