Canonical maps of surfaces defined by Abelian covers

TL;DR

This paper classifies surfaces with canonical maps as abelian covers over the projective plane, constructs a new Campedelli surface, and provides explicit equations for certain known surfaces.

Contribution

It classifies surfaces with canonical maps as abelian covers over , constructs a new Campedelli surface with a specific fundamental group, and explicitly describes equations for Perssson's and Tan's surfaces.

Findings

Classified surfaces with canonical maps as abelian covers over 

Constructed a new Campedelli surface with ^{} fundamental group

Provided explicit defining equations for Perssson's and Tan's surfaces

Abstract

In this paper, we classified the surfaces whose canonical maps are abelian covers over $\mathbb{P}^2$. Moveover, we construct a new Campedelli surface with fundamental group $\mathbb{Z}_2^{\oplus 3}$ and give defining equations for Perssson's surface and Tan's surfaces with odd canonical degrees explicitly.