On Tangency in Equisingular Families of Curves and Surfaces

Abstract

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.

On the fifth Whitney cone of a complex analytic curve

“…Since X is Cohen-Macaulay, then X t is reduced for all t. Therefore, ϕ t is a Puiseux parametrization of X t . Note that ChAM (X t , (0, 0, 0, t)) = {6, 9, 10} for all t ∈ T , thus by Proposition 4.11 the family of C 5 -generic plane projections Xt is a topologically trivial family of plane curves, the bi-Lipschitz equisingularity of p : X → T follows from [4,Cor. 3.6].…”

“…In the case of the C 3 -cone of curves, it is known that the number of irreducible components of the cone need no be constant in Whitney equisingular families of curves, see [4,Ex. 4.13].…”

“…4.13]. It has also been proved that in bi-Lipschitz equisingular families of curves, the number of irreducible components of the C 3 -cone is constant, see [4] and also [10] for a more general situation. Thus, we have a natural question: Is the number of planes of C 5 (X, 0) a bi-Lipschitz invariant for a curve (X, 0)?…”

“…In fact, another work on an implementation of this procedure in Singular program [7] is being carried out by the second author and Aldicio Miranda (from Universidade Federal de Uberlândia -Brazil) and we hope it will be available soon. (4) , 0) be the germ of curve in (C 3 , 0) where (X (i) , 0) is parametrized by the map ϕ i : (C, 0) → (X (i) , 0), defined respectively by: ϕ (1) (u) := (u 4 , u 6 , u 9 ), ϕ (2) (u) := (u 6 , u 11 − u 9 , u 11 + u 9 ), ϕ (3) (u) := (u, 2u, u) and ϕ (4) (u) := (u 4 , u 3 , u 5 ).…”

“…Recall that when p : X → T is bi-Lipschitz equisingular, then for each t, there exist a bi-Lipschitz homeomorphism from X 0 to X t . It has been proved that in bi-Lipschitz equisingular families of curves, the number of irreducible components of the C 3 -cone is constant, see [4] and also [10] for a more general situation. This implies that C 3 (X 0 , 0) and C 3 (X t , 0) are homeomorphic.…”