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](https://doi.org/10.1007/BFb0069274)’ without proof.

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.