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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Galois points for a normal hypersurface


Authors: Satoru Fukasawa and Takeshi Takahashi
Journal: Trans. Amer. Math. Soc. 366 (2014), 1639-1658
MSC (2010): Primary 14J70, 12F10
DOI: https://doi.org/10.1090/S0002-9947-2013-05875-8
Published electronically: October 23, 2013
MathSciNet review: 3145745
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Abstract: A Galois point for a hypersurface is a point from which the projection induces a Galois extension of function fields. The purpose of this article is to determine the set $ \Delta (X)$ of Galois points for a hypersurface $ X$ with $ \dim {\rm Sing}(X) \le \dim X-2$ in characteristic zero: In fact, if $ X$ is not a cone, we give a sharp upper bound for the cardinality of $ \Delta (X)$ in terms of $ \dim X$ and $ \dim {\rm Sing}(X)$, and completely classify $ X$ attaining the bound. We determine $ \Delta (X)$ also when $ X$ is a cone. To achieve our purpose, we prove a certain hyperplane section theorem on a Galois point in arbitrary characteristic. The hyperplane section theorem has other important applications: For example, we can classify the Galois group induced by a Galois point in arbitrary characteristic and determine the distribution of Galois points for a Fermat hypersurface of degree $ p^e+1$ in characteristic $ p>0$.


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Satoru Fukasawa
Affiliation: Department of Mathematical Sciences, Faculty of Science, Yamagata University, Kojirakawa-Machi 1-4-12, Yamagata 990-8560, Japan
Email: s.fukasawa@sci.kj.yamagata-u.ac.jp

Takeshi Takahashi
Affiliation: Division of General Education, Nagaoka National College of Technology, 888 Nishikatakai, Nagaoka, Niigata 940-8532, Japan
Email: takeshi@nagaoka-ct.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2013-05875-8
Keywords: Galois point, hypersurface
Received by editor(s): May 9, 2010
Received by editor(s) in revised form: March 29, 2012
Published electronically: October 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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