Spaces of geometrically generic configurations

Feler, Yoel. "Spaces of geometrically generic configurations." Journal of the European Mathematical Society 010.3 (2008): 601-624. <http://eudml.org/doc/277430>.

@article{Feler2008,
abstract = {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 a big enough $n$ any holomorphic map $f\colon \mathcal \{E\}^n(\mathbb \{CP\}^m,gp)\rightarrow \mathcal \{E\}^n(\mathbb \{CP\}^m,gp)$ commuting with the natural action of the symmetric group $\mathbf \{S\}(n)$ 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 $\tau \colon \mathcal \{E\}^n(\mathbb \{CP\}^m,gp) \rightarrow \{\mathbf \{PSL\}(m+1,\mathbb \{C\})\}$ is an $\mathbf \{S\}(n)$-invariant holomorphic map. A similar result holds true for mappings of the configuration space $\mathcal \{E\}^n(\mathbb \{C\}^m,gp)$.},
author = {Feler, Yoel},
journal = {Journal of the European Mathematical Society},
keywords = {configuration space; geometrically generic configurations; vector braids; points in general position; holomorphic endomorphism; configuration space; geometrically generic configurations; vector braids; points in general position; holomorphic endomorphism},
language = {eng},
number = {3},
pages = {601-624},
publisher = {European Mathematical Society Publishing House},
title = {Spaces of geometrically generic configurations},
url = {http://eudml.org/doc/277430},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Feler, Yoel
TI - Spaces of geometrically generic configurations
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 3
SP - 601
EP - 624
AB - 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 a big enough $n$ any holomorphic map $f\colon \mathcal {E}^n(\mathbb {CP}^m,gp)\rightarrow \mathcal {E}^n(\mathbb {CP}^m,gp)$ commuting with the natural action of the symmetric group $\mathbf {S}(n)$ 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 $\tau \colon \mathcal {E}^n(\mathbb {CP}^m,gp) \rightarrow {\mathbf {PSL}(m+1,\mathbb {C})}$ is an $\mathbf {S}(n)$-invariant holomorphic map. A similar result holds true for mappings of the configuration space $\mathcal {E}^n(\mathbb {C}^m,gp)$.
LA - eng
KW - configuration space; geometrically generic configurations; vector braids; points in general position; holomorphic endomorphism; configuration space; geometrically generic configurations; vector braids; points in general position; holomorphic endomorphism
UR - http://eudml.org/doc/277430
ER -