Surfaces with canonical map of maximum degree

Abstract

We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the $\mathbb{Z}/3$-invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth.

Ancillary files with Magma code included. Final version, to appear in J Algebraic Geom