Spaces of geometrically generic configurations

Abstract

Let X denote either CP^m or C^m. We study certain analytic properties of the space E^n 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 X=CP^m and n big enough any holomorphic map f:E^n-->E^n commuting with the natural action of the symmetric group S(n) in E^n is of the form f(q)=t(q)q=(t(q)q_1,...,t(q)q_n), for q in E^n, where t:E^n-->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 C^m.