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Zariski equisingularity of surface singularities in $\mathbb C^3$ by a local invariant

The paper introduces a new invariant called the multiplicity sequence, derived from the multiplicities of successive discriminants under generic projections, to characterize that an analytic family of surface singularities in $\mathbb{C}^3$ is generically Zariski equisingular if and only if this sequence remains constant.

Original authors: Adam Parusiński, Laurenţiu Păunescu

Published 2026-02-18

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Adam Parusiński, Laurenţiu Păunescu

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

Imagine you are an architect looking at a collection of strange, crumpled sculptures. Some are smooth, some have sharp points, some have deep valleys, and some are just a mess of tangled wires. In the world of mathematics, these sculptures are called surface singularities. They are points where a shape breaks its own rules of smoothness.

The paper you asked about is like a new rulebook for architects. It asks a very specific question: "How can we tell if two of these broken sculptures are essentially the same kind of brokenness, even if they look slightly different?"

Here is the breakdown of their discovery, using simple analogies.

1. The Problem: The "Shape-Shifting" Sculptures

Imagine you have a family of these sculptures, and you are slowly changing one into another (like a morphing animation).

The Goal: You want to know if the type of breakage stays the same throughout the animation.
The Old Way: Mathematicians used to check this by looking at a long list of numbers (like a fingerprint). If the numbers stayed the same, the shapes were considered "equisingular" (equally singular).
- The Catch: This old list of numbers only worked for sculptures that were broken in just one tiny spot (isolated singularities). If the breakage was a whole line or a messy cluster (non-isolated), the old numbers didn't exist. The math broke down.

2. The New Solution: The "Shadow Sequence"

The authors, Adam and Laurentiu, invented a new tool called the Multiplicity Sequence (let's call it the Shadow Sequence).

Here is how it works, using a flashlight analogy:

The Sculpture (V): Imagine your broken sculpture sitting in a dark room.
The First Flashlight (Projection 1): You shine a light on it from the side. The shadow it casts on the wall is a 2D shape.
- The Catch: The shadow might have its own weird kinks or breaks.
The Second Flashlight (Projection 2): You look at that shadow and shine a light on it from a different angle. This casts a second, smaller shadow (a line).
The Third Flashlight (Projection 3): You shine a light on that line.

The Magic: The authors realized that if you count the "thickness" or "density" (multiplicity) of the breaks in the original sculpture, the first shadow, the second shadow, and so on, you get a unique list of numbers (The Shadow Sequence).

The Discovery: If this list of numbers stays exactly the same as you morph one sculpture into another, then the two sculptures are fundamentally the same type of brokenness.
The Superpower: This new list works for every sculpture, even the messy ones where the old math failed.

3. The "Perfect Angle" (The Tricky Part)

There is a catch. If you shine your flashlight from a bad angle (like directly down a groove), the shadow might look weird and hide the true nature of the break.

The "ν-transverse" System: The authors had to invent a special rule for how to hold the flashlight. They call it a "nested uniformly transverse" system.
The Analogy: Imagine trying to take a photo of a tangled ball of yarn. If you take the photo from the top, it looks like a flat circle. If you take it from the side, you see the tangles. The authors proved that if you pick a "generic" (random but good) angle, the photo will always reveal the true structure. They showed that as long as you pick a "good" angle, the Shadow Sequence is a reliable, unchangeable fingerprint.

4. The Big Conclusion

The paper proves a beautiful symmetry:

If the Shadow Sequence (the list of numbers) stays constant while you morph the shapes...
Then the shapes are perfectly "equisingular" (they are the same type of breakage).
And vice versa.

This solves a puzzle that has been around for decades. It unifies the math for "clean" broken spots and "messy" broken spots into one single, elegant rule.

Summary in a Nutshell

Think of a surface singularity as a crumpled piece of paper.

Old Math: Could only count the crumples if the paper was crumpled in just one spot.
New Math (This Paper): Shines a series of lights on the paper, looks at the shadows of the shadows, and counts the "crumple density" at every level.
The Result: If the pattern of these counts doesn't change as you wiggle the paper, then the paper is changing in a perfectly controlled, predictable way.

The authors have given mathematicians a universal ruler to measure the "brokenness" of shapes in 3D space, no matter how messy they get.

1. Problem Statement and Context

The Core Problem:
The paper addresses the classification of Zariski equisingularity for families of surface singularities in $\mathbb{C}^3$ . Specifically, it seeks a necessary and sufficient condition, expressed via a single local invariant, to determine when an analytic family of surface singularities $(V_t, 0) \subset (\mathbb{C}^3, 0)$ is Zariski equisingular.

Background and Motivation:

Zariski Equisingularity: Originally proposed by O. Zariski, this notion defines equisingularity recursively using discriminants of generic projections. For isolated surface singularities, Briançon and Speder (and later Briançon and Henry) established that a family is Zariski equisingular if and only if a specific set of invariants remains constant: Teissier's numbers ( $\mu^*$ ), the number of ordinary double points ( $k$ ), and the number of cusps ( $\phi$ ) of a generic projection.
Limitations: These classical invariants ( $\mu^*, k, \phi$ ) are only well-defined for isolated singularities. They fail for non-isolated singularities, which are common in surface theory.
The Gap: There was no unified invariant capable of characterizing Zariski equisingularity for both isolated and non-isolated surface singularities in $\mathbb{C}^3$ . Furthermore, the existing criteria relied on multiple distinct invariants rather than a single structural sequence.

2. Methodology

The authors develop a methodology based on generic projections and discriminant loci to construct a unified invariant.

$\nu$ -Transverse Coordinates: The authors utilize the concept of a $\nu$ -transverse system of coordinates (introduced in their previous work [13]). A coordinate system $(x, y, z)$ is $\nu$ -transverse if:
1. The projection to the $(x, y)$ -plane is transverse to the surface $V$ .
2. The projection to the $x$ -axis is transverse to the discriminant locus $\Delta_0$ of the first projection.
3. The family of discriminants parameterized by the slope of the projection is equisingular.
  They prove that for any surface singularity, a generic linear change of coordinates yields a $\nu$ -transverse system.
Generalized Discriminants: To handle non-reduced discriminants (which occur when singularities are not isolated), the authors employ generalized discriminants (or subdiscriminants). Instead of relying solely on the standard discriminant, they use a sequence of generalized discriminants $D^i_f$ to capture the multiplicity structure even when the discriminant locus has embedded components or non-isolated singularities.
Construction of the Invariant: They define a new invariant, the Multiplicity Sequence ( $mult^*(V)$ ), which is constructed recursively by taking the multiplicities of successive discriminants of generic corank-1 projections.

3. Key Contributions

A. Definition of the Multiplicity Sequence ( $mult^*(V)$ )

The central contribution is the definition of the multiplicity sequence for a surface singularity $V = f^{-1}(0)$ in a $\nu$ -transverse coordinate system:
$mult^*(V) = (mult_0(V), mult_0(D_f), i_0, mult_0(D^{i_0}_{D_f}))$
Where:

$mult_0(V)$ is the multiplicity of the surface at the origin.
$D_f$ is the discriminant of the projection of $V$ .
$i_0 = \text{ind}(D_f)$ is the index of the first non-vanishing generalized discriminant of $D_f$ .
$D^{i_0}_{D_f}$ is that specific generalized discriminant.
$mult_0(\cdot)$ denotes the multiplicity at the origin.

Key Properties Proven:

Analytic Invariance: The sequence is independent of the choice of the $\nu$ -transverse coordinate system (Proposition 2.3).
Upper Semicontinuity: The sequence is upper semicontinuous in analytic families (Proposition 4.1).
Generalization: For isolated singularities, this sequence is explicitly expressible in terms of the classical invariants ( $\mu^*, k, \phi$ ).

B. Main Theorem (Theorem 0.2)

The paper establishes the equivalence of three conditions for an analytic family of surface singularities $V_t$ :

The family is $\nu$ -transverse Zariski equisingular (after a generic linear coordinate change).
The family is Zariski equisingular (after a generic linear coordinate change).
The multiplicity sequence $mult^*(V_t, 0)$ is constant with respect to $t$ .

This result unifies the theory: it shows that the constancy of a single, recursively defined local invariant is equivalent to the complex geometric condition of Zariski equisingularity.

C. Extension to Non-Isolated Singularities

The paper successfully extends the theory of Zariski equisingularity to non-isolated singularities. Since $mult^*(V)$ is defined via multiplicities of discriminants (which are well-defined even for non-isolated singularities), the authors provide a criterion for equisingularity where previous criteria involving Milnor numbers ( $\mu$ ) or cusp counts ( $\phi$ ) were inapplicable.

4. Key Results

Equivalence of Conditions: Theorem 0.2 proves that checking the constancy of $mult^*(V_t)$ is necessary and sufficient for Zariski equisingularity. This simplifies the verification process significantly compared to checking multiple invariants ( $\mu^*, k, \phi$ ) separately.
Recovery of Classical Results: Proposition 3.1 demonstrates that for isolated singularities, the multiplicity sequence $mult^*(V)$ encodes exactly the same information as the tuple $(\mu^*(V), k(V), \phi(V))$ . Specifically:
$mult^*(V) = (\mu_1 + \mu_0, \mu_2 + \mu_1, 1, \mu_3 + \mu_2 + 2k + 3\phi)$
This confirms that the new invariant is a true generalization of the classical theory.
Stability under Coordinate Changes: The authors prove that if a family is Zariski equisingular in one generic linear coordinate system, it is $\nu$ -transverse Zariski equisingular in any generic linear system (Corollary 0.3). This resolves ambiguity regarding the dependence on specific coordinate choices.

5. Significance

Unification: The paper unifies the theory of equisingularity for both isolated and non-isolated surface singularities in $\mathbb{C}^3$ under a single invariant framework.
Simplification: It replaces the need to calculate and track multiple distinct invariants (Teissier numbers, double points, cusps) with a single, algorithmically defined sequence of multiplicities.
Robustness: By relying on generalized discriminants and $\nu$ -transverse coordinates, the theory is robust against the complexities of non-isolated singularities, which often arise in deformation theory and algebraic geometry.
Foundational Impact: The results provide a solid foundation for further studies on the topology of surface singularities, polar varieties, and the construction of stratifications satisfying Whitney conditions, as the constancy of $mult^*$ implies Whitney's conditions (via Speder's results).

In summary, Parusiński and Păunescu have successfully generalized the criteria for Zariski equisingularity, providing a powerful, single-invariant tool that works universally for surface singularities in $\mathbb{C}^3$ , bridging the gap between isolated and non-isolated cases.