On the fifth Whitney cone of a complex analytic curve

Abstract

We present an explicit procedure to determine the C5-cone of a reduced complex analytic curve X ⊂ C n at a singular point 0 ∈ X, which is known to be a union of a finite number of planes. We also give upper bounds on the number of these planes. The procedure comes out with a collections of integers that we call auxiliary multiplicities and we prove they determine the Lipschitz type of complex curve singularities. We finish presenting an example which shows that the number of irreducible components of the C5-cone of a curve is not a bi-Lipschitz invariant.