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 $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$ 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.