Examples of Surfaces with Canonical Map of Maximal Degree

Abstract

It was shown by Beauville that if the canonical map $\\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\\operatorname{deg}(\\varphi_{|K_M|}) \\leq 36$. 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 $\\operatorname{Aut}(X) = C_7:C_3$ (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 Hacon and Cai.