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Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs

This paper classifies all weak contact conics to irreducible quartics with two nodes and one cusp by analyzing integral sections of associated rational elliptic surfaces, and utilizes these results to construct new Zariski pairs of degrees 7 and 8.

Original authors: Khulan Tumenbayar

Published 2026-05-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Khulan Tumenbayar

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 artist working on a canvas (the mathematical plane). You have drawn a specific, slightly messy four-sided shape called a quartic curve. This shape has two "kinks" (nodes) and one sharp "pointy tip" (a cusp). It's a bit like a distorted diamond with a jagged edge.

Now, you want to draw a conic (a smooth circle or oval shape) that touches this messy shape in a very special, polite way.

The "Polite Touch" (Weak Contact Conics)

The paper is about finding all the possible ovals that can touch your messy shape without "cutting" through it roughly.

The Rule: Wherever the oval touches the shape, they must meet in a way that feels like a perfect handshake—mathematically, the "intersection" must be an even number.
The Constraint: The oval must also touch the shape at a smooth, non-broken spot (a point we'll call $z_0$ ).

The author, Khulan Tumenbayar, asks: "How many different ovals can I draw that follow these rules?"

The Magic Map (Rational Elliptic Surfaces)

To answer this, the author doesn't just look at the drawing on the canvas. Instead, they use a magic map called a "Rational Elliptic Surface."

Think of this surface as a 3D landscape or a complex terrain that sits above your 2D drawing.

Every point on your messy shape corresponds to a path or a "section" on this 3D terrain.
The author translates the problem of "finding ovals" into the problem of "finding paths on this 3D map."
Just as you can add two numbers together, you can "add" two paths on this map to create a new path. This algebraic structure (the Mordell-Weil group) acts like a GPS system that tells the author exactly where the valid ovals are hiding.

The Four Scenarios

The author realizes the answer depends on how the "touching line" at the smooth point ( $z_0$ ) behaves. They categorize the situation into four scenarios:

The Standard Touch: The line hits the shape normally.
The Double Touch: The line is a "bitangent" (touching two spots at once) or hits very deeply.
The Cusp Touch: The line goes straight through the sharp pointy tip.
The Node Touch: The line goes straight through one of the two kinks.

For each scenario, the author creates a menu (a table) listing exactly how many ovals of different "types" exist.

Type 1: The oval touches only the sharp tip.
Type 2: The oval touches only one kink.
Type 3: The oval touches a kink and the tip.
...and so on.

The paper provides a precise count for every single type in every scenario. For example, in the "Standard Touch" scenario, there are exactly 3 ovals of Type 1, 4 of Type 2, etc.

The "Zariski Pairs" (The Optical Illusion)

The most exciting part of the paper is the application: creating Zariski Pairs.

Imagine you have two different paintings, Painting A and Painting B.

Painting A is made of your messy shape, a straight line, and one of the special ovals.
Painting B is made of the exact same messy shape, a different straight line, and a different special oval.

The Trick:
If you look at the "skeleton" of the paintings (how the lines cross, how many pieces there are, and where the kinks are), they look identical. They have the same "combinatorics."

However, the author proves that if you were to try to stretch or bend Painting A to make it look exactly like Painting B, you couldn't do it without tearing the canvas. They are topologically different. They are "twins" that look the same from a distance but have different internal structures.

The paper constructs specific examples of these "twins" (Zariski pairs) of degree 7 and degree 8. It shows that by swapping which specific oval you use (based on the math of the 3D map), you create a new shape that is mathematically distinct from the first, even though they look the same on paper.

Summary

In short, this paper is a guidebook for:

Counting specific types of touching ovals for a messy four-sided shape.
Using a 3D mathematical map to find them easily.
Using these ovals to build optical illusions (Zariski pairs)—pairs of shapes that look identical in their arrangement but are fundamentally different in their geometry.

The author doesn't claim this will cure diseases or build bridges; the value is purely in the beauty and logic of the geometry itself, solving a puzzle about how curves can touch and twist in the mathematical plane.

Technical Summary: Geometry of Weak Contact Conics to Irreducible Quartics with 2 Nodes and 1 Cusp

Problem Statement
This paper investigates the geometry of weak contact conics to an irreducible plane quartic curve $Q$ possessing exactly two nodes ( $x_1, x_2$ ) and one cusp ( $y$ ) as its singularities. A conic $C$ is defined as a weak contact conic to $Q$ if the intersection multiplicity $I_x(Q, C)$ is even for every point $x \in Q \cap C$ . The study specifically restricts attention to cases where $Q \cap C$ contains at least one smooth point $z_0$ of $Q$ . The primary objective is to classify and enumerate all such weak contact conics based on the configuration of singular points of $Q$ contained within $C$ (categorized into six types) and the local behavior of the tangent line $l_{z_0}$ at $z_0$ .

Methodology
The authors employ the theory of rational elliptic surfaces to translate the geometric problem of finding contact conics into an algebraic problem concerning sections of these surfaces.

Surface Construction: For a given quartic $Q$ and a smooth point $z_0 \in Q$ , the authors construct a rational elliptic surface $S_{Q, z_0}$ . This is achieved via a double cover of $\mathbb{P}^2$ branched over $Q$ (and an auxiliary curve), followed by a sequence of blow-ups to resolve singularities.
Mordell-Weil Group: The problem of finding weak contact conics is reduced to studying the sections of the fibration $\phi: S_{Q, z_0} \to \mathbb{P}^1$ . The set of sections, $MW(S_{Q, z_0})$ , forms the Mordell-Weil group, which is identified with the group of rational points on the generic fiber $E(\mathbb{C}(t))$ .
Height Pairing: The authors utilize Shioda's height pairing $\langle \cdot, \cdot \rangle$ on the Mordell-Weil group. By analyzing the singular fibers of $S_{Q, z_0}$ (determined by the intersection of the tangent line $l_{z_0}$ with $Q$ ), they compute the Gram matrices of the lattice generated by specific sections corresponding to lines passing through the singularities of $Q$ .
Classification: Using the height pairing values, the authors determine which linear combinations of basis sections correspond to conics satisfying the "weak contact" condition (even intersection multiplicities).

Key Contributions and Results
The paper provides a complete enumeration of weak contact conics for the specified quartic $Q$ , categorized by the position of the tangent line $l_{z_0}$ and the type of singular points the conic passes through.

Classification of Cases: The analysis is divided into four cases based on the intersection of $l_{z_0}$ $l_{z_{0}}$ with $Q$ $Q$ :
- (I) $l_{z_0}$ meets $Q$ transversely at other points (generic case).
- (II) $l_{z_0}$ is a bitangent or has 4-fold contact.
- (III) $l_{z_0}$ passes through the cusp.
- (IV) $l_{z_0}$ passes through a node.
Enumeration Tables: For each case, the authors derive the number of weak contact conics ( $n_i$ ) for each of the six types (defined by which subset of $\{x_1, x_2, y\}$ lies on $C$ ). For instance, in Case (I), there are 3 conics of Type 1 (passing through the cusp only), 4 of Type 2, 4 of Type 3, 1 of Type 4, 1 of Type 5, and 1 of Type 6.
Explicit Construction: The paper explicitly constructs the sections in the Mordell-Weil group corresponding to these conics. For Case (I), the conics correspond to elements such as $[2]P_1$ , $[2]P_2$ , and $[2]P_0$ , where $P_i$ are basis elements derived from lines connecting the singular points.
Zariski Pairs: As a significant application, the authors construct Zariski pairs of degree 7 and degree 8. A Zariski pair consists of two plane curves with the same combinatorial type (same singularities and intersection patterns) but non-homeomorphic embeddings in $\mathbb{P}^2$ $P^{2}$ .
- The constructed pairs consist of the quartic $Q$ , a line $L_i$ passing through a node and the cusp, and a weak contact conic $C_j$ .
- By selecting specific conics $C_0, C_1, C_2$ derived from the Mordell-Weil group, the authors demonstrate that pairs such as $(Q + L_1 + C_1, Q + L_2 + C_1)$ share the same combinatorics but are not homeomorphic. This non-homeomorphism is proven using the linear dependence/independence properties of the corresponding sections in the Mordell-Weil group (specifically utilizing Theorem 2.1 and 2.2 regarding homeomorphisms preserving the quartic component).

Significance
The paper extends previous work on contact conics for 3-nodal and 3-cuspidal quartics to the more complex case of quartics with mixed singularities (2 nodes, 1 cusp). It demonstrates that even under the restriction of having a smooth intersection point, non-trivial examples of Zariski pairs can be generated. The work establishes a rigorous link between the arithmetic of the Mordell-Weil group of rational elliptic surfaces and the topological classification of plane curve arrangements, providing explicit algebraic equations for these geometric objects. The construction of degree 7 and 8 Zariski pairs enriches the known catalog of such topological distinct pairs in algebraic geometry.