24.
$W(E_{7})$
-invariant
polynomial
of
degree
10
and
28
bitangents
of
plan
equartic
curves
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 $\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{r}$を利用する.\S 1.
The root system of type $E_{7}$.
Wefirst recall the definition of the root system $\triangle(E_{7})$ oftype $E_{7}$
.
We always denote it by $\triangle$for simplicity in this paper. Let $\overline{E}$
bean inner product spaceofdimension 8 with an orthonormal
orthogonalto $\epsilon_{7}+\epsilon_{8}$
.
As in [Se4],\S 4,
we define the following63
vectors of$E$: $\gamma_{1}=\epsilon_{8}-\mathcal{E}_{7}$, $\gamma j=\epsilon_{j-1}-\gamma 0+\gamma_{1}$ , $\gamma_{1j}=-\epsilon_{j-1}+\gamma_{\mathrm{O}}$, $(1 <j<8)$$\gamma_{jk}=\epsilon_{j-1}-\epsilon_{k-1}$, $\gamma 1jk=-\epsilon_{j-1}-\epsilon_{k-1}$,
$(1 <j<k<8)$
$\gamma ijk=-\epsilon:-1-\epsilon_{j-1}-\epsilon_{k-1}+\gamma 0$,$(1 <i<j<k<8)$
where$\gamma 0=\frac{1}{2}\sum_{j=1}\epsilon_{j}8-\epsilon_{7}$
.
The totality $\triangle \mathrm{o}\mathrm{f}\pm\gamma \mathrm{j},$ $\pm\gamma jk,$$\pm\gamma.jk$ isa rootsystem of type $E_{7}$ (cf. [B]).As a fundamental set of roots of $\triangle$, we may take
$\alpha_{1}=\gamma_{12}$, $\alpha_{2}=\gamma_{123},$ $\alpha_{3}=\gamma_{23},$ $\alpha_{4}=\gamma_{34},$ $\alpha_{5}=\gamma_{45}$, $\alpha_{6}=\gamma_{56}$, $\alpha_{7}=\gamma_{67}$.
Then the corresponding Dynkin diagram is:
$\alpha_{1}$ – $\alpha_{3}$ – $\alpha_{4}$ – $\alpha_{5}$ – $\alpha_{6}$ — $0_{7}$
$1$
$\alpha_{2}$
We denote by$\triangle^{+}$ theset ofpositiverootsin $\triangle$
.
It is easy tosee that $\triangle^{+}$ consists of$\gamma_{i},$$\gamma_{\iota_{J}},$$\gamma i\mathrm{J}k$.
If$gj$ is the reflection on $E$ with respect to the root $\alpha_{j}$, the group generated by $g_{1},$$\cdots$,$g_{7}$ is
the Weyl group $W(E_{7})$ of type $E_{7}$
.
In the sequel, we frequently identify $W(A_{6})\simeq\Sigma_{7}$ (resp. theWeylgroup $W(E_{6})$ oftype $E_{6}$) with the subgroup of $W(E_{7})$ generated by $g_{1},$$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
$h_{j}=\gamma j(t)$, $h_{jk}=\gamma jk(t)$, $hijk=\gamma_{1}jk(t)$, $(t\in E)$
.
\S 2.
The configuration space of7 points in $\mathrm{P}^{2}$.
Webriefly review thedefinition of the configuration spaceof7 points of$\mathrm{P}^{2}$ which we
denote by
$\mathrm{P}(2,7)$
.
We first define the vector space$M_{3,7}$ of$3\cross 7$ matrices. Then $M_{3,7}$ admits $GL(3)\cross GL(7)-$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 the $3\cross 3$ matrix consisting of the $i,j$,k-th column
vectors of$X\in M_{3.7}$. If$M_{3.7}’$ is the subset of$M_{3,7}$consisting of$X$ with D.jk$(X)\neq 0$ $\forall(i,j, k)(i<$
$j<k)$, we denote by $\mathrm{P}(2,7)$ the quotient of $M_{3,7}’$ by the action $GL(3)\mathrm{x}D(7)$
.
It is possible tochoose as a representative ofany element of$\mathrm{P}(2,7)$ a matrix ofthe form
In this way, $\mathrm{P}(2,7)$ is regarded as a quasi-affine subset of $\mathrm{C}^{6}$ by the correspondence
$arrow(_{X_{1},x_{2},x_{3}}, y1, y2, y3)$
.
In fact, $\mathrm{P}(2,7)$ is identified with $\mathrm{C}^{6}-^{s_{\mathit{0}}()}A_{6}$, where
So
$(A_{6})$ is theunion of the 28 hypersurfacesbelow:
$x_{i}=0$, $xi– l=0$, $y:=0$, $y$
.
$-1=0$, $x:-x_{\mathrm{j}}=0$, yi–yj $=0$, xi–yi $=0$,xiyj $-x.yj=0$,
$(1-x.)(1-yj)-(1-x_{j})(1-y:)=0$
,$\varphi\iota(X_{1}, x_{2}, x3, y1, y_{2}, y_{3})=\det=0$
.
We introduce the following seven birational transformations $s_{1},$$\cdots$ ,$s_{6},$$s_{R}$:
Sl : $(x_{1} , x_{2}, x_{3}, y1, y_{2}, y_{3})arrow(1/x_{1},1/x_{2},1/x_{3}, y_{1}/x_{1}, y_{2}/x_{2}, y_{3}/x_{3})$
$s_{2}$
:
$(_{X_{1},XX_{3,y1,y}}2,2, y3)arrow(y_{1}, y_{2}, y3, x_{1,2}x, x3)$$s_{3}$ : $(_{X\iota,x_{23}}, x, y1, y_{2}, y_{3})arrow(_{X_{1}^{\prime J}}, X_{2}, x_{3}, y1, y2 , y_{3})Jr\prime\prime$
$s_{4}$ : $(_{X_{1},x_{2},x_{3},y}1, y_{2}, y3)arrow(1/x_{1}, x_{2}/x_{1}, xs/x_{1},1/y_{1}, y_{2}/y_{1}, y_{3}/y_{1})$
$s_{5}$ : $(_{X_{1},X_{2}}, x_{3}, y1, y2, y3)arrow(x_{2}, x_{1}, x3, y_{2}, y_{1}, y_{3})$
$s_{6}$ ; $(_{X_{1},x_{2}x_{3},y,y_{2,y3}})1)arrow(_{X_{1},X_{3}}, x_{2}, y1, y3, y2)$
$s_{R}$
:
$(_{X_{12}}, X, X3, y_{1}, y2, y3)arrow(1/x_{1},1/x_{2},1/x_{3},1/y_{1},1/y_{2},1/y_{3})$where
$x_{\acute{J}}= \frac{x_{j}-yj}{1-yj}$, $y_{j}’= \frac{yj}{y_{\dot{J}}-1}$, $j=1,2,3$
.
The correspondence
$g_{1}arrow s_{1}$, $g_{2}arrow s_{R}$, $gjarrow s_{j-1}(j=3, \cdots, 7)$ induces a group isomorphismof $W(E_{7})$ to thegroup generated by $s_{1},$$$\cdot$
.
,$s_{6},$$s_{R}$
.
We introduce7 polynomials of($X_{1},$ $X_{2},$$X3,$$y_{1y2,ys)}$, defined by
$\sigma \mathrm{s}(X1,$$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 $Q_{j}$ be the hypersurface in $\mathrm{C}^{6}$ defined by
$\sigma_{j}=0$ $(j=1, \cdots , 7)$
.
Then it is easy to seethat $\Sigma_{7}$ acts on the set
$\{Q_{1}, \cdots , Q_{7}\}$ as a permutation group. If $\tilde{\sigma}_{7}=\sigma_{7}$ and $\tilde{\sigma}_{j}=\tilde{\sigma}_{j+1}\mathrm{o}s_{j}$
$(j=1,2, \cdots, 6)$, and $Q_{j}’$ is the hypersurface in $\mathrm{C}^{6}-S\mathit{0}(A_{6})$ defined by $\tilde{\sigma}_{j}=0$, then $Q_{j}$ is the
Zariski closure of $Q_{j}’$ in $\mathrm{C}^{6}$. A geometric
meaning of $Q_{j}$ will be given in
\S 6.
In the sequel, wedenote by
Po
$(2, 7)$ the complement of the union $S(E_{7})$ ofSo
$(A_{6})$ and $Q_{1},$ $\cdots$,$Q_{7}$.
Clearly all theelements of$W(E_{7})$ induce biregular transformations on
Po
$(2, 7)$.\S 3.
The crossratio
variety $c(\Delta(E_{7}), D_{4})$.
For any subroot system $\triangle_{1}$ of type $D_{4}$ in $\triangle$, we defined a
$D_{4}$-cross ratio map of the Zariski
open subset $Z(\triangle)$ ofthe projective space $\mathrm{P}^{6}=\mathrm{P}(E_{\mathrm{C}})$ associated to the complexification $E_{\mathrm{C}}$ of
$E$ to $CR(\mathrm{P})\simeq \mathrm{P}^{1}$
.
There are totally315
subroot systems of type$D_{4}$ in $\triangle$
.
The corresponding $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 :-h_{i\iota\cdot 4}h:_{\mathrm{t}^{ii}}34h_{i}2.5hi_{23}.:_{\mathrm{s}} :h_{12}..h_{1}.:_{2^{i}}3h:_{4^{i}}\mathrm{s}hi_{3}i4i\mathrm{S})$
$cr_{[\cdot\iota 2}^{2}.,,:_{345},,,i_{6}]=$ $(h_{i_{1^{\mathrm{i}}3}}.\cdot\cdot h\mathrm{s}^{h_{i_{2^{1\triangleleft^{i}52}}}}\cdot.:3i6h_{i_{1}}:_{46}i :-h_{23}.\cdot::_{\mathrm{s}\iota}h.\cdot:_{4}i\epsilon^{h.h}.\iota:3:6:_{2^{i_{46}}}.\cdot :h:\iota:2h.\cdot:34h.\cdot i_{67}h.\cdot)5$
$cr_{1ii]}^{3}i_{1234}i,=(h_{1^{i}2^{i}\tau}.\cdot h_{34}.\cdot i,\tau^{h:}\mathrm{s}\cdot.6h:\tau :-h_{i_{1}:_{2}}.\cdot 6h:_{\epsilon}:4:6h:_{67}.\cdot h_{6}.\cdot :h_{125}.\cdot:.\cdot h_{3}.\cdot.\cdot:_{5}h4:6’ 7h:_{\mathrm{s}})$
(cf. [Se4],
\S 4).
By taking the product of all the315
maps above, we obtain a map $cr_{D_{4},\Delta}$ of$Z(\triangle)$ to $CR(\mathrm{p})^{315}$
.
Let $C’(\triangle, D_{4})$ be the image $C\Gamma_{D_{4},\Delta}(Z(\triangle))$ and let $C(\triangle, D_{4})$ be its closurein $CR(\mathrm{P})s15$.
Forany subroot system $\triangle_{1}$ of$\triangle$,we defined a subvariety
$Y_{\Delta,D_{4}}(\triangle_{1})$ in [Se4],
\S 4.
There are fourkinds ofhypersurfaces of$C(\triangle, D_{4})$ defined as the form $Y_{\Delta,D_{4}}(\triangle_{1})$ for suitable subroot systems.
\S 4.
Hypersurfaces corresponding to subroot systems oftype $E_{6}$.
We introduce hypersurfacesof$C(\triangle, D_{4})$whichare fixedby$W(E_{\mathit{6}})$-actions(cf. [Se4],
\S 4,(4.15.10)).
If$\triangle_{1}$ is asubroot system oftype $E_{6}$ in $\triangle$, it is easy to show that
$C(\triangle, D_{4})$
.
Sucha hypersurfaceis called thatofthe$5^{th}$ kind. As abasic propertyof hypersurfacesof the $5^{th}$ 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 ratiovariety$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 $\mathrm{P}^{2}$ passing through 7 points. For simplicity, we take 7 points
$P_{1},$$\cdots$,$P_{7}$ of $\mathrm{P}^{2}$ asfollows:
$P_{1}=$ $($1 : $0$ : $0)$, $P_{2}=(0$ : 1 : $0)$, $P_{3}=(0$: $0$ : 1$)$, $P_{4}=(1$ : 1 : 1$)$,
$P_{5}=$ $($1 : $x_{1}$ : $y_{1})$, $P_{6}=(1 : x_{2} : y_{2})$, $P_{7}=(1 : x_{3} : y_{3})$
.
We assume that the 7 points above are in a general position which means the corresponding
matrix
is a representative of theconfiguration space $\mathrm{P}(2,7)$
.
Let $C(P_{1}, \cdots , P_{6} ; P_{7})$ be the cubic curve in $\mathrm{P}^{2}$ passing through
$P_{1},$ $\cdots$,$P_{7}$ such that $P_{7}$ is a
double point (cf. [M],$1\mathrm{L}]$). We now consider the case where $C(P_{1}, \cdots , P_{\mathit{6})}P_{7})$ 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 provable
that $\deg_{x\mathrm{s}}\Psi=\deg_{y\mathrm{s}^{\Psi}}=8$
.
Noting that $C(\triangle, D_{4})$ is a compactification of
Po
$(2, 7)$, we obtain a hypersurface $Y_{cu\iota}\mathrm{P}$ of$C(\triangle, D_{4})$ as the Zariski closure of the hypersurface of$\mathrm{P}_{0}(2,7)$ defined by $\Psi(x, y)=0$
.
Theorem 4.2. Ycusp $=Y_{\Delta,D_{4}}(\triangle_{1})\cap \mathrm{p}_{0^{(2,7)}}$.
The basic idea of the proof employed here is the comparison between the defining equations of
$Y_{\Delta,D_{4}}(\triangle \mathrm{l})$and $Y_{cu\iota p}$
.
Beforeenteringthe details ofitsoutline, we state aresulton the polynomial $\Psi(x, y)$.
Lemma 4.3. We put
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 $P_{7}$
.
Forthis purpose, weassume that $F(\xi 1, \xi 2, \xi_{3})=0$ isthedefiningequation 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 $\xi=$ $(\xi_{1} :\xi_{2} :\xi_{3})$as ahomogeneous coordinate of$\mathrm{P}^{2}$
.
Thecondition that $C(P_{1}, \cdots , P_{6} ; P_{7})$ passes through $P_{1},$$\cdots$,$P_{7}$ is equivalent to
(C.1) $F(P_{j})=0$, $j=1,$$\cdots,$$7$
.
The condition that $P_{7}$ is a double point of $C(P_{1}, \cdots , P_{6}; P_{7})$ is equivalent to
(C.2) $F_{\xi},(P_{7})=0$, $i=1,2,3$
.
The condition that $P_{7}$ is moreover a cusp point of $C(P_{1}, \cdots , P_{6} ; P_{7})$ is equivalent to
(C. 3) $F_{\xi_{1}\xi_{1}}(P_{7})F\epsilon_{2}\xi_{2}(P7)-F\xi_{1}\epsilon 2(P_{7})^{2}=0$
.
$i$From (C.1), (C.2), we conclude that
th.e
ratio of $c_{1},$$\cdots$,$c_{10}$ is uniquely determined.Substi-tuting such $c_{1},$$\cdot,$
.
$,$$c_{10}$ 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 reproducehere.
Our
next purpose istocomputethedefiningequation of the hypersurface$Y_{\triangle,D_{4}}(\triangle \mathrm{l})$ in $\mathrm{P}(2,7)$. For this purpose, we first recall the definition of the rational map of $\mathrm{P}^{6}$to $\mathrm{P}(2,7)$ in [Se4],
Lemma 4.2. Weput
$x_{1}(t)= \frac{h_{24}\cdot h_{234}\cdot h_{15}\cdot h135}{h_{14}\cdot h_{13}4h2\mathrm{s}\cdot h_{235}}.$, $x_{2}(t)= \frac{h_{24}\cdot h_{234}\cdot h_{1}6h_{13}6}{h_{14}\cdot h_{134}\cdot h_{26}\cdot h236}.$ , $x_{3}(t)= \frac{h_{24}\cdot h_{23}4h_{1}\gamma\cdot h_{1s7}}{h_{14}\cdot h_{134}\cdot h_{27}\cdot h237}.$,
$y_{1}(t)= \frac{h_{34}\cdot h_{234}\cdot h1\mathrm{s}.h125}{h_{\}4}\cdot h_{12}4h35h_{23}5}.\cdot$, $y_{2}(t)= \frac{h_{34}\cdot h_{234}\cdot h_{16}\cdot h126}{h_{14}\cdot h_{124}\cdot h_{36}\cdot h236}$, $y_{3}(t)= \frac{h_{34}\cdot h_{234}\cdot h_{1}7h_{12}7}{h_{14}\cdot h_{124}\cdot h_{37}\cdot h237}$
.
and define the map $F_{E_{7}}$ of$Z(\triangle)$ to the $(x, y)$-space by$F_{E_{7}}(t)=(x_{1}(t), X_{2}(t),$$x3(t),$$y_{1}(t),$$y2(t),$ $y3(t))$,
are linear forms on $E$ associated with the roots of$\Delta$
.
Nowwe put$(_{1}=h_{12}$, $\zeta_{2}=h_{123}$, $\zeta_{3}=h_{23}$, $\zeta_{4}=h_{34}$, $(\mathrm{s}=h_{45},$ $\zeta\epsilon=h_{56}$, $\zeta_{7}=h_{67}$
.
It is clear from the definition that linear forms in question corresponding to the roots of $\triangle(E_{6})$
are expressed as linear combinations of $\zeta_{j}$ $(j=1, \cdots , 6)$
.
We may take $\zeta=(\zeta_{1}$ :...
:
$\zeta_{\mathit{6}}$ : $\zeta_{7})$ as a homogeneous coordinate of $\mathrm{P}^{6}$. Now we write$\zeta_{j}=\zeta_{j}’\tau$ $(j=1, \cdots , 6)$
.
Then $(^{\sim}=((\zeta_{1}’$ :.. .
: $\zeta_{\mathit{6}}’),$$\tau)$ is also a local coordinate ofan affineopen subset defined by $\zeta_{7}\neq 0$ in $\mathrm{P}^{6}$Noting this, we write $\tilde{x}_{j}(\tilde{\zeta})=x_{j}(t)$, $\tilde{y}j(\tilde{\zeta})=yj(t)$ $(j=1,2,3)$
.
Now we put $u_{j}((_{1}’$ :.. .
: $\zeta_{6}’)=\lim_{\tauarrow 0}\tilde{x}j(\tilde{\zeta})$, $v_{j}(\zeta_{1}’$ :..
.
:
$(_{6}’)= \lim_{farrow \mathit{0}}\tilde{y}j(\tilde{\zeta})$ $(j=1,2,3)$.
We define $\mathrm{p}$olynomials
$f_{1}=x_{1}(\zeta_{1}’+\zeta_{2}+\zeta_{s}+\zeta_{4}+\zeta’\mathrm{s})\prime\prime\prime(\zeta’1+\zeta_{3}+\zeta_{4})\prime\prime(\zeta 2+\zeta_{3}^{J}’+(4’)(\zeta^{J}3+\zeta_{4}’+\zeta_{5}^{l})$
$-((_{1}^{J}+\zeta_{2}’+\zeta’3+\zeta_{4})’(\zeta_{1}’+\zeta_{3}+\zeta_{4}+\zeta_{\mathrm{s}}^{J}\prime\prime)(\zeta_{2}+\zeta_{s}’+\zeta’’\prime 4+\zeta \mathrm{s})((_{3}’+\zeta_{4}J)$,
$f_{2}=x_{2}((_{1}’+\zeta_{2}’+\zeta_{3}’+\zeta_{4}\prime l\zeta 6’+(\mathrm{s}+)(\zeta_{1}^{J}+\zeta_{3}’+(’4)(\zeta’2+\zeta_{3}’+(’4)((_{3}+(’\prime 4+(_{5}’+\mathrm{C}_{6}’)$
$-(\zeta_{1}’+(_{2}’+\zeta’3+\zeta_{4}^{J})(\zeta’1+\zeta_{3}^{\prime\prime\prime\prime}+\zeta_{4}+\zeta 5+\zeta_{\epsilon})((_{2}’+\zeta_{3}’+(4+\zeta_{5}+\zeta_{6}’\prime J)(\zeta_{3}’+\zeta 4’)$ ,
$f_{3}=x_{3}(\zeta_{1}’+\zeta_{3}’’+\zeta_{4})(\zeta_{2}’+\zeta_{3}+\zeta 4)\prime\prime-(\zeta_{1}’+\zeta_{2}’’’+\zeta_{3}+\zeta_{4})(\zeta_{3}+’\zeta 4’)$,
$f_{4}=y_{1}(\zeta_{1}’+(_{2}’+\zeta 3+\zeta_{4}\prime\prime\prime+\zeta_{5})(\zeta’1+\zeta_{3}+\zeta_{4}\prime\prime)(\zeta_{2}’+\zeta 4’)(\zeta’4+(_{5}’)$
$-((_{1}’+\mathfrak{c}_{2}’+\zeta J3+\zeta_{4}r)(\zeta_{1}l+(_{s}’+\zeta 4’+(_{5}’)(\zeta_{2}l\prime\prime+\zeta_{4}+(_{5})\zeta_{4}’$ ,
$f_{5}=y_{2}((_{1}^{J}+\zeta_{2}^{JJ\prime\prime}+\zeta_{3}+\zeta 4+\zeta_{5}+\zeta 6’)(\zeta_{1}’+\zeta_{3}’’+\zeta_{4})(\zeta 2J’+\zeta 4)(\zeta_{4}’+\zeta_{\mathrm{s}}’\zeta_{6}’+)$
$-(\zeta_{1}’+\zeta_{2}^{J\prime}+\zeta_{3}+(’4)(\zeta_{1}’+\zeta_{s}’+\zeta_{4}’+\zeta_{5}’’+\zeta_{6})((2\prime\prime+\zeta 4+\zeta_{5}’+\zeta’6)\zeta 4’$,
$f_{6}=y_{3}(\zeta_{1}’+\zeta_{3}’+\zeta_{4}’)(\zeta’’2+\zeta 4)-((_{1}’+(_{2}r+\zeta 3\prime J+\zeta 4)\zeta_{4}J$
.
Regarding(5) $f_{1}=\cdots=f_{6}=0$,
as a system of equations for $\zeta_{1}’,$$\cdots$,$\zeta_{6}’$ with coefficients in the function field $\mathrm{C}(x, y)$, we are going
to 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. $i^{\mathrm{F}\mathrm{r}}\mathrm{o}\mathrm{m}$ the construction, the hypersurface $\Psi’(x, y)=0$ in $\mathrm{P}(2,7)=C’(\triangle, D_{4})$ is nothing but the subvariety $Y_{\Delta,D_{4}}(\triangle(E_{6}))$ (cf.[Se4],
\S \S 1,4).
By a littlelengthy 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}))$.
Itis worthwhiletocompare similarities between hypersurfacesofthe$5^{\ell h}$kind
andthe subvariety $Y_{\Delta(E_{6}),D_{4}}(\triangle(D_{4}))$ of $c(\triangle(E_{6}), D_{4})$ introduced in [Se4],
\S 3
($\mathrm{c}\mathrm{f}.[\mathrm{N}],$ $[\mathrm{L}]$, [Sh2]).$\pi\urcorner$A$\mathrm{r}\supset\tau\bigcap_{}$ $\mathrm{T}$
We give here an explanation on TABLE I.
(6.1) Let $S$ be a non-singular cubic surface in $\mathrm{P}^{3}$
.
An Eckardt point on $S$ is the
intersection
of three lines on $S$ (cf. [N]). Every cubic surface does not have an Eckardt point.
On the other
hand, a flex of a non-singular plane quartic $C$ is a point $p\in C$ such that there is a line $l$ triply
tangent to $C$ at $p$ (cf. [Sh2]). A flex is ordinaryif$l\cap C$ consistsof two points and a flex is special
if$l\cap C=\{p\}$. Every plane quartic does not havea special flex.
(6.2) In [N], the parameter $\lambda$ was introduced. It was
shown,
in [Se3] (cf.$1\mathrm{H}]$) that $\lambda$ is regardedas a rational function on $\mathrm{P}(2,6)$
.
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 $Y_{\Delta(E_{6}).D_{\mathrm{a}}}(\triangle(D_{4}))$ as a cross ratio variety for the root system $\triangle(D_{4})$ of type $D_{4}$ ?
(6.4) If $\delta_{5}(t)$ is a $W(E_{6})$-invariant polynomial of degree 5 (which is unique up to a constant
factor), 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 thenotation there. Hence, by $W(F_{4})$-action, weobtain totally45 quintic
polyno-mials on thestandaxd representation space of $W(E_{6})$
.
For the $s$ake of convenience, wecall thesepolynomials associated quintics. There is a 1-1 correspondence between the set of associated
quintics and that of the 45 triple tangent planes.
to $\delta_{5}$
.
(Theconstruction of$\delta_{E_{7}}(t)$ will be givenlater.)
(6.5) (cf. [L]) Let $P_{1},$$\cdots$ ,$P_{6}$ be 6 points of $\mathrm{P}^{2}$
.
We considera conic $C$ passing through five
points $P_{1},$ $\cdots$,$P_{5}$ and a line $L$ passing through $P_{5},$$P_{6}$. The condition corresponding to $\lambda-1=0$
is that the line $L$ is also a tangent of $C$ at $P_{5}$
.
Let $P_{1},$ $\cdots,$$P_{7}$ be 7 points of $\mathrm{P}^{2}$.
The conditioncorresponding to $\Psi(x, y)=0$ is the main subject in the previous section. Namely, let $P_{1},$ $\cdots$,$P_{7}$
be7 points of$\mathrm{P}^{2}$
.
We consider a cubiccurve$C$ passing throughseven points$P_{1},$$\cdots,$$P_{7}$ such that $P_{7}$ is a double point. The condition corresponding to $\Psi(x, y)=0$ is that $C$ has a cusp at
$P_{7}$
.
We are going to explain the construction on $\delta_{E_{7}}(t)$
.
Let $\omega_{j}$ be afundamental weight of$\triangle(E_{7})$ such that $<\omega_{j},$$\alpha_{k}>=\delta_{\mathrm{j}k}$. Then $\omega_{7}$ belongs to the
set of weights of the 56 dimensional irreducible representation of the simple Lie algebra of type
$E_{7}$. By definition, $\omega_{7}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+6\alpha_{4}+5\alpha_{5}+4\alpha_{6}+3\alpha_{7}$
.
The totality$\Pi$ of $\alpha\in\triangle$
such that $<\omega_{7},$$\alpha>=0$ form a root system of type $E_{6}$
.
Let $\Omega_{27}$ be the set of weights of27 irreducible representation of the simple Lie algebra of type $E_{6}$ corresponding to the root system
$\Pi$
.
Then$z_{p}= \sum_{\omega\epsilon\Omega_{27}}\omega^{p}$ $(p=1,2, \cdots)$ are $W(\Pi)$-invariant polynomials. (From the definition, $W(\Pi)\simeq W(E_{6}).)$
Lemma 5.1. Under thenotation above,
$P(t)$ $=$ $43545600\omega_{7}^{1}\mathit{0}-3628800_{Z}2\omega_{7}^{8}+100800_{Z_{27}}2\omega^{6}+725760_{Z}5\omega_{7}\mathrm{s}$
$+(42\mathrm{o}\mathrm{o}_{Z^{3}}2-6048\mathrm{o}\mathrm{o}Z6)\omega_{7}^{4}$
–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 ofdegree
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 $\mathrm{P}^{6}=\mathrm{P}(E_{\mathrm{C}})$.
Similarly, $P(-2\omega 7, z2, Z5, z6, z8, z_{9})=0$ defines a hypersurface $H_{\omega_{7}}$ in $\mathrm{P}^{6}$.
Theorem 5.2. The closure of $cr_{D_{4},\Delta}(H_{w_{7}})$ is isomorphic to $Y_{\Delta(E_{7}),D_{4}}(\triangle(E_{6}))$ by $W(E_{7} )$
-action.
$i$From the theorem above, $\delta_{E_{7}}(t)=P(\omega_{7},$$z_{2,5}Z,$$z_{6},$$Z_{8,z)}9$ is a required $W(E_{7})$-invariant
poly-nomial. Therefore $P(-2\omega 7, Z2, Z\mathrm{s}, Z6, z8, z9)$is an associated$W(E_{7})$-inva$7\dot{\mathrm{t}}ant$polynomila
of
degree 10. There are totally 28 associated hypersurfaces in $C(\triangle, D_{4})$.
It is interesting to characterize the polynomial $\delta_{E_{7}}(t)$
among
$W(E_{7})$-invariant polynomials ofAppendix. Theorem 4.2, Theorem 5.2 $l^{\vee}.\supset\backslash T$ – $\Re \mathrm{R}\infty \mathrm{E}^{\backslash }J$ス$\overline{\tau}$ム$\circ\ovalbox{\tt\small REJECT}|\mathrm{J}\mathrm{f}\mathrm{f}\mathrm{l}$ –
(1)
\S 4
$\emptyset \mathrm{R}\Psi(X, y)g)_{\overline{\mathrm{p}}}^{-\ovalbox{\tt\small REJECT} x}=[] \mathrm{g}\theta\backslash \prime \mathfrak{h}\lambda_{\mathrm{A}}\ovalbox{\tt\small REJECT}$.
$\Re\pi\infty\Phi\backslash ’\backslash$ス$\overline{\tau}$ム REDUCE3.4 $k$Toshiba $\mathrm{J}3100T\ovalbox{\tt\small REJECT}^{1}\mathrm{J}$ $\#\mathrm{b}T_{\overline{\mathrm{Q}}}^{\approx}-\dagger \mathrm{g}_{\mathrm{b}}T’\mathrm{t}\mathrm{e}\mathrm{g}i\iota\gamma.\sim 21^{\backslash }\mathrm{E}\mathfrak{h}\emptyset\pi\theta^{\backslash \yen}\backslash \backslash \mathrm{b}\iota,,\backslash arrow\backslash \geq\epsilon_{\overline{\prime \mathrm{T}\backslash }}\mathrm{c}f.\vee$
.
$10,000\uparrow\prime \mathrm{r}\overline{T}\neq \mathrm{R}\emptyset\pi k$
E&hW
$\mathrm{b}T\#\ovalbox{\tt\small REJECT}_{6\S}$$t_{\sim}^{\wedge}$ 1,800$\mathrm{r}_{\overline{\mathrm{T}}}\mathrm{F}/\mathrm{R}\mathrm{f}\mathrm{o}\mathrm{R}\}^{\sim}.\prime x_{\mathcal{D}}t.arrow$
.
$\ovalbox{\tt\small REJECT}_{\mathrm{J}^{\backslash }}\not\subset,$ $\ovalbox{\tt\small REJECT}\pm 1^{\backslash }\underline{\mathfrak{F}}\circ?\#\mathrm{i}\Re \mathrm{b}$f.-risa/asir $\xi$ PC9801 $\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-$$\mathrm{b}\yen_{\mathrm{b}C;}\mathrm{e}\lambda r\vee.\theta^{\theta}\backslash \backslash$, $\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$
f.-.
(2) $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 $\geq*\backslash \iota’\Gamma^{\backslash }\llcorner\backslash \tau 6.56^{\backslash }\mathrm{A}\overline{\pi}\ovalbox{\tt\small REJECT}$
$\Re\emptyset$ weight }$\mathrm{f}$
$\pm(\mathcal{E}_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}+\epsilon \mathrm{s}+\epsilon_{\mathit{6}})$, $\pm(\mathcal{E}:_{1}+\mathcal{E}:_{2}+\epsilon:_{3}+\epsilon_{4}.\cdot-\epsilon:_{5}-\epsilon:_{\mathit{6}})$, $\pm(\gamma_{1}\pm 2\epsilon_{j})$
-C$b6$
.
$\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}$il
$68_{\mathrm{T}’\mathrm{f}\mathrm{f}\mathrm{i}}^{\underline{\ulcorner}}\overline{\backslash _{\underline{\backslash }}}$.
$\pm(\epsilon_{1}+\mathcal{E}_{2}+$
$\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 }$ Lemma$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}$
il
68\yen ffi. $rightarrow\emptyset_{\mathrm{O}}^{\equiv\ovalbox{\tt\small REJECT}\#}\vee- \mathrm{f}$ $\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{a}/\mathrm{a}s$ir }$=\ovalbox{\tt\small REJECT} \mathrm{F}\cdot 0f.\backslash$.
simple reflection $T$ weight $k^{\backslash }\lambda\star\}_{\sim}^{\sim}@\tau\emptyset$$k^{\underline{\backslash }}-\tau\prime^{J}4\ulcorner\overline{\mathrm{T}}\llcorner\tau$ 1,800$(^{\text{ノ_{}\overline{\mathrm{T}}}}\ovalbox{\tt\small REJECT}_{\mathrm{f}}\circ \mathrm{R}\sigma)\mathrm{R}ij\pi f\overline{l}$ $\Phi_{\wedge}^{\varpi}\backslash \mathrm{f}\mathrm{f}\mathrm{i}\tau\Phi i\mathrm{J}\frac{\pi}{\mathrm{J}\sigma}X\Re\wedge \mathrm{b}T\iota’\backslash []_{L^{\}\mathrm{f}^{\backslash }\backslash }}\mathfrak{u}\backslash$.
$*\mathrm{a}\mathrm{e}\epsilon\not\in_{\overline{\beta}\iota\backslash }^{-- \mathrm{z}t}-T6^{\backslash }\mathrm{c}\succeq$ }$\mathrm{f}*\ovalbox{\tt\small REJECT}\Leftrightarrow$.
(3) $x_{1},$ $x_{2},$$x3,$$y1,$$y_{2},$$y3kt\sigma)\#\backslash \mathrm{A}H\not\in x\tau\ovalbox{\tt\small REJECT} \mathrm{b}\tau,$ $\Phi(x, y)$
t\sim \tilde tt\mbox{\boldmath $\lambda$}\tau 6&t
$\emptyset \mathrm{f}\mathrm{i}\mathrm{E}\mathrm{R}\theta^{\backslash \prime}\backslash l\tau \mathrm{b}\backslash \mathrm{B}$il
6
$\phi\backslash \backslash \backslash$,$\mathrm{f}\sigma)k^{\backslash }\mp\sigma)\Xi \mathrm{B}fl\vee \mathrm{c}t_{J}\iota\prime 1\mathrm{E}\mp\}\mathrm{f}22\lambda\#\lambda x\tau \mathrm{a}\mathrm{e}\epsilon$
.
$arrow\emptyset\vee\pi\xi 8\mathrm{e}_{t_{\sim}^{\sim}}\mathrm{E}\mathrm{R}\mathrm{f}\mathrm{f}\mathrm{l}\# T6\succeq 1*\emptyset \mathrm{E}\mp t^{*}\backslash 12\Phi$ae
$\mathfrak{h}$, $F\gtrless \mathfrak{h}\emptyset \mathrm{E}\mp\phi^{\backslash ^{\backslash }}\backslash F(t)=P(-2\omega_{7}, z_{2,\mathrm{s}}z, Z6, Z8, Z9)t_{\sim}^{\sim\gamma}x6$
.
$-\backslash *\emptyset \mathrm{E}\mp[] \mathrm{f}\tau\wedge^{\backslash }\tau\backslash \mathrm{g}\Re\Phi\#^{\wedge}arrow\lambda\backslash \iota\Gamma \mathrm{L}\backslash \backslash T6$.
$F(t)\dagger \mathrm{g}_{t}\mathrm{o}\mathrm{R}\succeq h\epsilon\geq 5,000\cap^{\prime-}\ovalbox{\tt\small REJECT}\neq \mathrm{R}\mathrm{o}T\mathrm{a}6$
.
$\Phi \mathrm{g}\prime_{\zeta}.\pi\Re \mathrm{A}\wedge \mathrm{a}\mathrm{e}\pi\#_{\sim^{\mathit{1}}}^{\wedge}\supset\tau F(t)\theta^{\mathrm{Y}}6W(E\tau)$\tau‘\varpi^‘\mbox{\boldmath$\pi$}を$\ovalbox{\tt\small REJECT} \mathrm{g}$ $\mathrm{f}\mathrm{f}\mathrm{l}\mathcal{T}^{\sigma)\}}\mathrm{f}\Xi^{\mathrm{B}}\mathrm{f}\mathrm{l}\tau*\}\mathrm{g}\prime_{\mathrm{f}}‘\vee\backslash ‘ \mathrm{r}_{\grave{7}\}_{-\theta^{\backslash }h}^{\sim}}\mathrm{b}^{\mathrm{p}}$@.
$E_{6}\emptyset\ovalbox{\tt\small REJECT}_{\square }\mathrm{A}t\wedge\sim\epsilon\grave{\eta}\prime_{J}0\tau\iota’\backslash r.’\mathrm{k}\mathrm{t},\backslash \cdot,$$\emptyset\phi\backslash \gamma\backslash \mathrm{i}\not\in-\emptyset\Re \mathrm{j}\Phi$.
$\ll\cdot i\mathrm{t},$}$\vee.\mathrm{t}_{)}\succeq$$.\supset.\backslash \tau,$ $P(\mu v_{7}, z2, Z\mathrm{s}, z_{\mathit{6}}, z_{8}, z9)\theta^{\backslash }\backslash W(\backslash E7)\tau\backslash v\mathrm{r}\}\wedge\cdot f\sim p6X.\backslash ,$ $\prime x\hat{\pi}\Re p\epsilon*dbX,$ $\geq\backslash \vee^{\backslash }J7\mathfrak{o}7\mathrm{a}k\yen\check{X}_{-r-}.$
.
$\mathrm{b}\mathrm{I}_{\vee}\mathrm{j}\mathrm{E}\mathrm{b}\searrow kT6^{\prime_{p}}C\supset\{x,$ t&\llcorner T$6\% ffi_{\mathrm{L}}$を\mbox{\boldmath$\kappa$}\mbox{\boldmath$\lambda$}$\llcorner T\mathrm{b}\Re \mathfrak{h}\mathrm{x}\cdot\supset[] \mathrm{f}^{-r\tau k}6\ \yen\grave{\mathrm{x}},$ $1^{\backslash }\underline{\mathrm{i}5}\Leftrightarrow^{\gamma}p\mathrm{f}\llcorner \mathrm{g}\mathrm{g}\mathrm{f}\mathrm{t}\lambda\llcorner$
$T\ovalbox{\tt\small REJECT}\wedge^{\backslash }Th\gamma_{-}.$
.
$\mathrm{i}\mathrm{E}\llcorner 4^{\backslash }\theta\backslash \mathrm{i}\mathrm{E}\mathrm{b}$$\langle$ $f\mathrm{y}I^{1t}’\backslash$computer $l_{\sim \mathrm{p}}^{\vee\equiv}-\mathrm{f}\mathrm{f}\mathrm{l}6*r.\backslash \geqarrow 6\vee,$ $p=1r_{x6}\dagger X_{\vee}^{/}\grave{\mathrm{J}}\prime_{j}$ $\langle$ $\geq \mathrm{s}_{)}\mathrm{a}_{66}$$\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^{-}.$
.
$\not\in^{-};\iota\tau,$ $\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}$$\succeq t^{\sigma\overline{\sim}}\pi\sim_{\backslash }*\gamma_{-}.$
.
$<^{-}t\iota\tau.tk_{\wedge}^{m\Re\ }\mathrm{b}T_{\mathrm{p}}^{\equiv}\Phi-\llcorner T\mathrm{f}\mathfrak{F}\epsilon_{\overline{/\mathrm{T}}}\backslash \mathrm{b}\gamma.\sim$.
$\delta l^{J}$$\backslash t_{\zeta^{\vee}}\nearrow..k\gamma\approx.t\backslash P*(-2\omega 7, Z2, Z_{5}, Z6, z8, Z9)$)$[] \mathrm{f}t\frac{\backslash }{p_{\vee}}\ hT$ 5,000 $4^{\text{ノ}}\overline{\mathrm{T}}\mathrm{r}_{\mathrm{f}}\mathrm{B}\succeq\theta\backslash t‘\zeta \mathfrak{h}\mathrm{e}\iota\prime \mathrm{Y}\pi r_{arrow}\underline{\mathrm{p}}\phi\backslash P(\backslash ^{\backslash }\omega 7, Z2, Z5, Z_{6}, Z_{8,9}z)\}\mathrm{f}270\cap^{\prime-}\ovalbox{\tt\small REJECT}_{\neq}\mathfrak{o}_{\mathrm{R}\emptyset}i)kk\ddagger \mathrm{b}\wedge h\backslash ^{\backslash }$ $\}\mathrm{f}^{\backslash }t^{1}\gamma_{J\mathfrak{h}}\Phi\iota’\backslash \mathrm{R}l^{\sim}\sim^{\gamma}J\cdot\supset r.\sim$
.
\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 $\tilde{C}$ of cubic surfaces
over Naruki’s cross ratio variety
$C(=C(\triangle(E_{\mathit{6}}), D_{4}))$ so that the natural projection $\varpi$ : $\tilde{C}arrow C$ is $W(E_{\mathit{6}})$-equivariant.
In this section, we discuss a relation between $C(\triangle, D_{4})$ and $\tilde{C}$
.
For this purpose, we introduce the set$\tilde{\mathrm{P}}(2, k)$ of$k$ points $(P_{1}, \cdots , P_{k})$ in $\mathrm{P}^{2}$ such that
$P$
.
$\neq P_{j}$by $PGL(3)$
.
It is clear that $\mathrm{P}(2,7)$ is nothing but the one introduced in\S 2.
There is a naturalprojection $p$of$\mathrm{P}(2, k+1)$ to $\mathrm{P}(2, k)$ defined by$p((P_{1}, \cdots , P_{k}, P_{k+1}))=(P_{1},$$\cdot$
..
,$P_{k})$.
$i$From now on, we focus our attention to the cases $k=6,7$
.
It is known (cf. [Se4]) thatthere is a birational $W(E_{k})$-action on $\mathrm{P}(2, k)(k=6,7)$
.
This easily implies that the projection$p:\mathrm{P}(2,7)arrow \mathrm{P}(2,6)$ is $W(E_{6})$-equivariant. Denoting by $\tilde{p}$ the extension of$p$ to $C(\triangle, D_{4})$, we
obtain a biratioanl $W(E_{6})$-equivariant map $\tilde{p}$: $C(\triangle, D_{4})arrow C(\triangle(E_{6}), D4)$
.
We consider the $W(E_{6})$-orbits of the set of hypersurfaces of the $1*t$ kind in $C(\triangle, D_{4})$. There
are two orbits. The first one denoted by $\Omega_{1}$ consists ofthose correspondingto roots contained in
$\triangle(E_{6})$:
$Y_{7}$, $Y_{1j}$ $(1 \leq i<j<7)$, $Y_{jk}$
.
$(1 \leq i<j<k<7)$.The second one denoted by $\Omega_{2}$ consists of the remaining 27 hypersurfaces:
Y.
$(1 \leq i<7)$, $Y.\tau$ $(1 \leq i<7)$, $Y_{j7}$.
$(1 \leq i<j<7)$.
For any $(P_{1}, \cdots , P_{6})\in \mathrm{P}(2,6)$, the closure $S(P_{1}, \cdots , P_{\mathit{6}})=\overline{p^{-1}((P_{1},\cdots,P6))}$ of its fibre in
$C(\triangle, D_{4})$ is of dimension 2. The surface $S(P_{1}, \cdots , P_{\mathit{6}})$ intersects with all the hypersurfaces of $\Omega_{2}$
.
By the intersection relations among hypersurfaces of the $1^{s\ell}$ kind, we easily find that the intersection relationsamongthe27 curves $S(P_{1}, \cdots , P_{\mathit{6}})\cap Y(\forall Y\in\Omega_{2})$ on $S(P_{1}, \cdots , P_{6})$ aresameas those of the 27 lines on a non-singular cubicsurface.
If the interpretation of thework of Naruki and Matsuzawa iscorrect, $\tilde{C}$
coincides with $C(\triangle, D_{4})$
and $\tilde{p}$ : $C(\triangle, D_{4})arrow c(\triangle(E_{6}), D_{4})$ defined above is the natural projection $\varpi$
.
As an easy consequence (?), $S(P_{1}, \cdots , P_{6})$ is a cubic surface. Therefore it is hopeful that $S(P_{1}, \cdots , P_{6})\cap Y$$(\forall Y\in\Omega_{2})$are the 27 lines on it. If this is true, hypersurfaces of$\Omega_{2}$ are global sections of 27lines of cubic surfaces in the total space $\tilde{C}$
.
$i$From the definition, $\mathrm{P}(2,7)$ is identified with the open subset of $(x_{1}, X_{2}, x_{3}, y1, y2, y3)$-space
outside the union
So
$(A_{6})$ of28
hyperplanes (cf.\S 2).
Moreover, we introduced 7 hypersurfaces$Q_{1},$$\cdots$ ,$Q_{7}$ of the$(X_{1)}X_{2}, x_{3,y1y},2, y3)$-spacein order todefine $\mathrm{P}_{0}(2,7)$
.
It is clear that the closureof the hypersurface $D_{gk}.=0$ in $C(\triangle, D_{4})$ is nothing
but
$Y_{1jk}$ and that of$Q_{j}$ is $Y_{j}$.We now take seven points $P_{1},$$\cdots$,$P_{6},$$P_{7}$ as in
\S 4
and fix $P_{j}(j=1, \cdots, 6)$ for the moment.Then $P_{7}$ is regarded as a point on $\mathrm{P}^{2}-\{P_{1}, \cdots, P_{6}\}$
.
Therefore $(x_{3}, y_{3})$ are interpreted asan inhomogeneous coordinate of $\mathrm{P}^{2}$
.
Under this identification, the defining equation $D_{j7}.=0$
corresponds to the line on $\mathrm{P}^{2}$ passing through
$P_{i}$ and $P_{j}$ for $i,j(1\leq i<j<7)$
.
On the otherhand, thedefining equation of Q. corresponds to theconic on $\mathrm{P}^{2}$ passing
through the five points
$\{P_{\mathrm{j};}j=1, \cdots , 6, j\neq i\}$
.
This is a geometric interpretationof hypersurfaces $Q_{1},$$\cdots$ ,$Q_{7}$ (cf. [M],In [Se4] \S 3, we have studied the structure of subvarieties of the form $Y(M)$ in $C(\triangle, D_{4})$, where
$M$ is a subset of $\triangle$ consisting of mutually orthogonal positive roots. In particular, we now treat
the intersection $Y_{\alpha}\cap Y_{\beta}$ for $\alpha,$$\beta\in\triangle$ such that $Y_{\alpha},$$Y_{\beta}\in\Omega_{2}$
.
The intersection $Y_{\alpha}\cap Y_{\beta}$ may beregarded as aglobal section of the intersection oftwo lines of cubic surfaces $S(P_{1}, \cdots , P_{\mathit{6}})$, being
assumed 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 otmake clear a relation between cubic surfaces and the global section of the intersections of two
lines on them. References
[B] Bourbaki, N. Groupes et Alg\‘ebres de Lte, Chaps. 4, 5, 6, Herman, Paris (1968).
[DO] Dolgachev, I. and Ortland, D. ‘Point sets in projective spaces and theta functions’
Aster-isque 165 (1988)
[GHP] L. Gerritzen, F. Herrlich and M.van der Purt,
‘Stable
$n$-pointed trees of projective lines’Indag. Math. Proc. A. 91 (2) 50 (1988), 131-163.
[H] Hunt, B. ‘A remarkable quintic fourfold in $\mathrm{P}^{5}$ and its dual variety
(Update: 7.1.1992)’
preprint.
[L] Looijenga, E. ‘Moduli spaces of marked del Pezzo surfaces’ Appendix in
[N].
[M] Manin, Yu. I. Cubic Forms, Algebra, Geometry, Arithmetic, North-Holland, Amsterdam,
1974.
[N] Naruki, I. ‘Cross ratio variety as a moduli space of cubic surfaces’
45 (1982),
1-30.
[NS] Naruki, I. and Sekiguchi, J. ‘A modification of Cayley’s family of cubic surfaces and
birational action of$W(E_{6})$ over it’ 56 Ser.A (1980),
122-125.
[O] Oda, T. ‘The canonical compactification of the configuration space ofthe pure braid
group
of $\mathrm{P}^{1}$’ preprint.
[Sel] Sekiguchi, J. ‘The birational action of$S_{5}$ on $\mathrm{P}^{2}$ and the
icosahedron’ J. Math.
Soc.
Japan,44 (1992), 567-589.
[Se2] Sekiguchi, J. ‘Theconfiguration space of
6
points in $\mathrm{P}^{2}$, the moduli space of cubic surfacesand the Weylgroup of type $E_{6}$’
RIMS
Kokyuroku 848(1993),74-85.
[Se3] Sekiguchi, J. ‘The versal deformation of the $E_{6}$-singularity and a family of cubic surfaces’
J. Math. Soc. Japan, 46 (1994), 355-383.
[Se4] Sekiguchi, J. ‘Cross ratio varieties for root systems’ Kyusyu J. Math, 48 (1994),
123-168.
[Se4a] Sekiguchi, J. ‘Cross ratio varieties for root systems II –Subvarieties of$C(\triangle(E_{7}),$D4)
[Se5] Sekiguchi, J. ‘Hypergeometric function of type (3,6) and Naruki’s cross ratio variety’ preprint.
[Se6] Sekiguchi, J. ‘Hypergeometric function of type (3,7) and cross ratio variety of type $E_{7}$’ to
appear in
RIMS
Kokyuroku.[Se7] Sekiguchi, J. ‘Cross ratio varieties for root systems of type $A_{n}$ and the Terada model’
preprint.
[Shl] Shioda, T. ‘Constructionofelliptic curves with high rank via the invariants of the Weyl
groups’ J. Math. Soc. Japan 43 (1991), 673-719.
[Sh2] Shioda, T. ‘Plane quartics and Mordell-Weil lattices of type $E_{7}$’ preprint.
[T] Terada, T., ‘Fonctionhyperg\’eom\’etriques$F_{1}$ etfonctions automorph I’ J. Math. Soc. Japan
