Explicit equations of three pairs of fake projective planes

Source abstract

We find explicit polynomial equations in their bicanonical embeddings into $\mathbb C\mathbb P^9$ for the pairs of fake projective planes labeled by $(C18,p=3,\{2\},D_3), (C18,p=3,\{2\},(dD)_3)$ and $(C18,p=3,\{2\},(d^2D)_3)$ in the Cartwright-Steger classification. We do this by starting with the equations of the commensurable fake projective plane labeled by $(C18,p=3,\{2I\})$ that were found in arXiv:2512.03213.