Summary: Let \(X\) denote either \(\mathbb {CP}^m\) or \(\mathbb {C}^m\). 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 \(q=(q_{1},\dots ,q_{n}) \in X^{n}\) such that no \(m + 1\) of the points \(q_1,\dots ,q_n\) belong to a hyperplane in \(X\). In particular, we show that for large enough \(n\) any holomorphic map \(f : \mathcal {E}^n(\mathbb {CP}^m,gp) \to {\mathcal E}^n(\mathbb {CP}^m,gp)\) commuting with the natural action of the symmetric group \(S(n)\) in \({\mathcal E}^n(\mathbb {CP}^m,gp)\) is of the form \(f(q) = \tau(q)q = (\tau(q)q_{1},\dots ,\tau(q)q_{n})\), \(q \in {\mathcal E}^n(\mathbb {CP}^m,gp)\), where \(\tau:{\mathcal E}^n(\mathbb {CP}^m,gp) \to \text{PSL}(m+1,\mathbb C)\) is an S(\(n\))-invariant holomorphic map. A similar result holds true for mappings of the configuration space \({\mathcal E}^n(\mathbb {C}^m,gp)\).