대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제58권3호
  • /
  • Pages.617-635
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  • 2021
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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LINEAR AUTOMORPHISMS OF SMOOTH HYPERSURFACES GIVING GALOIS POINTS

  • 투고 : 2020.05.12
  • 심사 : 2021.01.29
  • 발행 : 2021.05.31
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초록

Let X be a smooth hypersurface X of degree d ≥ 4 in a projective space ℙn+1. We consider a projection of X from p ∈ ℙn+1 to a plane H ≅ ℙn. This projection induces an extension of function fields ℂ(X)/ℂ(ℙn). The point p is called a Galois point if the extension is Galois. In this paper, we will give necessary and sufficient conditions for X to have Galois points by using linear automorphisms.

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참고문헌

  1. E. Badr and F. Bars, Non-singular plane curves with an element of "large" order in its automorphism group, Internat. J. Algebra Comput. 26 (2016), no. 2, 399-433. https://doi.org/10.1142/S0218196716500168
  2. E. Badr and F. Bars, Automorphism groups of nonsingular plane curves of degree 5, Comm. Algebra 44 (2016), no. 10, 4327-4340. https://doi.org/10.1080/00927872.2015.1087547
  3. F. Bastianelli, R. Cortini, and P. De Poi, The gonality theorem of Noether for hypersurfaces, J. Algebraic Geom. 23 (2014), no. 2, 313-339. https://doi.org/10.1090/S1056-3911-2013-00603-7
  4. S. Fukasawa, K. Miura, and T. Takahashi, Quasi-Galois points, I: automorphism groups of plane curves, Tohoku Math. J. (2) 71 (2019), no. 4, 487-494. https://doi.org/10.2748/tmj/1576724789
  5. S. Fukasawa and T. Takahashi, Galois points for a normal hypersurface, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1639-1658. https://doi.org/10.1090/S0002-9947-2013-05875-8
  6. V. Gonzalez-Aguilera and A. Liendo, Automorphisms of prime order of smooth cubic nfolds, Arch. Math. (Basel) 97 (2011), no. 1, 25-37. https://doi.org/10.1007/s00013-011-0247-0
  7. V. Gonzalez-Aguilera and A. Liendo, On the order of an automorphism of a smooth hypersurface, Israel J. Math. 197 (2013), no. 1, 29-49. https://doi.org/10.1007/s11856-012-0177-y
  8. T. Harui, Automorphism groups of smooth plane curves, Kodai Math. J. 42 (2019), no. 2, 308-331. https://doi.org/10.2996/kmj/1562032832
  9. T. Harui, K. Miura, and A. Ohbuchi, Automorphism group of plane curve computed by Galois points, II, Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 6, 59-63. https://doi.org/10.3792/pjaa.94.59
  10. T. Hayashi, Smooth plane curves with freely acting finite groups, Vietnam J. Math. (2020). https://doi.org/10.1007/s10013-020-00398-z
  11. M. Kanazawa, T. Takahashi, and H. Yoshihara, The group generated by automorphisms belonging to Galois points of the quartic surface, Nihonkai Math. J. 12 (2001), no. 1, 89-99.
  12. J. Komeda and T. Takahashi, Relating Galois points to weak Galois Weierstrass points through double coverings of curves, J. Korean Math. Soc. 54 (2017), no. 1, 69-86. https://doi.org/10.4134/JKMS.j150593
  13. J. Komeda and T. Takahashi, Galois Weierstrass points whose Weierstrass semigroups are generated by two elements, arXiv:1703.09416
  14. H. Matsumura and P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1963/64), 347-361. https://doi.org/10.1215/kjm/1250524785
  15. K. Miura and A. Ohbuchi, Automorphism group of plane curve computed by Galois points, Beitr. Algebra Geom. 56 (2015), no. 2, 695-702. https://doi.org/10.1007/s13366-013-0181-3
  16. K. Miura and H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), no. 1, 283-294. https://doi.org/10.1006/jabr.1999.8173
  17. M. Noether, Zur Grundlegung der Theorie der algebraischen Raumcurven, J. Reine Angew. Math. 93 (1882), 271-318. https://doi.org/10.1515/crll.1882.93.271
  18. K. Oguiso and X. Yu, Automorphism groups of smooth quintic threefolds, Asian J. Math. 23 (2019), no. 2, 201-256. https://doi.org/10.4310/AJM.2019.v23.n2.a2
  19. T. Takahashi, Galois points on normal quartic surfaces, Osaka J. Math. 39 (2002), no. 3, 647-663. http://projecteuclid.org/euclid.ojm/1153492851
  20. H. Yoshihara, Degree of irrationality of an algebraic surface, J. Algebra 167 (1994), no. 3, 634-640. https://doi.org/10.1006/jabr.1994.1206
  21. H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), no. 1, 340-355. https://doi.org/10.1006/jabr.2000.8675
  22. H. Yoshihara, Galois points on quartic surfaces, J. Math. Soc. Japan 53 (2001), no. 3, 731-743. https://doi.org/10.2969/jmsj/05330731
  23. H. Yoshihara, Galois points for smooth hypersurfaces, J. Algebra 264 (2003), no. 2, 520-534. https://doi.org/10.1016/S0021-8693(03)00235-7
  24. Z. Zheng, On abelian automorphism groups of hypersurfaces, arXiv:2004.09008, 2020.