On equations of fake projective planes with automorphism group of order $21$

Imagine you are an architect trying to build a perfect, impossible house. In the world of mathematics, there is a famous blueprint called the "Projective Plane." It's a beautiful, symmetrical structure that mathematicians have studied for centuries.

But then, a few decades ago, mathematicians discovered "Fake Projective Planes." These are like hallucinations of the perfect house. They look exactly like the real thing from a distance (they have the same "Hodge numbers," which are like the house's energy efficiency rating or structural density), but if you walk inside and look at the walls, they are built from completely different materials. They are exotic, twisted, and impossible to build in our normal world, yet they exist in the complex mathematical universe.

For a long time, we knew these "fake houses" existed, but we only had their blueprints described in abstract, high-level theory. We knew that they existed, but we didn't have the actual construction manuals (the specific polynomial equations) to build them.

Lev Borisov's paper is the story of finally writing those construction manuals for two very special fake houses.

Here is the step-by-step journey of how he did it, explained simply:

1. The Starting Point: A Twisted Garden

To build these fake houses, Borisov didn't start from scratch. He started with a specific type of garden called a Dolgachev surface. Imagine a garden where the plants are arranged in rows (fibers). Most rows are normal, but this garden has two special, weird rows:

One row is "double" (like two vines twisted into one).
One row is "triple" (three vines twisted together).

Borisov wanted to find a very specific version of this garden that had a hidden symmetry: a group of 21 automorphisms. Think of this as a magical symmetry where if you rotate or flip the garden in specific ways, it looks exactly the same. There are only a few gardens in the universe that have this exact 21-way symmetry.

2. The Puzzle: Solving a Massive Equation Jigsaw

Borisov's first job was to figure out the rules for this garden. He knew the garden had to follow certain mathematical laws (like gravity). He set up a massive system of equations—imagine a jigsaw puzzle with 1,600 pieces and 92 unknown variables.

He used a computer (Mathematica) to try and fit the pieces together. It was like trying to solve a Sudoku puzzle where the grid is the size of a football field. He found a "family" of gardens that fit the basic rules. This family had 9 adjustable knobs (parameters).

3. The Hunt: Tuning the Knobs

Now, he had a family of 9-dimensional gardens, but he needed the one specific garden that would turn into a Fake Projective Plane. He had to turn the knobs until the garden developed specific "scars" or singularities at precise points.

The Analogy: Imagine you are tuning a radio. You have 9 dials. You need to find the exact frequency where the static clears up and a specific song plays.
The Method: He couldn't just guess. He used a clever trick called Finite-Field Reduction. He pretended the numbers in his equations were not infinite decimals, but just numbers on a clock (modulo a prime number, like 79). He searched through millions of combinations on this "clock" until he found a setting that created the right kind of "scars" on the garden.
The Breakthrough: Once he found the right setting on the "clock" (prime number 79), he used a mathematical "ladder" (Hensel's Lemma) to climb back up from the clock to the real numbers. This allowed him to find the exact algebraic numbers needed to define the garden.

4. The Transformation: From Garden to House

Once he had the equations for the special garden ( $Y_0$ ), he had to perform a magic trick to turn it into the Fake Projective Plane.

The 7-Step Ladder: He took the garden and created a "7-fold cover." Imagine taking a piece of paper, folding it 7 times, and gluing the edges together in a specific way. This creates a new, more complex surface.
The Result: This new surface is the Fake Projective Plane. It is a surface that looks like a sphere but has a very twisted, non-Euclidean interior.

5. The Identity Crisis: Which Fake House is This?

Borisov found two different gardens that worked.

The First One: He identified this as a new, previously unknown pair of Fake Projective Planes (Class C20). It's a brand new discovery!
The Second One: He found another set of equations that matched a garden previously discovered by a mathematician named J. Keum. This confirmed Keum's work with actual equations.

To prove he had the right house, he checked the "lockbox" (the Picard group) of the surface. He looked for "torsion line bundles," which are like hidden keys that only exist in certain types of houses. By counting these keys, he proved that his first discovery was indeed the new Class C20 house and not a copy of Keum's.

Why Does This Matter?

Before this paper, these "Fake Projective Planes" were like ghosts. We knew they haunted the mathematical landscape, but we couldn't see them clearly. We couldn't write down their coordinates.

Borisov has now pinned them to the map.

He gave us the exact recipe (the polynomial equations) to build them.
He showed us how to find them using a mix of deep theory and brute-force computer searching.
He opened the door for other mathematicians to study these surfaces in detail, perhaps leading to the discovery of the "Holy Grail" of this field: the equations for the very first Fake Projective Plane discovered by David Mumford.

In short: This paper is the story of a mathematician using a super-computer as a telescope to find the exact coordinates of two invisible, magical worlds, and then writing down the instructions so anyone can visit them.

Here is a detailed technical summary of the paper "On equations of fake projective planes with automorphism group of order 21" by Lev Borisov.

1. Problem Statement

The paper addresses the long-standing challenge in algebraic geometry of finding explicit polynomial equations for fake projective planes.

Definition: A fake projective plane is a smooth complex surface of general type with the same Betti numbers as the complex projective plane $\mathbb{CP}^2$ (specifically $c_1^2 = 3c_2 = 9$ ).
Context: While the classification of all such surfaces was completed by Cartwright and Steger (CS) in 2011, identifying them as free quotients of the complex 2-ball $B^2$ by arithmetic subgroups, no explicit polynomial equations were known for most of the 50 conjugate pairs.
Specific Goal: The author aims to construct explicit equations for two specific pairs of fake projective planes that possess an automorphism group of order 21. These are:
1. The surface constructed by J. Keum (class $(a=7, p=2, \{7\}, D_{327})$ ).
2. A new pair identified as $(C_{20}, p=2, \emptyset, D_{327})$ .

2. Methodology

The author employs a multi-step computational and geometric strategy, heavily relying on computer algebra systems (Mathematica, Magma, Macaulay2, PARI/GP, and C).

A. Geometric Framework: Dolgachev Surfaces

The construction begins not with the fake projective plane directly, but with its minimal resolution $Y$ after quotienting by the normal subgroup $C_7$ of the automorphism group.

Target Surface $Y$ : A Dolgachev surface (an elliptic surface over $\mathbb{CP}^1$ $CP^{1}$ ) with:
- A double fiber $2F_3 $and a triple fiber$ 3F_2$.
- A rational 6-section $S$ with $S^2 = -3$ .
- Specific singular fibers, including a fiber of type $I_9$ (a cycle of 9 rational curves).
Ring Construction: The author studies the graded ring $R = \bigoplus_{a,b \geq 0} H^0(Y, \mathcal{O}(aF + bS))$ $R = ⨁_{a, b \geq 0} H^{0} (Y, O (a F + b S))$ .
- A formula for the graded dimension is derived, suggesting $R$ is a free module of rank 6 over a polynomial subring generated by variables with specific weights.
- This implies the surface can be embedded in a weighted projective space defined by nine quadratic relations (quadrics).

B. Step-by-Step Construction

Family Construction (9 Parameters): The author postulates the free module structure and sets up a system of 9 quadrics with undetermined coefficients. Using associativity conditions of the ring, a system of over 1600 equations in 92 variables is solved to find a 9-parameter family of such surfaces.
Imposing Geometric Constraints:
- 7-Parameter Family: The condition that the special fiber contains one line is imposed.
- 5-Parameter Family: The condition that the special fiber contains two disjoint lines (corresponding to the geometry of the $I_9$ fiber) is imposed.
- 2-Parameter Family: Further conditions requiring the surface to have specific singularities (nodes) at intersection points of the fiber components reduce the family to 2 parameters ( $s_2, s_3, s_4$ ).
Finite Field Reduction & Lifting:
- Direct symbolic solving of the singularity conditions was computationally infeasible (equations grew to hundreds of MB).
- Strategy: The author searched for solutions over finite fields $\mathbb{F}_p$ (specifically looking for $p$ where $\sqrt{-7}$ exists). The first successful prime was $p=79$ .
- Solutions found in $\mathbb{F}_{79}$ were lifted to $p$ -adic numbers and then recognized as algebraic numbers of degree 12, eventually simplifying to the field $\mathbb{Q}(\sqrt{-7})$ .
Constructing the Fake Projective Plane:
- Once the specific Dolgachev surface $Y_0$ is identified, the fake projective plane $P^2_{fake}$ is constructed as a degree 7 cyclic cover of $Y_0$ .
- The cover is ramified along specific curves ( $S, B, C, S_1, \dots$ ).
- The equations of $P^2_{fake}$ are derived by computing the bicanonical linear system $|2K|$ , resulting in an embedding into $\mathbb{CP}^9$ defined by 84 cubic equations.

C. Identification

To distinguish between the two possible classes of surfaces with 21 automorphisms, the author analyzes the torsion subgroup of the Picard group:

The surface $(a=7, p=2, \{7\}, D_{327})$ (Keum's) has a torsion group $C_2^3$ .
The surface $(C_{20}, p=2, \emptyset, D_{327})$ has a larger torsion group $C_2^6$ .
By searching for non-reduced $C_3$ -invariant elements in $|2K|$ modulo a prime (29) and lifting them, the author found multiple 2-torsion line bundles, confirming the surface is of type $(C_{20}, p=2, \emptyset, D_{327})$ .
A second family of Dolgachev surfaces (found in Step 2) was processed similarly to yield the Keum surface.

3. Key Contributions

Explicit Equations: The paper provides the first explicit polynomial equations for the fake projective plane of type $(C_{20}, p=2, \emptyset, D_{327})$ .
Recovery of Keum's Surface: It provides explicit equations for J. Keum's previously known fake projective plane, which was constructed via different methods.
Methodological Advancement: The paper refines the "finite-field reduction and lifting" technique. This approach bypasses the intractability of solving massive symbolic systems by finding solutions in finite fields first, then lifting them to number fields.
Computational Infrastructure: The work demonstrates the necessity of high-performance computing (Amarel cluster) and a combination of symbolic (Mathematica) and numerical/algebraic (Magma, C) tools to solve problems in high-dimensional algebraic geometry.

4. Results

Two New Pairs: Two distinct pairs of fake projective planes with automorphism group order 21 are explicitly defined.
Equation Complexity: The surfaces are realized as intersections of 84 cubic hypersurfaces in $\mathbb{CP}^9$ . The coefficients involve algebraic numbers in $\mathbb{Q}(\sqrt{-7})$ .
Verification: The smoothness and correct topological invariants of the constructed schemes were verified computationally, confirming they are indeed fake projective planes.
Data Availability: Due to the extreme size of the equations (too large for the appendix), the author directs readers to a supplementary repository ([Bor22+]) containing the full code and equations.

5. Significance

Bridging Theory and Computation: This work bridges the gap between the abstract classification of fake projective planes (Cartwright-Steger) and concrete algebraic geometry. It moves these objects from "existence proofs" to "computable varieties."
Path to Mumford's Surface: The author notes that the Keum surface is commensurable with Mumford's original fake projective plane. Understanding the equations of Keum's surface is a critical step toward finding explicit equations for Mumford's surface, which remains the "holy grail" of this specific subfield.
Open Problems: The paper highlights that while the method works for surfaces with $C_3$ symmetry, extending it to surfaces with smaller or no automorphism groups remains a significant open problem. It also suggests future work on simplifying the equations using torsion line bundles.

In summary, Borisov's paper is a landmark computational achievement that successfully translates the abstract classification of fake projective planes into explicit algebraic models, utilizing a sophisticated blend of geometric insight and heavy-duty computation.