## Automorphism group of plane curve computed by Galois points. II.(English)Zbl 1396.14025

Summary: Recently, the first author [Proc. Japan Acad., Ser. A 94, No. 6, 59–63 (2018; Zbl 1396.14025)] classified finite groups obtained as automorphism groups of smooth plane curves of degree $$d\geq4$$ into five types. He gave an upper bound of the order of the automorphism group for each types. For one of them, the type (a-ii), that is given by $$\max \{2d(d-2),60d\}$$. In this article, we shall construct typical examples of smooth plane curve $$C$$ by applying the method of Galois points, whose automorphism group has order $$60d$$. In fact, we determine the structure of the automorphism group of those curves.
For part I, see [K. Miura and A. Ohbuchi, Beitr. Algebra Geom. 56, No. 2, 695–702 (2015; Zbl 1327.14145)].

### MSC:

 14H37 Automorphisms of curves 14H50 Plane and space curves

### Keywords:

icosahedral group; Galois point; plane curve; automorphism group

### Citations:

Zbl 1396.14025; Zbl 1327.14145
Full Text:

### References:

 [1] H. F. Blichfeldt, Finite collineation groups, with an introduction to the theory of groups of operators and substitution groups, Univ. of Chicago Press, Chicago, Ill., 1917. [2] S. Fukasawa, On the number of Galois points for a plane curve in positive characteristic. III, Geom. Dedicata 146 (2010), 9–20. · Zbl 1191.14035 [3] T. Harui, Automorphism groups of smooth plane curves, arXiv:1306.5842v2. · Zbl 1070.14035 [4] K. Miura, Field theory for function fields of singular plane quartic curves, Bull. Austral. Math. Soc. 62 (2000), no. 2, 193–204. · Zbl 0986.14016 [5] K. Miura and A. Ohbuchi, Automorphism group of plane curve computed by Galois points, Beitr. Algebra Geom. 56 (2015), no. 2, 695–702. · Zbl 1327.14145 [6] K. Miura and H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), no. 1, 283–294. · Zbl 0983.11067 [7] H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), no. 1, 340–355. · Zbl 1064.14023
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