Fundamental groups of projective discriminant complements

Abstract

We investigate the complement of the discriminant in the projective space ${\mathbf P}{\rm Sym}^d{\mathbf C}^{n+1}$ of polynomials defining hypersurfaces of degree $d$ in ${\mathbf P}^n$. 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 $n=1$ (i.e., of the spherical braid group on $d$ 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