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On linear deformations of Brieskorn singularities of two variables into generic maps. (English) Zbl 1376.57033
The authors study linear deformations of Brieskorn polynomials in two complex variables \((z,w)\) of the following type: \(z^p+w^q+a\overline{z}+b\overline{w}\), where \(p,q\geq 2\) and \(a,b \in \mathbb C\). In particular, they prove that for parameters \(a\) and \(b\) generic enough the corresponding polynomial maps have only indefinite fold and cusp singularities. They also show that the number of cusps of such maps belongs to the interval \([(p+1)(q-1),(p-1)(q+1)]\), so that this number for a complex Morse singularity is equal to 3, etc. It should be noted that in a different setting the same numerical invariant was studied by G.-M. Greuel [Manuscr. Math. 21, 227–241 (1977; Zbl 0359.32008)] for complex isolated hypersurface singularities.

MSC:
57R45 Singularities of differentiable mappings in differential topology
58K20 Algebraic and analytic properties of mappings on manifolds
58K25 Stability theory for manifolds
32S10 Invariants of analytic local rings
32S30 Deformations of complex singularities; vanishing cycles
14H50 Plane and space curves
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