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On Teissier's example of an equisingularity class that cannot be defined over the rationals

This paper corrects Teissier's example of a surface singularity derived from a real projective polygon cone and provides a complete proof demonstrating that this singularity is not Whitney equisingular to any singularity defined over the field of rational numbers.

Original authors: Adam Parusiński, Laurentiu Paunescu

Published 2026-02-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adam Parusiński, Laurentiu Paunescu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

The Big Picture: A Broken Puzzle Piece

Imagine you have a complex 3D sculpture made of glass (a mathematical "surface singularity"). Mathematicians love to study these shapes, but they also want to know if these shapes can be built using only "rational" materials—meaning numbers you can write down as simple fractions (like 1/2, 3/4, or 5).

The paper addresses a specific puzzle proposed by a mathematician named Bernard Teissier. Teissier claimed to have found a sculpture that, no matter how you try to reshape it or smooth it out, cannot be built using only rational numbers. It requires "irrational" numbers (like $\sqrt{2}$ or $\pi$ ) to exist.

However, the authors of this paper, Adam Parusiński and Laurențiu Păunescu, found two problems with Teissier's original story:

The Blueprint was slightly wrong: The specific shape Teissier used actually could be built with rational numbers.
The Proof was shaky: The mathematical rule Teissier used to prove his point was actually incorrect.

This paper fixes the blueprint and provides a new, solid proof to show that Teissier's main idea was right, even if his original example was flawed.

Part 1: The "Line Arrangement" Puzzle (Fixing the Blueprint)

To understand the shape, the authors start with a 2D drawing made of intersecting lines, similar to a starburst or a web. This comes from a famous geometry problem by Grünbaum.

The Original Mistake: Teissier used a specific arrangement of 9 lines. The authors realized that if you look closely, this specific arrangement can actually be drawn using rational numbers. It's like trying to prove a house is made of wood, but you accidentally picked a house made of plastic.
The Fix: The authors added one extra line to the drawing. This small change breaks the symmetry of the shape.
The Result: With this new 10-line arrangement, the shape becomes "rigid." It is mathematically impossible to rotate or shift this shape so that all its intersection points land on rational coordinates. It is stuck in a state that requires irrational numbers.

Analogy: Imagine trying to arrange 10 sticks on a table so they cross at specific points. With 9 sticks, you can arrange them so every crossing point is a "clean" fraction. But if you add a 10th stick in a specific way, the geometry forces at least one crossing point to be a messy, irrational number. You simply cannot build that specific configuration with "clean" math.

Part 2: The "Deformation" Problem (Fixing the Proof)

Teissier's original argument relied on a rule about how these shapes change over time. He claimed that if you slowly deform a shape (like melting ice into water), the "tangent cone" (the shape's sharp point or core) behaves in a predictable, smooth way.

The Broken Rule: The authors point out that the mathematical theorem Teissier used to make this claim is actually false. It's like using a map that says "all bridges are safe," when in reality, some bridges collapse.
The New Proof: Instead of using the broken map, the authors built a new path. They used a concept called "exceptional tangents."
- What are they? Imagine a sharp point on a surface. Usually, if you touch it with a flat sheet (a tangent plane), the sheet touches in a predictable way. But at certain "exceptional" points, the sheet can touch in weird, unpredictable directions.
- The Logic: The authors proved that for their specific 10-line shape, these "exceptional tangents" do not exist. Because they don't exist, the shape's core (the tangent cone) must stay exactly the same as the shape deforms.
- The Conclusion: Since the core is locked in a specific, irrational configuration, the whole shape cannot be deformed into a version built with rational numbers.

Part 3: The Counter-Example (Why the Old Rule Failed)

To prove that Teissier's old rule was broken, the authors created a "monster" example (Section 4).

They built a shape that has no exceptional tangents (it looks very smooth and well-behaved).
However, when they tried to deform it, the shape still broke apart in a weird way.
This proves that the old rule ("No exceptional tangents = Smooth deformation") is false. You can have a shape with no weird tangents that still behaves badly when you try to change it.

Summary

The Goal: Prove that some complex shapes cannot be built using simple rational numbers.
The Correction: The original shape Teissier used was too simple (it could be built rationally). The authors added a line to make it truly "irrational."
The New Proof: They discarded a faulty mathematical rule and replaced it with a rigorous argument about "exceptional tangents" and how shapes deform.
The Takeaway: There are indeed surface singularities that are "Whitney equisingular" (topologically stable) but cannot be defined over the rational numbers. Teissier was right about the existence of such shapes, but the paper provides the correct blueprint and the correct mathematical machinery to prove it.

In short: The paper is a "correction notice" that fixes a broken example and a broken proof, ultimately confirming that some mathematical shapes are fundamentally too complex to be constructed from simple fractions.

Technical Summary: On Teissier's Example of an Equisingularity Class That Cannot Be Defined Over the Rationals

Problem Statement
The paper addresses a specific question in singularity theory regarding the definability of equisingular families over the field of rational numbers ( $\mathbb{Q}$ ). While it is established that every complex singularity is equisingular (in a sense stronger than Whitney's) to a singularity defined by polynomial equations with coefficients in the field of algebraic numbers, it remains an open problem whether such a singularity can be deformed into one defined over $\mathbb{Q}$ via a topologically equisingular deformation.

Bernard Teissier, in a 1990 note [T90], proposed a counterexample: a surface singularity in $\mathbb{C}^3$ derived from a classical convex geometry example (Grünbaum's arrangement of lines) that is not Whitney equisingular to any singularity defined over $\mathbb{Q}$ . However, the original example contained two flaws:

The specific line arrangement cited from Grünbaum's work actually admits a polynomial equation with rational coefficients, failing to serve as a counterexample.
The proof relied on a theorem from Lê and Teissier [LT1] concerning the deformation of a surface to its tangent cone, which was stated incorrectly.

Methodology
The authors, Parusiński and Păunescu, employ algebraic geometry and stratification theory to correct the example and provide a rigorous proof of Teissier's underlying result.

Correction of the Geometric Example:
- The authors revisit Grünbaum's arrangements of 9 lines in the real projective space. They demonstrate that the original arrangement $C'$ (and its equiconfigurational variants) can indeed be defined over $\mathbb{Q}$ due to Galois invariance of the line set.
- To break this symmetry, they construct a new arrangement, denoted $C'$ , by adding a 10th line ($HE$) to the original configuration.
- Using Galois theory, they prove that the product of the linear forms defining this new 10-line arrangement is not invariant under the Galois group of the field extension generated by the coordinates. Consequently, this arrangement cannot be defined over $\mathbb{Q}$ .
Correction of the Analytic Proof:
- The authors identify that the original proof in [T90] relied on [LT1, Théorème (2.1.1)], which incorrectly claimed that a deformation to the tangent cone is equisingular if and only if the singularity has no "exceptional tangents."
- They provide a counterexample (Section 4) showing a singularity with no exceptional tangents whose deformation to the tangent cone is not equisingular (specifically, the singular locus splits topologically).
- Instead of relying on the flawed theorem, the authors provide a direct proof using the theory of Whitney stratifications and normal cones. They analyze the family of surface singularities $f(x, y)$ and its tangent cone $C_{X,Y}$ .

Key Contributions and Results

Refined Counterexample (Proposition 2.2): The authors prove that the cone over their corrected 10-line arrangement ( $C'$ ) yields a surface singularity $X_0$ in $\mathbb{C}^3$ that cannot be defined over $\mathbb{Q}$ .
Main Theorem (Theorem 3.5): The paper establishes a new result regarding the equisingularity of the normal cone. It proves that if a family of surface singularities $(X_{reg}, \Sigma \setminus Y, Y)$ $(X_{r e g}, Σ ∖ Y, Y)$ forms a Whitney stratification, then:
1. The fibers $X_y$ have no exceptional tangents.
2. The normal cone $C_{X,Y}$ satisfies Whitney's conditions.
3. The singular locus of the normal cone is exactly the normal cone of the singular locus ( $C_{\Sigma, Y} = (C_{X,Y})_{sing}$ ).
4. The pair $((C_{X,Y})_{reg}, C_{\Sigma, Y} \setminus Y, Y)$ forms a Whitney stratification.
Application to Teissier's Result: By applying Theorem 3.5, the authors show that if the original singularity $X_0$ (the cone over the corrected arrangement) were Whitney equisingular to a singularity defined over $\mathbb{Q}$ , then the tangent cones of the deformed family would also be equisingular. However, the tangent cones of the corrected arrangement are projectively equivalent to the arrangement itself, which cannot be defined over $\mathbb{Q}$ . This contradiction proves that the original singularity is not Whitney equisingular to any singularity defined over $\mathbb{Q}$ .
Counterexample to [LT1] (Section 4.1): The authors construct a specific singularity $f(x,y,z) = z(zx - y^2 + x^3)$ to demonstrate that the absence of exceptional tangents is insufficient to guarantee that the deformation to the tangent cone is equisingular. In this example, the singular locus splits from a cusp to a double line, violating topological triviality.

Significance
The paper's primary significance lies in its rigorous correction and completion of Teissier's 1990 result. It provides a valid, explicit example of a complex surface singularity that cannot be deformed into a rational singularity via a Whitney equisingular deformation. Furthermore, it clarifies the relationship between exceptional tangents and equisingularity of deformations to the normal cone, correcting a previously accepted but incorrect theorem in the field. The work ensures that the theoretical foundation for distinguishing equisingularity classes based on definability over $\mathbb{Q}$ is sound.