# zbMATH — the first resource for mathematics

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