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

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 ${\mathbb{P}}^9$.