On the bicanonical morphism of quadruple Galois canonical covers

TL;DR

This paper investigates the behavior of the bicanonical map of quadruple Galois canonical covers of minimal degree surfaces, revealing diverse properties including embeddings, degrees, and the structure of the canonical ring.

Contribution

It provides a detailed analysis of the bicanonical map's behavior and characterizes the canonical ring's generation, including a general formula for degree 2 generators applicable to arbitrary degrees.

Findings

The bicanonical map can be an embedding or have finite degree 1, 2, or 4.

The canonical ring is generated in degrees up to 3.

A general formula for degree 2 generators depends only on genus.

Abstract

In this article we study the bicanonical map $\phi_2$ of quadruple Galois canonical covers X of surfaces of minimal degree. We show that $\phi_2$ has diverse behavior and exhibit most of the complexities that are possible for a bicanonical map of surfaces of general type, depending on the type of X. There are cases in which $\phi_2$ is an embedding, and if so happens, $\phi_2$ embeds $X$ as a projectively normal variety, and cases in which $\phi_2$ is not an embedding. If the latter, $\phi_2$ is finite of degree 1, 2 or 4. We also study the canonical ring of X, proving that it is generated in degree less than or equal to 3 and finding the number of generators in each degree. For generators of degree 2 we find a nice general formula which holds for canonical covers of arbitrary degrees. We show that this formula depends only on the geometric and the arithmetic genus of X.