Fundamental Groups of Spaces of Smooth Projective Hypersurfaces

Abstract

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.

34 pages

References in corpus (3)

• Zariski theorems and diagrams for braid groups
• Braid Monodromy of Hypersurface Singularities
• Surfaces in the complex projective plane and their mapping class groups

Cited by in corpus (9)

• Discriminants in the Grothendieck Ring
• The vanishing cycles of curves in toric surfaces I
• Commuting symplectomorphisms and Dehn twists in divisors
• Kummer covers and braid monodromy
• Moduli spaces and braid monodromy types of bidouble covers of the quadric
• Monodromy and mapping class groups of 3-dimensional hypersurfaces
• A Zariski theorem for monodromy of $A$-hypergeometric systems
• Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large $b_2$
• Mapping Class Groups of Trigonal Loci