数理解析研究所講究録920 巻 1995 年 216-228

216

24.
-invariant polynomial of
degree 10 and 28 bitangents ofplan equartic curves$W(E_{7})$関口次郎 (電気通信大学)

序文

ルート系に対する複比多様体という概念を筆者は定義した$(\mathrm{C}\mathrm{f}.[\mathrm{s}_{\mathrm{e}}4])$

が,$E_{7}$型ルート系の場合にそれを詳しく調べる. 平面の非特異 4 次曲線には 28 本の複接線が存在するが, この古典的話題と関係がある.

.

本文の結果を説明する.
$E_{7}$型ルート系に対する複比多様体の$E_{6}$型部分ルート系に対して定義される部分多様体.

. 射影平面の 7 点の配置空間の特別な配置 (与えられた 7 点に対して, この中のある点で cuspになるようなこれらの 7 点を通る 3 次曲線が存在するような配置).

. 平面の非特異 4 次曲線には 28 本の複接線が存在するが, それらの複接線の接点は普通 2 点あるがそれらが–致するような複接線が存在する.
以上 3 つの条件がワイル群 $W(E_{7})$ のある 10 次の不変式を使って記述できる. この主張を示すことが本文の目的だが, 証明には数式処理システム\S 1. The root system of type

$E_{7}$$\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{r}$.

We first recall the definition of the root system
for simplicity in this paper. Let
basis ($\epsilon_{\mathrm{J}1}1\leq j\leq 8\}$$\overline{E}$を利用する.

$\triangle(E_{7})$of type$E_{7}$. We always denote it by$\triangle$

be an inner product space of dimension 8 with an orthonormal

with respect to an inner product $<.,$$\cdot>$and let $E$ be its linear subspace

217

orthogonal to$\epsilon_{7}+\epsilon_{8}$. As in [Se4], \S 4, we define the following 63 vectors of$\gamma_{1}=\epsilon_{8}-\mathcal{E}_{7}$,,$\gamma j=\epsilon_{j-1}-\gamma 0+\gamma_{1}$$\gamma_{jk}=\epsilon_{j-1}-\epsilon_{k-1}$,$\gamma_{1j}=-\epsilon_{j-1}+\gamma_{\mathrm{O}}$$\gamma 1jk=-\epsilon_{j-1}-\epsilon_{k-1}$$\gamma ijk=-\epsilon:-1-\epsilon_{j-1}-\epsilon_{k-1}+\gamma 0$

where$\gamma 0=\frac{1}{2}\sum_{j=1}\epsilon_{j}8-\epsilon_{7}$. The totality

As a fundamental set of roots of$\alpha_{1}=\gamma_{12}$,$\alpha_{2}=\gamma_{123},$$\triangle$$\triangle \mathrm{o}\mathrm{f}\pm\gamma \mathrm{j},$

,

$E$

:

$(1 <j<8)$$(1 <j<k<8)$,$(1 <i<j<k<8)$,$\pm\gamma.jk$$\pm\gamma jk,$

is a root system of type

$E_{7}$(cf. [B]).

, we may take$\alpha_{3}=\gamma_{23},$$\alpha_{5}=\gamma_{45}$$\alpha_{4}=\gamma_{34},$,$\alpha_{6}=\gamma_{56}$,$\alpha_{7}=\gamma_{67}$.

Then the corresponding Dynkin diagram is:$\alpha_{1}$––$\alpha_{3}$–$\alpha_{4}$$\alpha_{5}$–—$\alpha_{6}$$0_{7}$$1$$\alpha_{2}$We denote by$\triangle^{+}$

the set of positive roots in

If $gj$ is the reflection on. It is easy to see thatwith respect to the root$E$

the Weyl group $W(E_{7})$ of type
Weyl group $W(E_{6})$ of type$\triangle$$E_{6}$$E_{7}$$\alpha_{j}$consists of$\triangle^{+}$, the group generated by. In the sequel, we frequently identify$\gamma_{i},$$g_{1},$$W(A_{6})\simeq\Sigma_{7}$) with the subgroup of $W(E_{7})$ generated by$g_{1},$$\gamma_{\iota_{J}},$$\cdots$,$\gamma i\mathrm{J}k$$g_{7}$.

is(resp. the$gj(j=3,4,5,6,7)$(resp. $g\mathrm{j}(j=1,2,3,4,5,6)$ ).

Using the 63 positive roots defined above, we define linear forms on $E$ by$h_{j}=\gamma j(t)$,$h_{jk}=\gamma jk(t)$,$hijk=\gamma_{1}jk(t)$\S 2. The configuration space of 7 points in$\mathrm{P}^{2}$,$(t\in E)$.

We briefly review the definition of the configuration space of 7 points of$\mathrm{P}(2,7)$. We first define the vector space$M_{3,7}$of$3\cross 7$.

matrices. Then

$M_{3,7}$$\mathrm{P}^{2}$

which we denote byadmits $GL(3)\cross GL(7)-$action in a natural manner. Let $D(7)$ be the maximal torus of $GL(7)$ consisting of diagonal
matrices. Let Dijk (X) be the determinant of thevectors of $X\in M_{3.7}$ . If$j<k)$ , we denote by$M_{3.7}’$is the subset of$\mathrm{P}(2,7)$$M_{3,7}$

the quotient ofchoose as a representative of any element of$3\cross 3$

matrix consisting of the $i,j$ , k-th column

consisting of $X$ with D.jk $(X)\neq 0$$M_{3,7}’$$\forall(i,j, k)(i<$by the action $GL(3)\mathrm{x}D(7)$ . It is possible to$\mathrm{P}(2,7)$a matrix of the form

$X=$

218

In this way,$\mathrm{P}(2,7)$

is regarded as a quasi-affine subset ofby the correspondence$\mathrm{C}^{6}$$arrow(_{X_{1},x_{2},x_{3}}, y1, y2, y3)$ .
In fact,$\mathrm{P}(2,7)$

is identified with$\mathrm{C}^{6}-^{s_{\mathit{0}}()}A_{6}$below:$x_{i}=0$ ,$y:=0$ ,$xi– l=0$ ,xiyj $-x.yj=0$ ,$y$.

, where So $(A_{6})$ is the union of the 28 hypersurfaces$-1=0$ ,$x:-x_{\mathrm{j}}=0$,

yi–yj $=0$ ,xi–yi $=0$ ,$(1-x.)(1-yj)-(1-x_{j})(1-y:)=0$ ,$\varphi\iota(X_{1}, x_{2}, x3, y1, y_{2}, y_{3})=\det=0$.

We introduce the following seven birational transformationsSl,$s_{R}$$s_{6},$:: $(x_{1} , x_{2}, x_{3}, y1, y_{2}, y_{3})arrow(1/x_{1},1/x_{2},1/x_{3}, y_{1}/x_{1}, y_{2}/x_{2}, y_{3}/x_{3})$: $(_{X_{1},XX_{3,y1,y}}2,2, y3)arrow(y_{1}, y_{2}, y3, x_{1,2}x, x3)$$s_{2}$$s_{3}$$s_{4}$$\cdots$$s_{1},$:$(_{X\iota,x_{23}}, x, y1, y_{2}, y_{3})arrow(_{X_{1}^{\prime J}}, X_{2}, x_{3}, y1, y2 , y_{3})Jr\prime\prime$: $(_{X_{1},x_{2},x_{3},y}1, y_{2}, y3)arrow(1/x_{1}, x_{2}/x_{1}, xs/x_{1},1/y_{1}, y_{2}/y_{1}, y_{3}/y_{1})$$s_{5}$$s_{6}$$s_{R}$: $(_{X_{1},X_{2}}, x_{3}, y1, y2, y3)arrow(x_{2}, x_{1}, x3, y_{2}, y_{1}, y_{3})$; $(_{X_{1},x_{2}x_{3},y,y_{2,y3}})1)arrow(_{X_{1},X_{3}}, x_{2}, y1, y3, y2)$

: $(_{X_{12}}, X, X3, y_{1}, y2, y3)arrow(1/x_{1},1/x_{2},1/x_{3},1/y_{1},1/y_{2},1/y_{3})$where$x_{\acute{J}}= \frac{x_{j}-yj}{1-yj}$,$y_{j}’= \frac{yj}{y_{\dot{J}}-1}$,$j=1,2,3$ .

The correspondence$g_{1}arrow s_{1}$,$g_{2}arrow s_{R}$,$gjarrow s_{j-1}(j=3, \cdots, 7)$

induces a group isomorphism of $W(E_{7})$ to the group generated by
We introduce 7 polynomials of ($X_{1},$ $X_{2},$,$X3,$ $y_{1y2,ys)}$$s_{1},$$$\cdot$.,$s_{6},$$s_{R}$.

defined by$\sigma_{j}(x_{1}, X_{2}, X3, y_{1}, y2, y_{3})=\varphi\epsilon-j(x_{1,2}X, x_{3}, y_{1,y}2, y3)$,$(j=1,2,3,4)$ ,219$\sigma \mathrm{s}(X1,$$X2,$ $X_{3,y1,y_{2,y)=}}sx_{2y}s(1-X_{3})(1-y2)-x_{3}y2(1-X_{2})(1-y_{3})$,,$\sigma_{6}(_{X_{1},x_{2}}, x_{3}, y_{1}, y2, y_{3})=x_{1y}3(1-x_{3})(1-y1)-x_{3y1}(1-X_{1})(1-y_{3})$$\sigma_{7}(_{X_{1}x,x,y_{2},y3},23y1,)=x_{1y_{2}}(1-X_{2})(1-y1)-x_{2y1}(1-X_{1})(1-y_{2})$ ,where $\varphi j(j=2,3,4,5)$ are polynomials introduced in [Se4], \S 4. In particular,$\varphi_{2}(x_{1,2}X, x_{3}, y_{1}, y2, y_{3})=X1x2Xsy_{1y}2y3\varphi 1(1/x_{1},1/x_{2},1/x_{3},1/y_{1},1/y_{2},1/y_{3})$

Let
that

$Q_{j}$$\Sigma_{7}$be the hypersurface in$\mathrm{C}^{6}$

defined by $\sigma_{j}=0$ $(j=1, \cdots , 7)$ . Then it is easy to see

acts on the set $\{Q_{1}, \cdots , Q_{7}\}$ as a permutation group. If$(j=1,2, \cdots, 6)$ ,andZariski closure of$Q_{j}’$

is the hypersurface in$Q_{j}’$

in$\mathrm{C}^{6}$$\mathrm{C}^{6}-S\mathit{0}(A_{6})$. A geometric meaning of$Q_{j}$

and$\tilde{\sigma}_{7}=\sigma_{7}$defined by$\tilde{\sigma}_{j}=0$$\tilde{\sigma}_{j}=\tilde{\sigma}_{j+1}\mathrm{o}s_{j}$, thenis the$Q_{j}$

will be given in \S 6. In the sequel, we

denote by Po $(2, 7)$ the complement of the union $S(E_{7})$ of So

and$(A_{6})$$Q_{1},$$\cdots$,$Q_{7}$. Clearly all theelements of $W(E_{7})$ induce biregular transformations on Po $(2, 7)$ .

\S 3. The cross ratio variety $c(\Delta(E_{7}), D_{4})$ .
For any subroot systemopen subset$Z(\triangle)$$\triangle_{1}$of typeof the projective spaceto$D_{4}$-cross ratio maps are denoted by$cr_{[]}^{1}i_{3,67}ii=(h_{i_{24}}.h_{ii_{3}}i4h_{i\iota^{i}}2\mathrm{s}^{h}:1^{i\cdot \mathrm{s}}3 :$cr_{[\cdot\iota 2}^{2}.,,:_{345},,,i_{6}]=$in$\triangle$, we defined a$\mathrm{P}^{6}=\mathrm{P}(E_{\mathrm{C}})$$D_{4}$-cross ratio map of the Zariski

associated to the complexification. There are totally 315 subroot systems of type$E$

$CR(\mathrm{P})\simeq \mathrm{P}^{1}$$D_{4}$-h_{i\iota\cdot 4}h:_{\mathrm{t}^{ii}}34h_{i}2.5hi_{23}.:_{\mathrm{s}} :$(h_{i_{1^{\mathrm{i}}3}}.\cdot\cdot h\mathrm{s}^{h_{i_{2^{1\triangleleft^{i}52}}}}\cdot.:3i6h_{i_{1}}:_{46}i :$cr_{1ii]}^{3}i_{1234}i,=(h_{1^{i}2^{i}\tau}.\cdot h_{34}.\cdot i,\tau^{h:}\mathrm{s}\cdot.6h:\tau :$D_{4}$

in$\triangle$

h_{12}..h_{1}.:_{2^{i}}3h:_{4^{i}}\mathrm{s}hi_{3}i4i\mathrm{S})$h:\iota:2h.\cdot:34h.\cdot i_{67}h.\cdot)5$

h_{125}.\cdot:.\cdot h_{3}.\cdot.\cdot:_{5}h4:6’ 7h:_{\mathrm{s}})$(cf. [Se4], \S 4). By taking the product of all the 315 maps above, we obtain a map$Z(\triangle)$to$CR(\mathrm{p})^{315}$. Let

be the image$C’(\triangle, D_{4})$$C\Gamma_{D_{4},\Delta}(Z(\triangle))$of. The corresponding-h_{23}.\cdot::_{\mathrm{s}\iota}h.\cdot:_{4}i\epsilon^{h.h}.\iota:3:6:_{2^{i_{46}}}.\cdot :-h_{i_{1}:_{2}}.\cdot 6h:_{\epsilon}:4:6h:_{67}.\cdot h_{6}.\cdot :$E_{\mathrm{C}}$and let$C(\triangle, D_{4})$$cr_{D_{4},\Delta}$ofbe its closure

in $CR(\mathrm{P})s15$ .

For any subroot systemkinds of hypersurfaces of$\triangle_{1}$of$\triangle$, we defined a subvariety$C(\triangle, D_{4})$defined as the form

$Y_{\Delta,D_{4}}(\triangle_{1})$$Y_{\Delta,D_{4}}(\triangle_{1})$in [Se4], \S 4. There are four

for suitable subroot systems.

\S 4. Hypersurfaces corresponding to subroot systems of type
We introduce hypersurfaces ofIf$\triangle_{1}$$C(\triangle, D_{4})$is a subroot system of type

$E_{6}$

in$\triangle$

which are fixed by$W(E_{\mathit{6}})$, it is easy to show that

$E_{6}$.

-actions (cf. [Se4], \S 4,(4.15.10)).

$Y_{\Delta,D_{4}}(\triangle_{1})$is a hypersurface of220$C(\triangle, D_{4})$of the. Such a hypersurface is called that of the$5^{th}$$5^{th}$kind. As a basic property of hypersurfaces

kind, we have the lemma below.

Lemma 4.1. (cf.

$[\mathrm{S}\mathrm{e}4\mathrm{a}]$) $Y_{\Delta,D_{4}}(\triangle(E_{6}))\simeq C(\triangle(E_{6}), \{A_{3}, D_{4}\})$ .

Lemma 4.1 establishes an embedding of the cross ratio variety $C(\triangle(E_{6}), \{A_{3}, D_{4}\})$ into$C(\triangle, D_{4})$.

To show an identification of $C(\triangle(E_{6}), \{A_{3}, D_{4}\})$ with the variety defined [L], we need some

preparation on cubic curves in$P_{1},$$\cdots$,$P_{7}$

of$\mathrm{P}^{2}$$\mathrm{P}^{2}$

passing through 7 points. For simplicity, we take 7 points

as follows:$P_{1}=$

$($

1 :

$0$

:

$P_{5}=$

$0)$

$($

,

$P_{2}=(0$1 :$x_{1}$:$y_{1})$

,

: 1 :

$0)$

,

$P_{3}=(0$$P_{6}=(1 :x_{2} :

y_{2})$

:
,

$0$

: 1 ,

$P_{4}=(1$$)$$P_{7}=(1 :x_{3} :

: 1: 1 ,$)$y_{3})$.

We assume that the 7 points above are in a general position which means the corresponding

matrixis a representative of the configuration space$\mathrm{P}(2,7)$

Let $C(P_{1}, \cdots , P_{6} ; P_{7})$ be the cubic curve indouble point (cf. [M],$1\mathrm{L}]$.

passing through$\mathrm{P}^{2}$). We now consider the case where$P_{1},$$\cdots$,$P_{7}$$C(P_{1}, \cdots , P_{\mathit{6})}P_{7})$such that$P_{7}$is a

has a cusp at$P_{7}$(cf. [L]). This condition implies a relation $\Psi(x, y)=0$ among $(x, y)=(X_{1}, x_{2}, X_{3,y}1, y2, y3)$ .

The explicit form of the polynomial $\Psi(x, y)$ is too lengthy to write down here. It is provablethat$\deg_{x\mathrm{s}}\Psi=\deg_{y\mathrm{s}^{\Psi}}=8$

Noting that
$C(\triangle, D_{4})$$C(\triangle, D_{4})$.

is a compactification of Po $(2, 7)$ , we obtain a hypersurface

as the Zariski closure of the hypersurface ofTheorem 4.2.

Ycusp$=Y_{\Delta,D_{4}}(\triangle_{1})\cap \mathrm{p}_{0^{(2,7)}}$$\mathrm{P}_{0}(2,7)$$Y_{cu\iota}\mathrm{P}$ofdefined by $\Psi(x, y)=0$ .

.

The basic idea of the proof employed here is the comparison between the defining equations of$Y_{\Delta,D_{4}}(\triangle \mathrm{l})$$\Psi(x, y)$and$Y_{cu\iota p}$. Before entering the details of its outline, we state a result on the polynomial.

Lemma 4.3. We put$\Phi(x, y)=\Phi_{1}(_{X,y})^{2}-4\Phi_{2}(x, y)$

,

221

where
$\Phi_{1}(x, y)$$x_{1}x_{2}y_{1}-x_{1}X2y3-x_{1}x_{3}y1+x_{1}x3y2-x_{1}y_{1y_{2}}+X_{1}y1y3+x1y2$$=$$-x_{1}y_{3}+x2y1y3-x_{2}y_{1}-x_{3}y1y_{2}+x_{3}y_{1}$$\Phi_{2}(x, y)$,$(x_{1}-y_{1})(X_{2y-}3x3y_{2})(y_{1}-1)(y_{2}-y_{3})x_{1}$$=$Then there is $s\in W(E_{7})$ such that$\Phi \mathrm{o}s=\Psi$.

.

We are going to explain the outline of the proof of Theorem 4.2.
We first compute the condition that the cubic curve $C(P_{1}, \cdots , P_{6} ; P_{7})$ has a cusp at$P_{7}$. For

this purpose, we assume that $F(\xi 1, \xi 2, \xi_{3})=0$ is the defining equation of $C(P_{1}, \cdots , P_{6} ; P_{7})$ , where$F=c_{1}\xi 13+c_{2}\xi_{2}+33c_{3}\xi 3+c_{4}\xi_{1}\xi_{2}+2c\mathrm{s}\xi_{1}\xi 3+C\epsilon\xi 21\xi^{2}2+C_{7}\xi_{2\xi_{3}}2+c8\xi 1\xi^{2}3+c_{9}\xi 2\xi_{3}+C_{1}0\xi_{1}2\xi_{2}\xi 3$

In the discussion above, we have taken $\xi=$$(\xi_{1} :\xi_{2} :condition that $C(P_{1}, \cdots , P_{6} ; P_{7})$ passes through$P_{1},$(C. 1)$j=1,$$F(P_{j})=0$ ,The condition that

$P_{7}$$F_{\xi},(P_{7})=0$$P_{7}$,$P_{7}$$\cdots,$

is equivalent to
$7$

.

,

$i=1,2,3$ .

$F_{\xi_{1}\xi_{1}}(P_{7})F\epsilon_{2}\xi_{2}(P7)-F\xi_{1}\epsilon 2(P_{7})^{2}=0$

From (C. 1), (C.2), we conclude that th.e ratio oftuting such$c_{1},$$\cdot,$.

$,$$c_{10}$. The

is moreover a cusp point of $C(P_{1}, \cdots , P_{6} ; P_{7})$ is equivalent to(C. 3)$i$$\cdots$$\mathrm{P}^{2}$

is a double point of $C(P_{1}, \cdots , P_{6}; P_{7})$ is equivalent to(C.2)

The condition thatas a homogeneous coordinate of\xi_{3})$.

$c_{1},$$\cdots$,$c_{10}$.

is uniquely determined. Substi-

to the equation (C.3), we obtain an algebraic relation(4)$\Psi(x, y)=0$

if $(x, y)\in \mathrm{C}^{6}-S(A_{6})$ . We need a long computation to obtain (4) and it is hard to reproduce
here.

Our next purpose is to compute the defining equation of the hypersurface
For this purpose, we first recall the definition of the rational map of$Y_{\triangle,D_{4}}(\triangle \mathrm{l})$$\mathrm{P}^{6}$

toin$\mathrm{P}(2,7)$$\mathrm{P}(2,7)$.

in [Se4],

Lemma 4.2. We put$x_{1}(t)= \frac{h_{24}\cdot h_{234}\cdot h_{15}\cdot h135}{h_{14}\cdot h_{13}4h2\mathrm{s}\cdot h_{235}}.$,$y_{1}(t)= \frac{h_{34}\cdot h_{234}\cdot h1\mathrm{s}.h125}{h_{\}4}\cdot h_{12}4h35h_{23}5}.\cdot$

and define the map$F_{E_{7}}$of$x_{2}(t)= \frac{h_{24}\cdot h_{234}\cdot h_{1}6h_{13}6}{h_{14}\cdot h_{134}\cdot h_{26}\cdot h236}.$,$Z(\triangle)$,$x_{3}(t)= \frac{h_{24}\cdot h_{23}4h_{1}\gamma\cdot h_{1s7}}{h_{14}\cdot h_{134}\cdot h_{27}\cdot h237}.$,$y_{3}(t)= \frac{h_{34}\cdot h_{234}\cdot h_{1}7h_{12}7}{h_{14}\cdot h_{124}\cdot h_{37}\cdot h237}$$y_{2}(t)= \frac{h_{34}\cdot h_{234}\cdot h_{16}\cdot h126}{h_{14}\cdot h_{124}\cdot h_{36}\cdot h236}$.

to the $(x, y)$ -space by$F_{E_{7}}(t)=(x_{1}(t), X_{2}(t),$ $x3(t),$ $y_{1}(t),$ $y2(t),$ $y3(t))$

where$\mathrm{P}(2,7)$,

is identified with a Zariski open subset of the $(x, y)$ -space,$(\mathrm{c}\mathrm{f}.1^{\mathrm{s}\mathrm{e}}41, \S 4)$

and$h_{i},$$h:j$, hijk222

are linear forms on $E$ associated with the roots of$(_{1}=h_{12}$,$\zeta_{2}=h_{123}$,$\zeta_{3}=h_{23}$,$\Delta$$\zeta_{4}=h_{34}$. Now we put,$(\mathrm{s}=h_{45},$$\zeta\epsilon=h_{56}$,$\zeta_{7}=h_{67}$.

It is clear from the definition that linear forms in question corresponding to the roots ofare expressed as linear combinations ofa homogeneous coordinate of$\zeta_{\mathit{6}}’),$$\tau)$$\mathrm{P}^{6}$$\zeta_{j}$$(j=1, \cdots , 6)$. Now we write. We may take$\zeta_{j}=\zeta_{j}’\tau$$(j=1, \cdots , 6)$

is also a local coordinate of an affine open subset defined by

Noting this, we write$u_{j}((_{1}’$:. . . :$\tilde{x}_{j}(\tilde{\zeta})=x_{j}(t)$,$\zeta_{6}’)=\lim_{\tauarrow 0}\tilde{x}j(\tilde{\zeta})$$v_{j}(\zeta_{1}’$:. . . ::. .. :. Then$\zeta_{7}\neq 0$in$\zeta_{\mathit{6}}$:$(^{\sim}=((\zeta_{1}’$$\zeta_{7})$as:. . . :$\mathrm{P}^{6}$$(j=1,2,3)$ . Now we put$\tilde{y}j(\tilde{\zeta})=yj(t)$,$\zeta=(\zeta_{1}$$\triangle(E_{6})$$(_{6}’)= \lim_{farrow \mathit{0}}\tilde{y}j(\tilde{\zeta})$$(j=1,2,3)$ .

We define olynomials$\mathrm{p}$$f_{1}=x_{1}(\zeta_{1}’+\zeta_{2}+\zeta_{s}+\zeta_{4}+\zeta’\mathrm{s})\prime\prime\prime(\zeta’1+\zeta_{3}+\zeta_{4})\prime\prime(\zeta 2+\zeta_{3}^{J}’+(4’)(\zeta^{J}3+\zeta_{4}’+\zeta_{5}^{l})$$-((_{1}^{J}+\zeta_{2}’+\zeta’3+\zeta_{4})’(\zeta_{1}’+\zeta_{3}+\zeta_{4}+\zeta_{\mathrm{s}}^{J}\prime\prime)(\zeta_{2}+\zeta_{s}’+\zeta’’\prime 4+\zeta \mathrm{s})((_{3}’+\zeta_{4}J)$,$f_{2}=x_{2}((_{1}’+\zeta_{2}’+\zeta_{3}’+\zeta_{4}\prime l\zeta 6’+(\mathrm{s}+)(\zeta_{1}^{J}+\zeta_{3}’+(’4)(\zeta’2+\zeta_{3}’+(’4)((_{3}+(’\prime 4+(_{5}’+\mathrm{C}_{6}’)$$-(\zeta_{1}’+(_{2}’+\zeta’3+\zeta_{4}^{J})(\zeta’1+\zeta_{3}^{\prime\prime\prime\prime}+\zeta_{4}+\zeta 5+\zeta_{\epsilon})((_{2}’+\zeta_{3}’+(4+\zeta_{5}+\zeta_{6}’\prime J)(\zeta_{3}’+\zeta 4’)$$f_{3}=x_{3}(\zeta_{1}’+\zeta_{3}’’+\zeta_{4})(\zeta_{2}’+\zeta_{3}+\zeta 4)\prime\prime-(\zeta_{1}’+\zeta_{2}’’’+\zeta_{3}+\zeta_{4})(\zeta_{3}+’\zeta 4’)$,,$f_{4}=y_{1}(\zeta_{1}’+(_{2}’+\zeta 3+\zeta_{4}\prime\prime\prime+\zeta_{5})(\zeta’1+\zeta_{3}+\zeta_{4}\prime\prime)(\zeta_{2}’+\zeta 4’)(\zeta’4+(_{5}’)$,$-((_{1}’+\mathfrak{c}_{2}’+\zeta J3+\zeta_{4}r)(\zeta_{1}l+(_{s}’+\zeta 4’+(_{5}’)(\zeta_{2}l\prime\prime+\zeta_{4}+(_{5})\zeta_{4}’$$f_{5}=y_{2}((_{1}^{J}+\zeta_{2}^{JJ\prime\prime}+\zeta_{3}+\zeta 4+\zeta_{5}+\zeta 6’)(\zeta_{1}’+\zeta_{3}’’+\zeta_{4})(\zeta 2J’+\zeta 4)(\zeta_{4}’+\zeta_{\mathrm{s}}’\zeta_{6}’+)$,$-(\zeta_{1}’+\zeta_{2}^{J\prime}+\zeta_{3}+(’4)(\zeta_{1}’+\zeta_{s}’+\zeta_{4}’+\zeta_{5}’’+\zeta_{6})((2\prime\prime+\zeta 4+\zeta_{5}’+\zeta’6)\zeta 4’$$f_{6}=y_{3}(\zeta_{1}’+\zeta_{3}’+\zeta_{4}’)(\zeta’’2+\zeta 4)-((_{1}’+(_{2}r+\zeta 3\prime J+\zeta 4)\zeta_{4}J$.

Regarding$f_{1}=\cdots=f_{6}=0$ ,(5)

as a system of equations for$\zeta_{1}’,$$\cdots$,$\zeta_{6}’$with coefficients in the function field$\mathrm{C}(x, y)$, we are goingto solve the system (5). If $Xj,$ $yj(j=1,2,3)$ satisfy an algebraic equation $\Psi’(X, y)=0$ , thesystem (5) has a non-trivial solution.
$\mathrm{P}(2,7)=C’(\triangle, D_{4})$$i^{\mathrm{F}\mathrm{r}}\mathrm{o}\mathrm{m}$the construction, the hypersurface $\Psi’(x, y)=0$ in

is nothing but the subvariety$Y_{\Delta,D_{4}}(\triangle(E_{6}))$(cf.[Se4], \S \S 1,4). By a little

223

lengthy computation, we conclude that $\Psi’(x, y)$ coincides with $\Psi(x, y)$ up to a constant factor,
where $\Psi(x, y)$ is the polynomial introduced before.

In this way, we can prove Theorem 2.

\S 5. Comparison beween$Y_{\Delta(E_{6}),D_{4}}(\triangle(D_{4}))$

and$Y_{\Delta(E_{7}),D\mathrm{s}}(\triangle(E_{6}))$

It is worthwhile to compare similarities between hypersurfaces of the$Y_{\Delta(E_{6}),D_{4}}(\triangle(D_{4}))$of $c(\triangle(E_{6}), D_{4})$ introduced in [Se4], \S 3 ($\pi\urcorner$

A$\mathrm{r}\supset\tau\bigcap_{}$$\mathrm{c}\mathrm{f}.[\mathrm{N}],$$[\mathrm{L}]$.

$5^{\ell h}$kind and the subvariety, [Sh2]).

$\mathrm{T}$We give here an explanation on TABLE I.
(6.1) Let $S$ be a non-singular cubic surface inof three lines on$\mathrm{P}^{3}$. An Eckardt point on$S$

is the intersection(cf. [N]). Every cubic surface does not have an Eckardt point. On the other$S$

hand, a flex of a non-singular plane quartic $C$ is a point $p\in C$ such that there is a line

tangent to

$C$

at

$p$

(cf. [Sh2]). A flex is ordinary if$l\cap C$$l$triplyconsists of two points and a flex is special

if $l\cap C=\{p\}$ . Every plane quartic does not have a special flex.
(6.2) In [N], the parameteras a rational function on$\lambda$

was introduced. It was shown, in [Se3] (cf.

$\mathrm{P}(2,6)$$\triangle(D_{4})$of type(6.4) If$\delta_{5}(t)$$D_{4}$) that$\lambda$

is regarded. In fact, using the notation in [Se3], we have$\lambda=\frac{x_{2}(X_{1}-1)(y_{1}-y2)(y2-1)}{y_{2}(x_{1}-X2)(x_{2}-1)(y1-1)}$(6.3) Is it possible to regard$1\mathrm{H}]$$Y_{\Delta(E_{6}).D_{\mathrm{a}}}(\triangle(D_{4}))$.

as a cross ratio variety for the root system?

is a $W(E_{6})$ -invariant polynomial of degree 5 (which is unique up to a constantfactor), it is shown in [Se3] that the polynomial $P_{5}(t)=\delta_{\mathrm{s}}(t_{1}, t_{2}, t_{3}, t_{4}, t6, -3t5)$ is $W(F_{4})$ -semi-

invariant under the notation there. Hence, by $W(F_{4})$ -action, we obtain totally 45 quintic polynomials on the standaxd representation space of $W(E_{6})$ . For the ake of convenience, we call these$s$polynomials associated quintics. There is a 1-1 correspondence between the set of associatedquintics and that of the 45 triple tangent planes.

Similarly, there is a $W(E_{7})$ -invariant polynomial$\delta_{E_{7}}(t)$of degree 10 which plays a role analogues

224

to

$\delta_{5}$. (The construction of(6.5) (cf. [L]) Let

points$P_{1},$$\cdots$,$\cdots$,

and a line

$P_{5}$

is that the line

$P_{1},$$\delta_{E_{7}}(t)$$L$$P_{6}$$L$

will be given later.)be 6 points ofpassing throughis also a tangent of

$C$

at

$P_{5}$. We consider a conic$\mathrm{P}^{2}$$P_{5},$$P_{6}$. Let

passing through five$C$

. The condition corresponding to $\lambda-1=0$$P_{1},$$\cdots,$

be 7 points of$P_{7}$$\mathrm{P}^{2}$. The condition

corresponding to $\Psi(x, y)=0$ is the main subject in the previous section. Namely, let
be 7 points of$\mathrm{P}^{2}$. We consider a cubic curve$C$passing through seven points

is a double point. The condition corresponding to $\Psi(x, y)=0$ is that

$P_{7}$We are going to explain the construction onLet$\omega_{j}$be a fundamental weight of$\triangle(E_{7})$$\delta_{E_{7}}(t)$$C$$P_{1},$$\cdots,$$P_{7}$$P_{1},$,$\cdots$$P_{7}$

such thathas a cusp at$P_{7}$.

.

such that$<\omega_{j},$$\alpha_{k}>=\delta_{\mathrm{j}k}$. Then$\omega_{7}$belongs to theset of weights of the 56 dimensional irreducible representation of the simple Lie algebra of type
$E_{7}$. By definition, $\omega_{7}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+6\alpha_{4}+5\alpha_{5}+4\alpha_{6}+3\alpha_{7}$ . The totality

such that$<\omega_{7},$$\alpha>=0$

form a root system of type

$E_{6}$. Let

irreducible representation of the simple Lie algebra of type$\Pi$

. Then$z_{p}= \sum_{\omega\epsilon\Omega_{27}}\omega^{p}$$(p=1,2, \cdots)$$\Omega_{27}$

$\Pi$

of

$\alpha\in\triangle$

be the set of weights of 27

corresponding to the root system$E_{6}$

are $W(\Pi)$ -invariant polynomials. (From the definition,$W(\Pi)\simeq W(E_{6}).)$

Lemma 5.1.
$P(t)$$=$

Under the notation above,$43545600\omega_{7}^{1}\mathit{0}-3628800_{Z}2\omega_{7}^{8}+100800_{Z_{27}}2\omega^{6}+725760_{Z}5\omega_{7}\mathrm{s}$$+(42\mathrm{o}\mathrm{o}_{Z^{3}}2-6048\mathrm{o}\mathrm{o}Z6)\omega_{7}^{4}$–20160$z_{2}z_{5}\omega_{7}^{3}+(175Z_{2}4-40320_{z_{2Z6}}+18144\mathrm{o}z_{8})\omega_{7}^{2}$$+(3360z_{2}^{2}Z\mathrm{s}-57600_{Z)\omega_{7}}g+1008z_{5}^{2}$

is a $W(E_{7})$ -invariant polynomial of degree 10.
For simplicity, we put $P=P(\omega_{7,2,\mathrm{s},\mathit{6}z_{8,9}}ZZz,Z)$ .

$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$the definition,$P$

is defined on thestandard representation space $E$ of $W(E_{7})$ . Therefore $P=0$ is a hypersurface on

Similarly, $P(-2\omega 7, z2, Z5, z6, z8, z_{9})=0$ defines a hypersurface
Theorem 5.2. The closure of$cr_{D_{4},\Delta}(H_{w_{7}})$$H_{\omega_{7}}$is isomorphic toin$\mathrm{P}^{6}$$\mathrm{P}^{6}=\mathrm{P}(E_{\mathrm{C}})$.

.

$Y_{\Delta(E_{7}),D_{4}}(\triangle(E_{6}))$by $W(E_{7} )$-action.

$i$

From the theorem above,$\delta_{E_{7}}(t)=P(\omega_{7},$$z_{2,5}Z,$ $z_{6},$ $Z_{8,z)}9$

is a required $W(E_{7})$ -invariant poly-

nomial. Therefore $P(-2\omega 7, Z2, Z\mathrm{s}, Z6, z8, z9)$ is an associated $W(E_{7})$ -inva

10. There are totally 28 associated hypersurfaces in
It is interesting to characterize the polynomial

homogeneous degree 10.

$C(\triangle, D_{4})$$\delta_{E_{7}}(t)$$7\dot{\mathrm{t}}ant$

polynomila of degree.

among $W(E_{7})$ -invariant polynomials of

225

Appendix. Theorem 4.2, Theorem 5.2(1) \S 4$\emptyset \mathrm{R}\Psi(X, y)g)_{\overline{\mathrm{p}}}^{-\ovalbox{\tt\small REJECT} x}=[] \mathrm{g}\theta\backslash \prime \mathfrak{h}\lambda_{\mathrm{A}}\ovalbox{\tt\small REJECT}$$l^{\vee}.\supset\backslash T$.

–$\Re\pi\infty\Phi\backslash ’\backslash$$\Re \mathrm{R}\infty \mathrm{E}^{\backslash }J$ス.

$\#\mathrm{b}T_{\overline{\mathrm{Q}}}^{\approx}-\dagger \mathrm{g}_{\mathrm{b}}T’\mathrm{t}\mathrm{e}\mathrm{g}i\iota\gamma.\sim 21^{\backslash }\mathrm{E}\mathfrak{h}\emptyset\pi\theta^{\backslash \yen}\backslash \backslash \mathrm{b}\iota,,\backslash arrow\backslash \geq\epsilon_{\overline{\prime \mathrm{T}\backslash }}\mathrm{c}f.\vee$$t_{\sim}^{\wedge}$

1,800$\mathrm{r}_{\overline{\mathrm{T}}}\mathrm{F}/\mathrm{R}\mathrm{f}\mathrm{o}\mathrm{R}\}^{\sim}.\prime x_{\mathcal{D}}t.arrow$$\mathrm{b}\yen_{\mathrm{b}C;}\mathrm{e}\lambda r\vee.\theta^{\theta}\backslash \backslash$(2)$\Re\emptyset$,.

$\ovalbox{\tt\small REJECT}_{\mathrm{J}^{\backslash }}\not\subset,$$\ovalbox{\tt\small REJECT}\pm 1^{\backslash }\underline{\mathfrak{F}}\circ?\#\mathrm{i}\Re \mathrm{b}$$\overline{\tau}$ス$10,000\uparrow\prime \mathrm{r}\overline{T}\neq \mathrm{R}\emptyset\pi k$f.-risa/asir$\xi$$\epsilon_{3}+\epsilon_{4}-\epsilon_{5}-\epsilon_{6})\}_{\sim}^{\sim}\mathrm{x}\backslash \iota’\Gamma^{\backslash }\mathrm{b}\backslash \tau_{6\emptyset\phi\backslash }\backslash ^{\backslash }$$=\ovalbox{\tt\small REJECT} \mathrm{F}\cdot 0f.\backslash$.

$x_{1},$ $x_{2},$$\mathfrak{h}$$T$weight$k^{\backslash }\lambda\star\}_{\sim}^{\sim}@\tau\emptyset$}$*\mathrm{a}\mathrm{e}\epsilon\not\in_{\overline{\beta}\iota\backslash }^{-- \mathrm{z}t}-T6^{\backslash }\mathrm{c}\succeq$$x3,$ $y1,$ $y_{2},$ $y3kt\sigma)\#\backslash \mathrm{A}H\not\in x\tau\ovalbox{\tt\small REJECT} \mathrm{b}\tau,$.

$\mathrm{f}*\ovalbox{\tt\small REJECT}\Leftrightarrow$$\Phi(x, y)$$\geq*\backslash \iota’\Gamma^{\backslash }\llcorner\backslash \tau 6.56^{\backslash }\mathrm{A}\overline{\pi}\ovalbox{\tt\small REJECT}$.

$F(t)\dagger \mathrm{g}_{t}\mathrm{o}\mathrm{R}\succeq h\epsilon\geq 5,000\cap^{\prime-}\ovalbox{\tt\small REJECT}\neq \mathrm{R}\mathrm{o}T\mathrm{a}6$@.

$\mathrm{f}\mathrm{f}\mathrm{l}\mathcal{T}^{\sigma)\}}\mathrm{f}\Xi^{\mathrm{B}}\mathrm{f}\mathrm{l}\tau*\}\mathrm{g}\prime_{\mathrm{f}}‘\vee\backslash ‘ \mathrm{r}_{\grave{7}\}_{-\theta^{\backslash }h}^{\sim}}\mathrm{b}^{\mathrm{p}}$il$T\ovalbox{\tt\small REJECT}\wedge^{\backslash }Th\gamma_{-}.$.

t&\llcorner T$\mathrm{i}\mathrm{E}\llcorner 4^{\backslash }\theta\backslash \mathrm{i}\mathrm{E}\mathrm{b}$$\langle$$\Re \mathrm{E}\mathrm{L}T\}\mathrm{f}\mathrm{i}\mathrm{E}\mathrm{b}^{(}\prime 1_{\overline{arrow}}\geq i’\backslash bl\backslash _{\mathcal{D}}f^{-}.$$\succeq t^{\sigma\overline{\sim}}\pi\sim_{\backslash }*\gamma_{-}.$.

$f\mathrm{y}I^{1t}’\backslash$.

を\mbox{\boldmath $\kap a$}\mbox{\boldmath $\lambda$}

computer$\not\in^{-};\iota\tau,$

5,000

.

$\pm(\epsilon_{1}+\mathcal{E}_{2}+$il 68\yen ffi.
1,800

t\sim \tilde tt\mbox{\boldmath $\lambda$}\tau 6&t$rightarrow\emptyset_{\mathrm{O}}^{\equiv\ovalbox{\tt\small REJECT}\#}\vee- \mathrm{f}$$(^{\text{ノ_{}\overline{\mathrm{T}}}}\ovalbox{\tt\small REJECT}_{\mathrm{f}}\circ \mathrm{R}\sigma)\mathrm{R}ij\pi f\overline{l}$$\emptyset \mathrm{f}\mathrm{i}\mathrm{E}\mathrm{R}\theta^{\backslash \prime}\backslash l\tau \mathrm{b}\backslash \mathrm{B}$

.

il 6

$\phi\backslash \backslash \backslash$,.

$-\backslash *\emptyset \mathrm{E}\mp[] \mathrm{f}\tau\wedge^{\backslash }\tau\backslash \mathrm{g}\Re\Phi\#^{\wedge}arrow\lambda\backslash \iota\Gamma \mathrm{L}\backslash \backslash T6$\tau ‘\varpi ^‘\mbox{\boldmath $\pi$}を$\emptyset\phi\backslash \gamma\backslash \mathrm{i}\not\in-\emptyset\Re \mathrm{j}\Phi$\mathrm{k}\mathrm{t},\backslash \cdot,$$\prime x\hat{\pi}\Re p\epsilon*dbX,$$l_{\sim \mathrm{p}}^{\vee\equiv}-\mathrm{f}\mathrm{f}\mathrm{l}6*r.\backslash \geqarrow 6\vee,$.

$\ll\cdot i\mathrm{t},$}$\ovalbox{\tt\small REJECT} \mathrm{g}$$\vee.\mathrm{t}_{)}\succeq$$\geq\backslash \vee^{\backslash }J7\mathfrak{o}7\mathrm{a}k\yen\check{X}_{-r-}.$$\llcorner T\mathrm{b}\Re \mathfrak{h}\mathrm{x}\cdot\supset[] \mathrm{f}^{-r\tau k}6\ \yen\grave{\mathrm{x}},$.

$1^{\backslash }\underline{\mathrm{i}5}\Leftrightarrow^{\gamma}p\mathrm{f}\llcorner \mathrm{g}\mathrm{g}\mathrm{f}\mathrm{t}\lambda\llcorner$$p=1r_{x6}\dagger X_{\vee}^{/}\grave{\mathrm{J}}\prime_{j}$$\langle$$\geq \mathrm{s}_{)}\mathrm{a}_{66}$$\mathrm{t}\prime 1\langle\cdot\supset\theta^{1}6\mathfrak{R}\llcorner\S\epsilon \mathrm{f}\mathrm{f}\lambda \mathrm{b}\tau \mathrm{f}\mathrm{f}\mathrm{l}\theta^{\backslash \emptyset}f^{\sim}.\succeq-.6\iota’\backslash \tau \mathfrak{j}\iota \mathrm{s}_{)}\mathbb{E}\llcorner \mathrm{V}^{\backslash }\check{arrow}$$<^{-}t\iota\tau.tk_{\wedge}^{m\Re\ }\mathrm{b}T_{\mathrm{p}}^{\equiv}\Phi-\llcorner T\mathrm{f}\mathfrak{F}\epsilon_{\overline{/\mathrm{T}}}\backslash \mathrm{b}\gamma.\sim$$[] \mathrm{f}t\frac{\backslash }{p_{\vee}}\ hT$$68_{\mathrm{T}’\mathrm{f}\mathrm{f}\mathrm{i}}^{\underline{\ulcorner}}\overline{\backslash _{\underline{\backslash }}}$.

$E_{6}\emptyset\ovalbox{\tt\small REJECT}_{\square }\mathrm{A}t\wedge\sim\epsilon\grave{\eta}\prime_{J}0\tau\iota’\backslash r.’$6\% ffi_{\mathrm{L}}$$k^{\underline{\backslash }}-\tau\prime^{J}4\ulcorner\overline{\mathrm{T}}\llcorner\tau$$\pm(\gamma_{1}\pm 2\epsilon_{j})$$\Phi \mathrm{g}\prime_{\zeta}.\pi\Re \mathrm{A}\wedge \mathrm{a}\mathrm{e}\pi\#_{\sim^{\mathit{1}}}^{\wedge}\supset\tau F(t)\theta^{\mathrm{Y}}6W(E\tau)$$P(\mu v_{7}, z2, Z\mathrm{s}, z_{\mathit{6}}, z_{8}, z9)\theta^{\backslash }\backslash W(\backslash E7)\tau\backslash v\mathrm{r}\}\wedge\cdot f\sim p6X.\backslash ,$$\mathrm{b}\mathrm{I}_{\vee}\mathrm{j}\mathrm{E}\mathrm{b}\searrow kT6^{\prime_{p}}C\supset\{x,$,$arrow\emptyset\vee\pi\xi 8\mathrm{e}_{t_{\sim}^{\sim}}\mathrm{E}\mathrm{R}\mathrm{f}\mathrm{f}\mathrm{l}\# T6\succeq 1*\emptyset \mathrm{E}\mp t^{*}\backslash 12\Phi$$F\gtrless \mathfrak{h}\emptyset \mathrm{E}\mp\phi^{\backslash ^{\backslash }}\backslash F(t)=P(-2\omega_{7}, z_{2,\mathrm{s}}z, Z6, Z8, Z9)t_{\sim}^{\sim\gamma}x6$$.\supset.\backslash \tau,$$\mathrm{b}T\#\ovalbox{\tt\small REJECT}_{6\S}$weight$4.3-C_{\grave{\mathrm{i}}^{\mathrm{g}_{\mathrm{p}}}}\lambda \mathrm{b}f.’\Phi(x, y)\tau_{j\in}^{rightarrow}\ovalbox{\tt\small REJECT} \mathrm{s}$$\mathrm{f}\sigma)k^{\backslash }\mp\sigma)\Xi \mathrm{B}fl\vee \mathrm{c}t_{J}\iota\prime 1\mathrm{E}\mp\}\mathrm{f}22\lambda\#\lambda x\tau \mathrm{a}\mathrm{e}\epsilon$

ae ,

E&hW

$\mathrm{N}\mathrm{S}/\mathrm{R}T\ovalbox{\tt\small REJECT} \mathrm{I}\rfloor \mathrm{f}\mathrm{f}\mathrm{l}\llcorner T_{\mathrm{p}}^{\equiv}\Phi-$$\pm(\mathcal{E}:_{1}+\mathcal{E}:_{2}+\epsilon:_{3}+\epsilon_{4}.\cdot-\epsilon:_{5}-\epsilon:_{\mathit{6}})$Lemma. simple reflection$\Phi_{\wedge}^{\varpi}\backslash \mathrm{f}\mathrm{f}\mathrm{i}\tau\Phi i\mathrm{J}\frac{\pi}{\mathrm{J}\sigma}X\Re\wedge \mathrm{b}T\iota’\backslash []_{L^{\}\mathrm{f}^{\backslash }\backslash }}\mathfrak{u}\backslash$(3)$\mathrm{J}3100T\ovalbox{\tt\small REJECT}^{1}\mathrm{J}$PC9801$\omega_{7}=(\gamma_{1}-2\epsilon_{6})k^{\mathrm{Y}}t\sigma_{-}^{\backslash }\omega_{7}\}_{\sim}^{\sim}\lambda\backslash 1\Gamma\llcorner\backslash \backslash \tau 6\emptyset\theta^{\backslash }\backslash \Psi(x, y)\backslash =0\tau_{j\Xi}’\not\in \mathrm{g}$

ir }

–

$\mathrm{f}$,$\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{a}s$$\circ\ovalbox{\tt\small REJECT}|\mathrm{J}\mathrm{f}\mathrm{f}\mathrm{l}$f.-.

$\check{arrow}n^{\gamma}J\mathrm{b}i\backslash r_{J}\mathfrak{U}\ovalbox{\tt\small REJECT}\}^{\sim}.ffi\mathrm{p}\mathrm{N}\mathrm{g}-arrow\ovalbox{\tt\small REJECT}\}i$$\pm(\mathcal{E}_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}+\epsilon \mathrm{s}+\epsilon_{\mathit{6}})$-C $b6$ .

ム

REDUCE3.4 $k$ Toshibaム$Y_{\Delta,D_{4}}(\triangle(E_{6}))\emptyset\Psi/\nearrow \mathit{0})C(\triangle, D4)\emptyset_{\mathrm{O}}^{f}\beta t^{\backslash }\mathrm{J}\ovalbox{\tt\small REJECT}\Re\Phi\}\mathrm{f}56^{\backslash }\mathrm{A}\overline{\pi}\ovalbox{\tt\small REJECT}\Re\emptyset$

weight }$\overline{\tau}$.

)$\delta l^{J}$$\backslash

t_{\zeta^{\vee}}\nearrow..k\gamma\approx.t\backslash P*(-2\omega 7, Z2, Z_{5}, Z6, z8, Z9)$$4^{\text{ノ}}\overline{\mathrm{T}}\mathrm{r}_{\mathrm{f}}\mathrm{B}\succeq\theta\backslash t‘\zeta \mathfrak{h}\mathrm{e}\iota\prime \mathrm{Y}\pi r_{arrow}\underline{\mathrm{p}}\phi\backslash P(\backslash ^{\backslash }\omega 7, Z2, Z5, Z_{6}, Z_{8,9}z)\}\mathrm{f}270\cap^{\prime-}\ovalbox{\tt\small REJECT}_{\neq}\mathfrak{o}_{\mathrm{R}\emptyset}i)kk\ddagger \mathrm{b}\wedge h\backslash ^{\backslash }$$\}\mathrm{f}^{\backslash }t^{1}\gamma_{J\mathfrak{h}}\Phi\iota’\backslash \mathrm{R}l^{\sim}\sim^{\gamma}J\cdot\supset r.\sim$.

\S 6. Relations between$C(\triangle, D_{4})$and cubic surfaces.

At a meeting organized by H. Yamada held in RIMS, Kyoto University (December, 1993),
I. Naruki and J. Matsuzawa gave talks on a root system construction of universal cubic sur-faces.

They constructed a fibre space$C(=C(\triangle(E_{\mathit{6}}), D_{4}))$$\tilde{C}$of cubic surfaces over Naruki’s cross ratio variety

so that the natural projection$\varpi$:$\tilde{C}arrow C$

is

and

.

In this section, we discuss a relation between$C(\triangle, D_{4})$For this purpose, we introduce the setof

if $i\neq j$ . By definition,$\overline{\mathrm{P}}(2, k)$$\tilde{\mathrm{P}}(2, k)$$k$points$\tilde{C}$$W(E_{\mathit{6}})$$(P_{1}, \cdots , P_{k})$admits a $PGL(3)$ -action. Let$\mathrm{P}(2, k)$-equivariant.

in$\mathrm{P}^{2}$such thatbe the quotient of.

$P$$\neq P_{j}$$\tilde{\mathrm{P}}(2, k)$226

by $PGL(3)$ . It is clear thatprojection
$i$

$p$

is nothing but the one introduced in \S 2. There is a natural$\mathrm{P}(2,7)$

of $\mathrm{P}(2, k+1)$ to

defined by $p((P_{1}, \cdots , P_{k}, P_{k+1}))=(P_{1},$$\mathrm{P}(2, k)$$\cdot$.. ,$P_{k})$.

From now on, we focus our attention to the cases $k=6,7$ . It is known (cf. [Se4]) that

there is a birational $W(E_{k})$ -action on $\mathrm{P}(2, k)(k=6,7)$ . This easily implies that the projection$p:\mathrm{P}(2,7)arrow \mathrm{P}(2,6)$

is $W(E_{6})$ -equivariant. Denoting by

obtain a biratioanl $W(E_{6})$ -equivariant map$\tilde{p}$$\tilde{p}$the extension of$\triangle(E_{6})$$\Omega_{1}$, we$C(\triangle, D_{4})$. Thereconsists of those corresponding to roots contained in:$Y_{7}$,Y.

$\Omega_{2}$$(1 \leq i<j<k<7)$ .

$Y_{jk}$consists of the remaining 27 hypersurfaces:$(1 \leq i<7)$ ,.

$(1 \leq i<7)$ ,$Y.\tau$

For any $(P_{1}, \cdots , P_{6})\in \mathrm{P}(2,6)$ , the closure
is of dimension 2. The surface$C(\triangle, D_{4})$.

$(1 \leq i<j<7)$ ,$Y_{1j}$The second one denoted by$\Omega_{2}$$C(\triangle, D_{4})$: $C(\triangle, D_{4})arrow C(\triangle(E_{6}), D4)$ .

We consider the $W(E_{6})$ -orbits of the set of hypersurfaces of the $1*t$ kind inare two orbits. The first one denoted by

to

$p$

$(1 \leq i<j<7)$ .

$Y_{j7}$$S(P_{1}, \cdots , P_{\mathit{6}})$

intersects with all the hypersurfaces of. By the intersection relations among hypersurfaces of theintersection relations among the 27 curvesof its fibre in$S(P_{1}, \cdots , P_{\mathit{6}})=\overline{p^{-1}((P_{1},\cdots,P6))}$$1^{s\ell}$kind, we easily find that the$S(P_{1}, \cdots , P_{\mathit{6}})\cap Y(\forall Y\in\Omega_{2})$on $S(P_{1}, \cdots , P_{6})$ are sameas those of the 27 lines on a non-singular cubic surface.

If the interpretation of the work of Naruki and Matsuzawa is correct,

and$\tilde{p}$$\tilde{C}$coincides with

: $C(\triangle, D_{4})arrow c(\triangle(E_{6}), D_{4})$ defined above is the natural projection$C(\triangle, D_{4})$. As an easy$\varpi$consequence (?), $S(P_{1}, \cdots , P_{6})$ is a cubic surface. Therefore it is hopeful that $S(P_{1}, \cdots , P_{6})\cap Y$$(\forall Y\in\Omega_{2})$are the 27 lines on it. If this is true, hypersurfaces ofof cubic surfaces in the total space$i$

From the definition,

outside the union So$Q_{1},$$\cdots$,$Q_{7}$$\mathrm{P}(2,7)$$(A_{6})$.

is identified with the open subset of $(x_{1}, X_{2}, x_{3}, y1, y2, y3)$ -spaceof the $(X_{1)}X_{2}, x_{3,y1y},2, y3)$ -space in order to define

We now take seven points$P_{7}$$C(\triangle, D_{4})$$P_{1},$$\cdots$

is regarded as a point on

an inhomogeneous coordinate ofcorresponds to the line on$\mathrm{P}^{2}$$\mathrm{P}^{2}$,$P_{6},$

is nothing but$P_{7}$$Y_{1jk}$

Theorem 26.2).

$\mathrm{P}_{0}(2,7)$. It is clear that the closureand that ofis$Q_{j}$$Y_{j}$.

as in \S 4 and fix $P_{j}(j=1, \cdots, 6)$ for the moment.

$\mathrm{P}^{2}-\{P_{1}, \cdots, P_{6}\}$. Therefore$(x_{3}, y_{3})$are interpreted as. Under this identification, the defining equation $D_{j7}.=0$

passing through$P_{i}$

and$P_{j}$

for $i,j(1\leq i<j<7)$ . On the other

hand, the defining equation of Q. corresponds to the conic on$\{P_{\mathrm{j};}j=1, \cdots , 6, j\neq i\}$are global sections of 27 linesof 28 hyperplanes (cf. \S 2). Moreover, we introduced 7 hypersurfacesof the hypersurface $D_{gk}.=0$ inThen$\tilde{C}$$\Omega_{2}$$\mathrm{P}^{2}$

passing through the five points. This is a geometric interpretation of hypersurfaces$Q_{1},$$\cdots$,$Q_{7}$(cf. [M],227

In [Se4] \S 3, we have studied the structure of subvarieties of the form $Y(M)$ in$M$

is a subset of

the intersection$\triangle$$C(\triangle, D_{4})$, whereconsisting of mutually orthogonal positive roots. In particular, we now treat$Y_{\alpha}\cap Y_{\beta}$for$\alpha,$$\beta\in\triangle$

such that$Y_{\alpha},$$Y_{\beta}\in\Omega_{2}$. The intersection

regarded as a global section of the intersection of two lines of cubic surfaces$Y_{\alpha}\cap Y_{\beta}$

may be$S(P_{1}, \cdots , P_{\mathit{6}})$, beingassumed that $S(P_{1}, \cdots , P_{6})$ is a cubic surface for any $(P_{1}, \cdots , P_{6})\in \mathrm{P}(2,6)$ . It is interesting ot

make clear a relation between cubic surfaces and the global section of the intersections of twolines on them.
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