Galois points for a normal hypersurface
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- by Satoru Fukasawa and Takeshi Takahashi PDF
- Trans. Amer. Math. Soc. 366 (2014), 1639-1658 Request permission
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 $\dim \textrm {Sing}(X) \le \dim X-2$ 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 $\dim \textrm {Sing}(X)$, 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 $p>0$.References
- Emil Artin, Galois theory, 2nd ed., Dover Publications, Inc., Mineola, NY, 1998. Edited and with a supplemental chapter by Arthur N. Milgram. MR 1616156
- Satoru Fukasawa, On the number of Galois points for a plane curve in positive characteristic. II, Geom. Dedicata 127 (2007), 131–137. MR 2338521, DOI 10.1007/s10711-007-9170-8
- Satoru Fukasawa, Galois points for a plane curve in arbitrary characteristic, Geom. Dedicata 139 (2009), 211–218. MR 2481846, DOI 10.1007/s10711-008-9325-2
- Satoru Fukasawa, Complete determination of the number of Galois points for a smooth plane curve, Rend. Semin. Mat. Univ. Padova 129 (2013), 93–113. MR 3090633, DOI 10.4171/RSMUP/129-7
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Masaaki Homma, Galois points for a Hermitian curve, Comm. Algebra 34 (2006), no. 12, 4503–4511. MR 2273720, DOI 10.1080/00927870600938902
- Kei Miura, Galois points for plane curves and Cremona transformations, J. Algebra 320 (2008), no. 3, 987–995. MR 2427627, DOI 10.1016/j.jalgebra.2008.04.018
- Kei Miura and Hisao Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), no. 1, 283–294. MR 1749889, DOI 10.1006/jabr.1999.8173
- Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961
- Takeshi Takahashi, Galois points on normal quartic surfaces, Osaka J. Math. 39 (2002), no. 3, 647–663. MR 1932286
- Hisao Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), no. 1, 340–355. MR 1827887, DOI 10.1006/jabr.2000.8675
- Hisao Yoshihara, Galois points on quartic surfaces, J. Math. Soc. Japan 53 (2001), no. 3, 731–743. MR 1828978, DOI 10.2969/jmsj/05330731
- Hisao Yoshihara, Galois points for smooth hypersurfaces, J. Algebra 264 (2003), no. 2, 520–534. MR 1981419, DOI 10.1016/S0021-8693(03)00235-7
- Hisao Yoshihara, Rational curve with Galois point and extendable Galois automorphism, J. Algebra 321 (2009), no. 5, 1463–1472. MR 2494401, DOI 10.1016/j.jalgebra.2008.11.035
- Hisao Yoshihara, Private communications, January 2008.
Additional Information
- 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
- Received by editor(s): May 9, 2010
- Received by editor(s) in revised form: March 29, 2012
- Published electronically: October 23, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3145745