On the bicanonical morphism of quadruple Galois canonical covers

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.