Explicit Equations for Fake Projective Plane (a = 7, p = 2, \emptyset, D_3 X_7)

This repository contains supplementary code for the paper Explicit Equations for Fake Projective Plane (a = 7, p = 2, \emptyset, D_3 X_7) by Borisov, Ji, and Li.

Main Files

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

Magma Files

• Section2.txt provides relevant verification codes that occur in Section 2 of the manuscript.

Mathematica Files

• Section3.nb calculates the equations of W and the birational model of Z.
• Section4.nb computes the equations of the fake projective plane (a = 7, p = 2, \emptyset, D_3 X_7) in the number field and reduction modulo prime.

While all files can be executed independently on their own, the recommended order to run them are:

1. Section2.txt
2. Section3.nb
3. Section4.nb

The Mathematica files are written in Mathematica 12.0, but they should be compatible with Mathematica 13.0.

Other Directories

The folders for this repository are as follows:

1. 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 file 4HD_one_section.txt contains equations of one section of 4H + D from the FPP (a=7, p=2, \emptyset, D_3 2_7). This is from the paper On the Geometry of a Fake Projective Plane with 21 Automorphisms and its accompanied repository by Borisov, Ji, Li, and Mondal.

   • The other files are all computed as part of this project.

2. Equations: Contains relevant equations produced by the main Mathematica files:

   • ginvariant_section_c3.txt contains the 10 quadratics vanishing on the unique weight 0 section of 6H + D from the FPP (a=7, p=2, \emptyset, D_3 2_7). These 10 quadratics are used in the definition of the coordinates U10, U11, ..., U19 in the calculation of W.
   • minors_37.txt contains three 7 x 7 minors of the Jacobian of the equations of (a = 7, p = 2, \emptyset, D_3 X_7) modulo 37.
   • r-11_equations.txt contains the 56 quadratics vanishing on r_{-1,1}=0.
   • r00_equations.txt contains the 56 quadratics vanishing on r_{0,0}=0.
   • r11_equations.txt contains the 56 quadratics vanishing on r_{1,1}=0.
   • reduced_equations_37.txt contains the equations of the FPP (a = 7, p = 2, \emptyset, D_3 X_7) reduced modulo 37. The coefficients here also include a cubic root of unity (See Section4.nb for more details).
   • Target_FPP_simple.txt contains the 84 cubic equations defining the FPP (a = 7, p = 2, \emptyset, D_3 X_7) with coefficients in Q(\sqrt{-7}).
   • ten_cubics.txt contains the 10 cubic equations that defines the 10 sections of X = (a = 7, p = 2, \emptyset, D_3 X_7) in 2 K_X.
   • W_equations.txt contains the 100 equations of the double cover W. The first 45 equations are the odd equations, and the second 55 equations are the even equations.
   • Z_singular_equations.txt contains the 66 cubic equations defining a birational model of Z.

3. Verification: Contains code verifying the results produced by the main Mathematica files:

   • FPP_Hilbert.txt checks the Hilbert polynomial of the FPP (a = 7, p = 2, \emptyset, D_3 X_7).
   • FPP_M2.txt is a Macaulay2 file checking that the embedding of the FPP X = (a = 7, p = 2, \emptyset, D_3 X_7) is in 2 K_X.
   • FPP_smoothness_Hodge.txt checks that the equations of the FPP (a = 7, p = 2, \emptyset, D_3 X_7) is smooth and computes its Hodge numbers modulo prime.
   • r_11_check_Hilbert.txt checks the Hilbert polynomial of r_{1, 1}=0.
   • W_check_Hilbert.txt checks the Hilbert polynomial of W.
   • Z_singular_Hilbert.txt shows the Hilbert polynomial of the birational model of Z.