# On the Geometry of a Fake Projective Plane with 21 Automorphisms
This repository contains supplementary code for the paper "On the Geometry of a Fake Projective Plane with 21 Automorphisms".

## Main Files

The main files are laid out in the current directory as follows. Unless specified otherwise, files with suffix **.txt** are Magma files.

### Mathematica Files
- [3H-Torsion.nb](3H-Torsion.nb) : 
    1. Produces linear cuts representing $3H + T$ for all non-trivial torsion divisors $T$ from the outputs of [3H-Reduction.txt](3H-Reduction.txt). 
    2. Determines the group relations between the torsion divisors.
    3. Computes explicit quadratics vanishing on the sections $3H + D$, $3H + D_1$, and $3H + D + D_1$.
- [4H.nb](4H.nb) : 
    1. Produces relations on $s_3 = r_3^3$ and $d = r_3 r_5 r_6$ and finds a solution to $s_3$ and $d$.
    2. The intermediate calculation for solving the relations is done in the Magma file [4H-Quadratic.txt](4H-Quadratic.txt).
    3. Calculates the explicit quadratics vanishing on $4H$ based on this. 
- [5H-Torsion.nb](5H-Torsion.nb) :
    1. Produces an embedding of the fake projective plane into $\mathbb{C}P^5$ using sections of $5H + D$.
    2. Produces a map of the fake projective plane into $\mathbb{C}P^5$ using sections of $5H$. This image of the fake projective plane will turn out to be singular.
    3. Computes explicit quadratics vanishing on the basis of $6$ global sections of $5H$.
- [4H-Torsion.nb](4H-Torsion.nb) : Computes explicit quadratics vanishing on the sections $4H + D$, $4H + D_1$, and $4H + D + D_1$ and checks that $2H + D_1$ and $2H + D + D_1$ has no non-trivial sections.

THe Mathematica files are written in Mathematica 12.0, but they should be compatible with Mathematica 13.0. All Mathematica files involve some computations of finding random points on certain curves and surfaces. Please try re-running the file again if the results are inconsistent with what's in the folder [Equations](Equations). We also provide a standard set of pre-computed points in the folder [Dependency](Dependency) for consistency.

### Magma Files
- [3H-Reduction.txt](3H-Reduction.txt) finds two finite field solutions of linear cuts representing $3H + D_1$ and $3H + D + D_1$.
- [4H-Quadratic.txt](4H-Quadratic.txt) solves the relations on coefficients of $s_3 = r_3^3$ and $d = r_3 r_5 r_6$ produced by the file [4H.nb](4H.nb).
- [5H-is-prime.txt](5H-is-prime.txt) checks that the $59$ equations produced for $5H$ forms a prime ideal, hence they are no higher degree equations contributing to the image of the FPP.
- [5H-intersection.txt](5H-intersection.txt) checks that the $6$ sections of $5H$ produced in [5H-Torsion.nb](5H-Torsion.nb) does not have any common zeroes.
- [4H-Torsion-intersection.txt](4H-Torsion-intersection.txt) checks that $4H + D, 4H + D_1, 4H + D + D_1$ are basepoint free. It also double checks that $4H$ is not basepoint free.

While all files can be executed independently on their own, the recommended order to run them are:
1. [3H-Reduction.txt](3H-Reduction.txt)
2. [3H-Torsion.nb](3H-Torsion.nb)
3. [4H.nb](4H.nb)
4. [4H-Quadratic.txt](4H-Quadratic.txt)
5. [5H-Torsion.nb](5H-Torsion.nb)
6. [5H-is-prime.txt](5H-is-prime.txt)
7. [5H-intersection.txt](5H-intersection.txt)
8. [4H-Torsion.nb](4H-Torsion.nb)
9. [4H-Torsion-intersection.txt](4H-Torsion-intersection.txt)

This was the order of how we computed the results in this paper. 

## Other Folders
The folders for this repository are as follows:
1. [Dependency](Dependency) : Contains pre-computed points and other dependencies for the main Mathematica files. 
    - The file **EqsFPP.txt** contains equations of the fake projective plane **(a = 7, p = 2, \emptyset, D_3 2_7)** produced in the paper *Explicit equations of a fake projective plane* by Borisov and Keum.
    - The other files are all computed as part of this project.

2. [Equations](Equations) : Contains data produced by the main Mathematica files.
    - **3H_D.txt** contains the 28 quadratic equations vanishing on $3H + D$.
    - **3H_D+D1.txt** contains the 28 quadratic equations vanishing on $3H + D + D_1$.
    - **3H_D1.txt** contains the 28 quadratic equations vanishing on $3H + D_1$.
    - **4H_Coeffs_Vanishing.txt** contains the relations on the coefficients of $s_3 = r_3^3$ and $d = r_3 r_5 r_6$ produced by [4H.nb](4H.nb).
    - **4H_one_section.txt** contains the 21 quadratics vanishing on $r_3$.
    - **4HD_one_section.txt** contains the 21 quadratics vanishing on one section of $4H + D$.
    - **4HD.txt** contains the 3 equations for the 3 sections of $4H + D$.
    - **4HD1_one_section.txt** contains the 21 quadratics vanishing on one section of $4H + D_1$.
    - **4HD1.txt** contains the 3 equations for the 3 sections of $4H + D_1$
    - **4HDD1_one_section.txt** contains the 21 quadratics vanishing on one section of $4H + D + D_1$.
    - **4HDD1.txt** contains the 3 equations for the 3 sections of $4H + D + D_1$.
    - **5H_Equations.txt** contains the equations of the map of the fake projective plane into $\mathbb{C}P^5$ using sections of $5H$.
    - **5H+D_Embedding.txt** contains the equations of the embedding of the fake projective plane into $\mathbb{C}P^5$ using sections of $5H + D$.
    - **5H+D_Sections.txt** contains the $6$ global sections of $5H + D$.
    - **Quadratics_on_Zi.txt** contains the quadratics vanishing on each of the $6$ sections of $5H$.
    - **s3_and_D.txt** contains the equations for $s_3 = r_3^3$ and $d = r_3 r_5 r_6$.

3. [Verification](Verification) : Contains code verifying the results produced by the main Mathematica files.
    - The folder [5H+D](Verification/5H+D/) checks that the map produced by $5H + D$ is an embedding:
        - **check_Hilbert.txt** checks the Hilbert polynomial of the embedding.
        - **check_smoothness** checks that the image is smooth.
        - **check_M2** performs other checks on the equations. This file is in Macaulay2.
    - The folder [5H](Verification/5H/):
        - **check_Hilbert.txt** shows that the Hilbert polynomial of the equations is $-3$ off from the embedding.
        - **check_Z2.txt** checks the Hilbert polynomial of the quadratics vanishing on $Z_2$ (the main section of $5H$ we used in the paper and calculations).
    - The folders [3H+Torsion](Verification/3H+Torsion/) checks the Hilbert polynomial of sections produced on $3H + D, 3H + D_1,$ and $3H + D + D_1$. The files are in Macaulay2.
    - The folders [4H+Torsion](Verification/4H+Torsion/) checks the Hilbert polynomial of one section of $4H, 4H + D, 4H + D_1$, and $4H + D + D_1$ respectively. The files are in Macaulay2.