Versal deformation of algebraic hypersurfaces with isolated singularities Academic Article uri icon

abstract

  • We study deformations of projective hypersurfaces F⊂ Pn, n≥ 2, over an algebraically closed field of characteristic zero and pose the following questions. Given a hypersurface F∈ Pn of degree d with only isolated singular points, is the germ at F of the space| OPn (d)| of hypersurfaces of degree da versal deformation of the multisingularity of F? Is the germ at F of the equisingular stratumV⊂| OPn (d)| smooth of “expected” dimension? An affirmative answer means, in particular, that varying coefficients of the polynomial ofF, one can simultaneously and independently deform singular points of F in a prescribed (up to local diffeomorphisms) way. In other words, the geometry of the discriminant in| OPn (d)| in a neighborhood of F is determined by the local structure of singularities of F. Our goal is to find a sufficient condition for this via numerical invariants of hypersurfaces and their …

publication date

  • January 1, 1999