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Log MMP Constraints on Curves with Cuspidal Singularities

This paper utilizes the log minimal model program to establish sharp upper bounds on the self-intersection of cuspidal rational curves on smooth projective surfaces, thereby confirming a conjecture by Weimin Chen, extending results to arbitrary cuspidal singularities, and resolving a specific question by Evans regarding the non-existence of a degree 102 plane curve with a $(36,289)$ -cusp.

Original authors: Jenia Tevelev

Published 2026-08-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jenia Tevelev

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 a sheet of paper, perfectly smooth and flat. Now, draw a single line on it that loops back to touch itself, creating a sharp, pointed knot where the line crosses over. In the world of geometry, this knot is called a cusp. Mathematicians have long been fascinated by these sharp points, not just because they look interesting, but because they hide deep secrets about the shape of the space they inhabit. When a curve has such a knot, it forces the surface it lives on to behave in very specific ways. If the knot is too tight or the curve is too long, the surface might simply refuse to exist. This tension between the sharpness of a point and the size of the space around it is the central puzzle of this research.

The question at hand is deceptively simple: how large can the self-intersection of such a curve be? In plain terms, if you were to count how many times the curve crosses itself in a mathematical sense, what is the absolute maximum number allowed before the geometry breaks down? This isn't just an abstract game; these curves appear in the study of contact structures, which are mathematical models for how surfaces twist and turn in higher dimensions, and they are crucial for understanding how certain shapes can be filled in without tearing. For years, mathematicians have suspected a specific limit for curves with a single sharp knot, a limit that depends entirely on the complexity of that knot.

Jenia Tevelev has now provided a definitive proof of this limit, using a powerful set of tools known as the log minimal model program. Think of this program as a systematic way of simplifying a complex geometric shape, peeling away layers of unnecessary detail until only the essential structure remains. Tevelev applied this method to a surface containing a rational curve—a curve that can be traced out by a single continuous parameter—studded with sharp cusps. The goal was to see how the curve's self-intersection number changes as the surface is simplified. By carefully tracking how the curve and its surrounding "resolution" (the network of lines created when smoothing out the sharp points) interact during this simplification process, Tevelev demonstrated that the self-intersection cannot exceed a precise value determined by the knot's complexity.

The result is a sharp upper bound, a hard ceiling that no such curve can cross. For a curve with a single knot defined by two numbers, the maximum self-intersection is a specific calculation involving those numbers. Tevelev proved that this ceiling is not just a guess; it is a mathematical fact. Furthermore, the paper shows that this bound is "sharp," meaning there are actual examples of curves that reach this exact limit. The research also extends to curves with multiple knots, providing a new formula that accounts for the combined complexity of all the sharp points. In these cases, the limit is determined by the sum of the individual complexities, adjusted by a small correction factor depending on how the knots are arranged.

One of the most striking applications of this new understanding is the resolution of a specific, long-standing question about a curve of degree 102 with a genus of 10. This curve was suspected to exist in the context of a complex problem involving the embedding of four-dimensional shapes, specifically related to a structure known as the McDuff–Schlenk staircase. The staircase describes how certain shapes fit into others, with a series of "steps" where the rules change. The final step of this staircase involved a curve with a very specific, extremely tight knot. Using the new bounds derived from the minimal model program, Tevelev proved that such a curve cannot exist. The geometry simply does not allow for a curve of that degree and genus to have a knot of that specific type.

This finding is significant because it closes a door that many thought might be open. It confirms that the algebraic rules governing these curves are stricter than previously thought, ruling out a potential candidate for a shape that would have been a key piece in the puzzle of symplectic fillings. The proof relies on the fact that when you try to construct this impossible curve, the mathematical machinery of the minimal model program forces a contradiction. The curve would have to behave in a way that violates the fundamental laws of intersection on a smooth surface.

The paper also compares these new algebraic limits with older limits derived from symplectic geometry, a field that studies shapes using tools from physics and dynamics. In most cases, the new algebraic limits are tighter, meaning they rule out more possibilities than the symplectic ones. In a few specific cases, the two limits agree perfectly, reinforcing the idea that the underlying geometry is consistent across different mathematical approaches. The research highlights that while symplectic methods provide a good conceptual model, the algebraic approach offers a more precise and rigorous way to determine what is possible.

Ultimately, this work provides a clear map of the boundaries for a class of geometric objects. It tells us exactly how "large" a curve with sharp knots can be before it becomes impossible. By proving that a specific, highly complex curve cannot exist, the paper settles a question that had lingered in the study of weighted Seshadri constants and ellipsoid embeddings. The answer is a definitive no: the curve of degree 102 with the specified knot does not exist. This conclusion is not a suggestion or a probability; it is a proven fact derived from the rigorous application of algebraic geometry. The paper stands as a testament to the power of simplifying complex structures to reveal the hard limits of the mathematical universe.

Technical Summary: Log MMP Constraints on Curves with Cuspidal Singularities

Problem Statement
The paper addresses the problem of establishing sharp upper bounds on the self-intersection number ( $C^2$ ) of a rational curve $C$ with cuspidal singularities on a smooth projective complex algebraic surface $X$ . Specifically, it focuses on curves where the singularities are analytically irreducible (unibranch). The primary motivation stems from a conjecture by Weimin Chen regarding curves with a single Puiseux pair $(p, q)$ , which posits that $C^2$ is bounded by a value determined solely by the singularity type, independent of the ambient surface geometry. This conjecture arises from the study of contact structures and symplectic fillings, where the boundary of a regular neighborhood of such a curve corresponds to a surgery on a torus knot. While symplectic bounds exist (notably by Golla and Starkston), the paper seeks a purely algebraic proof using the Log Minimal Model Program (Log MMP) that matches or improves upon these symplectic estimates.

Methodology
The author employs the Log Minimal Model Program (Log MMP) for the pair $(X, C)$ , where $C$ is a rational curve with cusps. The methodology proceeds through the following steps:

Resolution and Multiplicity Sequences: The cusps are resolved via a sequence of blow-ups. The author utilizes the multiplicity sequence $(m_1, \dots, m_r)$ of each cusp to define invariants $M$ (sum of squared multiplicities) and $\ell$ (the last multiplicity).
Log Resolution and Forks: The resolution is extended to a normal-crossing (log) resolution. This process introduces a "fork" structure at each cusp, consisting of a central exceptional curve $E$ meeting two disjoint negative-definite configurations of curves (the "arms" or "sides").
Rooted Capacities: A key algebraic innovation is the introduction of "rooted capacities" ( $\lambda$ ). For a negative-definite configuration $\Gamma$ with a distinguished root component $A$ , the capacity is defined as $\lambda(\Gamma, A) = -1/(Q^{-1}_{\Gamma})_{AA}$ , where $Q_{\Gamma}$ is the intersection matrix. This quantity generalizes the fractions appearing in Chen's bound.
MMP Execution: The proof assumes a contradiction (that $C^2$ exceeds the proposed bound) and runs the $(K_X + C)$ -MMP. Since $K_X + C$ is not pseudoeffective (due to the adjunction formula on a rational curve), the program terminates in a Mori fiber space (either $\mathbb{P}^2$ or a Hirzebruch surface).
Tracking Invariants: The author tracks the behavior of the resolution divisors (specifically the exceptional curves of the cusp resolution) through the birational contractions of the MMP. A critical lemma establishes that intersection numbers between surviving curves can only increase under blow-downs. By analyzing the final state on the Hirzebruch surface, the author derives inequalities relating the initial self-intersection $C^2$ to the rooted capacities of the resolution arms.

Key Contributions and Results

Proof of Chen's Conjecture: The paper provides a short, alternative proof of Chen's conjecture (Theorem 1.1) for unicuspidal rational curves with one Puiseux pair $(p, q)$ . It confirms the sharp upper bound:
$C^2 \leq pq + \max\left\{ \left\lfloor \frac{p}{p-p'} \right\rfloor, \left\lfloor \frac{q}{q-q'} \right\rfloor \right\}$
where $p', q'$ are defined by $pq' + qp' = pq + 1$.
Generalization to Arbitrary Cusps: The main result (Theorem 1.3) extends the bound to rational curves with arbitrary cuspidal singularities (multiple cusps or multiple Puiseux pairs). The bound is given by:
$C^2 \leq M + 2\ell + \eta$
where $M$ is the sum of squared multiplicities, $\ell$ is the minimum of the last multiplicities of the cusps, and $\eta$ is an indicator (0 or 1) depending on whether the minimum is unique and if the corresponding "distinguished side" of the resolution fork is isolated.
Comparison with Symplectic Bounds: The algebraic bound derived via Log MMP is shown to be never weaker than the symplectic bound of Golla and Starkston. In the most typical cases (unique minimum $\ell$ with non-isolated distinguished side), the algebraic bound is strictly stronger (by 1) than the symplectic non-fillability threshold.
Sharpness and Examples: The paper demonstrates that the bounds are sharp by constructing explicit examples, including rational surfaces carrying curves that attain the bound for every coprime pair $(p, q)$ (Proposition 5.10). It provides a collection of examples covering tricuspidal quartics, Flenner–Zaidenberg curves, and families with multiple Puiseux pairs.
Non-Existence of Specific Plane Curves: As a significant application, the paper proves the non-existence of a plane curve of degree 102 and genus 10 with a $(36, 289)$ -cusp (Theorem 1.8). This answers a question posed by Evans regarding the "final post-Fibonacci step" of the McDuff–Schlenk staircase in symplectic topology. The proof involves a detailed analysis of the resolution of the singularity and the properties of the resulting divisor on a weighted blow-up of $\mathbb{P}^2$ , showing that the required line bundle conditions for the curve's existence cannot be met.

Significance
The paper establishes a rigorous algebraic framework for bounding the self-intersection of cuspidal curves, bridging the gap between symplectic topology and algebraic geometry. By utilizing the Log MMP, the author provides a conceptual model that mirrors McDuff's classification of symplectic manifolds containing symplectic spheres but extends it to handle the complexities of singular resolutions. The results clarify the relationship between algebraic constraints and symplectic fillability, showing that algebraic geometry can sometimes provide tighter constraints than symplectic topology alone. Furthermore, the negative result regarding the degree 102 curve resolves a specific open problem in the study of ellipsoid embeddings and weighted Seshadri constants, demonstrating the utility of these algebraic bounds in addressing questions arising from symplectic capacity problems.

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