# On the canonical map of some surfaces isogenous to a product

## TL;DR

This paper constructs new families of surfaces of general type with high-degree canonical maps, expanding understanding of their existence and properties, especially for surfaces isogenous to a product with elementary abelian group actions.

## Contribution

It introduces new examples of surfaces isogenous to a product with high-degree canonical maps and analyzes their properties, including the degree of the canonical map and moduli space components.

## Key findings

- Constructed families with canonical map degrees 48 and 56 for specific geometric genera.
- Identified a moduli component with surfaces having canonical map degree at least 2.
- Established foundational results on canonical maps of surfaces isogenous to a product using representation theory.

## Abstract

We give new contributions to the existence problem of canonical surfaces of high degree. We construct several families (indeed, connected components of the moduli space) of surfaces $S$ of general type with $p_g=5,6$ whose canonical map has image $\Sigma$ of very high degree, $d=48$ for $p_g=5$, $d=56$ for $p_g=6$. And a connected component of the moduli space consisting of surfaces $S$ with $K^2_S = 40, p_g=4, q=0$ whose canonical map has always degree $\geq 2$, and, for the general surface, of degree $2$ onto a canonical surface $Y$ with $K^2_Y = 12, p_g=4, q=0$. The surfaces we consider are SIP 's, i.e. surfaces $S$ isogenous to a product of curves $(C_1 \times C_2 )/ G$; in our examples the group $G$ is elementary abelian, $G = (\mathbb{Z}/m)^k$. We also establish some basic results concerning the canonical maps of any surface isogenous to a product, basing on elementary representation theory.