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Finding equations of the fake projective plane $(C18,p=3,\{2I\})$

This paper presents explicit equations for a new pair of fake projective planes labeled $(C18,p=3,\{2I\})$ by deriving them from known equations of a commensurable example through a sequence of cyclic covers and quotients.

Original authors: Lev Borisov, Bojue Wang

Published 2026-08-04

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Lev Borisov, Bojue Wang

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 the smooth curves of a beach ball or the sharp corners of a box, but as a vast, invisible landscape of "complex surfaces." These are mathematical objects that exist in higher dimensions, far beyond what our eyes can see. Some of these surfaces are like the familiar projective plane (think of a flat sheet where parallel lines meet at a horizon), while others are wild, twisted cousins that look exactly the same from a distance but have secret, hidden structures underneath. Mathematicians call these look-alikes "fake projective planes." They are rare, precious, and incredibly difficult to find. For decades, we knew they existed, but we couldn't write down their exact blueprints—like knowing a treasure is buried on an island but not having the map to dig it up. This paper is about drawing that map for a brand new pair of these hidden treasures.

The authors, Lev Borisov and Bojue Wang, have successfully found the explicit equations for a new pair of fake projective planes, specifically the ones labeled (C18, p = 3, {2I}) in a famous classification system. Think of these equations as the DNA of the shape; once you have them, you can build the surface on a computer and study its every twist and turn. They didn't find these shapes from scratch. Instead, they started with a "cousin" shape whose equations were already known. They then built a complex ladder of mathematical bridges—climbing up to a massive, shared "parent" shape and then climbing back down to the new target. Along the way, they had to solve a puzzle involving thousands of digits of precision and a massive group of symmetries (a mathematical dance of 648 different moves) to ensure the new shape was real and not just a computer glitch. In the end, they proved that their new equations describe a genuine fake projective plane, adding two more unique shapes to the known collection of 100.

The Story of the Hidden Twins

To understand what the authors did, imagine you have a very strange, intricate sculpture. You know it exists, and you know it looks like a perfect sphere from the outside, but if you zoom in, it has a secret, twisted core that makes it different. In the world of math, these are Fake Projective Planes (FPPs). They are surfaces that have the same "Hodge numbers" (a way of counting holes and loops) as a standard projective plane, but they are not actually the same shape. For a long time, mathematicians knew there were exactly 100 of these shapes (grouped into 50 pairs of mirror images), but for most of them, we only had a blurry description. We knew they were quotients of a "complex 2-ball" (a fancy, curved space), but we didn't have the explicit algebraic equations to draw them.

This paper is about finding those equations for a specific new pair. The authors call their new discovery $\widehat{P^2_{fake}}$ (the "hat" version) and the one they started with $P^2_{fake}$ (the "plain" version).

The Journey: Climbing the Ladder of Symmetry

The authors didn't just guess the equations. They used a clever strategy involving a "family tree" of shapes. Here is how they did it, step by step:

1. The Starting Point:
They began with the known equations of the "plain" fake projective plane ( $P^2_{fake}$ ). This shape lives in a 9-dimensional space and is defined by 84 cubic equations. The numbers in these equations were huge (about 100 digits long) and involved a specific type of complex number.

2. The First Climb (Finding the Cut):
To move from the plain shape to the new one, they needed to find a "nonreduced linear cut." Imagine the shape is a block of cheese. A normal cut slices through it cleanly. A "nonreduced" cut is like a slice that is so thin it barely exists, or a slice that is "doubled up." Finding this specific cut was like finding a secret key. They used a computer to search for this key by looking at the shape modulo a prime number (73), then used a technique called "lifting" to refine the key until it worked with perfect precision. This key allowed them to change the coordinate system, making the math much cleaner.

3. Building the Bridge (The Double Covers):
With the new coordinates, they built a series of "covers." Think of a cover as a sheet of paper wrapped around a shape. If you wrap it twice, you get a "double cover."

They first built a double cover (2-fold) of the original shape.
Then they built a quadruple cover (4-fold).
Then an octuple cover (8-fold).
Finally, they reached a massive 72-fold cover (let's call it the "Super Shape").

This Super Shape is the common ancestor of both the old fake plane and the new one they wanted to find. It has a huge symmetry group of 648 different moves (rotations and reflections) that keep it looking the same.

4. The Tricky Part (The 72-Fold Climb):
This was the hardest step. The authors had to figure out exactly how the "Super Shape" relates to the new target shape. They had to find a specific relationship between different parts of the Super Shape's equations. It was like trying to solve a 71-dimensional puzzle where the pieces were made of algebraic functions.
They used the symmetries of the Super Shape to find a "basis" (a set of building blocks) for the new shape. They computed values for these building blocks with incredible precision—hundreds or even thousands of decimal places—to ensure they were accurate enough to recognize the underlying algebraic numbers.

5. Coming Back Down:
Once they had the equations for the Super Shape, they "averaged" them out to get the equations for the new fake projective plane ( $\widehat{P^2_{fake}}$ ). This is like taking a complex, noisy signal and filtering it to find the clear message underneath.
Initially, the equations they found were messy, with huge coefficients and defined over a complicated field of numbers. But they used a clever trick involving "intersection points" (where lines cross on the shape) to simplify the equations. They swapped variables and used a "lattice reduction" algorithm (a method for finding the simplest possible numbers) to shrink the coefficients down.

6. The Final Result:
The final equations they found are much simpler than the starting point. They are defined over a smaller field of numbers ( $\mathbb{Q}(\sqrt{-2})$ ) and the coefficients are now only 20 to 30 decimal digits long. This makes the shape much easier to study.

Verification: Is it Real?

The authors didn't just stop at finding the equations. Because they used high-precision approximations (simulations) to get there, they had to prove the result was mathematically exact. They used a computer algebra system called Magma to perform exact calculations.

They checked that the shape has the correct "Hilbert polynomial" (a fingerprint of its size and shape).
They proved the shape is smooth (no sharp corners or tears) by checking the "Jacobian matrix" (a tool that measures how the shape bends).
They confirmed the shape has the right number of "holes" and loops to be a fake projective plane.

The paper concludes that they have successfully identified this new shape as the (C18, p = 3, {2I}) pair in the Cartwright-Steger classification.

Why This Matters

This isn't just about finding one more shape. The authors believe their method is a key that can unlock the remaining mysteries. They suspect that with this new technique, they can eventually find the equations for the three remaining pairs of fake projective planes that are related to the ones they just studied. It's like finding a new way to navigate a maze; once you know the path, you can explore the rest of the labyrinth.

In short, Borisov and Wang have taken a known mathematical object, climbed a steep, symmetrical ladder to a shared ancestor, and carefully descended to discover a brand new, previously invisible twin. They have turned a theoretical existence proof into a concrete, computable reality, giving mathematicians a new tool to explore the strange and beautiful world of complex surfaces.

Technical Summary: Finding Equations of the Fake Projective Plane (C18, p = 3, {2I})

Problem Statement
The paper addresses the challenge of finding explicit algebraic equations for "fake projective planes" (FPPs). FPPs are complex projective surfaces of general type with the same Hodge numbers as the complex projective plane $\mathbb{CP}^2$ (specifically $h^{1,0} = h^{2,0} = 0$ and $K_X^2 = 9$ ). While Cartwright and Steger provided a complete classification of 100 such surfaces (50 pairs of complex conjugates) as quotients of the complex 2-ball by arithmetic groups, these descriptions do not yield explicit polynomial equations for the surfaces embedded in projective space. Previous work by the authors and others has successfully found equations for several pairs, but the specific pair labeled $(C18, p = 3, \{2I\})$ in the Cartwright-Steger classification remained unsolved due to significant computational complexity.

Methodology
The authors employ a constructive approach based on the commensurability of FPPs. Rather than starting from scratch, they utilize the known equations of a "neighboring" FPP, denoted $P^2_{\text{fake}}$ (labeled $(C18, p = 3, \emptyset, d3D3)$ ), and traverse a chain of cyclic covers and quotients to reach the target surface, denoted $\widetilde{P^2_{\text{fake}}}$ (labeled $(C18, p = 3, \{2I\})$ ).

The process involves a common Galois cover, $72.P^2_{\text{fake}}$ , with an automorphism group $G_{648}$ of order 648. The methodology proceeds through seven distinct steps:

Lifting Non-Reduced Cuts: Starting with the bicanonical embedding of $P^2_{\text{fake}}$ (defined by 84 cubic equations in 10 variables with coefficients in $\mathbb{Z}[\sqrt{-2}]$ ), the authors identify a non-reduced linear cut corresponding to the square of a 2-torsion element in the Picard group. This is achieved via an exhaustive search modulo 73, followed by lifting to high-precision 73-adic approximations and recognizing the coefficients as algebraic numbers.
Basis Optimization: The identified cut is used to change the basis of the space of sections $H^0(P^2_{\text{fake}}, 2K)$ to one consisting of eigenvectors for the $C_3 \times C_3$ automorphism group. This shifts the coefficient field to $\mathbb{Q}(\sqrt{-2}, \sqrt{-3})$ .
Constructing Double Covers: Using the linear cut, the authors construct the double cover $2.P^2_{\text{fake}}$ and subsequently the surface $4.P^2_{\text{fake}}$ . Instead of computing full equations for these high-dimensional embeddings, they develop a method to construct points in the image of these surfaces with high numerical accuracy.
Lifting to the Quotient: The authors construct the surface $8.P^2_{\text{fake}}$ (a quotient of the common cover) and its singular quotient $8.P^2_{\text{fake}}/C_3$ . They determine the action of the group $Q_8$ and find equations for the singular quotient (4 cubic and 58 quartic equations in 7 variables). They then compute points in the bicanonical embedding of $8.P^2_{\text{fake}}$ into $\mathbb{CP}^{79}$ with high accuracy.
Determining the Common Cover: The most delicate step involves determining the $C_3 \times C_3$ cover $72.P^2_{\text{fake}} \to 8.P^2_{\text{fake}}$ . By finding relations among bicanonical sections on $8.P^2_{\text{fake}}$ (specifically products of sections from specific eigenspaces of the covering group), they construct a basis for the 71-dimensional space $H^0(72.P^2_{\text{fake}}, K)$ . They compute the action of the full group $G_{648}$ on this space.
Descending to the Target: By taking invariants under the appropriate subgroups, the authors descend to the target surface $\widetilde{P^2_{\text{fake}}}$ . They compute a basis for $H^0(\widetilde{P^2_{\text{fake}}}, 6H)$ and derive initial equations. To manage the complexity, they utilize intersection points of special curves to find a new basis defined over the smaller field $\mathbb{Q}(\sqrt{-2})$ . Finally, they apply lattice reduction algorithms to minimize the size of the coefficients, resulting in equations with coefficients of only 20–30 decimal digits.
Verification: The final equations are verified using exact arithmetic in Magma. The authors confirm the surface has the correct Hilbert polynomial, is smooth, possesses the correct cohomology dimensions ( $h^{1,0}=h^{2,0}=0$ ), and that the embedding is indeed the bicanonical one.

Key Contributions and Results

Explicit Equations: The primary result is the explicit set of equations defining the fake projective plane $(C18, p = 3, \{2I\})$ in its bicanonical embedding into $\mathbb{CP}^9$ .
Computational Framework: The paper details a robust computational pipeline for navigating the "ladder" of commensurable surfaces. It demonstrates how to handle surfaces with large degrees of common covers (order 648) by combining high-precision numerical point construction with algebraic number recognition.
Group-Theoretic Analysis: The authors provide detailed character table computations for the group $G_{648}$ to identify the specific representations corresponding to the spaces of holomorphic forms on the various covers.
Field Reduction: A significant technical achievement is the reduction of the defining field from $\mathbb{Q}(\sqrt{-2}, \sqrt{-3})$ to $\mathbb{Q}(\sqrt{-2})$ and the subsequent reduction of coefficient sizes from hundreds of digits to roughly 30 digits.

Significance and Claims
The authors state that this work continues the program of finding explicit equations for all FPPs in the Cartwright-Steger classification. They claim that the methods developed here, particularly the ability to traverse the chain of covers and handle the large Galois group $G_{648}$ , may enable the computation of the remaining three pairs of FPPs commensurable with the surfaces studied in this paper: $(C18, p = 3, \{2\}, D3)$ , $(C18, p = 3, \{2\}, (dD)^3)$ , and $(C18, p = 3, \{2\}, (d^2D)^3)$ .

The paper maintains a modest tone regarding its intermediate steps, acknowledging that the high-precision point calculations are not fully rigorous on their own and require the final verification step (Step 7) to confirm the geometric properties of the resulting surface. The work relies heavily on computational tools (Magma, Mathematica, GAP) and does not propose new theoretical conjectures, but rather provides the concrete algebraic realization of a previously abstractly classified object.

The Story of the Hidden Twins

The Journey: Climbing the Ladder of Symmetry

Verification: Is it Real?

Why This Matters

Technical Summary: Finding Equations of the Fake Projective Plane (C18, p = 3, {2I})

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