Equisingularity in families of double point curves

TL;DR

This paper explores the relationship between equisingularity in families of map germs from complex 2-space to 3-space and their double point curves, introduces new topologically trivial but non-Whitney equisingular families, and generalizes classical formulas to higher dimensions.

Contribution

It systematically compares equisingularity of unfoldings with their double point curves, constructs counterexamples, and extends double point formulas to higher-dimensional map germs.

Findings

Counterexamples to natural questions about equisingularity loci.

Introduction of Henry-type families that are topologically trivial but not Whitney equisingular.

Generalization of double point formulas to map germs from ( ext{C}^n, 0) to ( ext{C}^{2n-1}, 0) for n ≥ 3.

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.