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Defect of irreducible plane curves with simple singularities

This paper establishes lower bounds on the defect of irreducible plane curves with simple singularities (nodes, ordinary cusps, and ordinary triple points) and proves that such curves with sufficiently high Arnold exponents are never free.

Original authors: Piotr Pokora

Published 2026-08-27

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CC BY 4.0

Original authors: Piotr Pokora

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 the shapes formed by equations, known as algebraic geometry. Imagine drawing a line on a piece of paper; now imagine that line is not just a simple stroke but a complex curve defined by a specific mathematical rule. These curves can be smooth, or they can have sharp points, kinks, or self-intersections where the line crosses over itself. Mathematicians have long been fascinated by how these shapes behave, particularly when they are "free" in a specific technical sense. Being free is a special property that implies a curve has a high degree of symmetry and structural simplicity, making it easier to understand and predict. However, most curves are not free; they are messy, with defects that disrupt their perfect order. The question of how far a curve is from being free, and what causes that distance, is a central puzzle. Recently, a new way of measuring this distance, called the "defect," was introduced to help researchers quantify exactly how much a curve fails to be free. Understanding this defect is crucial because it connects to deep, unsolved problems about how the arrangement of lines and curves in space determines their fundamental nature.

A mathematician named Piotr Pokora has taken a fresh look at this problem, focusing on a specific type of curve: those that are irreducible, meaning they are made of a single continuous piece, and those that possess only simple, well-understood types of kinks. His work, published in September 2024, investigates whether these curves can ever achieve that elusive state of being free. The study centers on curves that have "nodes," which are simple crossings, "cusps," which are sharp points where the curve comes to a halt and turns back, and "triple points," where three branches meet at a single spot. Pokora's goal was to determine if curves with these specific features could ever be free, or if the presence of these singularities inevitably creates a defect that prevents them from ever reaching that perfect state.

The research reveals a clear and somewhat surprising boundary. Pokora found that for curves with a certain level of complexity, specifically those with high enough "Arnold exponents"—a measure of how severe the singularities are—the curves can never be free. In fact, he proved that for a wide class of these curves, the defect is not just a tiny imperfection but a substantial number that grows as the curve gets larger. For instance, when looking at curves made entirely of simple crossings, the defect is guaranteed to be at least a quarter of the square of the curve's degree minus one. This means that as the curve becomes more complex, the gap between it and being a "free" curve becomes arbitrarily large. The study also examined curves with triple points and found similar results, showing that the defect remains significant and follows a predictable lower bound based on the curve's size.

Perhaps the most striking finding concerns a specific family of curves constructed by a previous researcher, which feature only sharp cusps. Pokora calculated the exact defect for these curves and discovered that the defect is equal to the curve's genus, a number that describes the number of "holes" or loops in the shape. For the smallest curve in this family, a six-degree curve with nine cusps, the defect is exactly one. In the language of this field, a defect of one means the curve is "nearly free," which is the closest a curve can get to being perfect without actually being free. However, for larger versions of these curves, the defect grows, confirming that they are far from free. This suggests that rational curves with cusps are exceptionally special and rare, as it is incredibly difficult to construct them in a way that minimizes their defects.

The paper culminates in a powerful rule that applies to reduced plane curves with even degrees that have only simple singularities, provided their Arnold exponents are sufficiently high. Pokora demonstrated that if the singularities are severe enough to meet this specific threshold, the curve is mathematically guaranteed to have a defect of at least one. This means it is impossible for such a curve to be free. To prove this rule is the best possible, he provided a specific example of a curve made of four conic sections that meets the criteria exactly and has a defect of one, showing that the boundary he found cannot be pushed any further. This result effectively rules out the possibility of finding free curves within this specific subset of the category, addressing a key aspect of the broader open questions in the field. The work also applies to arrangements of straight lines, proving that no arrangement of twelve or more lines, intersecting only in double or triple points, can ever be free. By establishing these firm limits, the paper clarifies the structural landscape of plane curves, showing that for many natural classes of shapes, the imperfection known as the defect is not just a possibility, but an inevitability.

Problem Statement
This note investigates the "defect" of singular plane curves, a numerical invariant recently introduced by Dimca that measures the discrepancy between a reduced plane curve and the property of being "free." A reduced curve $C = \{f=0\} \subset \mathbb{P}^2_{\mathbb{C}}$ is defined as free if its module of derivations preserving the ideal $\langle f \rangle$ is a free graded $S$ -module. The defect, denoted $\nu(C)$ , is defined as the maximum dimension of the graded components of the Jacobian module $N(f) = I_f/J_f$ , where $J_f$ is the Jacobian ideal and $I_f$ is its saturation.

The primary motivation for this study stems from Conjecture 6, which posits that any rational cuspidal curve is either free or "nearly free" (where the defect $\nu(C)=1$ ). While this conjecture suggests a strong structural constraint on rational cuspidal curves, there is a lack of general predictions or results regarding the defect for other natural classes of irreducible plane curves. The paper aims to establish lower bounds on the defect for specific classes of irreducible curves and to demonstrate that curves with sufficiently high Arnold exponents are generally not free.

Methodology
The paper employs a combination of algebraic geometry tools and numerical invariants to analyze the defect:

Dimensional Analysis of Derivations: The author utilizes the minimal degree of derivations killing the defining polynomial, denoted $mdr(C)$, and the total Tjurina number $\tau(C)$ , which sums the local Tjurina numbers of all singular points.
Dimca's Theorem: The core computational tool is Theorem 2 (Dimca), which provides explicit formulas for $\nu(C)$ based on the relationship between the degree $d$ of the curve and $mdr(C)$. Specifically, if $mdr(C) \ge (d-2)/2$ , the defect is given by $\nu(C) = \lceil \frac{3}{4}(d-1)^2 \rceil - \tau(C)$ .
Arnold Exponents: To bound $mdr(C)$ from below, the author applies the Dimca-Sernesi theorem (Theorem 9), which states that $mdr(f) \ge \alpha_C \cdot d - 2$ , where $\alpha_C$ is the Arnold exponent of the curve (the minimum log canonical threshold over all singular points).
Singularities Classification: The analysis focuses on curves with specific types of singularities (nodes, ordinary triple points, ordinary cusps, and ADE singularities), for which the log canonical thresholds (and thus the Arnold exponents) are known or can be estimated.
Upper Bounds on Tjurina Numbers: To derive lower bounds for the defect, the author utilizes known upper bounds for the total Tjurina number $\tau(C)$ , such as the maximum Tjurina number for a given degree and $mdr$ derived by Du Plessis and Wall.

Key Contributions and Results
The paper presents four main theorems (A, B, C, and D) establishing lower bounds on the defect for various classes of irreducible plane curves:

Theorem A (Nodal Curves): For a nodal plane curve $C_d$ of degree $d \ge 4$ , the defect satisfies $\nu(C_d) \ge \frac{1}{4}(d^2 - 1)$ . This result demonstrates that the defect for nodal curves can be arbitrarily large, implying they are far from being free.
Theorem B (Genus Zero with Triple Points): There exist irreducible, reduced plane curves of genus zero with degree $d=3k$ ( $k \ge 3$ ) admitting exactly $2k$ ordinary triple points and nodes, such that $\nu(K_{3k}) \ge \frac{1}{4}(9k+1)(k-1)$ .
Theorem C (Ivinskis' Cuspidal Curves): For the family of curves $C_{6k}$ constructed by Ivinskis (degree $6k$ , $9k^2$ ordinary cusps), the defect is exactly calculated as $\nu(C_{6k}) = 9k^2 - 9k + 1$ . Notably, for $k=1$ , the curve $C_6$ is nearly free ( $\nu=1$ ). The paper observes that for this family, the defect coincides with the geometric genus $g(C_{6k})$ .
Theorem D (Non-freeness Criterion): This is the main result of the note. It provides a criterion for non-freeness: if a reduced plane curve $C$ of even degree $d=2m \ge 4$ admits only ADE singularities and has an Arnold exponent $\alpha_C \ge \frac{1}{2} + \frac{1}{m}$ , then $\nu(C) \ge 1$ . Consequently, such curves are never free. The paper includes an example of a curve arrangement of $2m$ conics where this bound is sharp ( $\nu=1$ ).

Significance and Claims
The paper claims that its results highlight the special nature of rational cuspidal plane curves. By showing that curves with "sufficiently high" Arnold exponents (specifically those satisfying the condition in Theorem D) are never free, the author justifies the heuristic phenomenon that constructing irreducible free or nearly free curves is extremely difficult.

The author explicitly states that their work continues the study of the defect for natural classes of irreducible plane curves, filling a gap where general predictions were previously absent. The results serve as a counterpoint to the potential simplicity suggested by Conjecture 6 for rational cuspidal curves, by demonstrating that for broader classes (like nodal curves or curves with high Arnold exponents), the defect is strictly positive and often large. The paper concludes that reduced simply singular plane curves with sufficiently high Arnold exponents are never free.

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