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\title $\Bbb C\Bbb R$-Geometry on the configuration space of 5 points on the projective line\endtitle
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%Dedicated to Professor Sadakazu AIZAWA
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\author
Nobuki Takayama and Masaaki Yoshida
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\address
Department of Mathematics, Faculty of Science, Kobe University, Kobe 657, Japan
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\address
Graduate School of Mathematics, Kyushu University 33, Fukuoka 812, Japan
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\bigskip\noindent
{\bf Introduction}\medskip\noindent
The configuration space $X$ of five distinct colored points on the complex projective line is studied by various authors; most of them are complex-algebraic-geometry oriented. In [Sek] and [ST] however, they divided $X$ into twenty simply connected parts, which are permuted by the symmetric group $S_5$, and studied their properties. The decomposition can be regarded as a generalization of the phenomenon that the complex projective line $\P^1$ is divided into two parts by the real projective line living in $\P^1$. This paper presents a polished version of [Sek], which is very hard to read, and hopefully answers Problem 6.3 there. In the text there are no {\it proofs}, because once the statements are well formulated and presented one can prove anyway.
If the reader wonders why this combinatorial-topology-like paper was submitted to this journal specialized to functional equations, see [ST].
\medskip
The second author is grateful to Professors K. Cho and M. Kato for valuable discussions.
\bigskip\noindent
{\bf 1. Preliminaries}\medskip\noindent
The configuration space $X$ of five distinct colored points on the complex projective line $\P^1$ is defined by
$$X=GL(2,\C)\backslash\{(\P^1)^5-\D\},$$
where the action is diagonal and
$$\D=\{(x_1,\dots,x_5)\in(\P^1)^5\mid x_i=x_j \text{ for some } i\ne j\}.$$
The symmetric group $S_5$ acts on $X$ as permutations of five points.
A smooth compactification of $X$ is given by
$$\overline X=GL(2,\C)\backslash\{(\P^1)^5-\D'\},$$
where
$$\D'=\{(x_1,\dots,x_5)\in \Delta\mid \text{no three points coincide}\}.$$
It is isomorphic to the surface obtained by blowing up four points in general position (i.e. no three points lie on a line) on the complex projective plane. Thus there are ten rational curves with self-intersection number $-1$; let us name them as follows:
$$L(ij)=L(ji)=\text{ the orbit of }\{x_i=x_j\},\quad 1\le i,j\le5,\quad i\ne j.$$
Two such distinct curves do not meet or meet normally at a point:
$$L(ij)\cap L(pq)=\text{ a point } \Longleftrightarrow \{i,j\}\cap\{p,q\}=\emptyset.$$
Notice that
$$X=\overline X-\cup L(ij).$$
The smooth action of $S_5$ on $\overline X$ induces a transitive action on the set $\{L(ij)\}$.
Let $X_\R,\ \overline X_\R$ and $L_\R(ij)$ be the manifolds consisting of real valued points of $X,\ \overline X$ and $L(ij)$, respectively. They are the sets of fixed points of the involution $c$ induced by the complex conjugation
$$(x_1,\dots,x_5)\mapsto (\overline x_1,\dots,\overline x_5).$$
\medskip
Let us consider an embedding $\overline X\to (\P^1)^5$:
$$(x_1,\dots,x_5)\mapsto (\varphi_1(x),\dots,\varphi_5(x)),$$
where $\varphi_j(x)$ is a cross-ratio of four points $\{x_i\mid 1\le i\le5,\ i\ne j\}$.
For four points on the projective line, if no three points coincide, one can define six cross-ratios, here we do not care which one we choose. One can readily check that this is well-defined and gives an embedding.
The first author found that
$$\overline X-\cup_{j=1}^5\{x\in \overline X\mid \Im \varphi_j(x)=0\},$$
where $\Im$ stands for imaginary part,
is the disjoint union of twenty simply connected open subsets (let us call them open chambers) of $\overline X$, and that the group $S_5$ acts transitively on the twenty open chambers. Note that the set $\{x\in \overline X\mid \Im \varphi_j(x)=0\}$ does not depend on the choice of a cross-ratio.
\proclaim{Proposition-Definition} For each curve $L(ij)$, there are exactly two open chambers that do not touch the curve, i.e. their closures do not intersect $L(ij)$. The two open chambers are permuted by the involution $c$. Let us call them $C(ij)^+=C(ji)^+,C(ij)^-=C(ji)^-$, and put
$$\Cal C=\{C(ij)^+,C(ij)^-\mid 1\le i