Quasi-Galois points. I: Automorphism groups of plane curves. (English) Zbl 1442.14098

Summary: We investigate the automorphism group of a plane curve, introducing the notion of a quasi-Galois point. We show that the automorphism group of several curves, for example, Klein quartic, Wiman sextic and Fermat curves, is generated by the groups associated with quasi-Galois points.


14H37 Automorphisms of curves
14H50 Plane and space curves
Full Text: DOI Euclid


[1] E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of algebraic curves, Vol. I. Grundlehren der Mathematischen Wissenschaften, 267, Springer-Verlag, New York, 1985. · Zbl 0559.14017
[2] W. Burnside, Theory of groups of finite order, Cambridge Univ. Press, Cambridge, 1911; reprinted by Dover, New York. · JFM 42.0151.02
[3] H. C. Chang, On plane algebraic curves, Chinese J. Math. 6 (1978), 185-189. · Zbl 0405.14009
[4] H. Doi, K. Idei and H. Kaneta, Uniqueness of the most symmetric non-singular plane sextics, Osaka J. Math. 37 (2000), 667-687. · Zbl 1063.14036
[5] S. Fukasawa, Galois points for a plane curve in arbitrary characteristic, Proceedings of the IV Iberoamerican conference on complex geometry, Geom. Dedicata 139 (2009), 211-218. · Zbl 1160.14304
[6] R. Hartshorne, Algebraic geometry, GTM 52, Springer-Verlag, New York-Heidelberg, 1977. · Zbl 0367.14001
[7] T. Harui, Automorphism groups of smooth plane curves, Kodai Math. J. 42 (2019), 308-331. · Zbl 1433.14024
[8] 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), 89-99. · Zbl 1077.14548
[9] F. Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann. 14 (1879), 428-471. · JFM 11.0297.01
[10] K. Miura, Galois points for plane curves and Cremona transformations, J. Algebra 320 (2008), 987-995. · Zbl 1159.14014
[11] K. Miura and A. Ohbuchi, Automorphism group of plane curve computed by Galois points, Beitr. Algebra Geom. 56 (2015), 695-702. · Zbl 1327.14145
[12] K. Miura and H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), 283-294. · Zbl 0983.11067
[13] K. Miura and H. Yoshihara, Field theory for the function field of the quintic Fermat curve, Comm. Algebra 28 (2000), 1979-1988. · Zbl 0978.14024
[14] A. Wiman, Ueber eine einfache Gruppe von 360 ebenen collineationen, Math. Ann. 47 (1896), 531-556. · JFM 27.0103.03
[15] H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), 340-355. · Zbl 1064.14023
[16] H. Yoshihara, Rational curve with Galois point and extendable Galois automorphism, J. Algebra 321 (2009), 1463-1472. · Zbl 1167.14017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.