On Tangency in Equisingular Families of Curves and Surfaces

We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the property that the $s$-invariant is constant along the parameter space, the number of tangents of each curve of the family is constant. In the context of families of isolated surface singularities, we show through examples that Whitney equisingularity is not sufficient to ensure that the tangent cones of the family are homeomorphic. We explain how the existence of exceptional tangents is preserved by Whitney equisingularity but their number can change.

Whitney equisingularity of families of surfaces in ℂ3

In this paper, we study families of singular surfaces in ℂ3 parametrised by $\mathcal {A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding F of a finitely determined map germ f : (ℂ2, 0) → (ℂ3, 0). We investigate the following question: topological triviality implies Whitney equisingularity of the unfolding F? We provide a complete answer to this question, by giving counterexamples showing how the conjecture can be false.

EULER OBSTRUCTION, POLAR MULTIPLICITIES AND EQUISINGULARITY OF MAP GERMS IN $\mathcal{O}(n,p), n < p$

There are two main goals in this article, one of them is to minimize the number of invariants needed to obtain Whitney equisingular one parameter families of finitely determined holomorphic map germs ft:(ℂn,0) → (ℂp,0), with n < p. The other is to show how to compute the local Euler obstruction of the stable types which appear in a finitely determined map germ in these dimensions. The polar multiplicities of all stable types and the 0-stable singularities are the invariants that guarantee the Whitney equisingularity of such families and the polar multiplicities are the numbers that also allow us to compute the local Euler obstruction. Therefore our first step is to describe all stable types which appear when n < p and show the relationship between the polar multiplicities in each stable type. Using the fact that these polar multiplicities are upper semi-continuous we minimize the number of invariants that guarantee Whitney equisingularity of such a family. We also apply the relationship between the polar multiplicities in each stable type and a result of Lê and Teissier to show how to compute the local Euler obstruction of the stable types which appear in these dimensions.