Let denote either or . We study certain analytic properties of the space of ordered geometrically generic -point configurations in . This space consists of all such that no of the points belong to a hyperplane in . In particular, we show that for a big enough any holomorphic map commuting with the natural action of the symmetric group in is of the form , , where is an -invariant holomorphic map. A similar result holds true for mappings of the configuration space .