Examples of surfaces with canonical maps of degree $12$, $13$, $15$, $16$ and $18$

TL;DR

This paper constructs explicit examples of complex algebraic surfaces with canonical maps of degrees 12, 13, 15, 16, and 18, expanding the known catalog of such surfaces.

Contribution

It provides the first known examples of surfaces with canonical maps of degrees 13, 15, and 18, using quotients of products of curves by specific group actions.

Findings

Examples of surfaces with canonical maps of degrees 12, 13, 15, 16, 18.

Construction method using quotients of product of curves.

No prior known examples for degrees 13, 15, 18.

Abstract

In this note we present examples of complex algebraic surfaces with canonical maps of degree $12$, $13$, $15$, $16$ and $18$. They are constructed as quotients of a product of two curves of genus $10$ and $19$ using certain non-free actions of the group $S_3\times \mathbb Z_3^2$. To our knowledge there are no other examples in literature of surfaces with canonical map of degree $13$, $15$ and $18$.