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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
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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  • [1] Emil Artin, Galois theory, 2nd ed., Dover Publications Inc., Mineola, NY, 1998. Edited and with a supplemental chapter by Arthur N. Milgram. MR 1616156 (98k:12001)
  • [2] Satoru Fukasawa, On the number of Galois points for a plane curve in positive characteristic. II, Geom. Dedicata 127 (2007), 131-137. MR 2338521 (2009f:14058),
  • [3] Satoru Fukasawa, Galois points for a plane curve in arbitrary characteristic, Geom. Dedicata 139 (2009), 211-218. MR 2481846 (2010d:14046),
  • [4] Satoru Fukasawa, Complete determination of the number of Galois points for a smooth plane curve, Rend. Sem. Mat. Univ. Padova 129 (2013), 93-113. MR 3090633
  • [5] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 #3116)
  • [6] Masaaki Homma, Galois points for a Hermitian curve, Comm. Algebra 34 (2006), no. 12, 4503-4511. MR 2273720 (2007i:14028),
  • [7] Kei Miura, Galois points for plane curves and Cremona transformations, J. Algebra 320 (2008), no. 3, 987-995. MR 2427627 (2009i:14044),
  • [8] Kei Miura and Hisao Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), no. 1, 283-294. MR 1749889 (2001f:14047a),
  • [9] Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961 (94k:14016)
  • [10] Takeshi Takahashi, Galois points on normal quartic surfaces, Osaka J. Math. 39 (2002), no. 3, 647-663. MR 1932286 (2003i:14049)
  • [11] Hisao Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), no. 1, 340-355. MR 1827887 (2002f:14038),
  • [12] Hisao Yoshihara, Galois points on quartic surfaces, J. Math. Soc. Japan 53 (2001), no. 3, 731-743. MR 1828978 (2002f:14048),
  • [13] Hisao Yoshihara, Galois points for smooth hypersurfaces, J. Algebra 264 (2003), no. 2, 520-534. MR 1981419 (2004c:14084),
  • [14] Hisao Yoshihara, Rational curve with Galois point and extendable Galois automorphism, J. Algebra 321 (2009), no. 5, 1463-1472. MR 2494401 (2010b:14060),
  • [15] Hisao Yoshihara, Private communications, January 2008.

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Additional Information

Satoru Fukasawa
Affiliation: Department of Mathematical Sciences, Faculty of Science, Yamagata University, Kojirakawa-Machi 1-4-12, Yamagata 990-8560, Japan

Takeshi Takahashi
Affiliation: Division of General Education, Nagaoka National College of Technology, 888 Nishikatakai, Nagaoka, Niigata 940-8532, Japan

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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