The Bicanonical Map of Fake Projective Planes with an Automorphism

We show, for several fake projective planes with a nontrivial group of automorphisms, that the bicanonical map is an embedding.

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].

Some bidouble planes with pg = q = 0 and 4 ≤ K2 ≤ 7

We give a list of possibilities for surfaces of general type with pg = 0 having an involution i such that the bicanonical map of S is not composed with i and S/i is not rational. Some examples with K2 = 4, …, 7 are constructed as double coverings of an Enriques surface. These surfaces have a description as bidouble coverings of the plane.

On Surfaces with pg = 0 and K2 = 5

We construct new examples of surfaces of general type with pg = 0 and K2 = 5 as ℤ2 × ℤ2-covers and show that they are genus three hyperelliptic fibrations with bicanonical map of degree two.