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A note on the self-intersection of rational unicuspidal curve with one Puiseux pair

This paper proves the existence of rational unicuspidal curves with a single Puiseux pair $(p,q)$ in algebraic surfaces that achieve a conjectured upper bound on self-intersection, utilizing recursive identities derived from studying bicuspidal curves to establish a new, more computable formula for this bound and a formula for the last entry of the singularity's multiplicity sequence.

Original authors: Weimin Chen, Yusupjan Ouyang, Sam Silver

Published 2026-08-10

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CC BY 4.0

Original authors: Weimin Chen, Yusupjan Ouyang, Sam Silver

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 the universe of shapes not as smooth, perfect spheres or cubes, but as a wild, tangled garden of mathematical surfaces. In this garden, there are special paths called "curves" that twist and turn. Sometimes, these paths hit a snag and form a sharp, pointy knot called a "cusp." Mathematicians love to study these knots because they hold secrets about the shape of the space they live in. But there's a tricky rule in this garden: how tightly can a path with a knot wrap around itself? If you pull the path too tight, it might snap or break the rules of the universe.

For a long time, mathematicians have been trying to figure out the absolute limit of how much a specific type of knotted path—a "rational unicuspidal curve" with just one special knot—can squeeze into a surface. They have a guess, a theoretical "speed limit" for how much this path can intersect itself. This limit depends entirely on the shape of the knot, described by two numbers, $(p, q)$ , which act like a secret code for the knot's geometry. Knowing this limit is like knowing the maximum weight a bridge can hold before it collapses; it helps us understand the fundamental laws of these mathematical worlds.

Now, enter a team of researchers who decided to test this speed limit. They didn't just guess; they built a massive, intricate machine to see if they could actually construct a path that hits this limit exactly. They looked at two different "gardens" (mathematical surfaces) and followed two specific families of twisted paths. By using a clever set of recursive rules—like a recipe that tells you how to bake a bigger cake based on a smaller one—they discovered something exciting. They proved that for any knot code $(p, q)$ , there really is a path that hits that theoretical maximum. They didn't just find a path that comes close; they found the "optimal" path that reaches the very edge of what is possible.

The authors also found a new, simpler way to calculate this maximum limit. Before, the formula was like a complex riddle with many steps. Now, they have a clearer map that connects the limit directly to the "multiplicity sequence" of the knot—a list of numbers that describes how the knot unravels when you zoom in. It's like realizing that the height of a building isn't just a random number, but is directly tied to the number of bricks in its foundation.

One of the most playful discoveries came from looking at "bicuspidal" paths—paths with two knots instead of one. The team wondered: if you take a path with two knots and "fix" one of them, how close does the resulting single-knot path get to the maximum limit? They found that the "gap" between the actual result and the perfect maximum is almost always zero or one. In fact, they mapped out exactly when the gap is zero, when it's one, and in one very rare, specific case, when it's two. This means that for almost every situation, you can get a path that is perfectly optimal just by fixing one knot on a double-knotted path.

So, what did they actually prove? They showed that the theoretical upper bound for the self-intersection of these curves is not just a dream; it is a reality that can be reached. They confirmed that for every pair of numbers $(p, q)$ , there exists a surface and a curve where the curve hits that exact maximum number. They also ruled out the idea that there might be cases where the limit is unreachable; their work shows that the limit is always attainable. The paper doesn't just suggest this; it provides a rigorous mathematical proof, using recursive formulas and specific constructions to demonstrate that these "optimal" curves exist for every single case they examined.

In the end, this paper is like finding the perfect key for every lock in a giant castle. The mathematicians had a theory about the size of the keys, and they built the keys to prove it. They also discovered that the keys are often made from slightly larger, double-keyed locks that are easier to find. This gives us a much deeper understanding of how these twisted, knotted paths behave and confirms that the mathematical universe is as orderly and predictable as the authors hoped, even in its most complex, pointy corners.

Technical Summary: Self-Intersection of Rational Unicuspidal Curves with One Puiseux Pair

Problem Statement
This paper addresses the problem of determining the upper bound for the self-intersection number ( $C \cdot C$ ) of a rational unicuspidal curve $C$ possessing a single Puiseux pair $(p, q)$ within an algebraic surface $X$ . In prior work [1], the first-named author conjectured an upper bound, denoted $m_{p,q}$ , which depends solely on the singularity type $(p, q)$ . The conjecture posits that for any such curve, $C \cdot C \leq m_{p,q}$ . While the existence of "optimal" symplectic rational unicuspidal curves realizing this bound was established in [1], the algebraic realization and a deeper computational understanding of the bound remained open. Specifically, the original definition of $m_{p,q}$ involved complex integer constraints ( $p', q'$ ) that were difficult to compute and offered limited insight into the topological structure of the singularity.

Methodology
The authors employ a combination of algebraic geometry and recursive analysis to refine the bound and verify its realizability. The methodology proceeds in three main stages:

Derivation of Recursive Identities: The authors introduce recursive relations for the multiplicity sequence of the singularity and the bound $m_{p,q}$ . By defining auxiliary quantities $R_{p,q}$ (related to the sum of squares of the multiplicity sequence) and $B_{p,q}$ (the deviation of the bound from $pq$), they establish how these values transform under the reduction $(p, q) \to (p, q-p)$ .
Formulation of Explicit Formulas: Using these recursive relations, the authors derive a new, computationally efficient formula for the last entry of the multiplicity sequence, $\ell(p, q)$ , and consequently for the bound $m_{p,q}$ . This new formula relies on the modular inverse of $p$ modulo $q$ .
Construction via Bicuspidal Families: To prove the existence of curves realizing the bound, the authors analyze two infinite families of rational bicuspidal curves:
- A family in $\mathbb{CP}^2$ parametrized by $[z^p : z^q : 1]$ .
- A family in $\mathbb{CP}^1 \times \mathbb{CP}^1$ parametrized by $([z^p : 1], [z^q + z^{\alpha p} : 1])$ .
  By resolving one of the two singularities in these bicuspidal curves, they generate rational unicuspidal curves. They calculate the "gap" (the difference between the conjectured bound $m_{p,q}$ and the actual self-intersection) for these resulting curves across all possible cases of $(p, q)$ .

Key Contributions and Results

New Formula for the Bound ( $m_{p,q}$ ): The paper provides a simplified formula for $m_{p,q}$ that connects the bound directly to the topological structure of the singularity. Specifically, $m_{p,q}$ is expressed in terms of $pq$, the last entry of the multiplicity sequence $\ell(p, q)$ , and a conditional term based on the relationship between $p$ and $q$ .
- If $q = p+1$ or ( $p > 2$ and $q \equiv -1 \pmod p$ ), then $m_{p,q} = pq + \ell(p, q) + 1$ .
- Otherwise, $m_{p,q} = pq + \ell(p, q)$ .
- The authors also provide a direct formula for $\ell(p, q) = \lfloor \frac{q-1}{\min(u, q-u)} \rfloor$ , where $up \equiv 1 \pmod q$ .
- This formulation reveals that $m_{p,q}$ is strictly smaller than the symplectic bound proposed by Golla and Starkston [4] by exactly 1 in most cases, coinciding only for specific pairs like $(m-1, m)$ or $(m, km-1)$.
Existence of Optimal Curves (Theorem 1.3): The authors prove that for any coprime pair $(p, q)$ , there exists a rational surface containing a rational unicuspidal curve with Puiseux pair $(p, q)$ such that its self-intersection exactly equals $m_{p,q}$ . This strengthens the symplectic existence result from [1] to the algebraic category.
Classification of Gap Patterns (Theorem 1.5): Through the analysis of the two bicuspidal families, the paper completely classifies the "gap" patterns (where gap $= m_{p,q} - C \cdot C$ ) for the unicuspidal curves obtained by resolving singularities.
- The gap is always non-negative (0, 1, or 2).
- In almost all cases, at least one of the two possible resolutions (at $z=0$ or $z=\infty$ ) yields a gap of 0, meaning the bound is realized.
- The only exception where a gap of 2 occurs is in the $\mathbb{CP}^1 \times \mathbb{CP}^1$ family when $q - p = 1$ .
- This classification confirms that the bicuspidal families do not produce counterexamples to the conjecture and serve as the constructive proof for Theorem 1.3.

Significance
The paper claims to strengthen the symplectic version of the result in [1] by establishing the existence of the optimal curves in the algebraic setting. By deriving a new formula for $m_{p,q}$ , the authors provide a tool that is "more amenable to computations" and offers "better insight concerning the nature of the bound" by linking it explicitly to the multiplicity sequence of the singularity. The work suggests that the set of singular rational curves achieving maximal self-intersection may be classified by a small number of infinite families, a hypothesis supported by the exhaustive gap analysis performed here. The results also clarify the precise relationship between the algebraic bound and the symplectic bound, showing they differ by exactly 1 in the cases where they are not equal.

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