On the bicanonical morphism of quadruple Galois canonical covers

F. J. Gallego; B. P. Purnaprajna

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arXiv · 1001.1128

On the bicanonical morphism of quadruple Galois canonical covers

F. J. Gallego·B. P. Purnaprajna

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Abstract

In this article we study the bicanonical map $ϕ_2$ of quadruple Galois canonical covers X of surfaces of minimal degree. We show that $ϕ_2$ 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 $ϕ_2$ is an embedding, and if so happens, $ϕ_2$ embeds $X$ as a projectively normal variety, and cases in which $ϕ_2$ is not an embedding. If the latter, $ϕ_2$ 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.

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BibTeX RIS

F. J. Gallego, B. P. Purnaprajna. 2010-01-07. On the bicanonical morphism of quadruple Galois canonical covers. https://arxiv.org/abs/1001.1128

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