Brian\c{c}on-Speder examples and the failure of weak Whitney regularity

TL;DR

This paper investigates whether Whitney regularity and weak Whitney regularity coincide in complex analytic varieties by systematically analyzing Briançon-Speder examples, finding that none are weakly Whitney regular and identifying specific curves where regularity fails.

Contribution

The paper provides the first systematic analysis of Briançon-Speder examples, showing they are not weakly Whitney regular and characterizing the curves where regularity fails in the complex case.

Findings

None of the Briançon-Speder examples are weakly Whitney regular.

Identified all complex curves where Whitney regularity fails.

Identified all complex curves where weak Whitney regularity fails.

Abstract

It is easy to find real algebraic varieties with weakly Whitney regular stratifications which are not Whitney regular, and we give such an example in section 3 below. No examples are known among complex analytic varieties however, so that the natural question arises : do Whitney regularity and weak Whitney regularity coincide in the complex case ? As a test, in this paper we study the well-known Brian\c{c}on-Speder examples. We investigate systematically all of the (infinitely many) Brian\c{c}on-Speder examples, and establish in particular that none of these examples are weakly Whitney regular. We determine all the complex curves along which Whitney regularity fails and all the complex curves along which weak Whitney regularity fails. It turns out that for each example there are a finite number of curves $\gamma_i$ such that weak Whitney regularity fails precisely along those curves tangent to one of the $\gamma_i$ at the origin.