We investigate the complement of the discriminant in the projective space of polynomials defining hypersurfaces of degree in . Following the ideas of Zariski, we are able to give a presentation for the fundamental group of the discriminant complement which generalises the well-known presentation in case (i.e., of the spherical braid group on strands).
In particular, our argument proceeds by a geometric analysis of the discriminant polynomial as proposed in [Be] and draws on results and methods from [L1] addressing a comparable problem for any versal unfolding of Brieskorn-Pham singularities
"Fundamental groups of projective discriminant complements." Duke Math. J. 150 (2) 357 - 405, 1 November 2009. https://doi.org/10.1215/00127094-2009-055