Fundamental Groups of Spaces of Smooth Projective Hypersurfaces

Details Fundamental Groups of Spaces of Smooth Projective Hypersurfaces Fundamental Groups of Spaces of Smooth Projective Hypersurfaces

2006-08-14

arxiv.org/abs/math/0608346v1

Mathematics

Algebraic Geometry

34 pages

Scientific paper

We investigate the complement of the discriminant in the projective space PSym^d C^{n+1} of polynomials defining hypersurfaces of degree d in 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 by Bessis and draws on results and methods from previous work of the author addressing a comparable problem for any versal unfolding of Brieskorn Pham singularities.

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