# On tangency in equisingular families of curves and surfaces

## TL;DR

This paper investigates the behavior of tangent limits in families of curves and surfaces, introducing the $s$-invariant for curves and analyzing tangent cone stability under Whitney equisingularity.

## Contribution

It introduces the $s$-invariant for curves and demonstrates its constancy implies constant tangent counts in Whitney equisingular families, while also showing limitations for surface singularities.

## Key findings

- The $s$-invariant remains constant in Whitney equisingular families with constant tangent counts.
- Whitney equisingularity does not guarantee homeomorphic tangent cones for surface singularities.
- The number of exceptional tangents can vary despite Whitney equisingularity.

## 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.