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Galois points for a plane curve in arbitrary characteristic. (English) Zbl 1160.14304
Summary: In 1996, Hisao Yoshihara introduced a new notion in algebraic geometry: a Galois point for a plane curve is a point from which the projection induces a Galois extension of function fields. Yoshihara has established various new approaches to algebraic geometry by using Galois point or generalized notions of it. It is an interesting problem to determine the distribution of Galois points for a given plane curve. In this paper, we survey recent results related to this problem.

MSC:
14H50 Plane and space curves
12F10 Separable extensions, Galois theory
14H05 Algebraic functions and function fields in algebraic geometry
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[1] Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves, vol. I. Grundlehren der Mathematischen Wissenschaften, vol. 267. Springer-Verlag, New York (1985)
[2] Chang H.C.: On plane algebraic curves. Chin. J. Math. 6, 185–189 (1978) · Zbl 0405.14009
[3] Cukierman F.: Monodromy of projections, 15th School of Algebra. Math. Contemp. 16, 9–30 (1999) · Zbl 1027.14006
[4] Duyaguit C., Miura K.: On the number of Galois points for plane curves of prime degree. Nihonkai Math. J. 14, 55–59 (2003) · Zbl 1080.14030
[5] Duyaguit C., Yoshihara H.: Galois lines for normal elliptic curves. Algebra Colloq. 12, 205–212 (2005) · Zbl 1076.14036
[6] Fukasawa S.: Galois points on quartic curves in characteristic 3. Nihonkai Math. J. 17, 103–110 (2006) · Zbl 1134.14307
[7] Fukasawa S.: On the number of Galois points for a plane curve in positive characteristic, II. Geom. Dedicata 127, 131–137 (2007) · Zbl 1186.14032 · doi:10.1007/s10711-007-9170-8
[8] Fukasawa S.: On the number of Galois points for a plane curve in positive characteristic. Commun. Algebra 36, 29–36 (2008) · Zbl 1186.14033 · doi:10.1080/00927870701649283
[9] Fukasawa, S.: Classification of plane curves with infinitely many Galois points (preprint) · Zbl 1211.14036
[10] Fukasawa, S., Hasegawa, T.: Singular plane curves with infinitely many Galois points (preprint) · Zbl 1185.14029
[11] Hefez A.: Non-reflexive curves. Compos. Math. 69, 3–35 (1989) · Zbl 0706.14024
[12] Hefez, A., Kleiman, S.: Notes on the duality of projective varieties. In: Geometry Today (Rome, 1984). Progress in Mathematics, vol. 60, pp. 143–183. Birkhäuser, Boston (1985) · Zbl 0579.14047
[13] Homma M.: A souped-up version of Pardini’s theorem and its application to funny curves. Compos. Math. 71, 295–302 (1989) · Zbl 0703.14017
[14] Homma M.: Galois points for a Hermitian curve. Commun. Algebra 34, 4503–4511 (2006) · Zbl 1111.14023 · doi:10.1080/00927870600938902
[15] Homma M.: The Galois group of a projection of a Hermitian curve. Int. J. Algebra 1, 563–585 (2007) · Zbl 1140.14304
[16] Iitaka S.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 76. , Springer-Verlag (1982) · Zbl 0491.14006
[17] Kanazawa M., Takahashi T., Yoshihara H.: The group generated by automorphisms belonging to Galois points of the quartic surface. Nihonkai Math. J. 12, 89–99 (2001) · Zbl 1077.14548
[18] Kleiman, S.L.: Tangency and duality. In: Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conference Proceedings, vol. 6, pp. 163–226. American Mathematical Society, Providence (1986) · Zbl 0601.14046
[19] Kleiman, S.L.: Multiple tangents of smooth plane curves (after Kaji). Algebraic Geometry: Sundance 1988. Contemporary Mathematics, vol. 116, pp. 71–84. American Mathematical Society, Providence (1991) · Zbl 0764.14020
[20] Miranda, R.: Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics, vol. 5. American Mathematical Society, Providence (1995) · Zbl 0820.14022
[21] Miura K.: Field theory for function fields of singular plane quartic curves. Bull. Aust. Math. Soc. 62, 193–204 (2000) · Zbl 0986.14016 · doi:10.1017/S0004972700018669
[22] Miura K.: Field theory for function fields of plane quintic curves. Algebra Colloq. 9, 303–312 (2002) · Zbl 1058.14044
[23] Miura K.: Galois points on singular plane quartic curves. J. Algebra 287, 283–293 (2005) · Zbl 1078.14031 · doi:10.1016/j.jalgebra.2005.02.015
[24] Miura K.: Galois points for plane curves and Cremona transformations. J. Algebra 320, 987–995 (2008) · Zbl 1159.14014 · doi:10.1016/j.jalgebra.2008.04.018
[25] Miura, K.: On dihedral Galois coverings arising from Lissajous’s curves. J. Geom. (to appear) · Zbl 1163.14308
[26] Miura K., Yoshihara H.: Field theory for function fields of plane quartic curves. J. Algebra 226, 283–294 (2000) · Zbl 0983.11067 · doi:10.1006/jabr.1999.8173
[27] Miura K., Yoshihara H.: Field theory for the function field of the quintic Fermat curve. Commun. Algebra 28, 1979–1988 (2000) · Zbl 0978.14024 · doi:10.1080/00927870008826940
[28] Namba M.: Geometry of Projective Algebraic Curves. Dekker, New York (1984) · Zbl 0556.14012
[29] Pardini R.: Some remarks on plane curves over fields of finite characteristic. Compos. Math. 60, 3–17 (1986) · Zbl 0607.14023
[30] Pirola G., Schlesinger E.: Monodromy of projective curves. J. Algebraic Geom. 14, 623–642 (2005) · Zbl 1084.14011
[31] Sakai H.: Infinitesimal deformation of Galois covering space and its application to Galois closure curve. Nihonkai Math. J. 14, 133–177 (2003) · Zbl 1052.14030
[32] Stichtenoth H.: Algebraic Function Fields and Codes. Universitext. Springer-Verlag, Berlin (1993) · Zbl 0816.14011
[33] Stöhr K.O., Voloch J.F.: Weierstrass points and curves over finite fields. Proc. Lond. Math. Soc. 52, 1–19 (1986) · Zbl 0593.14020 · doi:10.1112/plms/s3-52.1.1
[34] Takahashi T.: Minimal splitting surface determined by a projection of a smooth quartic surface. Algebra Colloq. 9, 107–115 (2002) · Zbl 1053.14047
[35] Takahashi T.: Galois points on normal quartic surfaces. Osaka J. Math. 39, 647–663 (2002) · Zbl 1036.14022
[36] Takahashi T.: Non-smooth Galois point on a quintic curve with one singular point. Nihonkai Math. J. 16, 57–66 (2005) · Zbl 1086.14027
[37] Watanabe S.: The genera of Galois closure curves for plane quartic curve. Hiroshima Math. J. 38, 125–134 (2008) · Zbl 1142.14021
[38] Yoshihara H.: Degree of irrationality of an algebraic surface. J. Algebra 167, 634–640 (1994) · Zbl 0834.14019 · doi:10.1006/jabr.1994.1206
[39] Yoshihara H.: Galois points on quartic surfaces. J. Math. Soc. Jpn. 53, 731–743 (2001) · Zbl 1067.14510 · doi:10.2969/jmsj/05330731
[40] Yoshihara H.: Function field theory of plane curves by dual curves. J. Algebra 239, 340–355 (2001) · Zbl 1064.14023 · doi:10.1006/jabr.2000.8675
[41] Yoshihara H.: Galois points for smooth hypersurfaces. J. Algebra 264, 520–534 (2003) · Zbl 1048.14025 · doi:10.1016/S0021-8693(03)00235-7
[42] Yoshihara H.: Families of Galois closure curves for plane quartic curves. J. Math. Kyoto Univ. 43, 651–659 (2003) · Zbl 1063.14035
[43] Yoshihara H.: Galois lines for space curves. Algebra Colloq. 13, 455–469 (2006) · Zbl 1095.14030
[44] Yoshihara H.: Galois embedding of algebraic variety and its application to abelian surface. Rend. Sem. Mat. Univ. Padova 117, 69–85 (2007) · Zbl 1136.14041
[45] Yoshihara H.: Galois points for plane rational curves. Far East J. Math. Sci. 25, 273–284 (2007) · Zbl 1132.14309
[46] Yoshihara, H.: Rational curve with Galois point and extendable Galois automorphism (preprint) · Zbl 1167.14017
[47] Yoshihara, H.: A note on minimal Galois embedding of abelian surface (preprint)
[48] Yoshihara, H.: Open questions. Available at: http://mathweb.sc.niigata-u.ac.jp/\(\sim\)yosihara/openquestion.html
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