## 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 $X$.## References

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## Additional Information

**Francisco Javier Gallego**- Affiliation: Departamento de Álgebra, Universidad Complutense de Madrid, Madrid, Spain
- Email: gallego@mat.ucm.es
**Bangere P. Purnaprajna**- Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-2142
- Email: purna@math.ku.edu
- Received by editor(s): July 3, 2009
- Received by editor(s) in revised form: February 9, 2010
- Published electronically: March 7, 2011
- Additional Notes: The first author was partly supported by Spanish Government grant MTM2006-04785 and by Complutense grant PR27/05-13876 and is part of the Complutense Research group 910772. He also thanks the Department of Mathematics of the University of Kansas for its hospitality

The second author thanks the General Research Fund of Kansas for partly supporting this research project. He also thanks the Department of Algebra of the Universidad Complutense de Madrid for its hospitality. - © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**363**(2011), 4401-4420 - MSC (2000): Primary 14J10, 14J29
- DOI: https://doi.org/10.1090/S0002-9947-2011-05353-5
- MathSciNet review: 2792993