A journey from the octonionic $\mathbb P^2$ to a fake $\mathbb P^2$

TL;DR

This paper constructs a new family of algebraic surfaces related to the octonionic projective plane, including a special case that is a quotient of a fake projective plane, using explicit equations and quotient techniques.

Contribution

It introduces a novel family of surfaces of general type derived from octonionic projective geometry, including explicit equations for a fake projective plane quotient.

Findings

Discovery of a family of surfaces with $K^2=3$, $p=q=0$

Identification of a special surface with three $A_2$ singularities

Explicit equations for the fake projective plane in its bicanonical embedding

Abstract

We discover a family of surfaces of general type with $K^2=3$ and $p=q=0$ as free $C_{13}$ quotients of special linear cuts of the octonionic projective plane $\mathbb O \mathbb P^2$. A special member of the family has $3$ singularities of type $A_2$, and is a quotient of a fake projective plane. We use the techniques of \cite{BF20} to define this fake projective plane by explicit equations in its bicanonical embedding.