数理解析研究所講究録
第 848 巻 1993 年 74-85

74

The configuration space of 6 points inthe moduli space of cubic surfaces
and the Weyl group of type$P^{2}$,$E_{6}$

関口 次郎 (Jiro, SEKIGUCHI)
電気通信大学 (Univ. Elec. Commun.)

1. Introduction
My first plan of the talk is to explain my study on the hypergeometric system $E(3,6)$
is the symmetricof type $(3,6)$ ([8]). The system in question admits -action, whereof 6group on 6 letters. This follows from that $E(3,6)$ lives in the configurations spacepoints inwhich admits -action as permutations of the 6 points. Recently M. Yoshida-action on the space(Kyushu Univ.) pointed out that theis naturally extended$W(E_{6})$$W(E_{6})$-action, where(cf. [3]). Moreover, hetois the Weyl group of type
$W(E_{6})$told me that B. Hunt studied relations between the-action in question and the$W(E_{6})$ -invariant quintic hypersurface of.
[4],
I felt that it is an interesting exercise for REDUCE user to show
Reading his note
whether his conjecture is true or not. For this reason, I changed the original plan and, namely, to the
I restrict my attention to the study on $W(E_{6})$ -actions on
and onbirational geometry related with the hypergeometric system $E(3,6)$ .
It is better for the readers who are interested in SYMBOLIC COMPUTATION to read
section 6 first.
$\Sigma_{6}$$\Sigma_{6}$$P_{2}^{6}$$P^{2}$$\Sigma_{6}$$P_{2}^{6}$$\Sigma_{6}$$E_{6}$$P^{5}$$P^{5}$$P_{2}^{6}$

2. The hypergeometric function of type $(3,6)$
Though I don’t treat it in this note, I begin this note with introducing the hypergeometric function of type $(3,6)$ :$E(a_{0}, a_{2}, a_{3}, a_{5}, a_{6};x_{1}, x_{2}, y_{1}, y_{2})= \sum_{m_{1}=0}^{\infty}\sum_{m_{2}=0}^{\infty}\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}A_{m_{1},m_{2},n_{1},n_{2}}x_{1}^{m_{1}}x_{2}^{m_{2}}y_{1}^{n_{1}}y_{2}^{n_{2}}$where$A_{m_{1},m_{2},n_{1},n_{2}}=\ovalbox{\tt\small REJECT}(a_{2}, m_{1}+m_{2})(a_{3},rt_{1}+n)(1-a_{5}, m+n)(1m_{1}!m_{2}!n_{1}!n_{2}!^{2}(a_{0},m_{1}+m_{2}^{1}+rt^{1_{1}}+n_{2}^{-})^{a_{6},m_{2}+n_{2})}$.

By definition,
has parameters $a_{j}(j=0,2,3,5,6)$ . Thisfunction is one of 4 variables generalizations of Gaussian hypergeometric function. It$E(a_{0}, a_{2}, a_{3}, a_{5}, a_{6};x_{1}, x_{2}, y_{1}, y_{2})$75
is known that the singularities of the system of differential equations whose solution is: $p_{j}=$is contained in the union of the 14 hypersurfaces$0(1\leq j\leq 14)$ , where$E(a_{0}, a_{2}, a_{3}, a_{5}, a_{6};x_{1}, x_{2}, y_{1}, y_{2})$$p_{1}=x_{1}y_{2}-x_{2}$$p_{4}=y_{2}-1$,$p_{9}=x_{2}-y_{2},$$T_{j}$

Yl $-x_{1}+x_{2}+y_{1}-y_{2}$ ,$p_{5}=x_{2}-1$,$p_{2}=y_{1}-1$$p_{6}=y_{1}-y_{2}$$p_{10}=x_{1}y_{2}-x_{2}y_{1},$,$p_{11}=x_{2},$:$s_{3}$:$C^{4}$$(x_{1},x_{2}, y_{1},y_{2})arrow(y_{1}, y_{2}, x_{1}, x_{2})$,,$p_{8}=x_{1}-y_{1}$,$p_{14}=y_{1}$$p_{13}=y_{2},$$p_{12}=x_{1},$.

by$(x_{1}, x_{2}, y_{1}, y_{2}) arrow(\frac{1}{x_{1}}, \frac{1}{x_{2}}, \frac{y_{1}}{x_{1}}, \frac{y_{2}}{x_{2}})$:$s_{2}$$p_{3}=x_{1}-1$$p_{7}=x_{1}-x_{2}$We define birational transformations $s_{j}(1\leq j\leq 5)$ on$s_{1}$,,,,$(x_{1}, x_{2}, y_{1},y_{2}) arrow(\frac{x_{1}-y_{1}}{1-y_{1}}, \frac{x_{2}-y_{2}}{1-y_{2}}, \frac{y_{1}}{y_{1}-1}, \frac{y_{2}}{y_{2}-1})$$s_{4}$:$(x_{1}, x_{2}, y_{1}, y_{2}) arrow(\frac{1}{x_{1}}, \frac{x_{2}}{x_{1}}, \frac{1}{y_{1}}, \frac{y_{2}}{y_{1}})$,

S5: $(x_{1}, x_{2}, y_{1},y_{2})arrow(x_{2}, x_{1}, y_{2},y_{1})$ .

Then the group generated by $s_{j}(1\leq j\leq 5)$ is identified with$s_{j}^{2}=id$. $(1\leq j\leq 5)$ ,$r$

because$s_{j}s_{k}=s_{k}s_{j}(|j-k|>1)$$s_{j}s_{k}s_{j}=s_{k^{S}j^{S}k}(|j-k|=1)$

Let$\Sigma_{6}$be a birational transformation on$C^{4}$,.

defined by$r:(x_{1}, x_{2}, y_{1}, y_{2})arrow(1/x_{1},1/x_{2},1/y_{1},1/y_{2})$.

, and is isomorphic to the Weyl group $W(E_{6})$Then the group generated byof typewhich will be seen later (cf. [3], [4]).
We define the hypersurface: $p_{15}=0$ , where$\tilde{G}$$s_{1},$$\cdots$$s_{5}$$r$$E_{6}$$T_{15}$$p_{15}=x_{1}y_{2}(1-y_{1})(1-x_{2})-x_{2}y_{1}(1-x_{1})(1-y_{2})$.

,It follows from the definition thatand therefore all the elements ofbiregular outside the union of the hypersurfaces $T_{j}(1\leq j\leq 15)$ .
$s_{1},$$\cdots$$s_{5},$$\tilde{G}$

$r$

are

$T$

3. The Weyl group $W(E_{6})$
Let

Let
are:

$E_{R}$

be a Cartan subalgebra of a compact Lie algebra of type , i.e.
be a coordinate system ofsuch that the roots of type
$E_{6}$$t=(t_{1},t_{2},t_{3}, t_{4}, t_{5},t_{6})$$E_{R}$$\pm(t_{i}\pm t_{j})$,$1\leq i<j\leq 5$$\pm\frac{1}{2}(\delta_{1}t_{1}+\delta_{2}t_{2}+\delta_{3}t_{3}+\delta_{4}t_{4}+\delta_{5}t_{5}+\delta_{6}t_{6})$$E_{R}\simeq R^{6}$.

$E_{6}$

76). Note that compared with the notation in [1], our variables
andin [1]. We now introduce, 5, while our coordinate is denoted$i=1,$:
the following linear forms on(where$\delta_{j}=\pm 1$$\Pi_{j}\delta_{j}=1$$\cdots$$t_{i}=\epsilon_{i},$$t_{6}$$\epsilon_{6}-\epsilon_{7}-\epsilon_{8}$$E_{R}$$h=- \frac{1}{2}(t_{1}+\cdots+t_{6})$$h_{1j}=-t_{j-1}+h_{0}$,$j=2,$$h_{jk}=t_{j-1}-t_{k-1}$,,$\cdots,$$6$$j,$ $k\neq 1$$h_{1jk}=-t_{j-1}-t_{k-1}$,$j,$ $k\neq 1$$h_{jkl}=-t_{j-1}-t_{k-1}-t_{l-1}+h_{0}$,$j,$ $k,$ $l\neq 1$

where$h_{0}= \frac{1}{2}(t_{1}+\cdots+t_{5}-t_{6})$.

)forms a set of positive roots of type . Let (resp.
with respect to the hyperplane $h=0$ (resp. $h_{ij}=0,$ $h_{ijk}=0$ ).
be the reflection onwhich is denoted by $W(E_{6})$ in this note is the group
Then the Weyl group of typegenerated by the 36 reflections defined above.
As a system of simple roots, we take

Then the totality of$h,$$h_{ij},$$E_{6}$$h_{ijk}$$s$$s_{ij},$$s_{ijk}$$E_{R}$$E_{6}$$\alpha_{1}=h_{12},$$\alpha_{2}=h_{123},$$\alpha_{3}=h_{23},$$\alpha_{4}=h_{34},$$\alpha_{6}=h_{56}$$\alpha_{5}=h_{45},$.

Then the Dynkin diagram is:$\alpha_{1}$–$\alpha_{3}$$\alpha_{4}$$\alpha_{6}$$\alpha_{5}$$1$$\alpha_{2}$Letbe the reflection on
the definition,$g_{j}$$g_{1}=s_{12},$$E_{R}$$g_{2}=s_{123},$

with respect to the root$g_{3}=s_{23},$$g_{4}=s_{34},$$(j=1, \cdots , 6)$$\alpha_{j}$$g_{5}=s_{45},$$g_{6}=s_{56}$. Then, from.

Let $E$ be the complexification of
and we extend the action of $W(E_{6})$ on
to$E$that on in a natural manner. Moreover let
be the projective space associated to $E$ .
$E$
Then the $W(E_{6})$ -action on induces a projective linear action of $W(E_{6})$ on.
$E_{R}$$E_{R}$$P^{5}$$P^{5}$

4. The configuration space of 6 points in$P^{2}$We have already defined a birational action of $W(E_{6})$ on
section, we explain that the birational transformations,
the study of the configuration space of 6 points in .
For this purpose, we first introduce the linear space $W$ of$s_{1},$$\cdots$$C^{4}$

in section 2. In thisnaturally arise from$s_{5},$ $r$$P^{2}$$3\cross 6$

matrices :77$W=\{X=(\begin{array}{llllll}x_{11} x_{12} x_{13} x_{14} x_{15} x_{16}x_{21} x_{22} x_{23} x_{24} x_{25} x_{26}x_{31} x_{32} x_{33} x_{34} x_{35} x_{36}\end{array}) ; x_{\dot{\epsilon}j}\in C(1\leq i\leq 3,1\leq j\leq 6)\}$.

Then $W$ admits a left $GL(3, C)$ -action and a right $GL(6, C)$ -action in a natural way. Fora moment, we identify
with the maximal torus of $GL(6, C)$ consisting of diagonal
matrices and consider the action of $GL(3, C)\cross(C^{*})^{6}$ on $W$ instead of that of $GL(3, C)\cross$$(C^{*})^{6}$$GL(6, C)$.

are
For simplicity, we write $X=(X_{1}, X_{2})$ for the matrix $X\in W$ , where bothwith the condition $y_{ij}\neq 0(1\leq$matrices. For any 3
matrix3$i,j\leq 3)$ , we define a 3
matrix$X_{1},$$\cross 3$$\cross 3$$X_{2}$$Y=(y_{ij})_{1\leq i,j\leq 3}$$\cross 3$$\sigma(Y)=(\frac{1}{y_{ij}}I_{1\leq i,j\leq 3}$

following a suggestion of M. Yoshida. Moreover, we put$D(i_{1}, i_{2}, i_{3})=\det(\begin{array}{lll}x_{1i_{1}} x_{1i_{2}} x_{1i_{3}}x_{2i_{1}} x_{2i_{2}} x_{2i_{3}}x_{3i_{1}} x_{3i_{2}} x_{3i_{3}}\end{array})$

for a given matrix $X\in W$ .
Using these notation, we define subsets $W’,$$W_{0}$

of $W$ by$W’=\{X\in W;D(i_{1}, i_{2}, i_{3})\neq 0(1\leq i_{1}<i_{2}<i_{3}\leq 6)\}$,$W_{0}=\{(X_{1}, X_{2})\in W’;(I_{3}, Cof(X_{1}^{-1}X_{2})), (I_{3}, \sigma(X_{1}^{-1}X_{2}))\in W’\}$,

where $Cof(Y)=(\det Y)^{t}Y^{-1}$ is the cofactor matrix of a given square matrix .
It is clear that the action of $GL(3, C)\cross(C^{*})^{6}$ on $W$ naturally induces that on each
. In the sequel, we mainly consider the quotient space ofof $W’,$
unde the action$GL(3,
C)\cross(C^{*})^{6}$, that is,of$Y$$W_{0}$$W_{0}$$W_{Q}=GL(3, C)\backslash W_{0}/(C^{*})^{6}$.

It is clear from the definition that for any element $X\in W_{0}$ , there are $(g, h)\in$$GL(3, C)\cross(C^{*})^{6}$ andsuch that$(x_{1}, x_{2}, y_{1}, y_{2})\in C^{4}$$gXh=(\begin{array}{llllll}1 0 0 1 1 10 1 0 1 x_{1} x_{2}0 0 1 1 y_{1} y_{2}\end{array})$

In particularis uniquely determined for $X\in W_{0}$ . In this sense, $W_{Q}=$is identified with an open subset of . Note thatisof section 2. Then $W_{Q}=C^{4}-T$ .

$(x_{1}, x_{2}, y_{1}, y_{2})$$GL(3, C)\backslash W_{0}/(C^{*})^{6}$the variables.

$(x_{1}, x_{2}, y_{1}, y_{2})$$C^{4}$$(x_{1}, x_{2}, y_{1}, y_{2})$

78
with
Changes of column vectors of $X\in W_{0}$ induce birational transformations on. The action $s_{j}(1\leq j\leq 5)$ introduced in section 2 iscoordinate systemcorresponding to the change of the j-th
nothing but the birational transformation on
admits an involution
column vector and $(j+1)$ -column vector of $X\in W_{0}$ . Moreoverdefined byinduced from the action on$C^{4}$$(x_{1}, x_{2}, y_{1}, y_{2})$$C^{4}$$W_{Q}$$W_{0}$: $(X_{1},X_{2})arrow(I_{3}, \sigma(X_{1}^{-1}X_{2}))$$\tilde{r}$for any $(X_{1},X_{2})\in W_{0}$ . The involution defined in section 2 is equal to that induced
from .
The following theorem which seems known shows a concrete correspondence between$W(E_{6})$ and $tbe$ group
introduced in section 2.
$r$$\tilde{r}$$\tilde{G}$Theorem 4.1. The correspondence$g_{1}arrow s_{1}$,$g_{2}arrow r$,$g_{3}arrow s_{2}$,$g_{4}arrow s_{3}$induces a group isomorphism of $W(E_{6})$ to the group,$\tilde{G}$$g_{5}arrow s_{4}$,$g_{6}arrow s_{5}$.

Remark. In [3], it is stated that there is a $W(E_{6})$ -action on$W_{Q}$. See also [4].

5. $W(E_{6})$ -equivariant maps

We first define rational functions on

$E$

by

$x_{1}(t)= \frac{h_{24}\cdot h_{234}\cdot h_{15}\cdot h_{135}}{h_{14}\cdot h_{134}\cdot h_{25}\cdot h_{235}}$,$x_{2}(t)= \frac{h_{24}\cdot h_{234}\cdot.h_{16}\cdot h_{136}}{h_{14}\cdot h_{134}h_{26}\cdot h_{236}}$,$y_{1}(t)= \frac{h_{34}\cdot h_{234}\cdot h_{15}\cdot h_{125}}{h_{14}\cdot h_{124}\cdot h_{35}\cdot h_{235}}$,$y_{2}(t)= \frac{h_{34}\cdot h_{234}\cdot.h_{16}\cdot.h_{126}}{h_{14}\cdot h_{124}h_{36}h_{236}}$,$\lambda(t)=\frac{h_{34}\cdot h_{345}\cdot h_{26}\cdot.h_{256}}{h_{24}\cdot h_{245}\cdot h_{36}h_{356}}\cdot\frac{h_{13}\cdot h_{136}\cdot.h_{24}\cdot h_{246}}{h_{12}\cdot h_{126}h_{34}\cdot h_{346}}$,,$\mu(t)=\frac{h_{456}\cdot h_{235}.\cdot h_{134}\cdot h_{126}}{h\cdot h_{15}h_{24}\cdot h_{36}}\cdot\frac{h_{16}\cdot h_{136}\cdot.h_{24}\cdot h_{234}}{h_{14}\cdot h_{134}h_{26}\cdot h_{236}}$,$\nu(t)=\frac{h_{25}\cdot.h_{235}\cdot.h_{46}\cdot.h_{346}}{h_{24}h_{234}h_{56}h_{356}}\cdot\frac{h_{15}\cdot.h_{156}\cdot.h_{24}\cdot.h_{246}}{h_{14}h_{146}h_{25}h_{256}}$$\rho(t)=\frac{h_{24}\cdot h_{245}\cdot h_{36}\cdot h_{356}}{h_{23}\cdot h_{235}\cdot h_{46}\cdot h_{456}}\cdot\frac{h_{14}\cdot h_{146}\cdot h_{23}\cdot h_{236}}{h_{13}\cdot h_{\dot{1}36}\cdot h_{24}\cdot h_{246}}$

,

where
denote linear functions on
introduced in section 3. Since all therational functions above are homogeneous of degree zero, they are regarded as functions. Therefore definingon$h,$$h_{ij},$$E$$h_{ijk}$$P^{5}$$F_{1}(t)=(x_{1}(t), x_{2}(t),$ $y_{1}(t),$ $y_{2}(t))$,$F_{2}(t)=(\lambda(t), \mu(t),$ $\nu(t),$ $\rho(t))$,

we obtain two maps
fromto
. The roles ofwill become clear in
Theorem 5.1 which will be given later. To define, I am indebted to [4]. We aregoing to explain the meaning offollowing [4].
$F_{1},$$F_{2}$$P^{5}$$C^{4}$$F_{1},$ $F_{2}$$F_{1},$$x_{j}(t),$$y_{j}(t)$$F_{2}$

79
We begin with defining the cross ratio. Let $\xi_{i}=[\xi_{1i} : \xi_{2i} : \xi_{3i}](1\leq i\leq 5)$ be five
. We denote bypoints of $pz$ and let : $q_{1}u_{1}+q_{2}u_{2}+q_{3}u_{3}=0$ be a generic line in
the intersection of and the line passing through the points and . Then
1 : :we put$P^{2}$$l$$[$$z_{i}$$l$$w_{i}]$(1)$\xi_{1}$$\xi_{i}$$CR( \xi_{2}, \xi_{3},\xi_{4},\xi_{5};\xi_{1})=\frac{(z_{2}-z_{4})(z_{3}-z_{5})}{(z_{2}-z_{5})(z_{3}-z_{4})}$.
which is in fact a cross ratio ofthe
Now we consider a matrix ofform$z_{2},$ $z_{3},$ $z_{4},$ $z_{5}$$X=(\begin{array}{llllll}l 0 0 1 1 10 1 0 1 x_{1} x_{2}0 0 1 1 y_{1} y_{2}\end{array})$

From the matrix $X$ , we define six points $\xi_{i}(i=1, \cdots, 6)$ in$\xi_{1}=[1 :0 :$\xi_{4}=[1;1:1]$

0]$

,

,

$\xi_{2}=[0$Then we can compute
On the other hand, we put$CR(\xi_{i_{2}},\xi_{i_{3}},\xi_{i_{4}}, \xi_{i_{5}} ; \xi_{i_{1}})$(2)

in a usual manner, that is,$P^{2}$$0]$,$\xi_{3}=[0$y_{1}]$,$\xi_{6}=[1:x_{2} :: 1 :$\xi_{5}=[1:x_{1} :

.

:

$0$

(3)

$CR’(i_{2}, i_{3}, i_{4}, i_{5};i_{1})$$]$y_{2}]$

explicitly for various$CR’(i_{2}, i_{3}, i_{4}, i_{5};i_{1})= \frac{h_{i_{2}i_{4}}h_{i_{1}i_{2}i_{4}}h_{i_{3}i_{5}}h_{i_{1}i_{3}i_{5}}}{h_{i_{3}i_{4}}h_{i_{1}i_{3}i_{4}}h_{i_{2}i_{5}}h_{i_{1}i_{2}i_{5}}}$

By definition,:1 ,is a function on$P^{5}$$i_{1},$.
$i_{2},$$i_{3},$$i_{4},$$i_{5}$.

.

. Then from the equation$CR(\xi_{i_{2}},\xi_{i_{3}},\xi_{i_{4}}, \xi_{i_{5}} ; \xi_{i_{1}})=CR’(i_{2}, i_{3}, i_{4}, i_{5};i_{1})$,we obtain various equalities. In particular, by computing the cases$(i_{1}, i_{2}, i_{3}, i_{4}, i_{5})=(3,2,1,4,5),$$(3,2,1,4,6),$ $(2,1,3,4,5),$ $(2,1,3,4,6)$ ,we have the definition of
Let be the birational transformation onwhere$x_{1}(t),$$F_{3}$$x_{2}(t),$$y_{1}(t),$$y_{2}(t)$$C^{4}$

at the beginning of this section.
defined by$F_{3}(x_{1}, x_{2}, y_{1}, y_{2})=(\lambda, \mu, \nu, \rho)$,$\lambda=\frac{x_{2}(x_{1}-1)(y_{1}-y_{2})(y_{2}-1)}{y_{2}(x_{1}-x_{2})(x_{2}-1)(y_{1}-1)}$$\mu=\ovalbox{\tt\small REJECT} x_{1}x_{2}y-x_{1}x_{2}y_{2}-xy_{1}^{-}y_{2}+x_{1}y_{2}+^{1}x_{2}y_{1}^{1}y_{2}-\{(y_{1^{1}}-1)(x_{2}-y_{2})_{1}(y_{2}-1)(x-y)\}x_{2}y_{x_{2}^{2}y_{1}}$,$\nu=-\frac{(x_{1}y_{2}-x_{2}y_{1})(x_{2}-1)(y_{2}-1)}{(x_{1}-x_{2})(x_{2}-y_{2})(y_{1}-y_{2})}$$\rho=\ovalbox{\tt\small REJECT}\{(x_{1}-1)(x_{2}-y_{2})-(x-y_{1}^{2})(x^{1_{2}}-(x_{1}-x_{2})(x_{2_{1}}-y)(y-1_{1})_{)\}(y_{2}-1)x_{2}}$

.

,

80
It is easy to show that

$F_{3}$

is birational, because its inverse is given by$F_{3^{-1}}( \lambda, \mu, \nu, \rho)=(\frac{(\lambda\rho-1)(\lambda\mu\nu\rho-1)}{(\lambda\mu\rho-1)(\lambda\nu\rho-1)},$ $\frac{(\lambda p-1)\mu}{\lambda\mu p-1},$$\frac{(\mu\nu\rho-1)(\rho-1)}{(\mu p-1)(\nu p-1)}\frac{(p-1)\mu}{\mu\rho-1})$.

-space implies that on the
By the map , the action of $W(E_{6})$ on the-space. In fact, we define the following six birational transformations on the-space (cf.[7]):$(x_{1}, x_{2}, y_{1}, y_{2})$$F_{3}$$(\lambda, \mu, \nu, \rho)$$(\lambda, \mu, \nu, \rho)$$\tilde{g}_{1}$:$\{\begin{array}{l}\lambdaarrow\lambda\mu\nu\rho^{2}(1-\lambda)/(\lambda\mu\nu\rho^{2}-l)\muarrow(\lambda\mu\rho-1)(\lambda\mu\nu\rho-1)/(\mu(\lambda\rho-l)(\lambda\nu p-1))\nuarrow(\lambda\nu p-1)(\lambda\mu\nu\rho-1)/(\nu(\lambda\rho-l)(\lambda\mu\rho-l))parrow(\lambda\rho-1)(\lambda\mu\nu\rho^{2}-l)/(\rho(\lambda-1)(\lambda\mu\nu p-1))\end{array}$$\tilde{g}_{2}$$\tilde{g}_{3}$$\tilde{g}_{4}$:$\tilde{g}_{5}$$\tilde{g}_{6}$Let$G_{1}$::$(\lambda, \mu, \nu, p)arrow(\lambda, 1/\mu, \nu, \mu p)$:$(\lambda,\mu, \nu, \rho)arrow(1/\lambda, \mu, \nu, \lambda\rho)$$(\lambda, \mu, \nu, \rho)arrow(\lambda\rho, \mu\rho, \nu\rho, 1/\rho)$:$(\lambda, \mu, \nu, \rho)arrow(\lambda, \mu, 1/\nu, \nu\rho)$$\{\begin{array}{l}\lambdaarrow(\lambda\nu\rho-1)(\lambda\mu\nu\rho-1)/(\lambda(\nu\rho-1)(\mu\nu p-1))\muarrow(\mu\nu\rho-l)(\lambda\mu\nu\rho-1)/(\mu(\nu\rho-l)(\lambda\nu p-1))\nuarrow\lambda\mu\nu p^{2}(l-\nu)/(\lambda\mu\nu p^{2}-l)\rhoarrow(\nu\rho-1)(\lambda\mu\nu\rho^{2}-1)/(p(\nu-l)(\lambda\mu\nu p-1))\end{array}$be the group generated by$\tilde{g}_{j}$$(j=1, \cdots , 6)$$g_{j}arrow\tilde{g}_{j}$$j=1,$. Then the correspondence$\cdots,$$6$.
is an isomorphism between $W(E_{6})$ andto the(resp.
) is regarded as a map from
Needless to say,-space.) Moreover,is regarded as a map from the(resp. the-space.
space to the$G_{1}$$F_{1}$$(\lambda,$$P^{5}$$F_{2}$$p)$$\mu,$ $\nu,$$(x_{1}, x_{2}, y_{1}, y_{2})$$F_{3}$-space$(x_{1}, x_{2}, y_{1}, y_{2})-$$(\lambda, \mu, \nu, \rho)$

Theorem 5.1. The three maps $F_{j}(j=1,2,3)$ are $W(E_{6})$ -equivariant and$F_{3}oF_{1}(g(t))=F_{2}(g(t))$$(\forall t\in P^{5},\forall g\in W(E_{6}))$.

The $W(E_{6})$ -equivariances ofare stated in [4] implicitly.
-space. In [2], A. Cayley defined a 4We now mention the meaning of thedimensional family of cubic surfaces. Modifying his family, we introduce a family of cubic
surfaces ofwith homogeneous coordinate (X : $Y:Z$ : $W$ ) depending on parameters
as follows (cf. [7]):$F_{1},$$F_{2}$$(\lambda, \mu, \nu, p)$$P^{3}$$(\lambda, \mu, \nu, \rho)$$pW[\lambda X^{2}+\mu Y^{2}+\nu Z^{2}+(\rho-1)^{2}(\lambda\mu\nu p-1)^{2}W^{2}$$+(\mu\nu+1)YZ+(\lambda\nu+1)ZX+(\lambda\mu+1)XY$

81$-(p-1)(\lambda\mu\nu\rho-1)W\{(\lambda+1)X+(\mu+1)Y+(\nu+1)Z\}]+XYZ=0$.

The family of cubic surfaces above admits a $W(E_{6})$ -action as given in [7]. In particular,
the $W(E_{6})$ -action in [7] preserves the parameter space. For this reason, we obtain a$W(E_{6})$ -action on the-space which actually coincides with the $W(E_{6})$ -action on-space explained before Theorem 5.1.
the$(\lambda,\mu, \nu, \rho)$$(\lambda, \mu, \nu, p)$

6. A Conjecture of B. Hunt
It is known (cf.[l]) that there is a unique $W(E_{6})$ -invariant homogeneous polynomialof degree 5 up to a constant factor. For example, we takebelow
of $t=$
as such a polynomial (cf. [4]):$Q_{5}(t)$$(t_{1}, \cdots , t_{6})$,$Q_{5}(t)=- \frac{5}{108}t_{6}^{5}+\frac{5}{18}\sigma_{1}t_{6}^{3}+\frac{5}{4}(\sigma_{1}^{2}-4\sigma_{2})t_{6}+30\sqrt{\sigma_{5}}$where$\sigma_{i}=\sigma_{i}(t_{1}^{2}, \cdots , t_{5}^{2})$$\sqrt{\sigma_{5}}=t_{1}\cdots t_{5}$.

is the i-th elementary symmetric polynomial in$t_{1}^{2},$$\cdots,t_{5}^{2}$anddefined by $Q_{5}(t)=0$ . Since
Let be the hypersurface inis $W(E_{6})$ -invariant,so is . Moreover, since $\dim I_{5}=4$ , the restrictions
are generically finite maps
from to . In [4], B. Hunt stated conjectures on these maps which tum out to be one
conjecture below.
$P^{5}$$I_{5}$$Q_{5}(t)$$F_{1}|I_{5},$$I_{5}$$I_{5}$$F_{2}|I_{5}$$C^{4}$

Conjecture 6.1.([4]) Both

$F_{1}|I_{5},$$F_{2}|I_{5}$are generically bijective.

How to attack Conjecture 6.1 with the help of REDUCE? In virtue of Theorem 5.1,,
it suffices to show Conjecture 6.1 for one of. Noting the definition of
we find that Conjecture 6.1 is rewritten as follows:$F_{1}|I_{5},$Problem 6.2. Let
outside the set . Usin$T$$x_{1},$ $x_{2},$ $y_{1},$$gx_{1},$$y_{2}$$x_{2},$ $y_{1},$$F_{1}(t)$$F_{2}|I_{5}$be constants. At least assume that
, we define four polynomials of by$(x_{1}, x_{2}, y_{1}, y_{2})$

is

$t$

$y_{2}$$f_{1}=h_{24}\cdot h_{234}\cdot h_{15}\cdot h_{135}-x_{1}\cdot h_{14}\cdot h_{134}\cdot h_{25}\cdot h_{235}$,$f_{2}=h_{24}\cdot h_{234}\cdot h_{16}\cdot h_{136}-x_{2}\cdot h_{14}\cdot h_{134}\cdot h_{26}\cdot h_{236}$,,$g_{1}=h_{34}\cdot h_{234}\cdot h_{15}\cdot h_{125}-y_{1}\cdot h_{14}\cdot h_{124}\cdot h_{35}\cdot h_{235}$,$g_{2}=h_{34}\cdot h_{234}\cdot h_{16}\cdot h_{126}-y_{2}\cdot h_{14}\cdot h_{124}\cdot h_{36}\cdot h_{236}$

whereare linear functions of defined in section 3. Then how many solutions
are there for the simultaneous equations of defined by$h,$$h_{ij},$$t$$h_{ijk}$

$t$

(4)

under the condition$fi=f_{2}=g_{1}=g_{2}=Q_{5}=0$$F_{1}(t)\not\in T$?

Needles to say, there is a gap between Conjecture 6.1 and Problem 6.2, that is, Conjecture 6.1 claims that for generic, equation (4) has a unique projective solution.
Since I don’t know whether Conjecture 6.1 is true or not, I reformulate it as a problem.
$x_{1},$ $x_{2},$ $y_{1},$$y_{2}$

82
I tried to solve Problem 6.2 directly by using REDUCE3.4 on TOSHIBA J3100 once
and at last abondaned to do because of out of capacity.
From now on, I am going to explain results related with Problem 6.2 and the moduli
defined by $\lambda(t)-1=0$ , that is,of cubic surfaces. We consider the hypersurface $H$ in$P^{5}$(5)$P(t)=h_{345}\cdot h_{26}\cdot h_{256}\cdot h_{13}\cdot h_{136}\cdot h_{246}-h_{245}\cdot h_{36}\cdot h_{356}\cdot h_{12}\cdot h_{126}\cdot h_{346}=0$.

Then it is easy to show that the polynomial $P(t)$ of equation (4) is decomposed into two
factors (up to a constant):$P(t)=h_{23}\cdot P_{5}(t)$ ,where $P_{5}(t)!s$ homogeneous of degree 5. Moreover$P_{5}$

is so taken that

$P_{5}(t_{1}, t_{2},t_{3},t_{4},t_{5},t_{6})=const.Q_{5}(t_{1}, t_{2},t_{3},t_{4},t_{6}, -3t_{5})$.

From this remarkable relation, we easily imply the following (cf. [4], [6]).

Proposition 6.3.
as the $W(E_{6})$ -orbit of $H$ . Moreover, the isotropy(i) There are 45 hypersurfaces insubgroup of $H$ in $W(E_{6})$ is isomorphic to the Weyl group of type .
is decomposed into two irreducible components. One is(ii) The intersection
defined by $t_{5}=t_{6}=0$ therefore is isomorphic to . The other is defined by an equationof degree 24.
(iii) If $t\in H$ , then $F_{2}(t)=(1,1,1,1)$ , that is, $\lambda(t)=\mu(t)=\nu(t)=\rho(t)=1$ . The
corresponding cubic surface has Eckard points.
$P^{5}$$F_{4}$$H\cap I_{5}$$P^{3}$

It follows from Proposition 6.3 (i) that there is a natural 1-1 correspondence between
the $W(E_{6})$ -orbit of $H$ and the 45 exceptional divisors of Naruki’s cross ratio variety [6].
We mention Proposition 6.3 (ii) in detail. We first introduce symmetric polynomialsbyof,$t_{1},t_{2},t_{3},$$t_{4}$$s_{2}=t_{1}^{2}+t_{2}^{2}+t_{3}^{2}+t_{4}^{2}$$s_{4}=t_{1}^{2}(t_{2}^{2}+t_{3}^{2}+t_{4}^{2})+t_{2}^{2}(t_{3}^{2}+t_{4}^{2})+t_{3}^{2}t_{4}^{2}$$s_{4}’=t_{1}t_{2}t_{3}t_{4}$Using$s_{2},$$s_{4},$$s_{4}’$, we define the polynomial

$h$

,

.

of degree 24 by,$h=c_{10}t_{5}^{20}+c_{9}t_{5}^{18}+c_{8}t_{5}^{16}+c_{7}t_{5}^{14}+c_{6}t_{5}^{12}+c_{5}t_{5}^{10}+c_{4}t_{5}^{8}+c_{3}t_{5}^{6}+c_{2}t_{5}^{4}+c_{1}t_{5}^{2}+c_{0}$where$c_{10}=1728s_{2}^{2}$,$c_{9}=432s_{2}(-21s_{2}^{2}+20s_{4})$,$c_{8}=27(4800s_{4^{2}}’+761s_{2}^{4}-1736s_{2}^{2}s_{4}+400s_{4}^{2})$,$c_{7}=8s_{2}(-46656s_{4^{2}}’-3217s_{2}^{4}+12852s_{2}^{2}s_{4}-10368s_{4}^{2})$,83$c_{6}=2(-190080s_{4^{2}}’s_{2}^{2}-336960s_{4^{2}}’s_{4}+9251s_{2}^{6}-55955s_{2}^{4}s_{4}$$+91368s_{2}^{2}s_{4}^{2}-28080s_{4}^{3})$,$c_{5}=2s_{2}(825360s_{4^{2}}’s_{2}^{2}-1582848s_{4}^{;2}s_{4}-3256s_{2}^{6}+27143s_{2}^{4}s_{4}$$-72496s_{2}^{2}s_{4}^{2}+61776s_{4}^{3})$,$c_{4}=-59833728s_{4^{4}}’-1370994s_{4^{2}}’s_{2}^{4}+5809680s_{4^{2}}’s_{2}^{2}s_{4}-4732128s_{4^{2}}’s_{4}^{2}-193s_{2}^{8}$$+3054s_{2}^{6}s_{4}-12981s_{2}^{4}s_{4}^{2}+10120s_{2}^{2}s_{4}^{3}+21168s_{4}^{4}$,$c_{3}=2s_{2}(-2191104s_{4^{4}}’+199476s_{4^{2}}’s_{2}^{4}-1263024s_{4^{2}}’s_{2}^{2}s_{4}+1990080s_{4^{2}}’s_{4}^{2}$,$+496s_{2}^{8}-7327s_{2}^{6}s_{4}+40443s_{2}^{4}s_{4}^{2}-98824s_{2}^{2}s_{4}^{3}+90160s_{4}^{4})$$c_{2}=-907200s_{4^{4}}’s_{2}^{2}+2491776s_{4^{4}}’s_{4}-54714s_{4^{2}}’s_{2}^{6}+554274s_{4^{2}}’s_{2}^{4}s_{4}$$-1854576s_{4^{2}}’s_{2}^{2}s_{4}^{2}+2051616s_{4}^{J2}s_{4}^{3}-256s_{2}^{1}0+4640s_{2}^{8}s_{4}-33505s_{2}^{6}s_{4}^{2}$$+120460s_{2}^{4}s_{4}^{3}-215600s_{2}^{2}s_{4}^{4}+153664s_{4}^{5}$,$c_{1}=6s_{4^{2}}’s_{2}(-4968s_{4^{2}}’s_{2}^{2}+14688s_{4^{2}}’s_{4}-26s_{2}^{6}+285s_{2}^{4}s_{4}-1032s_{2}^{2}s_{4}^{2}+1232s_{4}^{3})$$c_{0}=27s_{4^{4}}’(192s_{4^{2}}’+s_{2}^{4}-8s_{2}^{2}s_{4}+16s_{4}^{2})$,.

Moreover,$N=-2\{(5s_{2}^{5}-1602s_{2}^{4}t_{5}^{2}-34s_{2}^{3}s_{4}+4134s_{2}^{3}t_{5}^{4}+10037s_{2}^{2}s_{4}t_{5}^{2}-3005s_{2}^{2}t_{5}^{6}$$+56s_{2}s_{4}^{2}-12820s_{2}s_{4}t_{5}^{4}+828s_{2}t_{5}^{8}-15764s_{4}^{2}t_{5}^{2}+1980s_{4}t_{5}^{6}-360t_{5}^{10})t_{5}^{2}$$-(s_{2}^{2}+164s_{2}t_{5}^{2}-4s_{4}+7368t_{5}^{4})s_{4^{2}}’\}s_{4}’t_{5}$,$D=-\{3(31s_{2}^{3}+650s_{2}^{2}t_{5}^{2}-92s_{2}s_{4}+2320s_{2}t_{5}^{4}-1752s_{4}t_{5}^{2}+5648t_{5}^{6})s_{4^{2}}’t_{5}^{2}$$+2(2464s_{4}^{2}-2055s_{4}t_{5}^{4}+1S7t_{5}^{8})s_{2}^{2}t_{5}^{4}-4(1687s_{4}^{2}-415s_{4}t_{5}^{4}+12t_{5}^{8})s_{2}t_{5}^{6}$$-(1465s_{4}-1044t_{5}^{4})s_{2}^{4}t_{5}^{4}+15(269s_{4}-61t_{5}^{4})s_{2}^{3}t_{5}^{6}-16s_{4^{4}}’+144s_{2}^{6}t_{5}^{4}$$-599s_{2}^{5}t_{5}^{6}-5488s_{4}^{3}t_{5}^{4}+2072s_{4}^{2}t_{5}^{8}-120s_{4}t_{5}^{12}\}$.

Then from the equations$P_{5}=Q_{5}=0$

we obtain$t_{6}=N/D$ ,,$h=0$ .

The equation $h=0$ is the one stated in Proposition 6.3 (ii).
-space, we obtain a hyperIf we consider the equation $\lambda-1=0$ in thedefined bysurface$(x_{1}, x_{2}, y_{1}, y_{2})$$H_{0}$(6)$x_{2}(x_{1}-1)(y_{1}-y_{2})(y_{2}-1)-y_{2}(x_{1}-x_{2})(x_{2}-1)(y_{1}-1)=0$.

84
Now we formulate a problem simplified from Problem 6.2, noting Proposition 6.3 (ii).
Namely, we consider Problem 6.2 in the case $t_{5}=t_{6}=0$ and $t_{1}=1$ (The condition$t_{1}=1$ is not essential. From the homogeneity, we may assume $t_{j}=1$ for some $j.$ )Problem 6.2’. Define four polynomials of$t_{2},t_{3},$$t_{4}$by$f_{10}=(t_{2}+t_{3}-t_{4}+1)^{2}(t_{2}+t_{4})(t_{3}-1)-x_{1}(t_{2}+t_{3})(t_{2}-t_{3}+t_{4}+1)^{2}(t_{4}-1)$,$f_{20}=(t_{2}+t_{3}+t_{4}+1)(t_{2}+t_{3}-t_{4}+1)(t_{3}-1)t_{2}$$+x_{2}(t_{2}+t_{3})(t_{2}-t_{3}+t_{4}+1)(t_{2}-t_{3}-t_{4}+1)$,$g_{10}=(t_{2}+t_{3}-t_{4}+1)^{2}(t_{2}-t_{3})(t_{4}+1)-y_{1}(t_{2}-t_{3}+t_{4}+1)^{2}(t_{2}-t_{4})(t_{3}+1)$,$g_{20}=(t_{2}+t_{3}+t_{4}+1)(t_{2}+t_{3}-t_{4}+1)(t_{2}-t_{3})$$-y_{2}(t_{2}-t_{3}+t_{4}+1)(t_{2}-t_{3}-t_{4}+1)(t_{3}+1)t_{2}$

,

where
are constants with the condition (6) and(In
particular, we assume that is a rational function of) Then how many solutions
are there for the equations (7) of
below$x_{1},$$x_{2},$ $y_{1},$$(x_{1}, x_{2}, y_{1}, y_{2})\not\in T.$$y_{2}$$x_{2},$ $y_{1},$ $y_{2}.$$x_{1}$$t_{2},$ $t_{3},$ $t_{4}$(7)$f10=f_{20}=g_{10}=g_{20}=0$under the condition$t\not\in T$?

It is possible to give an answer to Problem 6.2’. In fact, erasing
obtain an equation for defined byfrom (7), we$t_{3},$ $t_{4}$$t_{2}$(8)$\sum_{j=0}^{9}b_{j}d_{2}=0$

,

where
$b_{9}=(x_{2}y_{1}y_{2}-x_{2}y_{2}-2y_{1}y_{2}+y_{1}+y_{2}^{2})^{2}(x_{2}y_{2}-2y_{2}+1)y_{2}^{4}$,$b_{8}=3(x_{2}y_{1}y_{2}-x_{2}y_{2}-2y_{1}y_{2}+y_{1}+y_{2}^{2})^{2}(x_{2}y_{2}-2x_{2}+1)y_{2}^{4}$,$b_{6}=-4(x_{2}^{2}y_{1}y_{2}-x_{2}^{2}y_{2}+x_{2}y_{1}^{2}+x_{2}y_{1}y_{2}^{2}-4x_{2}y_{1}y_{2}+x_{2}y_{1}+x_{2}y_{2}^{2}-y_{1}^{2}y_{2}+y_{1}y_{2})$$\cross(x_{2}y_{1}y_{2}-x_{2}y_{2}-2y_{1}y_{2}+y_{1}+y_{2}^{2})(x_{2}y_{2}-2x_{2}+1)y_{2}^{3}$,$b_{5}=-6(x_{2}y_{1}y_{2}-x_{2}y_{2}-2y_{1}y_{2}+y_{1}+y_{2}^{2})(x_{2}y_{1}+x_{2}y_{2}^{2}-2x_{2}y_{2}-y_{1}y_{2}+y_{2})$$\cross(x_{2}y_{2}-2y_{2}+1)x_{2}y_{1}y_{2}^{2}$,$b_{4}=6(x_{2}y_{1}y_{2}-x_{2}y_{2}-2y_{1}y_{2}+y_{1}+y_{2}^{2})(x_{2}y_{1}+x_{2}y_{2}^{2}-2x_{2}y_{2}-y_{1}y_{2}+y_{2})$$\cross(x_{2}y_{2}-2x_{2}+1)x_{2}y_{1}y_{2}^{2}$,$b_{3}=4(x_{2}^{2}y_{1}y_{2}-x_{2}^{2}y_{2}+x_{2}y_{1}^{2}+x_{2}y_{1}y_{2}^{2}-4x_{2}y_{1}y_{2}+x_{2}y_{1}+x_{2}y_{2}^{2}-y_{1}^{2}y_{2}+y_{1}y_{2})$$\cross(x_{2}y_{1}+x_{2}y_{2}^{2}-2x_{2}y_{2}-y_{1}y_{2}+y_{2})(x_{2}y_{2}-2y_{2}+1)x_{2}y_{1}y_{2}$$h=-3(x_{2}y_{1}+x_{2}y_{2}^{2}-2x_{2}y_{2}-y_{1}y_{2}+y_{2})^{2}(x_{2}y_{2}-2y_{2}+1)x_{2}^{2}y_{1}^{2}$

,

,

85
$b_{0}=-(x_{2}y_{1}+x_{2}y_{2}^{2}-2x_{2}y_{2}-y_{1}y_{2}+y_{2})^{2}(x_{2}y_{2}-2x_{2}+1)x_{2}^{2}y_{1}^{2}$$b_{7}=b_{2}=0$,.

Moreover, if is a solution of (8),are uniquely determined by (7).
I checked that equation (8) for is irreducible of degree 9 and that for generic(8) has no multiple factor. As a consequence, we obtain the following.
$t_{2}$$t_{3},t_{4}$$t_{2}$

Theorem 6.4. The restriction of$F_{1}$$x_{2},$ $y_{1},$$y_{2}$,

to the subspace $t_{5}=t_{6}=0$ is generically 9 to 1.

I am not sure whether Theorem 6.4 induces the invalidity of Conjecture 6.1 or not.

Acknowledgements
Last I mention that the note [4] and the communications with Prof. B. Hunt are
valuable when I formulate the maps $F_{j}(j=1,2,3)$ and solve Problem 6.2’. Moreover, I
am indebted to Prof. K. Okubo because without his help, I could not use REDUCE 3.4(not REDUCE 3.2 !) which is a powerful tool in obtaining the results of this note.
References[1] Bourbaki, N. “Groupes et Alg\‘ebres de Lie” Chaps. 4, 5, 6, Herman, Paris (1968).

[2] Cayley, A. “A memoir on cubic surfaces” Collected Papers VI.

[3] Dolgachev, I. and Ortland, D. “Point sets in projective spaces and theta functions” Asterisque 165 (1988)[4] Hunt, B. “A remarkable quintic fourfold in(unpublished)$P^{5}$and its dual variety (Update: 7.1.1992)”[5] Matsumoto, K., Sasaki T. and Yoshida, M. “The monodromy of the period map of a 4parameter family of K3 surfaces and the hypergeometric function of type (3,6) Int. J.
Math. 3 (1992), 1-164.
[6] Naruki, I. “Cross ratio variety as a moduli space of cubic surfaces” J. Lon don Math. Soc.

1982.
[7] Naruki, I and Sekiguchi, J. “A modification of Cayley’s family of cubic surfaces and birational action of $W(E_{6})$ over it” Proc. Japan Acad. 56 Ser. A (1980), 122-125.
[8] Sekiguchi, J. “A birational action of
prep aration.

$S_{6}$

on$C^{4}$and the hypergeometric system $E(3,$ 6) in