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On the Numerical Terao Conjecture

This paper proves the Numerical Terao Conjecture for even-degree conic-line arrangements with only ADE singularities, while simultaneously refuting its strongest formulation in the broader quasi-homogeneous setting by constructing a degree-nine counterexample consisting of seven lines and one conic that shares the same weak combinatorics but possesses different freeness properties.

Original authors: Piotr Pokora

Published 2026-08-07

📖 7 min read🧠 Deep dive

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

Imagine you are an architect designing a city made entirely of geometric shapes. In this city, the buildings are lines and curves, and the places where they crash into each other are the "singularities"—the traffic jams of the geometry world. For decades, mathematicians have been obsessed with a specific rule about these cities, known as the Numerical Terao Conjecture. The rule is surprisingly simple: if you have two different cities that look exactly the same on a map (meaning they have the same number of buildings and the same types of traffic jams), and one of those cities is "free" (a special mathematical state where the buildings are perfectly balanced and stable), then the other city must also be free. It's like saying if two houses have the exact same blueprint and the same number of cracks in the foundation, and one house is structurally sound, the other one has to be too. This idea matters because it suggests that the "soul" of a shape (its stability) is entirely determined by its "fingerprint" (its map of intersections). If the rule holds, you never have to do the hard math to check stability; you just look at the map.

However, this paper by Piotr Pokora takes a sledgehammer to that idea, but with a twist. The author first checks if the rule works for a specific type of city: one built only from straight lines and smooth circles (conics) that crash in very orderly, predictable ways. In this restricted neighborhood, the rule holds true. But then, the author decides to get messy. They introduce a new type of traffic jam called an "ordinary quadruple point," where four lines or curves meet at a single spot in a specific, slightly more complex way. By building a very specific, nine-degree city (a complex arrangement of seven lines and one smooth conic), the author proves that the Numerical Terao Conjecture is actually false in this broader setting. They constructed two cities that are identical twins on the map—both have seven lines, one conic, and the exact same pattern of eight double points, one triple point, and four quadruple points. Yet, one city is perfectly balanced (free), while the other is wobbling on the edge of collapse (nearly free). This discovery shows that the map isn't enough; you sometimes need to look at the actual bricks to know if the house will stand.

The Story of the Map and the House

In the world of algebraic geometry, mathematicians study shapes defined by equations. A "plane curve" is just a drawing on a flat sheet of paper that follows a specific mathematical recipe. Sometimes, these curves are made of straight lines, and sometimes they include smooth, round shapes like circles or ellipses (called conics). The most interesting parts of these drawings are the "singularities"—the spots where the lines cross, touch, or crash into each other.

Think of a singularity like a knot in a piece of string. Some knots are simple (two lines crossing), while others are complex (three or four lines meeting at a single point). Mathematicians have a way of classifying these knots, giving them names like $A_1$ , $D_4$ , or $X_9$ . The "weak combinatorics" of a curve is basically its ID card: it lists how many lines and curves are in the drawing and exactly how many of each type of knot they have.

The big question mathematicians have been asking is: Does this ID card tell us everything we need to know? Specifically, does it tell us if the curve is "free"? A "free" curve is a very special, stable type of shape. Imagine a mobile hanging from the ceiling; if it's free, every piece is perfectly balanced, and the whole thing moves smoothly without getting stuck. If it's not free, it's a bit wobbly or rigid in a bad way.

The Numerical Terao Conjecture was a hopeful guess that said: "If two curves have the same ID card (same weak combinatorics) and one of them is a perfectly balanced 'free' curve, then the other one must be free too." It was a beautiful idea because it meant you could skip the hard work of checking the balance and just look at the list of knots.

The Two Parts of the Discovery

Piotr Pokora's paper tackles this question in two distinct chapters, like a detective story with a twist.

Part 1: The Safe Neighborhood
First, the author looks at a very specific, well-behaved neighborhood. These are arrangements made of lines and smooth conics (like circles) that only have "ADE" singularities. These are the "good" knots—predictable and orderly. The paper proves that in this specific world, the Numerical Terao Conjecture is true. If you have two such arrangements with the same map of knots, and one is free, the other is definitely free too. The author even shows that for these shapes to be free, they have to be of a certain "even" size (degree). It's a solid, reassuring result for this specific corner of the mathematical universe.

Part 2: The Chaos Zone
Then, the author decides to break the rules. They ask: "What happens if we allow slightly more complex knots, specifically a triple point ( $D_4$ ) and a quadruple point ( $X_9$ )?" An $X_9$ is an "ordinary quadruple point," where four components meet at a single spot. It's still a nice, orderly knot, but it's more complicated than the ones in the first part.

Here is where the paper drops the bombshell. The author constructs a degree-nine counterexample. This is a specific mathematical object made of seven lines and one smooth conic.

The Twin Cities: They built two different curves, let's call them Curve F and Curve G.
The Identical Map: Both curves have the exact same weak combinatorics: 7 lines, 1 conic, 8 double points ( $A_1$ ), 1 triple point ( $D_4$ ), and 4 quadruple points ( $X_9$ ). Their total "Tjurina number" (a measure of the complexity of the knots) is exactly 48 for both.
The Different Reality:
- Curve F is free. It is perfectly balanced with exponents (4, 4). It's the stable, perfect house.
- Curve G is nearly free. It has exponents (3, 6). It is almost balanced, but not quite. It's the house that is wobbling.

Because these two curves have the exact same map but different stability, the Numerical Terao Conjecture is false in this broader setting. The map is not enough to predict the stability of the house.

Why This Matters

This isn't just a game of "gotcha." The paper proves that the Numerical Terao Conjecture, which was a strong hope for mathematicians, fails as soon as you allow these specific quadruple points (and the associated triple points). The author didn't just guess this; they used powerful computer software (SINGULAR) to calculate the exact equations and prove that one curve is free and the other is not.

The paper also highlights that this is the first time a counterexample has been found involving a non-linear component (the conic). Before this, people knew the conjecture could fail with just lines, but this shows it fails even when you mix lines and curves.

The author ends with a playful challenge: "Does this failure happen in the real world (over real numbers) or only in the imaginary world (over complex numbers)?" They leave that question open, inviting the next generation of mathematicians to build their own cities and see if the rules change again.

In short, the paper tells us that while the map of a geometric shape is incredibly useful, it doesn't tell the whole story. Sometimes, two shapes can look identical on paper but have completely different structural integrity, and that's a fascinating surprise for the architects of mathematics.

Technical Summary: On the Numerical Terao Conjecture

Problem Statement
The paper investigates the Numerical Terao Conjecture, which posits that the freeness of a reduced plane curve is determined solely by its "weak combinatorics" (the number of smooth irreducible components of each degree and the count of singular points of each quasi-homogeneous type). While it is established that this conjecture fails for line arrangements in the complex projective plane, the specific behavior of arrangements containing non-linear components (specifically conics) remains an open question. The author seeks to determine if the conjecture holds for even-degree conic-line arrangements with ADE singularities and, conversely, whether it fails when ordinary quadruple points ( $X_9$ ) are admitted.

Methodology
The author employs a combination of algebraic geometry techniques and symbolic computation:

Algebraic Bounds: The study utilizes the Dimca-Sernesi theorem to establish a lower bound for the minimal degree of a Jacobian relation ($mdr(f)$) based on the smallest Arnold exponent ( $\alpha_C$ ) of the singularities. Additionally, the du Plessis-Wall upper bound and its refined versions are used to characterize freeness and nearly freeness via the total Tjurina number ( $\tau$ ).
Combinatorial Rigidity Analysis: For the positive result, the author analyzes the constraints imposed by ADE singularities on conic-line arrangements to prove that the exponents of any free curve with a given weak combinatorics are uniquely determined.
Explicit Construction: For the negative result, the author constructs two specific degree-nine curves ( $C_F$ and $C_G$ ) consisting of seven lines and one smooth conic. These curves are defined by explicit homogeneous polynomials $F$ and $G$ .
Symbolic Computation: All computations regarding Jacobian syzygies, minimal degrees of relations, and Tjurina numbers were performed using the computer algebra system SINGULAR.

Key Contributions and Results

Positive Result (Even Degree, ADE Singularities):
The paper proves that the Numerical Terao Conjecture holds for conic-line arrangements of even degree ( $d=2m$ ) possessing only ADE singularities.
- Theorem 3.2: If such an arrangement is free, any other arrangement with the same weak combinatorics is also free.
- Mechanism: The proof demonstrates that for even degrees with ADE singularities, the minimal degree of a Jacobian relation ( $r$ ) is rigidly fixed to $m-1$ . Consequently, the total Tjurina number forces any curve with the same combinatorics to satisfy the equality condition for freeness. The author further notes that no free ADE conic-line arrangements of even degree exist for $d \ge 20$ .
Negative Result (Degree Nine, $X_9$ Singularities):
The paper constructs a counterexample to the conjecture in the broader setting of quasi-homogeneous singularities that include ordinary quadruple points ( $X_9$ ).
- Theorem 4.1: Two curves, $C_F$ and $C_G$ , are presented. Both share the same weak combinatorics: $W = (7, 1; 8A_1 + D_4 + 4X_9)$ , involving 7 lines, 1 conic, 8 nodes, 1 $D_4$ singularity, and 4 ordinary quadruple points.
- Divergence in Freeness:
  - $C_F$ is free with exponents $(4, 4)$ .
  - $C_G$ is nearly free with exponents $(3, 6)$ and is not free.
- Verification: The author explicitly constructs the Jacobian syzygies for both curves. For $C_G$ , a cubic relation exists ($mdr=3$), and the Tjurina number is $\tau = 48$ , which is exactly one less than the maximum possible for a free curve of degree 9 with $r=3$ ( $\tau_{max}=49$ ). For $C_F$ , the minimal degree is 4, and the Tjurina number satisfies the equality condition for freeness.

Significance and Claims
The paper claims to provide the first counterexample to the Numerical Terao Conjecture involving a non-linear irreducible component (a smooth conic).

The work refines the understanding of the conjecture's scope: it holds for even-degree conic-line arrangements restricted to ADE singularities but fails immediately upon the inclusion of ordinary quadruple points ( $X_9$ ).
The counterexample specifically addresses the "strongest known formulation" of the conjecture, demonstrating that even with fixed weak combinatorics and quasi-homogeneous singularities, freeness is not a combinatorial invariant when $X_9$ singularities are present.
The author concludes by posing an open problem regarding the existence of such counterexamples over the real numbers ( $\mathbb{P}^2_{\mathbb{R}}$ ).

The work is presented as the culmination of the author's study on arrangements of lines and conics with quasi-homogeneous ordinary singularities, building upon previous results in [10].

The Story of the Map and the House

The Two Parts of the Discovery

Why This Matters

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