Spaces of geometrically generic configurations

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Let X denote either CPᵐ or Cᵐ. We study certain analytic properties of the space Eⁿ of ordered geometrically generic n-point configurations in X. This space consists of all q=(q₁,...,qₙ) in Xⁿ such that no m+1 of the points q₁,...,qₙ belong to a hyperplane in X. In particular, we show that for X=CPᵐ and n big enough any holomorphic map f:Eⁿ-->Eⁿ commuting with the natural action of the symmetric group S(n) in Eⁿ is of the form f(q)=t(q)q=(t(q)q₁,...,t(q)qₙ), for q in Eⁿ, where t:Eⁿ-->PSL(m+1,C) is an S(n)-invariant holomorphic map. A similar result holds true for mappings of the space of ordered geometrically generic n-point configurations in Cᵐ.