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

TL;DR

This paper investigates the geometry of Keum's fake projective planes, proving the bicanonical map is an embedding and constructing a novel exceptional collection on a specific fake projective plane.

Contribution

It demonstrates that the bicanonical map of Keum's fake projective planes is always an embedding and introduces a new exceptional collection on a particular fake projective plane.

Findings

Bicanonical map is always an embedding for Keum's fake projective planes.

Constructed a nonstandard exceptional collection on a specific fake projective plane.

Provides new geometric insights into the structure 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 $H_{1}(X; \mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})^{4}$.