I have a very specific question. How does one check the following map 
${\cal C}^n-\cup \{z_i=\pm z_j\ for\ i\neq j\}\to {({\cal C}^*)}^{n-1}-\cup\{z_i= z_j\ for\ i\neq j\}$ defined by $(z_1,z_2,\ldots , z_n)\mapsto (z_n^2-z_1^2,\ldots , z_n^2-z_{n-1}^2)$ is a locally trivial fibration? This is stated in the paper of E. Brieskorn 
`Sur les groupes de tresses' without proof.

Actually, in the paper of Brieskorn there are few other above type maps  defined and stated to be locally trivial fibration. Probably, there is some standard technique to check.