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The Complete Intersection Discrepancy of a Curve II: Families of Curves

This paper characterizes Whitney equisingularity and topological equisingularity in families of reduced curves by establishing a fiberwise multiplicity criterion based on the discrepancy between the Jacobian ideal multiplicity and the complete intersection discrepancy, proving that these conditions are equivalent to the equidimensionality of specific exceptional loci and the emptiness of relative polar varieties, while also demonstrating the Zariski upper semicontinuity of the Milnor number.

Original authors: Andrei Benguş-Lasnier, Terence Gaffney, Antoni Rangachev

Published 2026-02-24

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CC BY 4.0

Original authors: Andrei Benguş-Lasnier, Terence Gaffney, Antoni Rangachev

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 designing a family of buildings. You have a blueprint for a "perfect" building (a smooth curve) and a blueprint for a "broken" building (a curve with a singularity, like a sharp corner or a self-intersection). Now, imagine you are creating a family of these buildings, where each building in the family is slightly different, perhaps due to a small shift in the foundation or a change in the weather.

The big question this paper asks is: How do we know if the entire family of buildings is "equally broken" or "equally smooth"?

In mathematics, this concept is called equisingularity. If a family is "equisingular," it means that as you walk from one building to the next, the type of damage (the singularity) doesn't suddenly get worse or change its nature. It's like a movie where the character's scar stays the same shape and size in every frame.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The Two Rulers: Measuring the "Brokenness"

To check if the family is stable, the authors use two different rulers to measure the "brokenness" of a singularity.

Ruler A: The Jacobian Multiplicity. Think of this as measuring how "messy" the equations are at the broken spot. If you have a sharp corner, the math describing it is very chaotic. This ruler counts the complexity of that chaos.
Ruler B: The Complete Intersection Discrepancy (CID). This is a clever "correction term." Imagine you try to fix the broken building by building a temporary scaffolding (a "complete intersection") around it. The CID measures the difference between the messy building and this clean scaffolding. It tells you how much "extra" mess exists that the scaffolding doesn't account for.

The Magic Formula: The authors discovered that if you take Ruler A and subtract Ruler B, you get a single, perfect number that describes the true "brokenness" of the curve. This number is closely related to the Milnor Number, a famous measure of how bad a singularity is.

2. The "Polar Variety": The Shadow of the Problem

How do we know if the brokenness stays the same as we move through the family?

The authors introduce a concept called the Relative Polar Variety. Imagine shining a light on your family of buildings from a specific angle. The shadow cast by the "critical points" (the spots where the building is most unstable) is the polar variety.

The Big Discovery: If the family is perfectly stable (Whitney equisingular), this shadow disappears (becomes empty).
The Metaphor: Think of a family of trees in a windstorm. If the trees are all swaying in the exact same rhythm (equisingular), the "shadow" of their instability vanishes. If one tree suddenly starts snapping or swaying wildly differently, a shadow appears. The paper proves that no shadow = perfect stability.

3. The "Nash Blowup": Smoothing the Rough Edges

Another tool they use is the Nash Blowup. Imagine taking a crumpled piece of paper (the singular curve) and trying to smooth it out by blowing it up like a balloon.

If the family is stable, the "balloon" (the exceptional divisor) grows in a very predictable, uniform way for every building in the family.
If the family is unstable, the balloon grows unevenly.

The paper proves that checking if this "balloon" grows uniformly is the same as checking if the "shadow" (polar variety) is empty.

4. The Main Result: A Simple Test for Stability

The authors provide a powerful test for families of curves of any size (not just simple lines, but complex shapes in high-dimensional space).

The Test:

Measure the "brokenness" (Jacobian minus CID) for the first building in the family.
Measure it for the next building.
If the numbers are the same, and the "shadow" (polar variety) is empty, then the whole family is topologically identical.

This means you can stretch, bend, or deform the family, and the "shape" of the break will never change. It's like having a rubber band with a knot; if you pull it, the knot stays a knot. If the knot suddenly turns into a loop, the family wasn't equisingular.

5. Why This Matters: Topology vs. Math

The paper also connects this to Topology (the study of shapes).

The Result: If the "brokenness" number (Milnor number) stays constant throughout the family, then the family is topologically equisingular.
In Plain English: If the mathematical "score" of the damage doesn't change, then the actual shape of the damage doesn't change either. You can't have a family where the math says "it's the same" but the shape actually changes from a sharp corner to a smooth loop.

Summary Analogy

Imagine a band playing a song.

The Singularity is a specific, tricky note the band plays.
The Family is the band playing that note in different rooms with different acoustics.
The Jacobian and CID are two different microphones measuring the "noise" of that note.
The Polar Variety is the echo.

The paper says: "If you measure the noise with both microphones, subtract the echo, and the result is the same in every room, AND there is no echo left over, then the band is playing the exact same song in every room, no matter how the room changes."

This allows mathematicians to predict the behavior of complex geometric shapes without having to simulate every single possible variation, using a simple "shadow check" to ensure stability.

1. Problem Statement and Context

The paper addresses the problem of characterizing equisingularity in families of reduced complex analytic curves over smooth parameter spaces of arbitrary positive dimension ( $k \geq 1$ ).

Background: In singularity theory, determining when a family of singular varieties is "equisingular" (i.e., topologically or analytically trivial) is a central challenge. For one-parameter families ( $k=1$ ), classical results by Buchweitz, Greuel, Teissier, and Lê established that Whitney equisingularity is equivalent to the constancy of the Milnor number ( $\mu$ ) and the multiplicity ( $m$ ).
The Gap: Extending these numerical characterizations to families with parameter spaces of dimension $k > 1$ has remained an open problem. Existing proofs for $k=1$ rely heavily on specific properties of one-parameter deformations (e.g., coherence of hypercohomology) that do not generalize directly.
The Goal: The authors aim to provide a unified, algebro-geometric framework to characterize Whitney equisingularity and topological equisingularity for curves in arbitrary dimensions using new analytic invariants.

2. Key Methodology and Tools

The authors employ a sophisticated blend of commutative algebra, intersection theory, and complex analytic geometry.

Complete Intersection Discrepancy ($cid$): Building on their previous work (Part I), they utilize the complete intersection discrepancy of a curve singularity. For a curve $X_0$ , this is defined as the intersection number $I_0(X_0, W_0)$ , where $W_0$ is the complement of $X_0$ in a generic complete intersection $Z_0$ containing $X_0$ . This invariant corrects the multiplicity of the Jacobian ideal for non-complete intersection singularities.
Relative Polar Varieties: They define the relative polar variety $\Gamma^k_h(X)$ of dimension $k$ associated with a linear functional restricted to the smooth part of the fibers. This object captures the "jump" in invariants between the central fiber and nearby fibers.
Buchsbaum–Rim Multiplicity: The paper utilizes the Buchsbaum–Rim (BR) multiplicity of modules (specifically relative Jacobian modules) rather than just the Hilbert–Samuel multiplicity of ideals. This allows for the handling of non-complete intersection curves.
Integral Closures and Conormal Spaces: The characterization of equisingularity relies on the geometry of the relative conormal space $C_h(X)$ and the relative Nash blowup $N_h(X)$ . The authors use the theory of integral closures of modules to relate the constancy of numerical invariants to the equidimensionality of the fibers of these geometric spaces.
Riemann–Hurwitz Formula for Singular Curves: A new Riemann–Hurwitz type formula is derived for singular curves projecting to a disc, relating the Euler characteristic, the degree of the map, critical points, and singularities.

3. Key Contributions and Results

A. The Main Formula (Theorem 1.2)

The authors establish a fundamental identity relating the change in analytic invariants to the degree of the relative polar variety. For a flat family of reduced curves $h: (X, 0) \to (Y, 0)$ :
$(e(\text{Jac}(X_0)) - \text{cid}(X_0)) - (e(\text{Jac}(X_y)) - \text{cid}(X_y)) = \deg_Y \Gamma^k_h(X)$
where:

$e(\text{Jac}(X_y))$ is the sum of Hilbert–Samuel multiplicities of the Jacobian ideals of the singular points.
$\text{cid}(X_y)$ is the sum of complete intersection discrepancies.
$\deg_Y \Gamma^k_h(X)$ is the number of points in the generic fiber of the relative polar variety.

Significance: This formula generalizes the classical Teissier–Lê–Greuel formula (which applies to complete intersections where $\text{cid}=0$ ) to arbitrary reduced curves. It shows that the difference in the "corrected" Jacobian multiplicity is exactly the degree of the polar variety.

B. Characterization of Whitney Equisingularity (Theorem 1.3)

The paper proves that for a family of reduced curves, the following conditions are equivalent:

The function $y \mapsto e(\text{Jac}(X_y, 0)) - \text{cid}(X_y, 0)$ is constant.
The relative polar variety $\Gamma^k_h(X)$ is empty (which is equivalent to the fibers of the relative conormal space $c_h^{-1}(Y) \to Y$ and the relative Nash blowup $b_h^{-1}(Y) \to Y$ being equidimensional).
The family is Whitney equisingular (satisfies Whitney condition B).
The family admits a strong simultaneous resolution (the normalization of the family is locally analytically trivial).

Significance: This resolves the open problem of extending the Buchweitz–Greuel characterization of Whitney equisingularity to arbitrary dimensions. It demonstrates that constancy of the corrected Jacobian multiplicity is the necessary and sufficient numerical condition.

C. Zariski Upper Semicontinuity (Theorem 6.5)

The authors prove that the Milnor number $\mu_y$ (summed over singular points) is Zariski upper semicontinuous for families of curves over parameter spaces of any dimension.

They establish the relation: $\mu_0 - \mu_y = 1 - \chi(X_y)$ for generic $y$ .
This generalizes previous results known only for $k=1$ .

D. Topological Equisingularity (Theorem 1.4)

They prove that for a family of reduced curves where the irreducible components intersect at a single point (and no other intersections), topological equisingularity is equivalent to the constancy of the Milnor number $\mu_y$ .

This result removes the need for a section parametrizing singular points, a requirement in earlier theorems.

4. Technical Significance and Applications

Generalization of Classical Formulas: The paper successfully generalizes the Teissier–Lê–Greuel formula and the Plücker formula to curves of arbitrary codimension and arbitrary parameter space dimension.
New Invariant for Equisingularity: The combination $e(\text{Jac}) - \text{cid}$ is shown to be the precise invariant controlling Whitney equisingularity. The paper provides a counterexample (Example 7.8) showing that $e(\text{Jac})$ alone is not sufficient for non-complete intersection curves, highlighting the necessity of the $cid$ correction term.
Algebro-Geometric Approach: Unlike previous approaches for $k=1$ that relied on topological arguments or hypercohomology, this paper uses purely algebro-geometric tools (integral closures, Rees algebras, polar varieties). This allows the results to hold for $k > 1$ .
Resolution of Open Problems: The work settles the question of whether the constancy of $\mu$ and $m$ implies Whitney equisingularity for $k > 1$ , confirming it does so under the specific geometric conditions of the fibers (irreducible components meeting at one point).

5. Conclusion

This paper represents a major advancement in the theory of equisingularity. By introducing the complete intersection discrepancy as a correction term to the Jacobian multiplicity, the authors provide a robust numerical criterion for Whitney equisingularity and topological equisingularity that works for families of curves over parameter spaces of any dimension. The results unify the theory of singular curves, extending classical one-parameter results to the multi-parameter setting through the lens of relative polar varieties and integral closure theory.