Abstract
Let
X
denote either
\Bbb{CP}^m
or
\Bbb{C}^m
. We study certain analytic properties of the space
\mathcal{E}^n(X,gp)
of ordered geometrically generic
n
-point configurations in
X
. This space consists of all
q=(q_1,...,q_n)\in X^n
such that no
m+1
of the points
q_1,...,q_n
belong to a hyperplane in
X
. In particular, we show that for a big enough
n
any holomorphic map
f\colon\mathcal{E}^n(\Bbb{CP}^m,gp)\to\mathcal{E}^n(\Bbb{CP}^m,gp)
commuting with the natural action of the symmetric group
\mathbf{S}(n)
in
\mathcal{E}^n(\Bbb{CP}^m,gp)
is of the form
f(q)=\tau(q)q=(\tau(q)q_1,...,\tau(q)q_n)
,
q\in \mathcal{E}^n(\Bbb{CP}^m,gp)
, where
\tau\colon\mathcal{E}^n(\Bbb{CP}^m,gp) \to{\mathbf{PSL}(m+1,\mathbb C)}
is an
\mathbf{S}(n)
-invariant holomorphic map. A similar result holds true for mappings of the configuration space
\mathcal{E}^n(\Bbb{C}^m,gp)
.