O quinto cone de Whitney de uma curva analítica complexa

Abstract

In 1965, in an attempt to de ne a tangent set for an analytic space at a singular point, Whitney proposed six objects that are now known in the literature as Whitney's tangent cones. Subsequently, in 1980, Briançon, Galligo, and Granger proved that the fth Whitney cone of a reduced singular complex curve is a nite union of 2- dimensional planes. Following that, Krasi«ski, and independently Giles-Flores, Silva, and Snoussi, developed an algorithm to describe this union as a set. Therefore, our main goal in this work was to study this procedure that describes the C5 cone as a set using only appropriate parametrizations of the irreducible branches of the curve. In the development of this algorithm, we also studied ordered sequences of multiplicities that determine and are determined by the Lipschitz type of the singular curve. Through these sequences, we also veri ed that the number of planes in the C5 cone of a singular curve is not a bi-Lipschitz invariant. However, we created a family of curve examples that have a pre-established number of planes in their C5 cone and presented some conjectures on this subject.