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 = (ft, t) of a finitely determined map germ f: (ℂ2, 0) → (ℂ3, 0) and the equisingularity of its associated families of double point curves: D(F), F(D(F)), D2(F), and D2(F)/S2. 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 the Whitney equisingularity conditions. Finally, we generalize classical formulas for double point curves, originally established for map germs from (ℂ2, 0) to (ℂ3, 0), to the higher-dimensional setting of map germs from (ℂn, 0) to (ℂ2n−1, 0) for n ≥ 3, thereby endowing the associated curves with a convenient analytic structure.
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BibTeX
@misc{silva2026equisingularity,
title = {{Equisingularity in families of double point curves}},
author = {Otoniel Nogueira da Silva and Manoel Messias da Silva Júnior},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.08507},
eprint = {2605.08507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.08507},
note = {CC0},
}
RIS
TY - UNPB
TI - Equisingularity in families of double point curves
AU - Otoniel Nogueira da Silva
AU - Manoel Messias da Silva Júnior
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2605.08507
UR - https://arxiv.org/abs/2605.08507
AB - 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: (ℂ^2, 0) → (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 the Whitney equisingularity conditions. Finally, we generalize classical formulas for double point curves, originally established for map germs from (ℂ^2, 0) to (ℂ^3, 0), to the higher-dimensional setting of map germs from (ℂ^n, 0) to (ℂ^2n−1, 0) for n ≥ 3, thereby endowing the associated curves with a convenient analytic structure.
ER -