Versality Properties of Projective Hypersurfaces

  • Andrew A. du Plessis
Part of the Trends in Mathematics book series (TM)


Let X be a hypersurface of degree d in P n(ℂ) with isolated singularities, and let f: ℂ n+1 → ℂ be a homogeneous equation for X.

The singularities of X can be simultaneously versally deformed by deforming the equation f, in an affine chart containing all of the singularities, by the addition of all monomials of degree at most r, for sufficiently large r; it is known (see, e.g., §1) that rn(d−2) suffices. Conversely, if the addition in the affine chart of all monomials of degree at most n(d−2)−1−a, a ≥ 0, fails to simultaneously versally deform the singularities of X, then we will say that X is a-non-versal.

The first main result of this paper shows that X is a-non-versal if, and only if, there exists a homogeneous polynomial vector field with coefficients of degree a, which annihilates f but is not Hamiltonian for f.

Our second main result is a sufficient condition for an a-non-versal hypersurface to be topologically a-versal.


Versal Deformation Versality Property Gorenstein Algebra Saturated Ideal Projective Hypersurface 
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© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Andrew A. du Plessis
    • 1
  1. 1.Department of Mathematical SciencesUniversity of AarhusAarhus CDenmark

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