I have a very specific question. How does one check the following map ${\mathbb C}^n-\bigcup \{\text{\(z_i=\pm z_j\) for \(i\neq j\)}\}\to {({\mathbb C}^*)}^{n-1}-\bigcup\{\text{\(z_i= z_j\) for \(i\neq j\)}\}$ defined by $(z_1,z_2,\dotsc , z_n)\mapsto (z_n^2-z_1^2,\dotsc , z_n^2-z_{n-1}^2)$ is a locally trivial fibration? This is stated in the paper of E. Brieskorn ‘Sur les groupes de tresses’.

Actually, in the paper there are few other above type maps are defined and stated to be locally trivial fibration. Probably, there is some standard technique to check. Any hint, how to prove it?