ON THE BICANONICAL MORPHISM OF QUADRUPLE GALOIS CANONICAL COVERS

In this article we study the bicanonical map phi(2) of quadruple Ga-lois canonical covers X of surfaces of minimal degree. We show that phi(2) has diverse behavior and exhibits 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 it so happens, phi(2) embeds X as a projectively normal variety, and there are 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.