On the bicanonical morphism of quadruple Galois canonical covers

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In this article we study the bicanonical map ϕ₂ of quadruple Galois canonical covers X of surfaces of minimal degree. We show that ϕ₂ 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 ϕ₂ is an embedding, and if so happens, ϕ₂ embeds X as a projectively normal variety, and cases in which ϕ₂ is not an embedding. If the latter, ϕ₂ 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.