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) .