Realizing a Fake Projective Plane as a Degree 25 Surface in $\mathbb P^5$

TL;DR

This paper constructs an explicit embedding of a specific fake projective plane into b5^5, simplifying its defining equations and providing new insights into its geometric realization.

Contribution

It presents a novel embedding of a fake projective plane into b5^5 and simplifies its defining equations, advancing understanding of its geometric structure.

Findings

Constructed a degree 25 surface embedding in b5^5

Simplified the equations defining the fake projective plane

Provided explicit equations for the embedding

Abstract

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in $\mathbb P^9$. In this paper, we study Keum's fake projective plane $(a=7, p=2, \{7\}, D_3 2_7)$ and use the equations of \cite{Borisov} to construct an embedding of fake projective plane in $\mathbb P^5$. We also simplify the 84 cubic equations defining the fake projective plane in $\mathbb P^9$.