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Automorphism group of plane curve computed by Galois points, II.

1. Introduction. The purpose of this article is to give typical examples of smooth plane curve of degree d whose automorphism group has order 60d. In fact, we study the structure of that group. Our method is based on the classification theorem of automorphism groups by the first author and the theory of Galois points for smooth plane curves.

First, we recall several definitions of Galois points in brief. Throughout the present article, we work over the complex number field C. The concept of Galois points was introduced by Yoshihara in 1996 (e.g. [6]). Let C [subset] [P.sup.2] be a smooth plane curve of degree d (d [greater than or equal to] 4) and C(C) the function field of C. Let P be a point of [P.sup.2]. Consider the morphism [[pi].sub.P]: C [right arrow] [P.sup.1], which is the restriction of the projection [P.sup.2] [right arrow] [P.sup.1] with the center P. Then we obtain the field extension induced by [[pi].sub.P], i.e., [[pi].sup.*.sub.P]: C([P.sup.1]) [right arrow] C(C). Putting [K.sub.P] = [[pi].sup.*.sub.P](C([P.sup.1])), we have the following definition.

Definition 1. The point P is called a Galois point for C if the field extension C(C)/[K.sub.P] is Galois. Furthermore, a Galois point is said to be inner (resp. outer) if P 2 C (resp. P [member of] [P.sup.2]\C). The group [G.sub.P] = Gal(C(C)/[K.sub.P]) is called the Galois group at P.

We denote by [delta](C) (resp. [delta]'(C)) the number of inner (resp. outer) Galois points for C. There are many known results on Galois points. We recall some of them.

Theorem 1 ([6], [7]). Suppose that C is a smooth plane curve of degree d (d [greater than or equal to] 4). Then,

(i) [delta]'(C) = 0,1 or 3. Further, [delta]'(C) = 3 if and only if C is projectively equivalent to the Fermat curve.

(ii) [delta](C) = 0,1 or 4 if d = 4. Further, [delta](C) = 4 if and only if C is projectively equivalent to the curve defined by [X.sup.4] + [Y.sup.4] + Y[Z.sup.3] = 0. When d [greater than or equal to] 5, we have [delta](C) =0 or 1.

Theorem 2 ([7]). Suppose that C is a smooth plane curve of degree d (d [greater than or equal to] 4). If P is an inner (resp. outer) Galois point, then Gp is isomorphic to the cyclic group of degree d - 1 (resp. d), i.e., [G.sub.P] [congruent to] [Z.sub.d-1] (resp. [Z.sub.d]).

Remark 1. If C has singularities, then the theorem above does not hold true. Namely, there exist a singular plane curve C and a Galois point P for C such that Gp is not cyclic. For example, see [4].

When C has a Galois point, we can give a concrete defining equation of C.

Proposition 3 ([7]). By a suitable change of coordinates, the defining equation of C with an outer Galois point can be expressed as [Z.sup.d] + [F.sub.d](X, Y) = 0, where [F.sub.d](X,Y) is a homogeneous polynomial of degree d without multiple factors.

Referring to [3], we may infer that plane curves with [delta](C) [not equal to] 0 or [delta]'(C) = 0 play an important role when we classify the automorphism group of smooth plane curves.

In [3], the first author classified finite groups obtained as automorphism groups of C into five types. First of all, we recall several definitions. Let G be a group of automorphisms of C. Then, it is well-known that G is considered as a subgroup of PGL(3, C) = Aut([P.sup.2]). Let [F.sub.d] be the Fermat curve [X.sup.d] + [Y.sup.d] + [Z.sup.d] = 0. We denote by [K.sub.d] a smooth curve defined by X[Y.sup.d-1] + Y[Z.sup.d-1] + Z[X.sup.d-1] = 0 (In [3], [K.sub.d] is called Klein curve of degree d). For a non-zero monomial c[X.sup.i][Y.sup.j][Z.sup.k] with c [member of] C \{0}, we define its exponent as max{i,j, k}. For a homogeneous polynomial F(X, Y, z), the core of F(X, Y, Z) is defined as the sum of all terms of F(X, Y, Z) with the greatest exponent.

Definition 2. Let [C.sub.0] be a smooth plane curve with defining equation [F.sub.0](X, Y, Z) = 0. Then a pair (C, G) of a smooth plane curve C and a subgroup G [subset] Aut(C) is said to be a descendent of [C.sub.0] if C is defined by a homogeneous polynomial whose core coincides with [F.sub.0](X, Y, Z) and G acts on [C.sub.0] in a suitable coordinate system.

Definition 3. We denote by PBD(2,1) the following subgroup of PGL(3, C):

[mathematical expression not reproducible]

We remark that there exists a natural group homomorphism [rho] : PBD(2,1) [right arrow] PGL(2, C), i.e., A [??] ([a.sub.ij]).

Using these concepts, the first author proved the following theorem.

Theorem 4 ([3]). Let C be a smooth plane curve of degree d [greater than or equal to] 4, G a subgroup of Aut(C). Then one of the following holds:

(a-i) G fixes a point on C and G is a cyclic group whose order is at most d(d - 1). Furthermore, if d [greater than or equal to] 5 and [absolute value of G] = d(d - 1), then C is projectively equivalent to the curve Y[Z.sup.d-1] + [X.sup.d] + [Y.sup.d] = 0 and Aut(C) [congruent to] [Z.sub.d(d-1)].

(a-ii) G fixes a point not lying on C and there exists a commutative diagram

[mathematical expression not reproducible]

where N is a cyclic group whose order is a factor of d and G' is a subgroup of PGL(2, C), i.e., a cyclic group [Z.sub.m], a dihedral group D2m, the tetrahedral group [A.sub.4], the octahedral group S4 or the icosahedral group [A.sub.5]. Furthermore, m [less than or equal to] d - 1 and if G' [congruent to] [D.sub.2m] then m | d - 2 or N is trivial. In particular, [absolute value of G] [less than or equal to] max{2d(d - 2), 60d}. (b-i) (C, G) is a descendant of the Fermat curve Fd : [X.sup.d] + [Y.sup.d] + [Z.sup.d] = 0. In this case [absolute value of G] [less than or equal to] 6[d.sup.2]. (b-ii) (C, G) is a descendant of the Klein curve [K.sub.d] : X[Y.sup.d-1] + Y[Z.sup.d-1] + Z[X.sup.d-1] = 0. In this case [absolute value of G] [less than or equal to] 3([d.sup.2] - 3d + 3) if d [greater than or equal to] 5. (c) G is conjugate to a finite primitive subgroup of PGL(3, C). Namely, the icosahedral group A5, the Klein group of order 168, the alternating group A6, the Hessian group H216 or its subgroup of order 36 or 72. In particular, [absolute value of G] [less than or equal to] 360.

2. Remark on (a-i). Let [P.sub.1], ... , [P.sub.m] be all inner and outer Galois points for C and G(C) denote the group generated by [mathematical expression not reproducible]. The group G(C) is called the group generated by automorphisms belonging to all Galois points for C. In [5], we have studied the difference between Aut(C) and G(C). Referring to [2], if [delta](C) [greater than or equal to] 1 and [delta]'(C) [greater than or equal to] 1, then C is projectively equivalent to the curve as in Theorem 4 (a-i). We denote the curve by C(d), i.e., C(d) : Y[Z.sup.d-1] + [X.sup.d] + [Y.sup.d] = 0. If d [greater than or equal to] 5, then P =(0:0:1) is the only inner Galois point and Q = (1 : 0 : 0) is the only outer Galois point for C(d). We put [G.sub.P] = <[sigma]> and [G.sub.Q] = <[tau]>. Then G(C(d)) = <[sigma],[tau]>. In [5], we obtain Aut(C(d)) = G(C(d)). Thus Galois points play an important role in studying the automorphism groups of smooth plane curves.

3. Main results. In this section, we first remark on Theorem 4 (a-ii). In general, we have 2d(d - 2) > 60d. However, clearly 2d(d - 2) < 60d if d < 32. Hence we consider the case d < 32, and try to construct C with [absolute value of Aut(C)] = 60d.

Let [F.sub.i](X,Y) (i = 1, 2, 3) be the homogeneous polynomials of X and Y defined by

[F.sub.30] = [X.sup.30] + 522 ([X.sup.25][Y.sup.5] - [X.sup.5][Y.sup.25]) - 10005([X.sup.20][Y.sup.10] + [X.sup.10][Y.sup.20]) + [Y.sup.30],

[F.sub.20] = [X.sup.20] - 228 ([X.sup.15][Y.sup.5] - [X.sup.5][Y.sup.15]) + 494[X.sup.10][Y.sup.10] + [Y.sup.20] and

[F.sub.12] = XY ([X.sup.10] + 11[X.sup.5][Y.sup.5] - [Y.sup.10]).

For these polynomials, we have well-known facts as follows:

Fact 1. Let [[zeta].sub.5] be a primitive 5th root of unity and put

[mathematical expression not reproducible]

and [bar.I] = <[alpha],[beta],[gamma]>. Then C[[X,Y].sup.[bar.I]] = C[[F.sub.30],[F.sub.20],[F.sub.12]]. Note that [bar.I] [congruent to] SL(2 ,5): the binary icosahedral subgroup of SL(2, C).

Under the situation above, our main results are stated as follows:

Theorem 5. Let [C.sub.30], [C.sub.20] and [C.sub.12] be the plane curves defined by

[C.sub.30]: [Z.sup.30] + [F.sub.30](X, Y) = 0,

[C.sub.20]: [Z.sup.20] + [F.sub.20](X, Y) = 0 and

[C.sub.12]: [Z.sup.12] + [F.sub.12](X, Y) = 0.

Then [absolute value of Aut([C.sub.d])] = 60d (d = 30 , 20, 12). Furthermore, the following hold:

Aut([C.sub.30]) [congruent to] [Z.sub.15] x SL(2,5),

Aut([C.sub.20]) [congruent to] [Z.sub.5] x (SL(2,5) [??] Z2) and

Aut([C.sub.12]) [congruent to] [Z.sub.3] x (SL(2,5) [??] [Z.sub.2]).

4. Proofs of Theorem 5. First of all, we review Theorem 4 (a-ii) from the viewpoint of Galois points. Let C be a smooth plane curve of degree d [greater than or equal to] 4 with a unique Galois point P, G a subgroup of Aut(C). Then by Proposition 3, we may assume that the defining equation of C is given by [Z.sup.d] + [F.sub.d](X,Y) = 0 for some homogeneous polynomial [F.sub.d](X, Y) of degree d and P = (0:0:1). Let [[pi].sub.P]: [P.sup.2] ... [right arrow] [P.sup.1] be the projection with the center P. Then [[pi].sub.P] is represented as [[pi].sub.P]((X : Y : Z)) = (X : Y). The Galois group Gp is represented by

[mathematical expression not reproducible]

where [[zeta].sub.d] is a primitive d-th root of unity. We denote by [[lambda].sub.d] this matrix generating [G.sub.P]. Then we get the following commutative diagram as in Theorem 4 (a-ii):

[mathematical expression not reproducible]

In this case N = [G.sub.P]. Thus we get the exact sequence

[mathematical expression not reproducible]

where G' [subset] PGL(2 C).

Now, we put

[mathematical expression not reproducible]

where [[xi].sup.12] = [[zeta].sub.5]. We also put

[mathematical expression not reproducible]

Referring to [1], we see that the image of [bar.I] under the natural homomorphism SL(2, C) [right arrow] PGL(2, C) is isomorphic to A5. Further, we define S(2,1) := <[alpha]', [beta]', [gamma]') = [bar.I].

First we deal with [C.sub.30]. Put [??] = <[sigma], [tau], [[lambda].sub.30]> [subset] GL(3 , C) and H = <[sigma], [tau]>. Then we can check that [mathematical expression not reproducible]. So we have [rho] [member of] H. Furthermore, since [alpha]' = [([[rho].sup.2][sigma][rho]).sup.2][rho], [beta]' = [sigma][([[rho].sup.2][sigma][rho]).sup.2][rho] and [gamma]' = [[sigma].sup.2][rho], we obtain H [subset] S(2, 1).

We also remark that [mathematical expression not reproducible]

and [mathematical expression not reproducible]. Thus we obtain [mathematical expression not reproducible].

Therefore, we have that

[mathematical expression not reproducible]

Hence [G.sub.0] [congruent to] SL(2,5) x [Z.sub.15]. In particular, [absolute value of [G.sub.0]] = 120 x 15 = 1800. On the other hand, we see that [absolute value of G] = 30 x 60 = 1800 by (#). Hence [G.sub.0] = G, which completes the proof of this case.

By a similar argument to the above, we can prove the other cases. So, we give the proofs in brief.

For the curve [C.sub.20], we put [mathematical expression not reproducible]. We see that

[mathematical expression not reproducible]

Since its center Z is

[mathematical expression not reproducible]

we obtain

[mathematical expression not reproducible]

where [G.sub.0] = [??]/Z [subset] G.

Further, since

[mathematical expression not reproducible]

we have the following exact sequence:

[mathematical expression not reproducible]

where [mathematical expression not reproducible]. The sequence is split by [mathematical expression not reproducible]

Hence [G.sub.0] [congruent to] (SL(2, 5) [??] [Z.sub.2]) x [Z.sub.5]. In particular, [absolute value of [G.sub.0]] = 120 x 2 x 5 = 1200. On the other hand, we see that [absolute value of G] = 20 x 60 = 1200 by (#). Hence [G.sub.0] = G, which completes the proof of this case.

Finally, for the curve [C.sub.12], we put [mathematical expression not reproducible], and K = <[sigma],[rho]>. We can check that K = S(2,1).

Putting [mathematical expression not reproducible], we obtain [mathematical expression not reproducible]. Furthermore we get

[mathematical expression not reproducible]

where [omega] is a cubic root of unity.

Hence [G.sub.0] [congruent to] (SL(2, 5) [??] [Z.sub.2]) x [Z.sub.3]. In particular, [absolute value of [G.sub.0]] = 120 x 2 x 3 = 720. On the other hand, we see that [absolute value of G] = 12 x 60 = 720 by (#). Hence [G.sub.0] = G, which completes the proof of this case.

doi: 10.3792/pjaa.94.59

Acknowledgments. The second author was partially supported by JSPS KAKENHI Grant Number JP26400057. The third author was partially supported by JSPS KAKENHI Grant Number JP15K04822.

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.

[3] T. Harui, Automorphism groups of smooth plane curves, arXiv:1306.5842v2.

[4] K. Miura, Field theory for function fields of singular plane quartic curves, Bull. Austral. Math. Soc. 62 (2000), no. 2, 193 204.

[5] K. Miura and A. Ohbuchi, Automorphism group of plane curve computed by Galois points, Beitr. Algebra Geom. 56 (2015), no. 2, 695 702.

[6] K. Miura and H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), no. 1, 283 294.

[7] H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), no. 1, 340 355.

Takeshi Harui, (*) Kei Miura, (**) Akira Ohbuchi (***)

(Communicated by Heisuke Hironaka, m.j.a., May 14, 2018)

2010 Mathematics Subject Classification. Primary 14H37; Secondary 14H50.

(*) Department of Core Studies, Kochi University of Technology, 185 Miyanokuchi, Tosayamada, Kami, Kochi 782-8502, Japan.

(**) Department of Mathematics, National Institute of Technology, Ube College, 2-14-1 Tokiwadai, Ube, Yamaguchi 7558555, Japan.

(***) Department of Mathematical Sciences, Faculty of Science and Technology, Tokushima University, 2-1 Minamijosanjima-cho, Tokushima 770-8502, Japan.
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Author:Harui, Takeshi; Miura, Kei; Ohbuchi, Akira
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
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Date:Jun 1, 2018
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