The Lê varieties. II. (English) Zbl 0727.32015

Let \(f: ({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a polynomial with singular locus of dimension s. In part I of this paper [ibid. 99, 357-376 (1990; Zbl 0712.32020)] the author defines a sequence of varieties \(\Lambda_ f^{(k)}\) using the polar varieties. The aim is to obtain similar informations in case \(s>0\) as in the case of isolated singularities from the Jacobian ideal. Especially the multiplicities of \(\Lambda_ f^{(k)}\) (the so called Lê-numbers) generalize the Milnor number.
In case of an isolated singularity \(\Lambda_ f^{(k)}=\emptyset\) if \(k>0\) and \(\Lambda_ f^{(0)}\) is the variety defined by the Jacobian ideal \((\frac{\partial f}{\partial z_ 0},...,\frac{\partial f}{\partial z_ n})\). The author investigates the Milnor fibration of f using his Lê varieties to extend the result of Lê and Ramanujan concerning the constancy of the Milnor fibration in a family of isolated singularities to nonisolated singularities: The constancy of the Lê-numbers in a family of singularities implies the constancy of the fibre-homotopy type of the Milnor-fibration.
Reviewer: G.Pfister (Berlin)


32S05 Local complex singularities
32S55 Milnor fibration; relations with knot theory
14B05 Singularities in algebraic geometry


Zbl 0712.32020
Full Text: DOI EuDML


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