A $W(E_6)$-equivariant projective embedding of the moduli space of cubic surfaces
An explicit projective embedding of the moduli space of marked cubic surfaces is given. This embedding is equivariant under the Weyl group of type $E_6$. The image is defined by a system of linear and cubic equations. To express the embedding in a most symmetric way, the target would be 79-dimensional, however the image lies in a 9-dimensional linear subspace.
math0002102.tex
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\begin{document}
\title{A $W(E_6)$-equivariant projective embedding of the moduli space of cubic surfaces}
\author{Masaaki YOSHIDA \thanks{Department of Mathematics, Kyushu University, Fukuoka 812 Japan}}
%\author{tex/40E6/text.tex}
\date{22 December 1999}
\maketitle
\begin{abstract}
An explicit projective embedding of the moduli space of marked cubic surfaces is given. This embedding is equivariant under the Weyl group of type $E_6$. The image is defined by a system of linear and cubic equations. To express the embedding in a most symmetric way, the target would be 79-dimensional, however the image lies in a 9-dimensional linear subspace.

\end{abstract}
\par\smallskip
% {\bf Key words.}
% \par\smallskip
% {\bf AMS subject classifications.} 33A35, 33C80, 32H20, 11F55, 14D07, 32G20
\section{The moduli space of cubic surfaces}
\subsection{The space $M$ and the action of the group $\underline G$}
We first fix some notation and recall a few known facts on the moduli space
of marked cubic surfaces.
The moduli space of marked cubic surfaces, which we denote by $M$,
is studied for example in \cite{Na} and \cite{SeYo}. 
Since any nonsingular cubic
surface can be obtained by blowing up the projective plane $\PP^2$ at
six points, it can be represented by a $3\times6$-matrix of which
columns give homogeneous coordinates of the six points. In order to
get a smooth cubic surface from six points, 
we assume that no three points are
collinear and the six points are not on a conic. 
On the set of $3\times 6$ matrices, we have a cannical action
of $GL_3$ on the left and the group $\C^\times$ acts
naturally on homogeneous coordinates. 
By killing such ambiguity of coordinates, we get the following expression
\[ x=\left( 
\begin{array}{cccccc}
1&0&0&1& 1& 1\\ 0&1&0&1&x_1&x_2\\ 0&0&1&1&x_3&x_4
\end{array}
\right); \] 
in this paper we use local coordinates $(x_1,x_2,x_3,x_4)$ on $M$.
The six points represented by the matrix above produces a non-singular cubic surface if and only if the following quantitiy does not vanish.
\begin{eqnarray*} D(x)&:=
&x_1x_2x_3x_4(x_1-1)(x_2-1)(x_3-1)(x_4-1)\\
&&\times(x_1-x_2)(x_1-x_3)(x_2-x_4)(x_3-x_4)D_1D_2Q,
\end{eqnarray*}
where
\begin{eqnarray*}
D_1&:=&x_1x_4-x_2x_3,\\ 
D_2&:=&x_1x_4-x_4+x_2-x_2x_3+x_3-x_1,\\
Q&:=&-x_2x_3x_1-x_2x_3x_4+x_2x_3+x_1x_4x_2+x_1x_4x_3-x_1x_4,\\
\end{eqnarray*}
Thus we can identify the moduli space $M$ 
with the affine open set $\{x=(x_1,\dots,x_4)\mid D(x)\not=0\}.$

Let us define as in \cite{SeYo} six bi-rational transformations $s_1,\dots,s_6$ in 
$x=(x_1,\dots,x_4):$
% \[
% \begin{array}{rcl}
\begin{eqnarray*}
s_1:(x_1,x_2,x_3,x_4) & \rightarrow &
 \left({1\over x_1},{1\over x_2},{x_3\over x_1},{x_4\over x_2}\right), \\
s_2:(x_1,x_2,x_3,x_4) & \rightarrow &
  (x_3,x_4,x_1,x_2) ,\\
s_3:(x_1,x_2,x_3,x_4) & \rightarrow &
  \left({x_1-x_3\over 1-x_3}, {x_2-x_4\over 1-x_4},
        {x_3\over x_3-1}, {x_4\over x_4-1}\right) ,\\
s_4:(x_1,x_2,x_3,x_4) & \rightarrow &
  \left({1\over x_1},{x_2\over x_1}, {1\over x_3}, {x_4\over x_3}\right),\\
s_5:(x_1,x_2,x_3,x_4) & \rightarrow &
  (x_2, x_1, x_4, x_3) ,\\
s_6:(x_1,x_2,x_3,x_4) & \rightarrow &
  \left({1\over x_1}, {1\over x_2}, {1\over x_3}, {1\over x_4}\right).
\end{eqnarray*}
If $M$ is regarded as the configuration space of six points in $\PP^2$, 
the transformation $s_1$, for example, corresponds to the interchange of 
the two points represented by the first two column vectors of the matrix $x$.
Each $s_i$ turns out to be a bi-regular involution on $M$, 
and they form a group $\underline G$ 
isomorphic to the Weyl group of type $E_6$; relation of the generators are
given by the Coxeter graph
$$
\begin{array}{ccccccccc}
s_1&\mbox{---}&s_2&\mbox{---}&s_3&\mbox{---}&s_4&\mbox{---}&s_5\\
                   &  &   &  &\mid& &   &  &   \\
                   &  &   &  &s_6&  &   &  &        
\end{array}$$
%\begin{rmk} 
\subsection{Root system $\Delta$ of type $E_6$}
We review the root system $\Delta$ of type $E_6$, following \cite{H}.
Consider an 8-dimensional Euclidean space 
${\tilde E}$ with
a standard basis $\varepsilon_1,\cdots,\varepsilon_8$.
Let $\langle\cdot,\cdot\rangle$ be the inner product on ${\tilde E}$
defined by
$\langle\varepsilon_j, \varepsilon_k\rangle = \delta_{jk}$
and let $E$ be the linear subspace of ${\tilde E}$
spanned by the six vectors 
\[\varepsilon_1,\cdots,\varepsilon_5,\>
{\tilde \varepsilon}=\varepsilon_6-\varepsilon_7-\varepsilon_8.\]
We introduce the 36 vectors:
\[
r=-{1 \over 2}(
\varepsilon_1+\varepsilon_2+\varepsilon_3+\varepsilon_4+\varepsilon_5
+{\tilde \varepsilon}),\]
\[r_{1j}=-\varepsilon_{j-1}+r_0, \quad 2 \le   j\le 6\]
\[r_{jk}= \varepsilon_{j-1}-\varepsilon_{k-1}, \quad 2 \le   j<  k\le 6\]
\[r_{1jk}= -\varepsilon_{j-1}-\varepsilon_{k-1}, \quad 2 \le   j<  k\le 6\]
\[r_{ijk}= -\varepsilon_{i-1}-\varepsilon_{j-1}-\varepsilon_{k-1}+r_0, 
\quad 2 \le  i<j<k\le 6\]
where $r_{ij}=r_{ji}, r_{ijk}=r_{jik}=r_{ikj},$
\[r_0={1 \over 2}(\varepsilon_1+\varepsilon_2+\varepsilon_3
+\varepsilon_4+\varepsilon_5-{\tilde \varepsilon}).\]
Note that 
$$r\perp r_{ij},\quad r_{ij}\perp r_{kl},\quad r_{ij}\perp r_{ijk},\quad r_{ij}\perp r_{klm},\quad r_{ijk}\perp r_{ilm}.$$
The set 
$$\Delta=\{\pm r,\ \pm r_{ij},\ \pm r_{ijk}\}$$
forms a root system of type $E_6$.
For example,
$$r_{12},\quad r_{123},\quad r_{23},\quad r_{34},\quad r_{45},\quad r_{56}$$
can serve as a system  of positive simple roots; its extended Dynkin diagram is given as

\begin{center}\begin{tabular}
{p{5mm} p{10mm} p{5mm} p{10mm} p{5mm} p{10mm} p{5mm} p{10mm} p{5mm}}
$r_{12}$ &------&$r_{23}$&------&$r_{34}$&------&$r_{45}$
&------&$r_{56}$\\
      &      &       &      & $\mid$&      &       &      & \\
      &      &       &      & $r_{123}$ &  &       &      & \\
      &      &       &      & $\mid$&      &       &      & \\
      &      &       &      & $r$ &  &       &      & \\
\end{tabular}\end{center}
The set 
$\{r$, $r_{jk}$, $r_{ijk}\}$ is the totality of
positive roots of $\Delta$.
%and $r$ is the longest root.

Let $s_r, s_{ij}$ and $s_{ijk}$ be the reflections on $E$ with respect to $r, r_{ij}$ and $r_{ijk}$. These reflections act on $\Delta$ as 
$$\begin{array}{ll}\quad\ \  s_r: &r_{ij}\leftrightarrow r_{ij},\quad r_{ijk}\leftrightarrow r_{lmn},\quad 
\{i,j,k,l,m,n\}=\{1,\dots,6\},\\
\quad\ \  s_{ij}: &{\rm permutation\ of\ the\ indices\ }i {\rm \ and\ }j,\end{array}$$
$$\begin{array}{llllllllll}
s_{123}:&r_{12}&\leftrightarrow &r_{12},\quad &r_{14}&\leftrightarrow &r_{234},\quad &r_{56}&\leftrightarrow &r_{56},\\
&r_{145}&\leftrightarrow &r_{145},\quad &r&\leftrightarrow &r_{456},&&&\end{array}$$
modulo signs. Let us define two reflection group
$$
G_1=\langle s_{12},s_{23},s_{34},s_{45},s_{56}\rangle\cong S_6,\quad
G=\langle G_1,s_{123}\rangle\cong W(E_6),$$
where $S_6$ is the symmetric group on six numerals $\{1,\dots,6\}$, $W(E_6)$ the Weyl group of type $E_6$, and $\langle a,b,\dots\rangle$ denotes the group generated by $a,b,\dots$   Note that
$$G_1\subset \langle G_1,s_r\rangle= G_1\times\langle s_r\rangle
\subset G,$$
and $G$ acts transitively on $\Delta$.
\subsection{Naruki's cross-ratio variety}
A smooth compactification of $M$ known as Naruki's cross-ratio
variety $\mathcal C$ (\cite{Na}, \cite{SeYo}), embedded in $(\PP^1)^{45}$, is the union of $M$ and the 76 divisors. The 36 of them correspond 
to the positive roots of $\Delta$ (they are said to be of the first kind), the other 40 divisors (said to be of the second kind, and are isomorphic to $(\PP^1)^3$) can be represented
as follows:
Take three subsets
$\Delta_1$, $\Delta_2$, $\Delta_3$ of $\Delta$ satisfying the following conditions:
\begin{itemize}
\item Each of 
$\Delta_1$, $\Delta_2$, $\Delta_3$ is a root system of type $A_2$.
\item $\Delta_1$, $\Delta_2$, $\Delta_3$ are mutually orthogonal.
\item The vectors in
$\Delta_1\cup\Delta_2 \cup \Delta_3$ span $E$.
\end{itemize}
Note that each one of such three root systems determines the other two.

 Such a triple  $\{\Delta_1,\Delta_2,\Delta_3\}$ dertermines a divisor. 
According to the naming of the roots, we must use two different expressions:
The first one is of the form
\[
\{\pm r_{12},\pm r_{23},\pm r_{13}\},
\quad
\{\pm r_{45},\pm r_{56},\pm r_{46}\},
\quad
\{\pm r,\pm r_{123},\pm r_{456}\},
\]
(the corresponding divisor is denoted by $Z_{123,456}$ in \cite{SeYo},) and
the second one is of the form
\[
\{\pm r_{12},\pm r_{234},\pm r_{134}\},
\quad
\{\pm r_{34},\pm r_{356},\pm r_{456}\},
\quad
\{\pm r_{56},\pm r_{125},\pm r_{126}\},
\]
(the corresponding divisor is denoted by $Z_{12,34,56}$ in \cite{SeYo}).
Note that
 $$Z_{123,456}=Z_{456,123},\quad
Z_{12,34,56}=Z_{56,12,34}\not=Z_{12,56,34}.
$$
Thus, permuting the indices under $G_1$, we have 40 ($=10+30$) such divisors.
The group $G$ acts transitively on these 40 divisors.
\par\medskip\noindent
{\bf Remark:}
These divisors (of the second kind) are disjoint to each other, and can be blown-down to points. In fact they correspond bijectively to the cusps of the modular group studied in
 \cite{ACT} (see also \cite{Ma2} and \cite{SaYo2}). 
\section{Embedding $\varphi:M\to \PP^{80-1}$}
\subsection{Coordinates on $\PP^{80-1}$ and the action of $G$}
Let $\mathcal A$ be the set of 40 labels $(123,456)$ and $(12,34,56)$
with the following identification
\begin{eqnarray*}
(123,456)&=&(213,456)=(132,456)=(456,123),\\
(12,34,56)&=&(21,34,56)=(56,12,34).
\end{eqnarray*}
Since $G$ acts on the set $\Delta$ of roots, it acts also on the set of
40 divisors above, and so that it also acts on the set $\mathcal A$ of
40 labels.

We introduce 80 homogeneous coordinates 
$$y_\alpha,\quad y_{-\alpha},\qquad \alpha\in\mathcal A$$
on $\PP^{80-1}$. I define an action of $G$ on $\PP^{80-1}$ by the
following action of the generators $s_{12},\dots,s_{56}$ and $s_{123}$
on the coordinates. Let $s$ be one of the generators and $\alpha\in
\mathcal A$; we assign
\begin{eqnarray*}
s(y_\alpha)&=&y_\alpha,\quad s(y_{-\alpha})=y_{-\alpha}\qquad \mbox{if
 $s\alpha=\alpha$,}\\
s(y_\alpha)&=&y_{-\beta},\quad s(y_{-\alpha})=y_{\beta}\qquad \mbox{if
 $s\alpha=\beta\not=\alpha$.}
\end{eqnarray*}

\subsection{Definition of $\varphi$}
In this section we define a map $M\to \PP^{80-1}$. For a $3\times6$
matrix $x=(x_{ij})$, we consider 80 polynomials  of degree 18 as follows:
\begin{eqnarray*}
y_{(123,456)}(x)&=& D_{123}(x)D_{456}(x)Q(x),\\
y_{(12,34,56)}(x)&=& D_{134}(x)D_{234}(x)D_{356}(x)D_{456}(x)D_{512}(x)D_{612}(x),
\end{eqnarray*}
and $y_{-\alpha}(x)=-y_\alpha\ (\alpha\in\mathcal A)$, where $Q(x)$ is the determinant of the $6\times6$-matrix with columns 
$$(x_{1j}x_{2j},x_{2j}x_{3j},x_{3j}x_{1j}, x_{1j}^2,x_{2j}^2,x_{3j}^2)\quad j=1,\dots,6.$$
Since we have
$$y_\alpha(gxh)=(\det g)^6y_\alpha(x)(\det h)^3,$$
the correswpondence above defines a map $\varphi:M\to \PP^{80-1}$.
For later use, we present 40 polynomials $y_\alpha(x)$ in terms of the
coordinates  $(x_1,x_2,x_3,x_4)$ introduced in \S1; the remaining 40
polynomoals are given by $y_{-\alpha}(x)=-y_\alpha(x)$. In the following 
table, $y_\alpha(x)$ is denoted simply by $\alpha$, and we number them as $y_1,\dots,y_{40}$:
\begin{eqnarray*}
y_{1}=(156,234)&:=&D_1Q:\quad y_{2}=(123,456):=D_2Q:\\
y_{3}=(124,356)&:=&(x_2-x_1)Q:\quad y_{4}=(145,236):=(x_3-x_1)Q:\\
y_{5}=(146,235)&:=&(x_4-x_2)Q:\quad y_{6}=(134,256):=(x_4-x_3)Q:\\
y_{7}=(135,246)&:=&x_1(x_4-1)Q:\quad y_{8}=(136,245):=x_2(x_3-1)Q:\\
y_{9}=(125,346)&:=&x_3(x_2-1)Q:\quad y_{10}=(126,345):=x_4(x_1-1)Q:\\
%\end{eqnarray*}
%\begin{eqnarray*}
y_{11}=(12,56,34)&:=&D_1(x_1-1)(x_2-1)(x_4-x_3):\\
y_{12}=(16,23,45)&:=&D_1(x_1-1)(x_3-1)(x_4-x_2):\\
y_{13}=(15,23,46)&:=&D_1(x_2-1)(x_4-1)(x_3-x_1):\\
y_{14}=(13,56,24)&:=&D_1(x_3-1)(x_4-1)(x_2-x_1):\\
%\end{eqnarray*}
%\begin{eqnarray*}
y_{15}=(15,24,36)&:=&-D_1x_1(x_2-1)(x_3-1):\\
y_{16}=(16,24,35)&:=&-D_1x_2(x_1-1)(x_4-1):\\
y_{17}=(15,34,26)&:=&-D_1x_3(x_1-1)(x_4-1):\\
y_{18}=(16,34,25)&:=&-D_1x_4(x_2-1)(x_3-1):\\
%\end{eqnarray*}
%\begin{eqnarray*}
y_{19}=(12,36,45)&:=&D_2x_2x_3(x_1-1):\\
y_{20}=(12,35,46)&:=&D_2x_1x_4(x_2-1):\\
y_{21}=(13,26,45)&:=&D_2x_1x_4(x_3-1):\\
y_{22}=(13,25,46)&:=&D_2x_2x_3(x_4-1):\\
%\end{eqnarray}
%\begin{eqnarray}
y_{23}=(13,24,56)&:=&D_2x_1x_2(x_4-x_3):\\
y_{24}=(15,46,23)&:=&D_2x_1x_3(x_4-x_2):\\
y_{25}=(16,45,23)&:=&D_2x_2x_4(x_3-x_1):\\
y_{26}=(12,34,56)&:=&D_2x_3x_4(x_2-x_1):\\
%\end{eqnarray*}
%\begin{eqnarray*}
y_{27}=(13,46,25)&:=&x_1(x_2-1)(x_3-1)(x_4-x_2)(x_4-x_3):\\
y_{28}=(13,45,26)&:=&x_2(x_1-1)(x_4-1)(x_3-x_1)(x_4-x_3):\\
y_{29}=(12,46,35)&:=&x_3(x_1-1)(x_4-1)(x_2-x_1)(x_4-x_2):\\
y_{30}=(12,45,36)&:=&x_4(x_2-1)(x_3-1)(x_2-x_1)(x_3-x_1):\\
%\end{eqnarray*}
%\begin{eqnarray*}
y_{31}=(14,35,26)&:=&-x_1(x_1-1)(x_4-x_2)(x_4-x_3):\\
y_{32}=(14,36,25)&:=&-x_2(x_2-1)(x_3-x_1)(x_4-x_3):\\
y_{33}=(14,25,36)&:=&-x_3(x_3-1)(x_2-x_1)(x_4-x_2):\\
y_{34}=(14,26,35)&:=&-x_4(x_4-1)(x_2-x_1)(x_3-x_1):\\
%\end{eqnarray*}
%\begin{eqnarray*}
y_{35}=(16,25,34)&:=&-x_2x_3(x_1-1)(x_4-x_2)(x_4-x_3):\\
y_{36}=(15,26,34)&:=&-x_1x_4(x_2-1)(x_3-x_1)(x_4-x_3):\\
y_{37}=(16,35,24)&:=&-x_1x_4(x_3-1)(x_2-x_1)(x_4-x_2):\\
y_{38}=(15,36,24)&:=&-x_2x_3(x_4-1)(x_2-x_1)(x_3-x_1):\\
%\end{eqnarray*}
%\begin{eqnarray*}
y_{39}=(14,56,23)&:=&D_1D_2:\\
y_{40}=(14,23,56)&:=&(x_2-x_1)(x_3-x_1)(x_4-x_2)(x_4-x_3):\\
\end{eqnarray*}


\subsection{$G$-Equivariance of $\varphi$}
Recall that the group $\underline G$ acts on $M$, and that $G$ acts on $\PP^{80-1}$. Let us identify the groups $\underline G$ and $G$ by
$$\iota:s_{12}\mapsto s_1,\dots,s_{56}\mapsto s_5, \ s_{123}\mapsto
s_6.$$ Then we have 
\begin{thm}The map $\varphi:M\to\PP^{80-1}$ is $G$-equivariant:
$$g(\varphi(x))=\varphi(\iota(g)x),\quad g\in G,\ x\in M,$$
that is,
$$(gy_\alpha)(x)=c_gy_\alpha(\iota(g)x),\quad g\in G,\ 
\alpha\in\pm\mathcal A,\ x\in M,$$
where $c_g$ is a rational function in $(x_1,x_2,x_3,x_4)$.\end{thm}
\par\noindent
{\bf Convention:} Once this theorem is established, we ignore the redundant ones $y_{-\alpha}(x)=-y_{\alpha}(x)$ and regard $\varphi$ as the map
$$M\ni x\longmapsto :y_\alpha(x):\in \PP^{40-1}.$$
The group $G$ still acts on $\PP^{40-1}$ by the transformations given in\S2.3.
\par\smallskip
In order to prove the theorem, we have only to check the identity for a set of gnerators of $G$.
Under $s_1$, the fourty polynomials are transformed as follows:
$$\begin{array}{lllll}
y_{1}\to -c_1y_{6},&
y_{2}\to c_1 y_{2},&
y_{3}\to c_1 y_{3},&
y_{4}\to -c_1y_{8},&
y_{5}\to -c_1y_{7},\\
y_{6}\to -c_1y_{1},&
y_{7}\to -c_1y_{5},&
y_{8}\to -c_1y_{4},&
y_{9}\to c_1 y_{9},&
y_{10}\to c_1 y_{10},\\
y_{11}\to c_1 y_{11},&
y_{12}\to -c_1y_{28},&
y_{13}\to -c_1y_{27},&
y_{14}\to -c_1y_{40},&
y_{15}\to -c_1y_{32},\\
y_{16}\to -c_1y_{31},&
y_{17}\to -c_1y_{35},&
y_{18}\to -c_1y_{36},&
y_{19}\to c_1 y_{19},&
y_{20}\to c_1 y_{20},\\
y_{21}\to -c_1y_{25},&
y_{22}\to -c_1y_{24},&
y_{23}\to -c_1y_{39},&
y_{24}\to -c_1y_{22},&
y_{25}\to -c_1y_{21},\\
y_{26}\to c_1 y_{26},&
y_{27}\to -c_1y_{13},&
y_{28}\to -c_1y_{12},&
y_{29}\to c_1 y_{29},&
y_{30}\to c_1 y_{30},\\
y_{31}\to -c_1y_{16},&
y_{32}\to -c_1y_{15},&
y_{33}\to -c_1y_{38},&
y_{34}\to -c_1y_{37},&
y_{35}\to -c_1y_{17},\\
y_{36}\to -c_1y_{18},&
y_{37}\to -c_1y_{34},&
y_{38}\to -c_1y_{33},&
y_{39}\to -c_1y_{23},&
y_{40}\to -c_1y_{14},
\end{array}$$
where $c_1=(x_1x_2)^{-3}$, 
under $s_{2}$,
$$\begin{array}{lllll}
y_{1}\to y_{1},&
y_{2}\to y_{2},&
y_{3}\to -y_{6},&
y_{4}\to y_{4},&
y_{5}\to y_{5},\\
y_{6}\to -y_{3},&
y_{7}\to -y_{9},&
y_{8}\to -y_{10},&
y_{9}\to -y_{7},&
y_{10}\to -y_{8},\\
y_{11}\to -y_{14},&
y_{12}\to y_{12},&
y_{13}\to y_{13},&
y_{14}\to -y_{11},&
y_{15}\to -y_{17},\\
y_{16}\to -y_{18},&
y_{17}\to -y_{15},&
y_{18}\to -y_{16},&
y_{19}\to -y_{21},&
y_{20}\to -y_{22},\\
y_{21}\to -y_{19},&
y_{22}\to -y_{20},&
y_{23}\to -y_{26},&
y_{24}\to y_{24},&
y_{25}\to y_{25},\\
y_{26}\to -y_{23},&
y_{27}\to -y_{29},&
y_{28}\to -y_{30},&
y_{29}\to -y_{27},&
y_{30}\to -y_{28},\\
y_{31}\to -y_{33},&
y_{32}\to -y_{34},&
y_{33}\to -y_{31},&
y_{34}\to -y_{32},&
y_{35}\to -y_{37},\\
y_{36}\to -y_{38},&
y_{37}\to -y_{35},&
y_{38}\to -y_{36},&
y_{39}\to y_{39},&
y_{40}\to y_{40},\end{array}$$
under $s_3$,
$$\begin{array}{lllll}
y_{1}\to c_3y_{1},&
y_{2}\to -c_3y_{3},&
y_{3}\to -c_3y_{2},&
y_{4}\to -c_3y_{7},&
y_{5}\to -c_3y_{8},\\
y_{6}\to c_3y_{6},&
y_{7}\to -c_3y_{4},&
y_{8}\to -c_3y_{5},&
y_{9}\to c_3y_{9},&
y_{10}\to c_3y_{10},\\
y_{11}\to c_3y_{11},&
y_{12}\to -c_3y_{16},&
y_{13}\to -c_3y_{15},&
y_{14}\to -c_3y_{39},&
y_{15}\to -c_3y_{13},\\
y_{16}\to -c_3y_{12},&
y_{17}\to c_3y_{17},&
y_{18}\to c_3y_{18},&
y_{19}\to -c_3y_{29},&
y_{20}\to -c_3y_{30},\\
y_{21}\to -c_3y_{34},&
y_{22}\to -c_3y_{33},&
y_{23}\to -c_3y_{40},&
y_{24}\to -c_3y_{38},&
y_{25}\to -c_3y_{37},\\
y_{26}\to c_3y_{26},&
y_{27}\to -c_3y_{32},&
y_{28}\to -c_3y_{31},&
y_{29}\to -c_3y_{19},&
y_{30}\to -c_3y_{20},\\
y_{31}\to -c_3y_{28},&
y_{32}\to -c_3y_{27},&
y_{33}\to -c_3y_{22},&
y_{34}\to -c_3y_{21},&
y_{35}\to c_3y_{35},\\
y_{36}\to c_3y_{36},&
y_{37}\to -c_3y_{25},&
y_{38}\to -c_3y_{24},&
y_{39}\to -c_3y_{14},&
y_{40}\to -c_3y_{23},\end{array}$$
where $c_3=(1-x_3)^{-3}(1-x_4)^{-3}$, under $s_4$,
$$\begin{array}{lllll}
y_{1}\to -c_4y_{5},&
y_{2}\to c_4y_{2},&
y_{3}\to -c_4y_{9},&
y_{4}\to c_4y_{4},&
y_{5}\to -c_4y_{1},\\
y_{6}\to -c_4y_{7},&
y_{7}\to -c_4y_{6},&
y_{8}\to c_4y_{8},&
y_{9}\to -c_4y_{3},&
y_{10}\to c_4y_{10},\\
y_{11}\to -c_4y_{29},&
y_{12}\to c_4y_{12},&
y_{13}\to -c_4y_{40},&
y_{14}\to -c_4y_{27},&
y_{15}\to -c_4y_{33},\\
y_{16}\to -c_4y_{35},&
y_{17}\to -c_4y_{31},&
y_{18}\to -c_4y_{37},&
y_{19}\to c_4y_{19},&
y_{20}\to -c_4y_{26},\\
y_{21}\to c_4y_{21},&
y_{22}\to -c_4y_{23},&
y_{23}\to -c_4y_{22},&
y_{24}\to -c_4y_{39},&
y_{25}\to c_4y_{25},\\
y_{26}\to -c_4y_{20},&
y_{27}\to -c_4y_{14},&
y_{28}\to c_4y_{28},&
y_{29}\to -c_4y_{11},&
y_{30}\to c_4y_{30},\\
y_{31}\to -c_4y_{17},&
y_{32}\to -c_4y_{38},&
y_{33}\to -c_4y_{15},&
y_{34}\to -c_4y_{36},&
y_{35}\to -c_4y_{16},\\
y_{36}\to -c_4y_{34},&
y_{37}\to -c_4y_{18},&
y_{38}\to -c_4y_{32},&
y_{39}\to -c_4y_{24},&
y_{40}\to -c_4y_{13},\end{array}$$
where $c_4=(x_1x_3)^{-3}$, under $s_5$,
$$\begin{array}{lllll}
y_{1}\to y_{1},&
y_{2}\to y_{2},&
y_{3}\to y_{3},&
y_{4}\to -y_{5},&
y_{5}\to -y_{4},\\
y_{6}\to y_{6},&
y_{7}\to -y_{8},&
y_{8}\to -y_{7},&
y_{9}\to -y_{10},&
y_{10}\to -y_{9},\\
y_{11}\to y_{11},&
y_{12}\to -y_{13},&
y_{13}\to -y_{12},&
y_{14}\to y_{14},&
y_{15}\to -y_{16},\\
y_{16}\to -y_{15},&
y_{17}\to -y_{18},&
y_{18}\to -y_{17},&
y_{19}\to -y_{20},&
y_{20}\to -y_{19},\\
y_{21}\to -y_{22},&
y_{22}\to -y_{21},&
y_{23}\to y_{23},&
y_{24}\to -y_{25},&
y_{25}\to -y_{24},\\
y_{26}\to y_{26},&
y_{27}\to -y_{28},&
y_{28}\to -y_{27},&
y_{29}\to -y_{30},&
y_{30}\to -y_{29},\\
y_{31}\to -y_{32},&
y_{32}\to -y_{31},&
y_{33}\to -y_{34},&
y_{34}\to -y_{33},&
y_{35}\to -y_{36},\\
y_{36}\to -y_{35},&
y_{37}\to -y_{38},&
y_{38}\to -y_{37},&
y_{39}\to y_{39},&
y_{40}\to y_{40},\end{array}$$
and under $s_6$,
$$\begin{array}{lllll}
y_{1}\to-c_6y_{39},&
y_{2}\to c_6y_{2},&
y_{3}\to-c_6y_{26},&
y_{4}\to-c_6y_{25},&
y_{5}\to-c_6y_{24},\\
y_{6}\to-c_6y_{23},&
y_{7}\to-c_6y_{22},&
y_{8}\to-c_6y_{21},&
y_{9}\to-c_6y_{20},&
y_{10}\to-c_6y_{19},\\
y_{11}\to c_6y_{11},&
y_{12}\to c_6y_{12},&
y_{13}\to c_6y_{13},&
y_{14}\to c_6y_{14},&
y_{15}\to-c_6y_{18},\\
y_{16}\to-c_6y_{17},&
y_{17}\to-c_6y_{16},&
y_{18}\to-c_6y_{15},&
y_{19}\to-c_6y_{10},&
y_{20}\to-c_6y_{9},\\
y_{21}\to-c_6y_{8},&
y_{22}\to-c_6y_{7},&
y_{23}\to-c_6y_{6},&
y_{24}\to-c_6y_{5},&
y_{25}\to-c_6y_{4},\\
y_{26}\to-c_6y_{3},&
y_{27}\to c_6y_{27},&
y_{28}\to c_6y_{28},&
y_{29}\to c_6y_{29},&
y_{30}\to c_6y_{30},\\
y_{31}\to-c_6y_{35},&
y_{32}\to-c_6y_{36},&
y_{33}\to-c_6y_{37},&
y_{34}\to-c_6y_{38},&
y_{35}\to-c_6y_{31},\\
y_{36}\to-c_6y_{32},&
y_{37}\to-c_6y_{33},&
y_{38}\to-c_6y_{34},&
y_{39}\to-c_6y_{1},&
y_{40}\to c_6y_{40},
\end{array}$$
where $c_6=(x_1x_2x_3x_4)^{-2}$.
Maybe it is interesting to see what happens under the operation of the 
involution $s_r$ (classically called the association) which sends $(x_1,x_2,x_3,x_4)$ to
$$\left(
{(x_4-1)D_1\over(x_4-x_2)(x_4-x_3)},
{(x_3-1)D_1\over(x_3-x_1)(x_4-x_3)},
{(x_2-1)D_1\over(x_4-x_2)(x_2-x_1)},
{(x_1-1)D_1\over(x_3-x_1)/(x_2-x_1)}\right):$$
$$\begin{array}{lllll}
y_{1}\to c_ry_{1},&
y_{2}\to c_ry_{2},&
y_{3}\to c_ry_{3},&
y_{4}\to c_ry_{4},&
y_{5}\to c_ry_{5},\\
y_{6}\to c_ry_{6},&
y_{7}\to c_ry_{7},&
y_{8}\to c_ry_{8},&
y_{9}\to c_ry_{9},&
y_{10}\to c_ry_{10},\\
y_{11}\to-c_ry_{26},&
y_{12}\to-c_ry_{25},&
y_{13}\to-c_ry_{24},&
y_{14}\to-c_ry_{23},&
y_{15}\to-c_ry_{38},\\
y_{16}\to-c_ry_{37},&
y_{17}\to-c_ry_{36},&
y_{18}\to-c_ry_{35},&
y_{19}\to-c_ry_{30},&
y_{20}\to-c_ry_{29},\\
y_{21}\to-c_ry_{28},&
y_{22}\to-c_ry_{27},&
y_{23}\to-c_ry_{14},&
y_{24}\to-c_ry_{13},&
y_{25}\to-c_ry_{12},\\
y_{26}\to-c_ry_{11},&
y_{27}\to-c_ry_{22},&
y_{28}\to-c_ry_{21},&
y_{29}\to-c_ry_{20},&
y_{30}\to-c_ry_{19},\\
y_{31}\to-c_ry_{34},&
y_{32}\to-c_ry_{33},&
y_{33}\to-c_ry_{32},&
y_{34}\to-c_ry_{31},&
y_{35}\to-c_ry_{18},\\
y_{36}\to-c_ry_{17},&
y_{37}\to-c_ry_{16},&
y_{38}\to-c_ry_{15},&
y_{39}\to-c_ry_{40},&
y_{40}\to-c_ry_{39},
\end{array}
$$
where 
$$c_r=\left({D_1D_2\over(-x_4+x_2)(x_4-x_3)(x_1-x_3)(x_1-x_2)}\right)^3.$$
\subsection{$\varphi$ embeds $M$}
It is known in \cite{Yo3} that the map
$$M\ni x\longrightarrow y_1(x):y_3(x):y_4(x):y_5(x):y_7(x)\in \PP^4$$
is two-to-one, and induces an embedding of
the quotient space $M/\langle s_r\rangle$.
Thus the composite of $\varphi$ and the projection
$$M\longrightarrow\PP^{40-1}\longrightarrow\PP^4$$
is a two-to-one map.
This fact together with the equivariance of $\varphi$ under the involution $s_r$ shown just above
implies 
\begin{thm} $\varphi$ embeds $M$ into $\PP^{40-1}$. \end{thm}
\subsection{Prolongation of $\varphi$ to degenerate arrangements}
Let us consider degenerate arrangements of six points on the plane. Since arrangements with three collinear points can be transformed under $G$ to those with six points on a conic, we assume, Without loss
of generality, that our arrangements represented by $x=(x_1,\dots,x_4)$ satisfies $Q=0$, that is, the six points are on a conic. Since a (nonsingular) conic is isomorphic to a line, such arrangements form the configuration space  
$$X(2,6)=GL(2)\backslash\{Mat(2,6)\mid \mbox{any }2\times2\mbox{ minor}\not=0\}/(\C^\times)^6$$ of six points on the projective line: if we 
represent a point of $X(2,6)$ by a matrix of the form
\[ z=\left( 
\begin{array}{cccccc}
1&0&1&1& 1& 1\\ 0&1&1&z_1&z_2&z_3
\end{array}\right), \] 
where 
$$\prod_{i=1}^3z_i(z_i-1)\prod_{1\le i<j\le3}(z_i-z_j)\not=0,$$
then the degenerate arrangements in question  can be parametrized by $z=(z_1,z_2,z_3)$ as
$$x_1=(1-z_1)/(1-z_2),\quad x_2=(1-z_1)/(1-z_3),\quad x_3=z_1/z_2,\quad x_4=z_1/z_3.$$
Among the two $S_6$-equivariant projective embedding of $X(2,6)$ presented in 
\cite{Yo1}, let us recall the following one given by the fifteen polynomials
$$D_{ij}(z)D_{kl}(z)D_{mn}(z),\quad\{i,j,k,l,m\}=\{1,\dots,6\},$$
where $D_{ij}(z)$ is the $(i,j)$-minor of the $2\times6$-matrix $z$. Their actual forms are given by
$$(z_i-1)(z_j-z_k), \quad z_j-z_k,\quad z_i(z_j-z_k),\quad z_i(z_j-1).$$
It is known and easy to show that the image is projectively equivalent
to the so-called Segre cubic defined by
$$t_0+\cdots,t_5=0,\quad (t_0)^3+\cdots+(t_5)^3=0.$$

On the other hand, let us prolong the domain of definition of the map $\varphi$
on these degenerate arrangements by the same forty polynomials. 
Then the map $\varphi$ in $z$-coodinates is given by $y_{1}=\cdots y_{10}=0$ and
$$\begin{array}{lll}
cy_{11}=-z_1(z_2-z_3),& cy_{12}=-z_2+z_1,& cy_{13}=z_1-z_3,\\
cy_{14}=-(-1+z_1)(z_2-z_3),& cy_{15}=(-1+z_1)z_3,& cy_{16}=(-1+z_1)z_2,\\
cy_{17}=z_1(-1+z_3),& cy_{18}=z_1(-1+z_2),& cy_{19}=-(-z_2+z_1)z_3,\\
cy_{20}=-(z_1-z_3)z_2,& cy_{21}=-(-z_2+z_1)(-1+z_3),&cy_{22}=-(z_1-z_3)(-1+z_2),\\ 
cy_{23}=-(-1+z_1)(z_2-z_3), &cy_{24}=z_1-z_3,& cy_{25}=-z_2+z_1,\\
cy_{26}=-z_1(z_2-z_3),& cy_{27}=-(z_1-z_3)(-1+z_2),& cy_{28}=-(-z_2+z_1)(-1+z_3),\\
cy_{29}=-(z_1-z_3)z_2,& cy_{30}=-(-z_2+z_1)z_3,& cy_{31}=(-1+z_3)z_2,\\
cy_{32}=(-1+z_2)z_3,& cy_{33}=(-1+z_2)z_3,& cy_{34}=(-1+z_3)z_2,\\
cy_{35}=z_1(-1+z_2),& cy_{36}=z_1(-1+z_3),& cy_{37}=(-1+z_1)z_2,\\
cy_{38}=(-1+z_1)z_3,& cy_{39}=z_2-z_3,& cy_{40}=z_2-z_3,\end{array} $$
where
$$c={(z_1-1)(z_1-z_3)(z_2-z_3)(z_1-z_2)z_1\over(1-z_2)^2z_3^2(1-z_3)^2z_2^2}.$$
This shows that the prolonged $\varphi$ gives exactly the embedding of the arranged arragements given by $\{x\mid Q(x)=0\}$, isomorphic to $X(2,6)$, given above.
\par\medskip
Further put
$$x_1=t\xi_1,\quad x_2=t\xi_2,\quad x_3=t\xi_3,\quad x_4=1+t\xi_4,$$
and let $t$ tends to zero in $\varphi(x)$. We can easily see that the limit is a point whchi is independ of $(\xi_1,\dots,\xi_4)$.
\par\medskip
These results together with the facts on Naruki's cross-ratio $\mathcal C$ 
variety reviewd in \S1.3, we can readily show 
\begin{thm}The closure of the
image of $M$ under $\varphi$ is isomorphic to the variety obtained from Naruki's cross-ratio $\mathcal C$ by
blowing down the 40 exceptional divisors of the second kind to points. \end{thm}
\noindent
{\bf Remark:} This variety is isomorphic to the Stake compactification of the modular variety obtained in \cite{ACT}. Note that the 40 cusps correspond to the 40 points obtained by blowing down the divisors of the second kind.
\section{Defining equations}
We will find a set of generators of the ideal defining the closure $\overline{\varphi(M)}$ of the image of $M$ in $\PP^{40-1}.$ 
Among the forty polynomials $y_1(x),\dots,y_{40}(x)$, we can find some relations. The Pr\"ucker relations (\cite{Yo3}) of the $3\times3$-minors of a $3\times6$-matrix yield linear relations among $y_{1}(x),\dots,y_{10}(x)$; for example,
$$(124,356)-(145,236)+(146,235)-(134,256)=y_3(x)-y_4(x)+y_5(x)-y_6(x)=0.$$
On the other hand, it is easy to check the cubic relation
\begin{eqnarray*}&&(124,356)(15,23,46)(13,26,45)-(145,236)(13,56,24)(12,35,46)
\\&&=y_{3}(x)y_{13}(x)y_{21}(x) - y_{4}(x)y_{14}(x)y_{20}(x) =0.\end{eqnarray*}
Let $V$ be the subvariety of $\PP^{40-1}$, coordinatized by $y_1,\dots,y_{40}$,
defined  by the $G$-orbits of the linear and the cubic equations:
$$y_3-y_4+y_5-y_6=0,\quad y_{3}y_{13}y_{21} - y_{4}y_{14}y_{20}=0.$$
\begin{thm}$\overline{\varphi(M)}=V.$\end{thm}
\subsection{An outline of the proof}
By operating the group $G$ to the linear equation, we get many linear relations among $y_1,\dots,y_{40}$. 
These relations form a system of rank 30, that is, all $y$'s can be expressed linearly in terms of ten chosen ones. For example, in terms of 
$$y_{1},\ y_{3},\ y_{4},\ y_{5},\ y_{7},\ y_{11},\ y_{12},\ y_{13},\ y_{15},\ y_{19},$$
the remaining thirty ones are expressed as 
\begin{eqnarray*}
y_{2} &=& y_{1} - y_{5} + y_{4},\quad y_{6} = -y_{4} + y_{5} + y_{3},\quad y_{8} = -y_{3} - y_{1} + y_{7},\\
y_{9} &=& -y_{1} + y_{7} - y_{4},\quad y_{10} = y_{7} - y_{5} - y_{3},\quad y_{14} = y_{11} + y_{13} - y_{12},\\ 
y_{16} &=& -y_{1} + y_{12} - y_{13} - y_{11} + y_{15},\quad y_{17} = y_{15} - y_{1} - y_{13},\\
y_{18}&=&-y_{13} - y_{11} + y_{15},\quad y_{20} = y_{19} + y_{3} + y_{11},\quad y_{21} = y_{4} + y_{19} + y_{12},\\
y_{22} &=& y_{19} + y_{3} + y_{11} + y_{13} + y_{5},\quad y_{23} = y_{1} - y_{12} + y_{13} + y_{3} + y_{11},\\ 
y_{24} &=& y_{4} + y_{1} + y_{13},\quad y_{25} = y_{12} + y_{5} - y_{1},\\ 
y_{26}&=&-y_{4} + y_{5} - y_{1} + y_{3} + y_{11},\quad y_{27}=y_{19} + y_{4} + y_{1} + y_{3} - y_{7} + y_{11} + y_{13},\\ y_{28} &=& -y_{7} + y_{19} + y_{12} + y_{5} + y_{3},\quad
y_{29} = y_{3} - y_{7} + y_{11} + y_{19} + y_{5},\\
y_{30} &=& y_{3} + y_{1} - y_{7} + y_{19} + y_{4},\quad 
y_{31}= -y_{1} - y_{13} + y_{19} + y_{15} + y_{12},\\
y_{32} &=& y_{19} + y_{3} + y_{15},\quad 
y_{33} = y_{19} + y_{4} + y_{15},\\
y_{34} &=& y_{19} - y_{13} + y_{15} + y_{12} + y_{5} - y_{1} + y_{3},\\
y_{35} &=& -y_{5} - y_{1} - y_{3} + y_{7} - y_{11} + y_{15} - y_{13},\\
y_{36} &=& -y_{1} + y_{7} - y_{4} - y_{13} + y_{15},\\
y_{37} &=& -y_{3} + y_{7} - y_{11} + y_{15} - y_{1} + y_{12} - y_{13},\\ 
y_{38} &=& -y_{1} + y_{7} + y_{15},\quad
y_{39} = y_{1} - y_{12} + y_{13},\\ 
y_{40} &=& -y_{5} + y_{4} + y_{1} + y_{13} - y_{12}.
\end{eqnarray*}

By operating the group $G$ to the cubic equation, we get many such relations.
Substituting the expressions of the $y$'s obtained above, 
we get cubic relations in terms of the chosen ten $y$'s; 
let us here rename the ten coordinates as:
\begin{eqnarray*}&&g_1=y_{1},\ g_2=y_{3},\ g_3=y_{4},\ g_4=y_{5},\ g_5=y_{7},\\
&& g_6=y_{11},\ g_7=y_{12},\ g_8=y_{13},\ g_9=y_{15},\ g_0=y_{19}.\end{eqnarray*}
Among these cubic equations we can find exactly thirty linearly independent ones. 
Therefore the variety $V$ is isomorphic to a subvariety of $\PP^9$, coodinatized by $g_1,\dots,g_9,g_0$, defined by thirty cubic equations, say $cub_1,\dots, cub_{30}.$  Some of them which are relatively simple are shown below:
\begin{eqnarray*}
cub_{1}&:=&   g_2g_8g_0+g_2g_8g_7-g_3g_6g_0-g_2g_3g_6-g_3g_6^2-g_8g_0g_3-g_3g_6g_8\\
&&+g_3g_7g_0+g_2g_3g_7+g_6g_7g_3,\\
cub_{2}&:=&   g_0^2g_1+g_3g_1g_0+g_1^2g_0+g_2g_0g_1-g_5g_1g_0+g_0g_1g_6+g_8
g_1g_0+g_2g_3g_6\\
&&+g_2g_6g_1+g_2g_8g_6+g_3g_1g_6+g_1^2g_6+g_8g_1g_6-g_5g_3g_6
-g_5g_1g_6-g_5g_8g_6,\\
cub_{3}&:=&   -g_8g_0g_3+g_8g_0g_4+g_2g_8g_0-g_3g_6g_0-g_2g_3g_6-g_3g_6^2-g_3g_6g_8-g_3g_6g_4,\\
cub_{4}&:=&   g_2g_0g_7+g_2^2g_7+g_2g_6g_7+g_2g_8g_7+g_2g_7g_4-g_0g_4g_6
-g_8g_0g_4+g_0g_4g_7,\\
cub_{5}&:=&   -g_5g_3g_2-g_2g_5g_1-g_5g_8g_2+g_2g_3g_0+g_2g_0g_1+g_2g_8g_0+
g_2g_3g_7\\
&&+g_2g_1g_7+g_2g_8g_7+g_2g_3g_4+g_2g_1g_4+g_2g_8g_4+g_2^2g_3+g_2^2g_1+g_2^2g_8+g_5g_4g_0\\
&&-g_5g_3g_0-g_5g_1g_0-g_5g_8g_0+g_5g_7g_0,\\
cub_{6}&:=&   g_5g_8g_1-g_8g_1g_4-g_2g_8g_1-g_5g_8g_7+g_4g_7g_8+g_2g_8
g_7\\
&&+g_5g_8^2-g_8^2g_4-g_2g_8^2+g_5g_8g_2-g_2g_8g_4-g_2^2g_8+g_5g_8g_6-g_8g_4
g_6-g_2g_8g_6\\
&&-g_5g_7g_6-g_5g_4g_6+g_5g_1g_6,\\
cub_{7}&:=&   -g_9g_2g_3+g_3g_9g_5-g_3g_9g_6-g_9g_3g_0-g_9g_3g_4+2g_9g_2
g_4-g_4g_9g_5\\
&&+g_4g_9g_6+g_9g_0g_4+g_9g_4^2+g_9g_2^2-g_9g_5g_2+g_9g_2g_6+g_9
g_0g_2\\
&&-g_5g_6g_0-g_5g_3g_6-g_5g_6g_9,\\
cub_{8}&:=&   -g_5g_3g_7+g_5g_4g_7+g_2g_5g_7+g_5g_7g_6-g_2g_3g_1-g_2g_0g_1-
g_2g_1g_7+g_5g_3g_1\\
&&+g_5g_1g_0-g_3g_1g_6-g_0g_1g_6-g_1g_7g_6-g_3g_1g_0-g_0^2
g_1-g_1g_7g_0\\
&&-g_1g_4g_3-g_1g_4g_0-g_1g_4g_7,\\
cub_{9}&:=&   -g_8g_1g_3-g_3g_1g_6+g_3g_8g_7+g_6g_7g_3-g_9g_3g_7-g_3g_8
^2-2g_3g_6g_8\\
&&+g_9g_3g_8-g_3g_2g_8-g_2g_3g_6+g_9g_2g_3-g_3g_6^2+g_3g_9g_6-
g_3g_1g_0-g_2g_3g_1\\
&&-g_0^2g_1-g_2g_0g_1-g_9g_0g_1-g_1g_7g_0-g_2g_1g_7-g_9g_1
g_7,\\
cub_{10}&:=&   g_5g_8g_0-g_8g_0g_4-g_2g_8g_0+g_5g_8g_9-g_9g_8g_4-g_2g_9
g_8+g_3g_6g_8\\
&&+g_8g_0g_3+g_2g_3g_6-g_5g_3g_6+g_3g_6^2+g_3g_6g_0+g_3g_6g_4-
g_9g_2g_3+g_3g_9g_5\\
&&-g_3g_9g_6-g_9g_3g_0-g_9g_3g_4,\\
cub_{11}&:=&   g_2g_5g_1+g_5g_1^2-g_5^2g_1+g_5g_1g_0+g_5g_3g_1-g_2g_5g_7-g_5g_1
g_7+g_5^2g_7\\
&&-g_5g_7g_0-g_5g_3g_7+g_5g_8g_2+g_5g_8g_1-g_5^2g_8+g_5g_8g_0+g_5
g_3g_8\\
&&-g_3g_2g_8-g_2g_8g_0-g_2g_8g_7,\\
&\vdots&\\
cub_{19}&:=&   -g_5g_1g_0-g_2g_5g_1+g_0^2g_1+2g_2g_0g_1+g_0g_1g_6+g_1g_7
g_0+g_2g_1g_7+g_1g_7g_6\\
&&+g_1g_4g_0+g_2g_1g_4+g_1g_4g_6+g_2^2g_1+g_2g_6g_1-g_5g_7
g_6-g_5g_4g_6,\\
&\vdots&
\end{eqnarray*}

We first study this system over the field $K:=\C(g_1,\dots, g5)$. 
Geometrically speaking, we project the variety $V$ onto the 4-dimensional 
projective space coordinatized by $g_1:\cdots:g_5$, and study the generic 
fibre of the projection
$$\pi: \PP^9\supset V\ni g_1:\cdots:g_0\longmapsto g_1:\cdots:g_5\in \PP^4.$$
We shall prove that $\pi$ is generically two-to-one. 
This implies $\pi$ is two-to-one on
$$V^\circ:=V-V\cap\cup_{j=1}^{40}\{y_j=0\}.$$
Thus the argument in \S2.4 shows that $\varphi:M\to V^\circ$ is an isomorphism.

\par
We next study the intersection $V\cap\{y_1=0\}$, and prove that $V\cap\cup_{j=1}^{40}\{y_j=0\}$ is the totality of the $G$-orbit of the closure of the image of 
$X(2,6)$ under the prolonged $\varphi$, stated in \S2.5.
\subsection{Computation over $K$}
 From $cub_{3}=0$ and $cub_{7}=0,$ we can solve $g_8$ and $g_9$ as:
\begin{eqnarray*}
g_8 &=& g_3g_6(g_2+g_6+g_0+g_4)/(g_0g_2-g_0g_3+g_0g_4-g_3g_6),\\
g_9 &=& g_5g_6(g_0+g_3)/(-g_2g_3+g_5g_3-g_3g_6-g_0g_3-g_4g_3+2g_2g_4-g_5g_4+g_4g_6+g_0g_4\\
&&+g_4^2+g_2^2-g_5g_2+g_2g_6+g_0g_2-g_5g_6).
\end{eqnarray*}
Substituting these into $cub_{19}$, we can solve $g_6$:
$$g_6=-{g_1(-g_5g_0-g_5g_2+g_0^2+2g_0g_2+g_0g_4+g_7g_0+g_2g_7+g_2^2+g_2g_4)\over g_0g_1+g_1g_4+g_1g_7+g_2g_1-g_5g_7-g_5g_4}.$$
Substituting these expressions into $cub_{1},\dots,cub_{30}$, we have $cub_{3}=cub_{7}=cub_{19}=0$ of course and $cub_{10}=0$; though most of the remaining ones are complicated, $cub_{8}$ is relatively simple:
$$cub_{8}=-(g_1-g_5)qq/(g_0g_1+g_1g_4+g_1g_7+g_2g_1-g_5g_7-g_5g_4),$$
where 
\begin{eqnarray*}
qq&=&g_1g_4t^2+(-g_2g_5-g_5g_4+g_1g_4+g_5g_3)s^2+2g_1g_4st\\
&+&(-g_5g_3g_1+g_2g_1g_5+g_1g_4g_3-g_4^2g_5+g_1g_2g_4+g_5g_3g_4-g_2g_5g_4+g_4^2g_1)s\\
&+&(-g_4g_5g_1+g_4^2g_1+g_1g_4g_3+g_1g_2g_4)t
+g_3g_2g_1g_4-g_5g_3g_1g_4+g_1g_4^2g_3
\end{eqnarray*}
is a quadratic form in 
$$t:=g_0,\quad s:=g_7.$$
By this quadratic equation, we reduce the degree with respect to $t$ of the 
numerators of the $cub$'s to 1. In this way we get 26 $(=30-4)$ equations of the form
$$cube_j:a_jt+b_j=0,\quad j=1,\dots,30,\quad j \not=3,7,10,19,$$
where $a_j,b_j\in K[s].$ The polynomials
$$D_{jk}=a_jb_k-a_kb_j\in K[s],\quad j<k$$
have a unique common factor $dd$, which is quadratic in $s$; its actual form is given by
\begin{eqnarray*}dd&:=&
(g_1^2g_4^2+2g_2g_1^2g_4+g_3^2g_5^2+g_2^2g_3^2+g_2^2g_1^2+g_5^2g_4^2+2g_3g_2g_1g_4-2g_2g_1g_5g_4\\
&&-2g_3g_5^2g_4-2g_1g_5g_4^2+2g_2^2g_3g_1-2g_3^2g_5g_2+2g_5g_3g_2g_4
+2g_5g_3g_1g_4-2g_5g_3g_1g_2)s^2\\
&+&
(-g_2^2g_1^3-2g_1^3g_2g_4+2g_1^2g_2g_4^2+2g_1^2g_5g_4^2+g_1^2g_4^3-g_1^3g_4^2-2g_1g_5g_4^3\\
&&-2g_3g_5^2g_4^2-2g_3g_2^2g_1^2+2g_3g_2g_1g_4^2+g_3^2g_2^2g_4+g_5^2g_4^3+
2g_1^2g_2g_5g_4\\
&&+2g_2^2g_3g_1g_4-4g_3g_5g_1g_2g_4+g_5^2g_3^2g_4+g_2^2g_1^2g_4-g_1
g_5^2g_4^2-g_2^2g_3^2g_1\\
&&-g_3^2g_5^2g_1+2g_5g_3g_2g_4^2+2g_5g_1^2g_2g_3+2g_3^2g_5
g_1g_2-2g_1g_2g_5g_4^2\\
&&+2g_3g_5^2g_1g_4+2g_3g_1g_5g_4^2-2g_3g_5g_1^2g_4-2g_3^2
g_5g_2g_4-2g_1^2g_2g_3g_4)s\\
&-&g_1^2g_3^2g_4^2+g_3g_1^2g_4^3-g_1^3g_3g_4^2-g_1^2g_2g_3^2g_4-g_1^3g_3g_2g_4-g_3^2g_1g_2^2g_4\\
&&-g_3^2g_1g_2g_4^2+g_3g_1g_2g_4^3+g_2^2g_3g_1g_4^2-g_3g_5^2g_1^2g_4+g_1^3
g_3g_5g_4\\
&&+g_3^2g_5g_1g_4^2-g_3^2g_5^2g_1g_4+g_3g_5^2g_1g_4^2-g_3g_5g_1g_4^3-g_3g_1^2
g_2^2g_4\\
&&+g_3^2g_5g_1^2g_4-2g_3g_1g_2g_5g_4^2+2g_3g_1^2g_2g_5g_4+2g_3^2g_1g_2g_5
g_4.\end{eqnarray*}
So $dd(s)=0$ is the compatibility condition of the system of equations $cube_j:a_j(s)t+b_j(s)=0$, among which 
$cube_{11}:a_{11}(s)t+b_{11}(s)=0$
has the minimal degree of coefficients; actually, degrees of $a_{11}$ and $b_{11}$ with respect to $s$ are 3 and 4.

Now we get three equations $dd=0, qq=0$ and $cube_{11}=0$. Substituting the expression $t=-b_{11}/a_{11}$ into $qq$, we get a polynomial in $s$. We can prove that this polynomial has $dd$ as a factor. 
\par\medskip
We have proved that for a given generic point $g_0:\cdots:g_5\in\PP^4$, the inverse under $\pi$ consists of two points. 
They are obtained as follows:  solve the quadratic equation $dd=0$ with respect to $s(=g_7)$. For each root, $t(=g_0)$ is uniquely determined by the linear equation $cube_{11}$. And $g_6,g_8,g_9$ are uniquely determined by $cub3,cub7,cub19$ as stated above. Then all the relations are satisfied.
\subsection{Intersection of $V$ and $\{y_1=0\}$}
We should better work on $V$ living in $\PP^{40-1}$. In terms of the forty coordinates $y_1,\dots,y_{40}$, the cubic equations are 2-term equations. Thus the vanishing of $y_1$ implies that of some other coordinates. Thanks to $G$-action, we can assume that $y_{10}=0$. The vanishing of these two coordinates forces the
vanishing of other coordinates. Tedious case-by-case study shows that every component of $V\cap\{y_1=0\}$ is included in the $G$-orbit of 
$$V\cap\{y_1=\cdots=y_9=y_0=0\},$$
which is the closure of the image of $\{x\mid Q(x)=0\}$ under the prolonged $\varphi$.
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math0002102.extracted_bib
\begin{thebibliography}{MSTY}
\bibitem{ACT}{\sc D. Allcock, J. Carlson and D. Toledo,} 
A complex hyperbolic structure for Moduli of cubic
 surfaces. C.R. Acad. Sci. 326(1998), 49--54.
%\bibitem{CT}{\sc J. Carlson and D. Toledo,} Discriminant compliments and kernels of monodromy representations, alg-geom/9708002, version 3, 11 May 1998.
\bibitem{Ma1}{\sc K. Matsumoto} Theta constants associated with the triple covering of the complex projective line branching at 6 points, preprint, 1999, September.
\bibitem{Ma2}{\sc K. Matsumoto} in preparation, 1999, September.
\bibitem{MSY} K. Matsumoto, T. Sasaki and M. Yoshida:
The monodromy of the period map of a 4-parameter family of K3 surfaces
and the hypergeometric system of type (3,6).
Intern. J. Math. {\bf 3} (1992), 1-164.
\bibitem{Na}{\sc I. Naruki,} Cross ratio variety as a moduli space of cubic surfaces, Proc. london Math. Soc. 45(1982), 1--30
%\bibitem{SaYo1}{\sc T. Sasaki and M. Yoshida,} Linear differential equations in two variables of rank 4, I, II, Math. Ann. 282(1988), 69--93, 95--111.
\bibitem{H}  B. Hunt, A remarkable quintic fourfold in ${\bf P}^5$
and its dual variety (private note 7.1.1992).
\bibitem{SaYo2}{\sc T. Sasaki and M. Yoshida,} 
The uniformizing differential equation of the complex hyperbolic
structure on the moduli space of marked cubic surfaces,
Proc. Japan Acad. {\bf75}(1999), 129--133.
\bibitem{SeYo}{\sc J. Sekiguchi and M. Yoshida,} $W(E_6)$-orbits of the
 configurations space of 6 lines on the real projective space, Kyushu J.
 Math. 51(1997), 1--58.
\bibitem{Yo1}{\sc M. Yoshida,} The real loci of the configuration space
 of six points on the projective line and a Picard modular 3-fold,
 Kumamoto J. of Math. 11(1998), 43--67.
\bibitem{Yo3}{\sc M. Yoshida,} {\em Hypergeometric Functions, My Love,} Vieweg
 Verlag, 1997.
\end{thebibliography}