# Surfaces with canonical map of maximum degree

## TL;DR

This paper constructs examples of algebraic surfaces with maximal degree canonical maps, demonstrating new geometric properties and providing explicit equations over finite fields for certain fibers.

## Contribution

It introduces surfaces with canonical maps of degrees 36 and 27, using equations from fake projective planes and the Cartwright-Steger surface, advancing understanding of surface mappings.

## Key findings

- Existence of a regular surface with canonical map degree 36
- Existence of an irregular surface with canonical map degree 27
- Explicit equations for certain fibers over finite fields

## 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.