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

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 ${\mathbb{P}}_{\mathrm{fake}}^{}=(a=7,p=2,\mathrm{\varnothing },D_3{2}_7)$ in the Cartwright-Steger classification. In this paper, we exploit the large symmetries given by $\mathrm{Aut}({\mathbb{P}}_{\mathrm{fake}}^{})=C_7⋊C_3$ 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 $T\in \mathrm{Pic}({\mathbb{P}}_{\mathrm{fake}}^{})$, we also compute and study the linear systems $\mathrm{\mid }nH+T\mathrm{\mid }$ with small $n$, where $H$ is an ample generator of the Néron-Severi group.