On the bicanonical morphism of quadruple Galois canonical covers

Gallego, Francisco; Purnaprajna, Bangere

doi:10.1090/S0002-9947-2011-05353-5

On the bicanonical morphism of quadruple Galois canonical covers
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by Francisco Javier Gallego and Bangere P. Purnaprajna PDF

Trans. Amer. Math. Soc. 363 (2011), 4401-4420 Request permission

Abstract:

In this article we study the bicanonical map $\varphi _2$ of quadruple Galois canonical covers $X$ of surfaces of minimal degree. We show that $\varphi _2$ 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 $\varphi _2$ is an embedding, and if it so happens, $\varphi _2$ embeds $X$ as a projectively normal variety, and there are cases in which $\varphi _2$ is not an embedding. If the latter, $\varphi _2$ 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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