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

We apply the recent results of Galkin et al. [Derived categories of Keum’s fake projective planes, Adv. Math. 278 (2015) 238–253] 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 [Formula: see text] with [Formula: see text].