Equisingularity in families of double point curves

Abstract

In this paper, we provide a systematic comparison between the equisingularity of a 1-parameter unfolding F = (f_t, t) of a finitely determined map germ f: (\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0) and the equisingularity of its associated families of double point curves: D(F), F(D(F)), D^2(F), and D^2(F)/S_2. We also construct explicit counterexamples to several natural questions concerning the equisingularity of these loci. As a key application, we introduce new families of complete intersection curves - referred to as Henry-type families - which are topologically trivial but fail to satisfy Whitney equisingularity conditions. Finally, we generalize classical double point curve formulas, originally established for map germs from (\mathbb{C}^2, 0) to (\mathbb{C}^3, 0), to the higher-dimensional setting of map germs from (\mathbb{C}^n, 0) to (\mathbb{C}^{2n-1}, 0) for n \geq 3, providing the associated curves with a convenient analytic structure.