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Quintic surfaces with 18 cusps

This paper constructs quintic surfaces in $\mathbb{P}^3$ with 18 ordinary cusps by leveraging a specific moduli component of surfaces admitting two Barth–Rams decompositions to find examples over finite fields and subsequently lifting one to characteristic zero via Newton–Hensel lifting and LLL reconstruction.

Original authors: Lev Borisov, Carlos Rito

Published 2026-08-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lev Borisov, Carlos Rito

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 sculpture out of mathematical clay. In the world of algebraic geometry, this clay is shaped into surfaces floating in a four-dimensional space (though we can only see three dimensions at a time). For centuries, mathematicians have been fascinated by the "cracks" and "dents" that appear on these sculptures. Some dents are simple bumps called "nodes," while others are sharper, more dramatic points called "cusps," which look like the tip of a needle or the point where two sheets of paper meet and fold into a sharp spike.

The big question in this field is: How many of these sharp spikes can you pack onto a single sculpture before it falls apart? For simpler shapes, we know the answers. But for a specific type of sculpture called a "quintic surface" (which is defined by a complex equation with terms up to the fifth power), the limit has been a mystery. We know you can't have more than 20, but no one has ever successfully built a perfect one with that many. The record holders so far have only managed 16 or 17. It's like trying to stack the most unstable blocks possible; the more you add, the more likely the whole thing is to collapse into a mess.

This paper is about two mathematicians, Lev Borisov and Carlos Rito, who decided to tackle this stacking problem. They didn't just guess; they built a new kind of scaffolding to hold the blocks in place. Their main discovery is that they successfully constructed a family of these quintic surfaces that usually has 17 sharp spikes. More impressively, by tweaking their design, they found specific examples that manage to hold 18 spikes without breaking. They didn't just find these in a computer simulation; they used a clever mix of digital detective work and high-precision math to lift one of these examples out of the world of simple numbers and into the real, complex mathematical world, proving it exists with absolute certainty.

The Blueprint: Finding the Hidden Pattern

To understand how they did it, think of the quintic surface as a giant, invisible puzzle. The mathematicians knew that if you have a lot of spikes (cusps), they aren't just scattered randomly; they follow a secret code. This code is like a pattern in a quilt where certain patches must align perfectly.

They started with a known trick called the "Barth–Rams decomposition." Imagine you have a complex shape, and you can describe it by combining two simpler shapes (like two cubes) and subtracting a third. In this case, the "spikes" appear where these shapes touch in a very specific way. Usually, this trick gives you a surface with 12 spikes. The authors realized that if you could force the surface to have two of these patterns happening at the same time, you might get more spikes.

They set up a system where the surface had to satisfy two different "Barth–Rams" rules simultaneously. When they did this, the two sets of 12 spikes overlapped in 7 places. Instead of having 24 spikes, they had 17 unique ones (12 + 12 - 7). This created a massive, 6-dimensional family of surfaces, all of which naturally came with 17 spikes. It was like building a machine that automatically produced 17-cusp sculptures.

The Breakthrough: Cracking the 18th Spike

Having a machine that makes 17-cusp surfaces was great, but the goal was 18. The authors knew that adding one more spike only required two extra conditions (like tightening two specific screws). They tried to find these 18-cusp surfaces by looking at the problem through the lens of "finite fields."

Think of a finite field as a video game world where numbers wrap around. Instead of counting 1, 2, 3, ... forever, you might count 1, 2, 3, 4, 5, 1, 2... again. In these small, wrapped-around worlds, the math is much faster to compute. The team wrote a computer program to search through these tiny digital worlds. They found several examples where the surface had exactly 18 spikes.

However, finding a surface in a "wrapped-around" video game isn't the same as proving it exists in the real, infinite mathematical world. To bridge this gap, they used a technique called "Newton–Hensel lifting." Imagine you have a blurry photo of a face. You know it's a person, but the details are fuzzy. Lifting is like taking that blurry photo and using a super-powerful zoom lens to sharpen it, pixel by pixel, until the face is crystal clear. They started with their 18-cusp example from the finite field and used this method to "zoom in" with extreme precision, refining the numbers until they could describe the surface exactly.

The Final Proof: A Surface of Pure Math

Once they had the exact numbers, they used a powerful tool called LLL (a type of mathematical detective work that finds hidden patterns in numbers) to figure out the exact algebraic recipe for the surface. They discovered that the surface could be described using numbers from a "number field" of degree 22. This is a fancy way of saying the numbers involved are complex, but they are well-defined and real.

Finally, they put the surface under a microscope. They checked every single point to make sure there were exactly 18 ordinary cusps and nothing else—no extra bumps, no weird cracks, no hidden surprises. They confirmed that the surface was stable and exactly as they predicted.

So, while the ultimate dream of finding a quintic with 20 cusps remains a "maybe" for the future, this paper proves that 18 is definitely possible. They didn't just suggest it might be true; they built the surface, lifted it from a digital approximation to an exact mathematical object, and showed us exactly what it looks like. It's a solid step forward in understanding the limits of how many sharp points a mathematical sculpture can hold.

Technical Summary: Quintic Surfaces with 18 Cusps

Problem Statement
The paper addresses the extremal problem of determining the maximum number of ordinary cusp singularities ( $A_2$ ) possible on a complex quintic surface in $\mathbb{P}^3$ . While the moduli space of quintic surfaces has dimension 40, and a cusp is expected to impose two conditions (suggesting a theoretical maximum of 20), the sharpness of the bound of 20 remains an open question. Prior to this work, the largest known examples were isolated surfaces with 16 or 17 cusps. The existence of a quintic with 20 cusps is of particular interest due to its potential connection to surfaces of general type with canonical maps of degree 9, which would satisfy $K^2_X = 9\chi(\mathcal{O}_X)$ .

Methodology
The authors employ a combination of algebraic geometry, coding theory, and computational number theory.

Cusp Code and Constraints: The study utilizes the "cusp code," a ternary linear code associated with the singularities of the surface. By applying Tan's estimates and the Griesmer bound, the authors prove that any quintic with at least 19 cusps must contain a 3-divisible set of 12 cusps. This theoretical restriction directs the search toward surfaces admitting a specific geometric structure.
Barth–Rams Decomposition: The core geometric tool is the Barth–Rams description of quintics containing a 3-divisible set of 12 cusps. Such surfaces admit a decomposition $F = (AB - C^3)/w$ , where $A$ and $B$ are contact cubics and $C$ is a quadric.
Two Approaches to Construction:
- Specialization: The authors first explore a specialization where the contact cubics are singular along two skew lines. This yields a family with 16 cusps generally, and examples with 18 cusps were found over small finite fields. However, this family is restrictive.
- Main Construction: The primary method involves requiring a single quintic to admit two distinct Barth–Rams decompositions. The authors analyze the locus of such surfaces, showing that the two associated sets of 12 cusps intersect in 7 points, resulting in a union of 17 cusps.
Dimensional Analysis: The authors prove that the locus of quintics admitting two Barth–Rams decompositions contains a 6-dimensional component in the moduli space. The general member of this component has exactly 17 cusps. Finding an 18th cusp requires imposing only two additional conditions, making the search for 18-cusp examples efficient over finite fields.
Lifting to Characteristic Zero: To obtain an explicit example over a number field, the authors:
- Found an 18-cusp example over $\mathbb{F}_{23}$ using the specialized family (singular along skew lines) for computational stability.
- Applied Newton–Hensel lifting to lift the parameters to high $p$ -adic precision.
- Reduced the system to four essential parameters and used the LLL algorithm (Lenstra–Lenstra–Lovász) to reconstruct polynomial relations over $\mathbb{Q}$ .
- Computed a Gröbner basis to determine the exact algebraic values of the parameters, defining the surface over a number field of degree 22.

Key Results

Existence of 18-Cusp Surfaces: The paper constructs a 6-dimensional family of quintic surfaces where the general member has 17 cusps. Within this family, the authors successfully identify members with 18 ordinary cusps.
Explicit Construction: An explicit quintic surface with 18 ordinary cusps and no other singularities is constructed over a number field of degree 22.
Theoretical Bounds: The paper confirms that a quintic with 19 or 20 cusps must admit a Barth–Rams decomposition (a 3-divisible set of 12 cusps), reinforcing the structural constraints on extremal cases.
Verification: The reconstructed surface is rigorously verified. The singular subscheme is shown to have degree 36, corresponding exactly to 18 ordinary cusps (each contributing length 2), with no other singularities.

Significance and Claims
The paper claims to provide the first known examples of quintic surfaces with 18 cusps, surpassing the previous record of 17. The construction is significant not only for increasing the known lower bound for the maximum number of cusps but also for demonstrating a method to lift such high-singularity examples from finite fields to characteristic zero using modern computational algebraic geometry techniques.

The authors note that while they have not reached the theoretical bound of 20, the existence of 18-cusp surfaces brings the problem closer to the extremal case. They emphasize that the existence of a quintic with 20 cusps remains an open problem, but their work establishes a robust framework for searching for such surfaces by leveraging the Barth–Rams decomposition and the cusp code. The paper does not claim to solve the 20-cusp problem but provides a concrete step forward in the classification of singular quintic surfaces.

The Blueprint: Finding the Hidden Pattern

The Breakthrough: Cracking the 18th Spike

The Final Proof: A Surface of Pure Math

Technical Summary: Quintic Surfaces with 18 Cusps

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