Examples of surfaces with canonical maps of maximal degree

TL;DR

This paper constructs and classifies certain complex surfaces with maximal canonical map degree, confirming the upper bound of 36 and exploring related threefolds, advancing understanding of algebraic surface geometry.

Contribution

It demonstrates that specific degree four Galois étale covers of fake projective planes attain the maximal canonical degree of 36 and classifies all such covers.

Findings

35 surfaces have canonical degree 36

The base locus of the canonical map is finite for these surfaces

Confirmed the upper bound of 36 for canonical degree in certain threefolds

Abstract

It was shown by A. Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then ${\rm deg}(\varphi_{|K_M|})\leq 36$. The first example of a surface with canonical degree 36 was found by the second author. In this article, we show that for any surface which is a degree four Galois \'etale cover of a fake projective plane $X$ with the largest possible automorphism group ${\rm Aut}(X)=C_7:C_3$ (the unique non-abelian group of order 21), the base locus of the canonical map is finite, and we verify that 35 of these surfaces have maximal canonical degree 36. We also classify all smooth degree four Galois \'etale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of C. Hacon and J.-X. Cai.