Exceptional collections and the bicanonical map of Keum's fake projective planes

Abstract

We apply the recent results of Galkin et al. [GKMS15] to study some geometrical features of Keum's fake projective planes. Among other things, we show that the bicanonical map of Keum's fake projective planes is always an embedding. Moreover, we construct a nonstandard exceptional collection on the unique fake projective plane $X$ with $H_{1}(X; \mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})^{4}$.