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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the bicanonical morphism of quadruple Galois canonical covers


Authors: Francisco Javier Gallego and Bangere P. Purnaprajna
Journal: Trans. Amer. Math. Soc. 363 (2011), 4401-4420
MSC (2000): Primary 14J10, 14J29
Published electronically: March 7, 2011
MathSciNet review: 2792993
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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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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05353-5
PII: S 0002-9947(2011)05353-5
Keywords: Surfaces of general type, bicanonical map, quadruple Galois canonical covers, canonical ring, surfaces of minimal degree
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.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.