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

Lev Borisov, Bojue Wang

TL;DR

The paper furnishes explicit equations for a new fake projective plane in the Cartwright–Steger class (C18,p=3,{2I}) by lifting from a commensurable model through a carefully controlled chain of Galois covers and quotients. It combines group-theoretic and representation-theoretic analysis with high-precision, multi-system computations (GAP, Magma, Mathematica) to build invariants and reduce coefficients, culminating in a verifiable bicanonical model. The contribution provides not only explicit equations for this FPP but also a general framework for deriving equations of commensurable FPPs, potentially enabling construction of the remaining CS pairs. The work highlights the interplay between arithmetic group data, torsion in Picard groups, and explicit projective embeddings in proving the existence and shape of these rare surfaces.

Abstract

We find explicit equations of a new pair of fake projective planes, labeled by $(C18,p=3,\{2I\})$ in the Cartwright-Steger classification. Our method involves starting with known equations of a commensurable fake projective plane $(C18,p=3,\emptyset,d_3 D_3)$ and working through a chain of cyclic covers and quotients to get to the new one.

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

TL;DR

The paper furnishes explicit equations for a new fake projective plane in the Cartwright–Steger class (C18,p=3,{2I}) by lifting from a commensurable model through a carefully controlled chain of Galois covers and quotients. It combines group-theoretic and representation-theoretic analysis with high-precision, multi-system computations (GAP, Magma, Mathematica) to build invariants and reduce coefficients, culminating in a verifiable bicanonical model. The contribution provides not only explicit equations for this FPP but also a general framework for deriving equations of commensurable FPPs, potentially enabling construction of the remaining CS pairs. The work highlights the interplay between arithmetic group data, torsion in Picard groups, and explicit projective embeddings in proving the existence and shape of these rare surfaces.

Abstract

We find explicit equations of a new pair of fake projective planes, labeled by in the Cartwright-Steger classification. Our method involves starting with known equations of a commensurable fake projective plane and working through a chain of cyclic covers and quotients to get to the new one.

Paper Structure

This paper contains 12 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction.
  2. General description of the process.
  3. Disclaimers, acknowledgements and further directions.
  4. Technical details: group-theoretic calculations.
  5. Technical details: computing the equations.
  6. Step 1.
  7. Step 2.
  8. Step 3.
  9. Step 4.
  10. Step 5.
  11. Step 6.
  12. Step 7.

Key Result

Proposition 1.2

There is a surface $72.{{\mathbb P}^2_{fake}}$ with an automorphism group $G_{648}$ of order $648 = 2^3 3^4$ which is isomorphic to the direct product of $C_3$ and the semidirect product of $SL(2,{\mathbb Z}/3{\mathbb Z})$ and $C_3\times C_3$ (with the canonical action of the former on the latter) The fake projective planes ${{\mathbb P}^2_{fake}}$ and $\widehat{{{\mathbb P}^2_{fake}}}$ are the q

Theorems & Definitions (14)

  • Proposition 1.2
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 4 more