On the Geometry of a Fake Projective Plane with $21$ Automorphisms

TL;DR

This paper investigates a specific fake projective plane with 21 automorphisms, leveraging its symmetries to embed it into projective space and analyze associated linear systems, providing explicit algebraic descriptions.

Contribution

It constructs an explicit embedding of the fake projective plane into 5 using its automorphisms and computes linear systems for various torsion line bundles.

Findings

Embedded the surface as 56 sextics in 5 with rational coefficients.

Analyzed linear systems |nH + T| for torsion line bundles T.

Provided explicit algebraic equations for the surface.

Abstract

A fake projective plane is a complex surface with the same Betti numbers as $\mathbb{C} P^2$ but not biholomorphic to it. We study the fake projective plane $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$ in the Cartwright-Steger classification. In this paper, we exploit the large symmetries given by $\operatorname{Aut}(\mathbb{P}_{\operatorname{fake}}^2) = C_7 \rtimes C_3$ to construct an embedding of this surface into $\mathbb{C} P^5$ as a system of $56$ sextics with coefficients in $\mathbb{Q}(\sqrt{-7})$. For each torsion line bundle $T \in \operatorname{Pic}(\mathbb{P}_{\operatorname{fake}}^2)$, we also compute and study the linear systems $|nH + T|$ with small $n$, where $H$ is an ample generator of the N\'eron-Severi group.