Spaces of geometrically generic configurations

Abstract

Let $X$ denote either $\Bbb{CP}^m$ or $\Bbb{C}^m$. We study certain analytic properties of the space $\Cal{E}^n(X,gp)$ 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 $f\colon\Cal{E}^n(\Bbb{CP}^m,gp)\to\Cal{E}^n(\Bbb{CP}^m,gp)$ commuting with the natural action of the symmetric group $\mathbf{S}(n)$ in $\Cal{E}^n(\Bbb{CP}^m,gp)$ is of the form $f(q)=\tau(q)q=(\tau(q)q_1,...,\tau(q)q_n)$, $q\in \Cal{E}^n(\Bbb{CP}^m,gp)$, where $\tau\colon\Cal{E}^n(\Bbb{CP}^m,gp) \to{\mathbf{PSL}(m+1,\mathbb C)}$ is an $\mathbf{S}(n)$-invariant holomorphic map. A similar result holds true for mappings of the configuration space $\Cal{E}^n(\Bbb{C}^m,gp)$.