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Equisingularity in families of double point curves

This paper systematically compares the equisingularity of 1-parameter unfoldings of finitely determined map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$ with that of their associated double point curves, providing counterexamples to natural conjectures, introducing topologically trivial but non-Whitney equisingular "Henry-type" families, and generalizing classical double point formulas to higher dimensions.

Original authors: Otoniel Nogueira da Silva, Manoel Messias da Silva Júnior

Published 2026-05-12

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CC0 1.0

Original authors: Otoniel Nogueira da Silva, Manoel Messias da Silva Júnior

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect designing a complex, multi-layered sculpture. In mathematics, this sculpture is a "map" that takes points from a flat, two-dimensional sheet (like a piece of paper) and folds them into a three-dimensional space. Sometimes, when you fold the paper, different parts of the sheet land on top of each other in the 3D space. These overlapping spots are called double points.

This paper is like a detective story where mathematicians investigate what happens when they gently wiggle or "unfold" this sculpture. They want to know: If the main sculpture stays the same shape (topologically) while we wiggle it, do the hidden patterns of overlaps (the double points) also stay the same?

Here is a breakdown of their findings using everyday analogies:

1. The Setup: The Sculpture and the Wiggle

Think of the original map as a piece of paper being crumpled into a ball.

The Map ( $f$ ): The way the paper is crumpled.
The Unfolding ( $F$ ): A slow-motion video of the crumpling process, where time ( $t$ ) is the parameter.
The Double Point Curves ( $D$ ): If you shine a light through the crumpled paper, the "shadows" where the paper touches itself form specific lines or curves. The paper studies four different versions of these shadow lines:
1. The lines on the original paper ( $D$ ).
2. The lines where the paper actually overlaps in 3D space ( $F(D)$ ).
3. A complex "double" view of the overlaps ( $D^2$ ).
4. A simplified, symmetrical version of that double view ( $D^2/S_2$ ).

2. The Big Question: "If the main shape is stable, are the shadows stable?"

Mathematicians have different ways of defining "stable" (or equisingular):

Topologically Trivial: The shape doesn't tear or glue together; it just stretches. Like a rubber band being pulled.
Whitney Equisingular: The shape is stable and its sharpness (multiplicity) doesn't change.
Bi-Lipschitz Equisingular: The shape is stable in a very strict geometric sense; distances don't get distorted too wildly.

The authors asked: If the main sculpture ( $F$ ) is stable in one of these ways, are the shadow lines ( $D, F(D), D^2, D^2/S_2$ ) also stable?

3. The Findings: It's Complicated!

The authors found that the answer is a mix of "Yes," "No," and "It depends."

The Good News: If the main sculpture is Topologically Trivial (just a gentle stretch), then all the shadow lines are also topologically trivial. They don't tear or break.
- Analogy: If you stretch a rubber sheet, the shadows it casts on the wall might change size, but they won't suddenly snap into two separate pieces.
The Bad News (The Counterexamples):
- Even if the main sculpture is Whitney Equisingular (very stable), the shadow line $D^2/S_2$ can suddenly become "sharper" or change its geometry. It's like a shadow that looks smooth one second and suddenly develops a jagged edge the next, even though the object casting it is perfectly stable.
- Even if the main sculpture is Bi-Lipschitz Equisingular (the strictest stability), the shadow line $D^2/S_2$ can still fail to be stable.
- Analogy: Imagine a perfectly smooth, stable spinning top. You might expect its shadow to be a perfect, unchanging circle. But the authors found cases where the top is perfectly stable, yet its shadow suddenly changes from a circle to a star shape, or changes its "sharpness," defying intuition.

4. The "Henry-Type" Families: The Tricksters

The paper introduces a new family of curves called "Henry-type" families.

The Story: These are curves that look like they are doing nothing (they are topologically trivial—they don't change shape). However, if you look closely at their geometry (Whitney conditions), they are actually changing their "texture" or sharpness.
The Metaphor: Think of a chameleon that changes its skin texture (smooth to rough) without changing its overall shape or color. To a casual observer, it looks the same, but a scientist measuring the skin texture sees a change. The authors used their double-point curves to build new examples of these "chameleons."

5. The High-Dimensional Upgrade

Finally, the authors took formulas that were previously only known for 2D-to-3D maps and upgraded them for higher dimensions (like mapping a 3D space into a 5D space).

They provided a new, convenient "ruler" (analytic structure) to measure these complex overlaps in higher dimensions, ensuring that the mathematical formulas for counting these overlaps work correctly even in these abstract, high-dimensional worlds.

Summary

In short, this paper shows that stability is not contagious. Just because the main mathematical object (the map) is perfectly stable and unchanging, it does not guarantee that the hidden patterns of its overlaps (the double point curves) will remain stable in the same way. The authors mapped out exactly where this stability holds and where it breaks, providing new tools and counterexamples to help mathematicians navigate these tricky geometric landscapes.

Technical Summary: Equisingularity in Families of Double Point Curves

Problem Statement
This paper investigates the relationship 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. Specifically, the authors analyze four associated loci: the double point curve in the source $D(F)$ , the image of the double point curve $F(D(F))$ , the double point space $D_2(F)$ , and the quotient space $D_2(F)/S_2$ .

The central inquiry addresses whether specific notions of equisingularity for the unfolding $F$ (topological triviality, Whitney equisingularity, and bi-Lipschitz equisingularity) imply the same properties for these associated curve families. While it is established that topological triviality of $F$ implies topological triviality of all four associated families, the behavior under stronger conditions (Whitney and bi-Lipschitz) remains less understood. The paper seeks to determine if the converse implications hold or if counterexamples exist where $F$ satisfies a strong equisingularity condition, but the associated curves do not.

Methodology
The authors employ tools from singularity theory, utilizing the analytic structures of multiple point spaces defined via Fitting ideals and the construction of double point spaces as complex spaces. Key methodological components include:

Analytic Structures: The paper rigorously defines the analytic structures for $D(f)$ , $f(D(f))$ , and $D_2(f)/S_2$ . For $D_2(f)/S_2$ , the authors utilize the action of the permutation group $S_2$ on $D_2(f)$ and provide explicit computational methods (including Singular code) to derive generators for the quotient ideal.
Invariants and Criteria: The study relies on classical invariants such as the Milnor number ( $\mu$ ), multiplicity ( $m$ ), and the number of cross-caps ( $C$ ) and triple points ( $T$ ) in stabilizations. Equisingularity is tested via the constancy of these invariants and the satisfaction of Whitney's condition (b).
Counterexample Construction: The authors construct explicit families of map germs $f_t$ to test the limits of equisingularity implications. These examples often involve quasihomogeneous or homogeneous map germs where the deformation adds terms of specific weighted degrees.
Generalization: The methodology extends classical formulas for double point curves from the $(\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0)$ setting to the higher-dimensional case of map germs from $(\mathbb{C}^n, 0) \to (\mathbb{C}^{2n-1}, 0)$ for $n \geq 3$ .

Key Contributions and Results

Refinement of Equisingularity Implications:
The paper provides a systematic comparison of equisingularity types. It confirms that if $F$ is topologically trivial, all four associated families are topologically trivial. However, the authors present counterexamples showing that:
- If $F$ is Whitney equisingular, $D_2(F)/S_2$ may fail to be Whitney equisingular.
- If $F$ is Whitney equisingular, $D_2(F)$ may fail to be bi-Lipschitz equisingular.
- If $F$ is bi-Lipschitz equisingular, $D_2(F)/S_2$ may fail to be Whitney or bi-Lipschitz equisingular.
- Even if $F$ is Whitney equisingular, $F(D(F))$ may fail to be bi-Lipschitz equisingular.
Sufficient Conditions for Bi-Lipschitz Equisingularity:
The authors prove that if $f$ is a homogeneous map germ and $F$ is topologically trivial, then $D_2(F)$ is bi-Lipschitz equisingular. This result relies on the specific geometric structure of the double point space for homogeneous maps, where components are unions of lines.
Henry-Type Families:
The paper introduces and constructs new families of curves termed "Henry-type." These are defined as families of quasihomogeneous complete intersection curves that are topologically trivial but fail Whitney equisingularity conditions (due to the non-constancy of multiplicity). The authors demonstrate that the families $D_2(F)/S_2$ derived from specific unfoldings constitute such Henry-type families, providing new sources of counterexamples in equisingularity theory.
Generalization of Double Point Formulas:
Addressing a question regarding higher-dimensional map germs ( $n \geq 3$ ), the authors propose a convenient analytic structure for the curves $D(f)$ and $f(D(f))$ using reduced Fitting ideals. They establish that for $f: (\mathbb{C}^n, 0) \to (\mathbb{C}^{2n-1}, 0)$ , the Milnor number of $D(f)$ equals that of $D_2(f)$ . They also provide a formula relating $\mu(D(f)_{red})$ to $\mu(D_2(f))$ and the epsilon invariant of $D(f)$ .

Significance
The paper's significance lies in its clarification of the hierarchy of equisingularity conditions for double point loci. It demonstrates that the equisingularity of an unfolding $F$ does not automatically transfer to all its associated double point curves under Whitney or bi-Lipschitz conditions, thereby refining the understanding of the geometric constraints in singularity theory.

By constructing explicit counterexamples, the authors challenge the assumption that topological triviality or Whitney equisingularity of the source map implies similar properties for the quotient double point space $D_2(F)/S_2$ . The introduction of new Henry-type families expands the catalog of counterexamples available for testing fundamental conjectures in the field. Furthermore, the generalization of double point formulas to higher dimensions ( $n \geq 3$ ) provides a necessary analytic framework for studying map germs in codimension $n-1$ , extending classical results that were previously limited to the $(\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0)$ case.