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Equivalence of Curve Singularities and delta-Invariants

This paper establishes that curve singularities are determined by their finite-order truncations modulo sufficiently high powers of the maximal ideal, providing new, improved bounds for the equivalence of parameterizations and the isomorphism of completions that strengthen previous results by Greuel, Pfister, and Hironaka.

Original authors: Reinhold Hübl, Irena Swanson

Published 2026-08-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Reinhold Hübl, Irena Swanson

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

In the vast landscape of mathematics, there is a branch dedicated to understanding the shapes of curves, not just the smooth, perfect lines drawn in geometry textbooks, but the messy, jagged points where lines cross themselves or come to a sudden, sharp halt. These are known as singularities, and they are the rough edges of the mathematical world. For over a century, mathematicians have tried to classify these rough spots, asking a fundamental question: how much do you need to know about a singularity to be absolutely certain of what it is? Imagine trying to identify a unique, complex knot just by looking at a tiny, magnified fragment of it. If the fragment is too small, it might look like any other knot. But if you zoom out just a little further, the unique pattern of the knot becomes undeniable. The challenge has always been figuring out exactly how much "zoom" is required to distinguish one singular point from another.

This question is not merely about abstract shapes; it is about the precise language of algebra that describes these points. Mathematicians use a tool called a parameterization, which is essentially a way of tracing the curve with a single moving point, to study these singularities. They also rely on a specific number, known as the delta-invariant, which acts like a measure of the complexity or the "roughness" of the point. The higher this number, the more tangled the singularity is. For decades, researchers have known that if two singularities look identical when you ignore the very finest details—meaning they match up to a certain level of precision—they are likely the same. However, the exact threshold for this precision has been a matter of debate, with previous estimates suggesting you needed to look very deeply into the details to be sure.

In a recent paper, Reinhold Hüb and Irena Swanson have sharpened this understanding, proving that you do not need to look as deeply as previously thought to confirm that two singularities are the same. They focused on a specific type of curve singularity that is "unibranch," meaning it looks like a single line that has folded over on itself, rather than multiple lines crossing. The researchers demonstrated that if two different descriptions of such a curve match each other up to a power of the maximal ideal that is just slightly larger than twice the complexity number, then the two curves are mathematically identical. This is a significant improvement over earlier work, which suggested a much higher threshold was necessary. By refining the bounds, they showed that the "fingerprint" of a singularity becomes unique much sooner than anyone had calculated before.

The paper also addresses a broader scenario involving two different curves that might be isomorphic, or structurally the same, even if they are described in different ways. The authors proved that if these two curves agree on a sufficiently high level of detail, then their complete mathematical structures are not just similar, but actually isomorphic. Furthermore, they showed that the transformation connecting the full, complete versions of these curves can be constructed to match the original partial agreement. This means that a local agreement, observed at a high enough level of precision, guarantees a global agreement. The researchers provided a new, tighter bound for this agreement, improving upon a famous result by Heisuke Hironaka from the 1960s. While Hironaka had shown that agreement up to a certain high power implied isomorphism, Hüb and Swanson proved that the required power can be reduced, making the condition for equivalence less restrictive and more efficient.

To reach these conclusions, the authors utilized a deep connection between the geometry of the curve and a set of numbers called the value semigroup, which tracks the orders of vanishing of functions on the curve. They used a specific sequence of numbers, known as the Herzog–Kunz sequence, which acts like a set of coordinates for the singularity. By analyzing how these sequences behave when the curves are truncated at different levels, they were able to construct a precise argument showing that if the curves match up to a certain point, the differences between them must be zero. They also examined cases where the curves have multiple branches, showing that while the situation is more complex, similar principles apply, allowing them to recover and sometimes improve upon existing bounds for these multi-branched cases.

The significance of this work lies in its precision. In mathematics, knowing the exact limit of a rule is often as important as the rule itself. By lowering the threshold for when two singularities are considered equivalent, the authors have provided a more efficient tool for classification. They showed that for a curve with a complexity measure of $\delta$ , checking agreement up to a power of $2\delta + 1$ is sufficient to guarantee that the curves are the same. This is a concrete, proven fact, not a suggestion. They even provided examples where this new bound is the absolute best possible, meaning it cannot be lowered any further without losing accuracy. This work strengthens the foundation of singularity theory, offering a clearer, more efficient path for mathematicians to navigate the intricate world of curve singularities.

Technical Summary: Equivalence of Curve Singularities and Delta-Invariants

Problem Statement
The paper addresses the classification of local curve singularities over an algebraically closed field $k$ . Specifically, it investigates the conditions under which two parameterizations of a curve singularity, or two singularities themselves, are equivalent (isomorphic) based on their behavior modulo high powers of their maximal ideals. The central question is to determine the minimal power $N$ such that if two objects agree modulo $\mathfrak{m}^N$ , they are guaranteed to be equivalent. Previous work by Hironaka, Greuel, and Pfister established bounds for this "determinacy" depending on the delta-invariant ( $\delta$ ) and the conductor degree ( $c$ ) of the singularity. This paper aims to refine and strengthen these bounds.

Methodology
The authors employ a combination of valuation theory, the theory of value semigroups, and the study of blowups (quadratic transforms) of curve singularities.

Value Semigroups and Herzog-Kunz Sequences: For unibranch (analytically irreducible) curves, the authors utilize the value semigroup $V$ of the normalization $\bar{R} \cong k[[t]]$ . They rely on the Herzog-Kunz sequence, which characterizes the minimal generators of the local ring $R$ in terms of their valuations.
Inductive Arguments on Blowups: For the case of multiplicity $e(R)=2$ , the authors utilize quadratic transforms (blowups) of the singularity. They establish that the delta-invariant decreases by 1 under such a transform, allowing for an inductive proof strategy.
Lifting Isomorphisms: A core technical tool is the construction of isomorphisms between blowup algebras. By proving that an isomorphism between truncated rings $R/\mathfrak{m}^{j+1}$ induces an isomorphism between the corresponding blowup algebras modulo lower powers, the authors can "lift" the equivalence from the singular curve to its normalization or a transformed version, eventually reconstructing the full isomorphism.
Characteristic-Specific Analysis: The proof for the case $e(R)=2$ in characteristic 2 requires specific classification results for algebroid curves, distinguishing between different normal forms (e.g., $Y^2 + X^{2\delta+1}$ vs. $Y^2 + YX^i + X^{2\delta+1}$ ).

Key Contributions and Results

1. Improved Determinacy Bounds for Parameterizations (Section 3)
The paper improves upon the bounds established by Greuel and Pfister regarding when a parameterization is determined by its truncation.

Theorem 3.1: For a unibranch complete curve $R$ with conductor degree $c(R)$ and Herzog-Kunz sequence $a_1 < \dots < a_n$ , any primitive parameterization is $(d_R - 1)$ -L-determined, where $d_R = \max\{c(R), a_n + 1\}$ .
Corollary 3.3: This result is sharpened to depend explicitly on the delta-invariant. Any parameterization of a unibranch curve with delta-invariant $\delta(R)$ is $(2\delta(R) + 1)$ -L-determined.
Sharpness: The authors provide examples (Example 3.4) demonstrating that the bound $2\delta(R) + 1$ is sharp in certain cases where the previous bound $2c(R) - 1$ (from Greuel and Pfister) was not.
Multibranched Curves: The authors extend these results to multibranched curves (Theorem 3.7), recovering the Greuel-Pfister bound of $2c(R)-1$ (Corollary 3.8) and improving it to $c(R)-1$ for plane multibranched curves (Corollary 3.9).

2. Strengthened Equivalence Theorem for Singularities (Section 4)
The paper refines Hironaka's Theorem B regarding the equivalence of two reduced and irreducible curve singularities.

Theorem 4.8: Let $(R, \mathfrak{m})$ and $(R', \mathfrak{m}')$ be complete reduced and irreducible curve singularities over an algebraically closed field. If there exists a $k$ -algebra isomorphism $\phi: R/\mathfrak{m}^{j+1} \to R'/\mathfrak{m}'^{j+1}$ with $j \ge 2\delta(R) + 1$ , then there exists an isomorphism $\psi: \hat{R} \to \hat{R}'$ of the completions such that $\psi \equiv \phi \pmod{\mathfrak{m}^{j+1-2\delta(R)}}$ .
Improvement: This replaces Hironaka's original bound of $3\delta(R) + 1$ with $2\delta(R) + 1$ .
Proof Strategy: The proof proceeds by induction on the delta-invariant. For $e(R) \ge 3$ , the authors follow Hironaka's strategy but utilize the improved bound derived from the conductor properties. For $e(R) = 2$ , they utilize the quadratic transform to reduce the delta-invariant, applying the inductive hypothesis to the transformed curves.
Sharpness: Remark 4.9 notes that the bound $2\delta(R) + 1$ is sharp, as demonstrated by curves of the form $k[[t^2, t^{2n+1}]]$ .

Significance and Claims
The authors claim that their work provides a "new and better bound" on the power of the maximal ideal required to guarantee the equivalence of curve singularities and their parameterizations.

Strengthening Existing Bounds: The paper explicitly states it strengthens bounds from Greuel and Pfister [3] regarding parameterization equivalence and Hironaka [5] regarding singularity equivalence.
Optimality: The authors demonstrate through examples that their bounds are sharp in specific cases, indicating that further general improvement without additional invariants is not possible.
Modesty: The paper acknowledges that while the bound $2\delta(R) + 1$ is an improvement, even better bounds involving more invariants (specifically $6\delta(R)/e(R) + 1$ ) can be derived from their arguments (Remark 4.9). The assumption of an algebraically closed field is noted as necessary only for the specific classification of multiplicity 2 curves in characteristic 2; otherwise, an infinite residue field suffices.

In summary, the paper establishes that the delta-invariant is a more precise measure for the determinacy of curve singularities than previously utilized, providing tighter, often sharp, bounds for when local isomorphisms modulo high powers of the maximal ideal extend to global isomorphisms of the completions.

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