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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
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