Singularities of polar curves. (English) Zbl 0806.14021

Casas-Alvero, E.

Compos. Math. 89, No. 3, 339-359 (1993).

Let \(\xi\) be a plane algebroid (reduced) curve defined over \(\mathbb{C}\), recall that two local curves \(C\), \(C'\) have the same topological type if there exists a local homeomorphism of \(\mathbb{C}^ 2\) which exchanges \(C\) and \(C'\). To determine the topological type of the polar curves of \(\xi\) has been an open problem since M. Noether. In fact the topological type of the polar curves depends on the analytical type of \(\xi\). The author solves this problem for a generic curve in the equisingular class giving the answer via the theory of infinitely near points with imposed singularities. More precisely he defines a cluster of infinitely near points with virtual multiplicities and show that the polar curve of such a generic curve goes through this cluster with effective multiplicities equal to the virtual ones and has not singularities outside this cluster.

Reviewer: M.Morales (Saint-Martin-d’Heres)

MSC:

14H20	Singularities of curves, local rings
14B10	Infinitesimal methods in algebraic geometry
14E15	Global theory and resolution of singularities (algebro-geometric aspects)
14B05	Singularities in algebraic geometry

Keywords:

equisingularity; topological type of the polar curves of algebroid curve; infinitely near points

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