Orders of automorphisms of smooth plane curves for the automorphism groups to be cyclic

Tarō Hayashi1
1Faculty of Agriculture, Kindai University, Nakamaticho 3327-204, Nara, Nara, 631-8505, Japan

Tóm tắt

AbstractFor a fixed integer $$d\ge 4$$ d 4 , the list of groups that appear as automorphism groups of smooth plane curves whose degree is d is unknown, except for $$d=4$$ d = 4 or 5. Harui showed a certain characteristic about structures of automorphism groups of smooth plane curves. Badr and Bars began to study for certain orders of automorphisms and try to obtain exact structures of automorphism groups of smooth plane curves. In this paper, based on the result of T. Harui, we extend Badr–Bars study for different and new cases, mainly for the cases of cyclic groups that appear as automorphism groups.

Từ khóa


Tài liệu tham khảo

Bars, F.: On the automorphisms groups of genus 3 curves. Surv. Math. Sci. 2(2), 83–124 (2012)

Badr, E.; Bars, F.: Automorphism groups of non-singular plane curves of degree 5. Commun. Algebra 44, 4327–4340 (2016)

Badr, E.; Bars, F.: Non-singular plane curves with an element of "large" order in its automorphism group. Int. J. Algebra Comput. 26, 399–434 (2016)

Blichfeldt, H.: Finite Collineation Groups: With an Introduction to the Theory of Groups of Operators and Substitution Groups. Univ. of Chicago Press, Chicago (1917)

Fukasawa, S.; Miura, K.; Takahashi, T.: Quasi-Galois points, I: automorphism groups of plane curves. Tohoku Math. J. (2) 71(4), 487–494 (2019)

Harui, T.: Automorphism groups of smooth plane curves. Kodai Math. J. 42(2), 308–331 (2019)

Harui, T.; Miura, K.; Ohbuchi, A.: Automorphism group of plane curve computed by Galois points, II. Proc. Jpn. Acad. Ser. A Math. Sci. 94(6), 59–63 (2018)

Harui, T.; Kato, T.; Komeda, J.; Ohbuchi, A.: Quotient curves of smooth plane curves with automorphisms. Kodai Math. J. 33(1), 164–172 (2010)

Hayashi, T.: Smooth plane curves with freely acting finite groups, Vietnam. J. Math. https://doi.org/10.1007/s10013-020-00398-z

Hayashi, T.: Linear automorphisms of hypersurfaces giving Galois points. arXiv:2101.04797

Henn, P.: Die Automorphismengruppen dar algebraischen Functionenkorper vom Geschlecht 3. Inagural-dissertation, Heidelberg (1976).

Komeda, J.; Takahashi, T.: Relating Galois points to weak Galois Weierstrass points through double coverings of curves. J. Korean Math. Soc. 54(1), 69–86 (2017)

Komeda, J.; Takahashi, T.: Galois Weierstrass points whose Weierstrass semigroups are generated by two elements. arXiv:1703.09416

Kuribayashi, A.; Komiya, K.: On Weierstrass points of non-hyperelliptic compact Riemann surfaces of genus three. Hiroshima Math. J. 7, 743–786 (1977)

Miura, K.; Ohbuchi, A.: Automorphism group of plane curve computed by Galois points. Beitr. Algebra Geom. 56(2), 695–702 (2015)

Miura, K.; Yoshihara, H.: Field theory for function fields of plane quartic curves. J. Algebra 226, 283–294 (2000)

Namba, M.: Geometry of Projective Algebraic Curves. Marcel Dekker, New York (1984)

Pambianco, F.: Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve. Math. Z 277, 975–993 (2014)

Tzermias, P.: The group of automorphisms of the Fermat curve. J. Num. Theory 53(1), 173–178 (1995)

Yoshihara, H.: Function field theory of plane curves by dual curves. J. Algebra 239, 340–355 (2001)