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Invariants of equidimensional corank-1 maps. (English) Zbl 1072.32019
Janeczko, Stanisław (ed.) et al., Geometry and topology of caustics – CAUSTICS ’02. Proceedings of the Banach Center symposium, Warsaw, Poland, June 17–29, 2002. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 62, 239-248 (2004).
This paper concerns the singularity theory for complex-analytic equidimensional corank-1 germs \(f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)\). The \(\mathcal{A}\)-finiteness property is introduced. It is defined by a set of integer invariants (called \(\mathcal{A}\)-invariants) \(v_{\mathbf{k}}(f),\) \(\mathbf k=(k_{1},\dots,k_{s})\), \(1\leq\sum k_{i}\leq n\). It is shown that the finiteness only of the known invariants \(r_{\mathbf{k}}(f)\) does not ensure the \(\mathcal{A}\)-finiteness of \(f\).
In a paper of W. L. Marar, J. A. Montaldi and M. A. S. Ruas [Lond. Math. Soc. Lect. Note Ser. 263, 353–367 (1999; Zbl 0947.32017)] formulae had been given for \(r_{\mathbf{k}}(f)\), \(\sum k_{i}=n\) in the case of weighted homogeneous corank-1 germs \(f\). New formulae for the number of transverse singular points which appear in generic deformations of \(\mathcal{A}\)-finite corank-1 germs are given. The obtained general theorem says that the mentioned germs \(f\) are \(\mathcal{A}\)-finite iff the associated set of \(\mathcal{A}\)-invariants is finite. The author’s statement of this theorem is very precise (formulated in 4 points) and the proof is quite technical
For the entire collection see [Zbl 1031.14001].

MSC:
32S05 Local complex singularities
32S10 Invariants of analytic local rings
58K60 Deformation of singularities
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