数理解析研究所講究録
第 1922 巻 2014 年 8-20

8On automorphism groups and plane models of
Riemann surfaces$*$

Takeshi Harui
Academic Support Center, Kogakuin University
This article is a summary of several results of the author on automorphism groupsof Riemann surfaces. It is based on his talk at RIMS Conference “The Topology
and the Algebraic Structures of Transformation Groups”’ on May 26, 2014.

1

Introduction

The group of automorphisms of a Riemann surface is an old subject of research inalgebraic geometry. In this article we investigate automorphism groups of Riemannsurfaces via its plane models, i.e., plane algebraic curves birationally equivalent toit.
Except for the last section we consider smooth plane curves, in other words, Riemann surfaces embedded into a projective plane. Automorphism groups of smoothplane curves of degree at most three are classically well understood. So we study
the cases of higher degree and consider the following problems:

Problem. (1) Classify automorphism groups of smooth plane curves.
(2) Give a sharp upper bound for the order of automorphism groups of such curves.
(3) Determine smooth plane curves with the group of automorphisms of large order.

We shall give a complete answer for each problem in Main Theorem 1, Main
Theorem 2 and Main Theorem 3, respectively. These are the main results of thisarticle.
Roughly speaking, Main Theorem 1 gives a classification of automorphism groupsof smooth plane curves as follows: They are divided into five kinds. Groups of thefirst kind are cyclic and have a fixed point on the curves. The second kind consistsof the central extension of finite subgroups of M\"obius group $PGL(2, \mathbb{C})=Aut(\mathbb{P}^{1})$by cyclic groups. Groups of the third (resp. the fourth) kind are subgroups of thefull automorphism group of Fermat (resp. Klein) curves. The fifth kind consists ofprimitive subgroups of $PGL(3, \mathbb{C})$ .
$*$Automorphism groups of smooth plane curves, $arXiv:$ math. $AG/1306.5842$ [H1]

9For the order of automorphism groups of smooth plane curves, it is natural toexpect that there exists a stronger upper bound than Hurwitz’s one. We obtain sucha bound in Main Theorem 2, which is a by-product of Main Theorem 1. We showthat the order of the full automorphism group of a smooth plane curve $d\neq 4$ , 6 is at
most 6 , which is attained by Fermat curves only. Moreover, a smooth plane curvewith the full automorphism group of maximum order is unique up to projective
equivalence for each degree.
Main Theorem 3, which is another by-product of Main Theorem 1, is a classification of smooth plane curves whose full automorphism group has large order interms of defining equations.
In the last section we investigate automorphisms of Riemann surfaces induced
by projective transformations of their plane models and we propose a theorem on
the order of linear automorphism groups of irreducible plane curves, which is a
generalization of Main Theorem 2.
$d^{2}$

2

Main results

In this article denotes a smooth plane curve of degree $d\geq 4$ and is a subgroupof $Aut(C)$ unless otherwise mentioned. First note that has a unique embedding
up to projective equivalence, which implies thatis naturally considered
into
as a subgroup of $PGL(3, \mathbb{C})=Aut(\mathbb{P}^{2})$ .
be Fermat curve $X^{d}+Y^{d}+Z^{d}=0$ of degree . In this article we denoteLetbya smooth plane curve defined by the equation $XY^{d-1}+YZ^{d-1}+ZX^{d-1}=0,$
which is called Klein curve of degree
For a non-zero monomial
we define its exponent as $\max\{i, j, k\}$ . For ahomogeneous polynomial , the core of $F$ is defined as the sum of all terms of $F$
with the greatest exponent. A term of $F$ is said to be low if it does not belong to
the core of $F.$Let
be a smooth plane curve of degree at least four with a defining polynomial
ifis defined by a
. Then the pair $(C, G)$ is said to be a descendant ofunder ahomogeneous polynomial whose core coincides withandacts on
if $(C, Aut(C)$ ) issuitable coordinate system. We simply call a descendant ofa descendant ofthat consists of all elements
We denote by $PBD(2,1)$ the subgroup ofrepresentable by acomplex matrix of the form
$G$

$C$

$C$

$G$

$\mathbb{P}^{2}$$d$$F_{d}$$K_{d}$$d.$$cX^{i}Y^{j}Z^{k}$$F$$C_{0}$$C_{0}$$F_{0}$$F_{0}$

$C$

$C$

$G$

$C_{0}$$C_{0}$$C_{0}.$$PGL(3, \mathbb{C})$$A$$3\cross 3$$(_{00}^{A’}$$\alpha 00)$$(A’ is a$ regular $2\cross 2$ matrix,$\alpha\in \mathbb{C}^{*})$.

There exists a natural group homomorphism : $PBD(2,1)arrow PGL(2, \mathbb{C})([A]\mapsto$$[A’])$ , where $[M]$ denotes the equivalence class of a regular matrix $M$ . Using theseconcepts we state our first main result as follows:$\rho$

10
Main Theorem 1. Let be a smooth plane curve of degree $d\geq 4,$ a subgroupof $Aut(C)$ . Then one of the following holds:(a-i)
fixes a point on and is a cyclic group whose order is at most $d(d-1)$ .
is projectively equivalent toFurthermore, if $d\geq 5$ and $|G|=d(d-1)$ , then
$YZ^{d-1}+X^{d}+Y^{d}=0$$G=Aut(C)$.
andthe curve(a-ii)
fixes a point not lying on and there exists a commutative diagram
$G$

$C$

$G$

$C$

$G$

$C$

$C$

$G$

$1arrow \mathbb{C}^{*}arrow PBD(2,1)arrow^{\rho}PGL(2, \mathbb{C})arrow 1$$\uparrow$$\uparrow$$\uparrow$$1arrow N arrow G arrow G’ arrow 1$

whose rows are exact sequences, where $N$ is a cyclic group whose order is afactor of and $G’$ is conjugate to a cyclic group , a dihedral groupor the icosahedral groupthe tetrahedral group , the octahedral group
Furthermore, $m\leq d-1$ and if $G’\simeq D_{2m}$ then $m|d-2$ or $N$ is trivial. $In$
particular $|G| \leq\max\{2d(d-2), 60d\}.$(b-i) $(C, G)$ is a descendant of Fermat curve $F_{d}:X^{d}+Y^{d}+Z^{d}=0$ . In this case$d$$D_{2m},$$\mathbb{Z}_{m}$$A_{5}.$$S_{4}$$A_{4}$$|G|\leq 6d^{2}.$(b-ii) $(C, G)$ is a descendant of Klein curve $K_{d}:XY^{d-1}+YZ^{d-1}+ZX^{d-1}=0$ . In
this case $|G|\leq 3(d^{2}-3d+3)$ if $d\geq 5$ . On the other hand, $|G|\leq 168$ if $d=4.$(c)
is conjugate to a finite primitive subgroup of $PGL(3, \mathbb{C})$ , namely, the icosa, the, the altermating group, the Klein grouphedral group$G$$PSL(2, \mathbb{F}_{7})$$A_{5}$

Hessian groupular $|G|\leq 360.$$H_{216}$$A_{6}$of order 216 or its subgroup of order 36 or 72. In partic-

We make some remarks on this theorem.

fixes a point, say $P$ . In fact,
Remark 2.1. (1) In the cases (a-i) and (a-ii),
also fixes a line not passing through $P$ (cf. Theorem 3.5).
from to a(2) A point inis called a Galois point for if the projection$P$line with center is a Galois covering. A Galois point for is said to be inner(resp. outer) if $P\in C$ (resp. $P\not\in C$ ). In the case (a-ii), if $|N|=d$ then the fixed
point of is an outer Galois point for $C.$
(3) The Klein group in the case (c) is the full automorphism group of Klein quarticis that of Wiman sextic (see Main Theorem 2). The
and the alternating group
Hessian group of order 216 is generated by the four elements $h_{i}(i=1,2,3,4)$
represented by the following matrices (cf. [B1]):$G$$G\subset PGL(3, \mathbb{C})$
$P$

$C$

$\mathbb{P}^{2}$$\pi_{P}$

$C$

$C$

$P$

$G$

$A_{6}$$(\begin{array}{lll}0 1 00 0 11 0 0\end{array}),$$(\begin{array}{lll}1 0 00 \omega 00 0 \omega^{2}\end{array}),$$(\begin{array}{lll}1 1 11 \omega \omega^{2}1 \omega^{2} \omega\end{array})$

and$(\begin{array}{lll}1 0 00 \omega 00 0 \omega\end{array}),$where is a primitive third root of unity. This group is the full automorphism groupof a smooth plane sextic (see Remark 2.2 (2)). Its primitive subgroups of order 36, whereandand 72 are respectively equal to$\omega$$\langle h_{1},$$h_{2},$$h_{3}\rangle$$\langle h_{1},$$h_{2},$$h_{3},$$u\rangle$$u=h_{1}^{-1}h_{4}^{2}h_{1}.$11
As a corollary of Main Theorem 1, we obtain a sharp upper bound for the orderof automorphism groups of smooth plane curves and classify the extremal cases.
Main Theorem 2.
be a smooth plane curve of degree $d\geq 4$ .
Let$|Aut(C)|\leq 6d^{2}$ except the following cases:
$C$

Then

(i) $d=4$ and $C$ is projectively equivalent to Klein quartic $XY^{3}+YZ^{3}+ZX^{3}=0.$
In this case $Aut(C)$ is the Klein group, which of order 168.
$d=6$
(ii)
and is projectively equivalent to the sextic$PSL(2, \mathbb{F}_{7})$$i\mathcal{S}$$C$$10X^{3}Y^{3}+9X^{5}Z+9Y^{5}Z-45X^{2}Y^{2}Z^{2}-135XYZ^{4}+27Z^{6}=0.$

In this case Aut(C) is equal to$A_{6}$, a simple group of order 360.

Furthermore, for any $d\neq 6$ , the equality $|Aut(C)|=6d^{2}$ holds if and only if $C$
is projectively equivalent to Fermat curve $F_{d}:X^{d}+Y^{d}+Z^{d}=0$ , in which case
$Aut(C)$ is a emidirect product of. In particular, for each $d\geq 4,$
acting on
there exists a unique smooth plane curve with the full group of automorphisms of
maximum order up to projective equivalence,$S_{3}$$\mathcal{S}$$\mathbb{Z}_{d}^{2}$

Remark 2.2. (1) It is classically known that Klein quartic has the Klein groupas its group of automorphisms. For the sextic in the above theorem,
Wiman [W] proved that its group of automorphisms is isomorphic to . In [DIK]
Doi, Idei and Kaneta called this curve Wiman sextic and showed that it is the onlysmooth plane sextic whose full automorphism group has the maximum order 360.
We shall give a simpler proof on the uniqueness of Klein quartic (resp. Wiman sextic)
as a smooth plane curve of degree four (resp. six) with the group of automorphismsof maximum order (see the proof of Proposition 5.1).
(2) When $d=6$ , the smooth plane sextic defined by the equation$PSL(2, \mathbb{F}_{7})$$A_{6}$$X^{6}+Y^{6}+Z^{6}-10(X^{3}Y^{3}+Y^{3}Z^{3}+Z^{3}X^{3})=0$

satisfies $|Aut(C)|=216=6^{3}$ . In this case Aut(C) is equal to the Hessian group oforder 216, therefore this curve is not a descendant of Fermat curve.
(3) In fact, a more general theorem holds true for linear automorphism groups ofirreducible plane curves (see Theorem 6.1).

As another by-product of Main Theorem 1, we also give a stronger upper boundfor the order of automorphism groups of smooth plane curves and classify the exceptional cases when $d\geq 60$ :Main Theorem 3.
$|Aut(C)|\leq d^{2}$ unless

Let $C$ be a smooth plane curve of degree $d\geq 60$ . Then
$C$ is:projectively equivalent to one of the following$curve\mathcal{S}$(i) Fermat curve $F_{d}:X^{d}+Y^{d}+Z^{d}=0(|Aut(F_{d})|=6d^{2})$ .
(ii) Klein curve $K_{d}:XY^{d-1}+YZ^{d-1}+ZX^{d-1}=0(|Aut(K_{d})|=3(d^{2}-3d+3$12(iii) The smooth plane curve defined by the equation$Z^{d}+XY(X^{d-2}+Y^{d-2})=0,$

in which case $|Aut(C)|=2d(d-2)$ .
(iv) The descendant of Fermat curve defined by the equation$X^{3m}+Y^{3m}+Z^{3m}-3\lambda X^{m}Y^{m}Z^{m}=0,$

where $d=3m$ and$\lambda$

is a non-zero number with$\lambda^{3}\neq 1$. In this $case|Aut(C)|=$$2d^{2}.$(v) The descendant of Fermat curve defined by the equation$X^{2m}+Y^{2m}+Z^{2m}+\lambda(X^{m}Y^{m}+Y^{m}Z^{m}+Z^{m}X^{m})=0,$

where $d=2m$ and$\lambda\neq 0,$$-1,$ $\pm 2$ .

In this case $|Aut(C)|=6m^{2}=(3/2)d^{2}.$Preliminary results

3

Notation and Conventions
In this article we say that a group$G$

acting on a set$\Omega$fixes a subset$S\subset\Omega$

if$GS=S.$

We identify a regular matrix with the projective transformation represented by
it if no confusion occurs. When a planar projective transformation fixes a smoothplane curve, it is also identified with the automorphism obtained by its restriction
to the curve.
a planar projective transformation defined by (X : :We denote by$Z)\mapsto(H_{1}(X, Y, Z) : H_{2}(X, Y, Z) : H_{3}(X, Y, Z))$ for homogeneous linear polynomials
andA projective transformation of finite order is classically called a homology if it isdefined by $[X, Y, \zeta Z]$ under a suitable coordinate system, where is a root of unity.
A nontrivial homology fixes a unique line pointwise and a unique point not lying
the line.
A triangle means a set of three non-concurrent lines. Each line is called an edgeof the triangle.
The line defined by the equation $X=0$ resp. $Y=0, Z=0)$ will be denoted(resp.
and ) the point $(1: 0:0)$. We also denote byresp.
by(resp. $(0:1:0)$ and $(0:0:1$) a cyclic group of order $m$(resp.
For a positive integer $m$ , we denote by).
(resp. the direct product of copies of$Y$$[H_{1}, H_{2}, H_{3}]$$H_{1},$$H_{2}$$H_{3}.$$\zeta$$($$L_{1}$$($$P_{1}$$L_{2}, L_{3})$$\mathbb{Z}_{m}$$r$$P_{2}$$P_{3}$$\mathbb{Z}_{m}^{r}$$\mathbb{Z}_{m}$In this section $C$ denotes a smooth irreducible projective curve of genus $g\geq 2$
defined over the field of complex numbers. Then the full group of its automorphisms
is a finite group and we have a famous upper bound of its order, which is known as
Hurwitz bound:13
Theorem 3.1. (Hurwitz) Let
More precisely,$\frac{|G|}{g-1}=84$$G$

be a subgroup of $Aut(C)$ . Then $|G|\leq 84(g-1)$ .

, 48, 40, 36, 30 oror$\frac{132}{5}$$\frac{|G|}{g-1}\leq 24.$

Oikawa [O] and Arakawa [A] gave possibly stronger upper bounds under theassumption that fixes finite subsets of :Theorem 3.2. $([O,$ Theorem $1], [A,$ Theorem $3])$ Let be a subgroup of $Aut(C)$ .
$C$

$G$

$G$

(1) (Oikawa’s inequality) If$G$fixes a finite subset

$S$

of

$C$

with $|S|=k\geq 1$ , then$|G|\leq 12(g-1)+6k.$(2) (Arakawa’s inequality) If fixes three finite disjoint subsets $S_{i}(i=1,2,3)$ of$C$with $|S_{i}|=k_{i}\geq 1$ , then $|G|\leq 2(g-1)+k_{1}+k_{2}+k_{3}.$$G$As an application of the former inequality, we can describe the structure of thefull automorphism groups of Fermat curves and Klein curves, though the former wasdetermined in a diﬀerent way and the latter is also probably known.
Proposition 3.3. Let be an integer with $d\geq 4$ . Then the full group of autoof Fermat curve is generated by the four transformations $[\zeta X, Y, Z],$$[X, \zeta Y, Z],$ $[Y, Z, X]$ and $[X, Z, Y]$ , whereis a primitive d-th root of unity. It isisomorphic to a semidirect product of
acting on , in other words, there exists
asplit short exact sequence of groups$d$$morphi\mathcal{S}m\mathcal{S}$$F_{d}$$\zeta$$S_{3}$$\mathbb{Z}_{d}^{2}$$1arrow \mathbb{Z}_{d}^{2}arrow Aut(F_{d})arrow S_{3}arrow 1.$

In particular $|Aut(F_{d})|=6d^{2}.$Proposition 3.4. If $d\geq 5$ then the full group of automorphisms of Klein curve$K_{d}:XY^{d-1}+YZ^{d-1}+ZX^{d-1}=0$

is generated by the two transformations $[\xi^{-(d-2)}X, \xi Y, Z]$ and $[Y, Z, X]$ , where isa primitive $(d^{2}-3d+3)-rd$ root of unity. It is isomorphic to a semidirect product, in other words, there exists a split short exact sequence ofof acting on$\xi$$\mathbb{Z}_{d^{2}-3d+3}$$\mathbb{Z}_{3}$$group_{\mathcal{S}}$$1arrow \mathbb{Z}_{d^{2}-3d+3}arrow Aut(K_{d})arrow \mathbb{Z}_{3}arrow 1.$

In particular $|Aut(K_{d})|=3(d^{2}-3d+3)$ . On the other hand, $|Aut(K_{4})|=168.$
In the end of this section, we refer to a theorem on finite groups of planar
projective transformations. It is a basic tool to prove Main Theorem 1.
Theorem 3.5. $([M,$ Section $1- 10], [DI,$ Theorem $4.8])$ Let be a finite subgroup of$PGL(3, \mathbb{C})$ .
Then one of the following holds:(a)fixes a line and a point not lying on the line;(b)fixes a triangle; $or$(c)
, the Klein groupis primitive and conjugate to the icosahedral group(of order 168), the alternating group , the Hessian groupoforder 216 or its subgroup of order 36 or 72.
$G$

$G$

$G$

$G$

$PSL(2, \mathbb{F}_{7})$$A_{5}$$A_{6}$$H_{216}$

14

4An outline of our proof of Main Theorem 1
denotes a smooth plane curve of degree $d\geq 4$ defined
In the following sections
by a homogeneous polynomial $F$ and let be a subgroup of Aut (C) , which is also
considered as a subgroup of $PGL(3, \mathbb{C})$ . We identify an element a of with thecorresponding planar projective transformation, which is also denoted byThis section is wholly devoted to give a sketch of our proof of Main Theorem 1.
From Theorem 3.5 there are three cases:
$C$

$G$

$G$

$\sigma.$

(A)
(B)
(C)

$G$

$G$

$G$

fixes a line and a point not lying on the line.
fixes a triangle and there exists neither a line nor a point fixed by $G.$
is primitive and conjugate to a group described in Theorem 3.5.

Note that the last case leads us to the statement (c) in Main Theorem 1. We argue
the other cases one by one.

Case (A): fixes a line and a point not lying on
We prove that the statement (a-i) (resp. (a-ii)) in Main Theorem 1 holds if $P\in C$). For the sake of simplicity, we omit to estimate the order of $G.$
(resp.
If passes through , then is cyclic. In this case (a-i) in Main Theorem 1
holds.
Assume that does not pass through . We may assume that is defined by
is a subgroup of PBD $(2,1)$ . Hence there exists$Z=0$ and $P=$ $(0 : 0:1)$ . Thena short exact sequence of groups
$L.$

$P$

$L$

$G$

$P\not\in C$

$G$

$P$

$C$

$L$

$P$

$C$

$G$

$1arrow Narrow Garrow^{\rho}G’arrow 1,$

where$\rho$: $PBD(2,1)arrow PGL(2, \mathbb{C})$ is a natural homomorphism, $N=Ker\rho$ and$G’={\rm Im}\rho.$

We show that $N$ is a cyclic group. For each element of $N$ , there exists a
unique diagonal matrix of the form diag $(1, 1, \zeta)$ that represents . Hence we have an, which implies that $N$ is isomorphic
injective homomorphism :to a finite subgroup of . It follows that $N$ is cyclic.
, is
On the other hand, it is well known that $G’$ , a finite subgroup of. Thus (a-ii) in Main Theorem 1 holds.
orisomorphic to$\eta$$\eta$$\varphi$$Narrow \mathbb{C}^{*}(\eta\mapsto\zeta)$$\mathbb{C}^{*}$$PGL(2, \mathbb{C})$$\mathbb{Z}_{m},$$D_{2_{7}n},$$A_{4},$$S_{4}$$A_{5}$

Next we show the statement (b-i) or (b-ii) in Main Theorem 1 holds in Case (B).

Case (B):$G$

fixes a triangle$\triangle$

and there exists neither a line nor a point fixed by$G.$: $Y=0$ and: $X=0,$consists of three lines
We may assume that
: $Z=0$ . Let be the set of vertices of , i.e., $V=\{P_{1}, P_{2}, P_{3}\}$ . Note thatfixes a line or a point, which conflicts
transitively. Indeed, otherwise
acts on$C$
and are disjoint or $C$ contains $V.$
eitherwith our assumption. It follows that
$L_{1}$$\triangle$$V$$L_{3}$$V$$L_{2}$$G$$\triangle$

$G$

$V$

15
We note a trivial but useful observation:
Observation. Each element of gives a permutation of the set $\{X, Y, Z\}$ of thecoordinate functions up to constants.
$G$the tangent line to at $P_{i}(i=1,2,3)$ . Note
If contains , we denote bythat these lines are distinct and not concurrent by our assumption. Furthermore,fixes the set
and acts on it transitively. Thus Case (B) is divided into
three subcases:
$C$

$V$

$G$

$C$

$T_{i}$$\{T_{1}, T_{2}, T_{3}\}$(B-1) $C$ and $V$ are disjoint.
(B-2) $C$ contains and each of $T_{i}’ s(i=1,2,3)$ is an edge of(B-3) $C$ contains and none of $T_{i}’ s(i=1,2,3)$ is an edge of
$V$

$V$

$\triangle.$$\triangle.$

and are disjoint.
Subcase (B-1):$(C,G)$ is a descendant of Fermat curve: $X^{d}+Y^{d}+Z^{d}=0$ in
We show that
this subcase. By our assumption the defining polynomial $F$ of is of the form
$C$

$V$

$F_{d}$$C$$F=aX^{d}+bY^{d}+cZ^{d}+$ (low terms)$(a, b, c\neq 0)$.

Furthermore we may assume that $a=b=c=1$ after a suitable coordinate change ifnecessary. Then the core of $F$ is $X^{d}+Y^{d}+Z^{d}$ , which is fixed by up to a constantfrom the above observation. It follows that also acts on Fermat curve , in other. Thus we conclude that $(C, G)$ is a descendant
words,
is a subgroup of Aut
of
$G$

$G$

$G$

$F_{d}$$(F_{d})$$F_{d}.$

Subcase (B-2):
contains and each $T_{i}(i=1,2,3)$ is an edge of: $XY^{d-1}+YZ^{d-1}+$
We show that $(C, G)$ is a descendant of Klein curve$ZX^{d-1}=0$ in this subcase. Without loss of generality we may assume that $T_{1}=L_{3},$$T_{2}=L_{1}$ and $T_{3}=L_{2}$ . Then the defining polynomial $F$ of
is of the form
$V$

$C$

$\triangle.$$K_{d}$$C$$F=aXY^{d-1}+bYZ^{d-1}+cZX^{d-1}+$ (low terms)$(a, b, c\neq 0)$.

Again we may assume that $a=b=c=1$ after a suitable coordinate change ifnecessary. Then the core of is $XY^{d-1}+YZ^{d-1}+ZX^{d-1}$ , which is fixed by up
also acts on Klein curve
to a constant from the above observation. Hence. Thus $(C, G)$ is a descendant ofthat is to say, is a subgroup of Aut
$G$

$F$

$G$

$G$

$K_{d},$$(K_{d})$$K_{d}.$

Subcase (B-3):
contains and no $T_{i}(i=1,2,3)$ is an edge ofWe exclude this subcase. Let $V’=\{P_{1}’, P_{2}’, P_{3}’\}$ be the set of the intersection
andwith $\{i, j, k\}=$points ofand , where is the intersection pointareand, 2, 3 . These points are pairwise distinct. Indeed, otherwise
with
ourconcurrent andfixes the intersection point of them, which conflicts
assumption. Thus
andconstitute a triangle , which is fixed by and$V’$ is the set of its vertices. Furthermore,
and $V’$ are disjoint by our assumption.
$V$

$C$

$T_{1},$$\{1$$T_{2}$$T_{3}$$\triangle.$$ofT_{j}$$P_{i}’$$T_{k}$$T_{1},$$\}$$T_{2}$$T_{3}$$G$$T_{1},$$T_{2}$$\triangle’$$T_{3}$

$V$

$G$

16
for some
can be written in the form
Every element$X_{3}=Z.$$\{i,j,k\}=\{1$$X_{2}=Y$$X_{1}=X,$
2,
and
,
3
where
,
constants
andHence we have a natural homomorphism : $Garrow S_{3}$ defined by$\sigma\in G$$\alpha,$$\sigma=[\alpha X_{i}, \beta X_{j}, \gamma X_{k}]$$\},$$\beta$$\gamma$$\rho$$\rho(\sigma)=(\begin{array}{lll}1 2 3i j k\end{array}).$or , since there exists neither a line nor a point fixedThen
is isomorphic to$Ker\rho$
is trivial. Indeed, any element of $Ker\rho$ can be written in theby . Furthermore,. Hence it fixes pointwise, which implies that it fixes
form$V’$ also pointwise. It is easy to show that such a planar projective transformation isortrivial. Thus
If is isomorphic to , then fixes a line, which contradicts our assumption.
Thus
is isomorphic to . Hence is generated by $\eta=[Y, Z, X]$ and anotherafter a suitable coordinate change if necessary.
element of order two with$\tau=[\omegaY,\omega^{-1}X,Z](\omega^{3}=1)$ . Bothand fixes the samemay
Then weassume that
also fixes this point, which is a contradiction. It. Therefore
point
follows that this subcase is excluded.
Thus we complete the proof of Main Theorem 1 thoroughly.
$S_{3}$$\mathbb{Z}_{3}$${\rm Im}\rho$

$G$

$V$

$[\alpha X, \beta Y, Z](\alpha, \beta\neq 0)$$G\simeq{\rm Im}\rho\simeq \mathbb{Z}_{3}$$G$$S_{3}.$$G$$\mathbb{Z}_{3}$

$G$

$G$

$S_{3}$$\tau\eta\tau=\eta^{-1}$$\tau$$\tau$$\eta$$(1 : \omega^{2} :5\omega)$$G$

An outline of our proof of Main Theorem 2 and3

In this section we describe an outline of our proof of Main Theorem 2 and Main
Theorem 3. First we consider primitive groups acting on smooth plane curves.
Proposition 5.1. Let $C$ be a smooth plane curve of degree $d\geq 4,$
of $Aut(C)$ . If is primitive, then $|G|\leq 6d^{2}$ except the following

$G$

$G$

a finite subgroup$ca\mathcal{S}es$:(i) $d=4$ and is projectively equivalent to Klein quartic $XY^{3}+YZ^{3}+ZX^{3}=0$
and $G\simeq Aut(K_{4})\simeq PSL(2,\mathbb{F}_{7})$ .
, which is defined(ii) $d=6$ andis projectively equivalent to Wiman sextic
$C$

$C$

$W_{6}$$by$$10X^{3}Y^{3}+9ZX^{5}+9Y^{5}Z-45X^{2}Y^{2}Z^{2}-135XYZ^{4}+27Z^{6}=0$

and $G\simeq Aut(W_{6})\simeq A_{6}.$Proof. First note that Aut (C) is also primitive, which implies that $|G|\leq|Aut(C)|\leq$by Theorem 3.5. Hence $|G|<6d^{2}$ if $d\geq 8.$
Assume that $d\leq 7$ . If $d=5$ or 7, then we have the inequality $|G|<6d^{2}$ except
for $(d, |G|)=(5,168)$ , $(5,216)$ , $(5,360)$ or $(7,360)$ again by Theorem 3.5. It is easy
to check by Theorem 3.1 that these four exceptional cases do not occur.
Assume that $d=4$ . We then have the inequality $|G|\leq 168$ by Hurwitz’s theorem.
If $|G|<168$ , then $|G|\leq 72<6d^{2}$ holds by Theorem 3.5. Suppose that $|G|=168$
$360$

17
. Then is conjugate toandis not projectively equivalent to Klein quartic
acts on both and. Hence we may assume that
the Klein group$C\capK_{4}$
subset of of order at
non-empty.
This is a
In particular
is fixed bymost $4^{2}=16$ by virtue of B\’ezout’s theorem. It follows from Oikawa’s inequalitythat $168=|G|\leq 12\cdot 2+6\cdot 16=120$ , a contradiction.
Next assume that $d=6$ . If $|G|<360$ , then $|G|\leq 216=6d^{2}$ by Theorem 3.5.
Suppose that $|G|=360$ and is not projectively equivalent to Wiman sextic. Hence we may assume that acts on both andThen is conjugate toB\’ezout’s
It follows fromtheorem again that $C\cap W_{6}$ is a non-empty subset of oforder at most $6^{2}=36$ , which is fixed by . Again applying Oikawa’s inequality wecome to the conclusion that $360=|G|\leq 12\cdot 9+6\cdot 36=324$ , a contradiction.
$C$$K_{4}$$G$$G$$PSL(2, \mathbb{F}_{7})$$C$$K_{4}.$$C$$W_{6}.$

$C$

$G$

$C$

$G$

$W_{6}.$$G$$A_{6}$$C$$G$$\square$

We show Main Theorem 2 by using the above proposition, Theorem 3.5 andOikawa’s inequality.

Proof of Main Theorem 2. We may assume that $Aut(C)$ is not primitive by virtueof Proposition 5.1. Then it follows from Theorem 3.5 that $Aut(C)$ fixes a line or atriangle. First suppose that $Aut(C)$ fixes a line . Then $:=C\cap L$ is a non-empty
set of order at most , which is also fixed by $Aut(C)$ . Hence Oikawa’s inequality
implies that
$S$

$L$

$d$

$|Aut(C)|\leq 12(9-1)+6|S|\leq 6d(d-3)+6d=6d(d-2)<6d^{2}$

is a non-empty set
Next suppose that $Aut(C)$ fixes a triangle . Then
$3d$
$Aut(C)$. Thus we have the inequalityof order at most , which is also fixed by$|Aut(C)|\leq 6d^{2}$ by the same argument as above.
Finally assume that $|Aut(C)|=6d^{2}$ and $d\neq 6$ . From Proposition 5.1 andthe above argument $Aut(C)$ fixes a triangle and does not fix a line. Then $C$ is aby virtue of Main Theorem 1. Comparing the orderdescendant of Fermat curveof two groups we know that $G=Aut(F_{d})$ . Let $cX^{i}Y^{j}Z^{k}(c\neq 0, i+j+k=d)$
be a term of $F$ . Note that $[\zeta X, Y, Z]$ and $[X, \zeta Y, Z](\zeta$ is a primitive d-th root ofunity), which are elements of , preserve $F$ . Hence they also preserve the monomialholds, which implies that $(i, j, k)=(d, 0,0)$ , $(0, d, 0)$ or. Then$(0,0, d)$ . It follows that $F=X^{d}+Y^{d}+Z^{d}.$$C\cap\triangle$$\triangle$$F_{d}$$G$$cX^{i}Y^{j}Z^{k}$$\zeta^{i}=\zeta^{j}=1$$\square$In the rest of this section we give a sketch of our proof of Main Theorem 3. First
we describe the full automorphism groups of curves in three exceptional cases (iii),(iv) and (v) in the theorem without proof.
Proposition 5.2. Assume that $d\geq 4$ andthe equation $Z^{d}+XY(X^{d-2}+Y^{d-2})=0.$$C$

is the smooth plane curve defined by(i) If $d\neq 4$ , 6, then $Aut(C)$ is a central extension of $D_{2(d-2)}$ by . In particular$|Aut(C)|=2d(d-2)$ .
by . In particular $|Aut(C)|=$(ii) If $d=6$ , Aut(C) is a central extension of$\mathbb{Z}_{d}$$S_{4}$$144.$$\mathbb{Z}_{6}$18(iii) If $d=4$ , then$C$

is isomorphic to Fermat quartic$\mathbb{Z}_{4^{\lambda}}^{2}S_{3}(|Aut(C)|=96)$$F_{4}$. In particular $Aut(C)\simeq$.

Proposition 5.3. For a positive integer $d=3m\geq 6$ , letcurve defined by$F_{d}’$

be the smooth plane$X^{3m}+Y^{3m}+Z^{3m}-3\lambda X^{m}Y^{m}Z^{m}=0,$. It is a descendant of Fermat curve
where is a non-zero number withand $Aut(F_{d}’)$ is generated by the five transformations $[\zeta^{3}X, Y, Z],$ $[X, \zeta^{3}Y, Z],$$[Y, Z, X]$ and $[Y, X, Z]$ , whereis a primitive d-th root of unity. $In$
this case $|Aut(C)|=2d^{2}.$$\lambda^{3}\neq 1$$\lambda$$F_{d}$$[\zeta X, \zeta^{-1}Y, Z],$$\zeta$Proposition 5.4. For a positive even integer $d=2m\geq 8$ , letplane curve defined by$F_{d}"$

be the smooth$X^{2m}+Y^{2m}+Z^{2m}+\lambda(X^{m}Y^{m}+Y^{m}Z^{m}+Z^{m}X^{m})=0,$$-1,$. It is a descendant ofFermat curve and $Aut(F_{d}")$ is generated
where$[Y, Z, X]$ and $[Y, X, Z]$ , whereby the four transformations
is a primitive d-th root of unity. It is isomorphic to a semidirect product of. there exists a split short exact sequence of groups, in otheracting on$\pm 2$$\lambda\neq 0,$$F_{d}$$[\zeta^{2}X, Y, Z],$$[X, \zeta^{2}Y, Z],$

$S_{3}$$\zeta$$word_{\mathcal{S}},$$\mathbb{Z}_{m}^{2}$$1arrow \mathbb{Z}_{m}^{2}arrow Aut(F_{d}")arrow S_{3}arrow 1.$

In particular $|Aut(F_{d}’)|=6m^{2}=(3/2)d^{2}.$Furthermore we obtain a characterization of these descendants of Fermat curve.

are the only descendants of Fermat curveandLemma 5.5. Two curves
whose group of automorphisms has order greater than up to projective equivalence,
exceptitself.

$F_{d}$$F_{d}"$$F_{d}’$$d^{2}$$F_{d}$We also need the uniqueness of smooth plane curves of degree
automorphism group is of order $3(d^{2}-3d+3)$ .

$d$whose fulla subgroup of
Proposition 5.6. Let be a smooth plane curve of degree $d\geq 5,$$Aut(C)$ .
is projectively equivalent tothat $|G|=3(d^{2}-3d+3)$ . Then
and $G=Aut(K_{d})$ .
Klein curve
$G$

$C$

$C$

$As\mathcal{S}ume$$K_{d}$

By using these facts we can prove Main Theorem 3.

be a smooth plane curve of degree $d\geq 60,$ $Fa$
defining homogeneous polynomial of . Assume that a subgroup of $Aut(C)$ is oforder greater thanSince $d\geq 60$ , we have the inequalities $|G|>60d$ and $|G|>360$ . Then there are
only three possibilities from Main Theorem 1:

Proof of Main Theorem 3. Let

$C$

$C$

$d^{2}.$$G$19
fixes a point
not lying on and is isomorphic to a central extension$D_{2(d-2)}$
byof(ii) $(C, G)$ is a descendant of Fermat curve: $X^{d}+Y^{d}+Z^{d}=0.$(iii) $(C, G)$ is a descendant of Klein curve $K_{d}:XY^{d-1}+YZ^{d-1}+ZX^{d-1}=0.$
(i)

$G$

$P$

$C$

$G$

$\mathbb{Z}_{d}.$$F_{d}$

Case (i) In this casealso fixes a line not containing $P$ . We may assume that
$P=$ $(0 : 0 : 1)$ andis defined by $Z=0$ . Then is generated by the three elements$\eta=[X, Y, \zeta Z],$ $\sigma=[X, \omega Y, \omega’Z]$ and $\tau=[\gamma Y, \gamma X, Z]$ , where
and are rootsof unity and the order of (resp. ) is (resp. $d-2$ ). Since preserves $F$ up toa constant, $F$ is written as $F=Z^{d}+\hat{F}(X, Y)$ , where $\hat{F}(X, Y)$ is a homogeneouspolynomial of $X$ and $Y$ without multiple factors. Furthermore, we can show thatintersects transversally at $P_{1}=$ $(1 : 0:0)$ and $P_{2}=(0:1 : O)$ , respectively. It
follows that $\hat{F}(X, Y)$ has a factor of the form $X-cY(c\neq 0)$ . Since preserves$\hat{F}(X, Y)$
up to a constant, we conclude that $\hat{F}(X, Y)=\lambda XY\Pi_{k=0}^{d-3}(X-\omega^{k}cY)=$$\lambda XY(X^{d-2}-c^{d-2}Y^{d-2})(\lambda\in \mathbb{C}^{*})$ . Thus it is clear thatis projectively equivalent
to the curve defined by $Z^{d}+XY(X^{d-2}+Y^{d-2})=0.$
$G$

$L$

$G$

$L$

$\zeta,$$\zeta$$C$$\omega$$d$$\omega’$$\omega,$$\gamma$$\eta$$L$$\sigma$$C$

Case (ii) From Lemma 5.5 we see thatin this case.

$C$

is projectively equivalent to$F_{d},$$F_{d}’$

or$F_{d}"$

Case (iii) In this case is a subgroup of $Aut(K_{d})$ . Since $Aut(K_{d})$ has an odd order$3(d^{2}-3d+3)$ , we see that $G=Aut(K_{d})$ by our assumption that $|G|>d^{2}$ . It follows
from Proposition 5.6 that is projectively equivalent to Klein curve
$G$

$C$

6

$\square$$K_{d}.$

On linear automorphism groups of plane curves

Let $X$ be a Riemann surface of genus at least two, a plane model of $X$ , i.e., an
irreducible plane curve birationally equivalent to $X$ . Then $d:=\deg\Gamma\geq 4.$
We denote by
the group of linear automorphisms of , that is to say, the$PGL(3,\mathbb{C})$subgroup of
consisting of projective transformations preserving . It isnaturally considered as a subgroup of Aut(X). In particular it is a finite group by
our assumption. We close this article by proposing a recent result on the order of(cf. [H2]) without proof:$\Gamma$$Lin(\Gamma)$$\Gamma$$\Gamma$$Lin(\Gamma)$

Theorem 6.1. The order of the groupis at mostunless is projectively
equivalent to Klein quartic $K_{4}(d=4)$ or Wiman sextic $W_{6}(d=6)$ . Furthermore,$|Lin(\Gamma)|=6d^{2}$or
if and only if is projectively equivalent to Fermat curve
Hessian sextic $H_{6}:X^{6}+Y^{6}+Z^{6}-10(X^{3}Y^{3}+Y^{3}Z^{3}+Z^{3}X^{3})=0(d=6)$ .
$Lin(\Gamma)$$\Gamma$$6d^{2}$$\Gamma$$F_{d}$

20

References[A] T. Arakawa, Automorphism groups of compact Riemann surfaces with invariant
subsets, Osaka J. Math. 37, No. 4 (2000), 823-846.
[B1] H. Blichfeldt, Finite Collineation Groups: With an Introduction to the Theory of Groups of Operators and Substitution Groups, Univ. of Chicago Press,
Chicago (1917).

[DI] I. Dolgachev and V. Iskovskikh, Finite subgroups of the plane Cremona group,
Algebra, Arithmetic, and Geometry, Progress in Mathematics Volume 269(2009), 443-548.
[DIK] H. Doi, K. Idei and H. Kaneta, Uniqueness of the most symmetric nonsingular plane sextics. Osaka J. Math. 37 no. 3 (2000), 667-687.
[H1] T. Harui, Automorphism groups of smooth plane curves,(arXiv: math. $AG/1306.5842)$ .

preprint[H2] T. Harui, On linear automorphism groups of irreducible plane curves, preprint.

[M] H. H. Mitchell, Determination of the ordinary and modular ternary linear
groups, Trans. Amer. Math. Soc. 12, no. 2 (1911), 207-242.
[O] K. Oikawa, Notes on conformal mappings of a Riemann surface onto itself,
Kodai Math. Sem. Rep. 8, no. 1 (1956), 23-30.

[W] A. Wiman, Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math.
Ann. 47 no. 4 (1896), 531-556.
Takeshi Harui
Academic Support Center, Kogakuin University
Nakano, Hachioji, Tokyo 192-0015, Japan-mail address: takeshi@cwo.zaq.ne.jp
kt13459@ns.kogakuin.ac.jp$e$