Real and Complex Singularities pp 289-298 | Cite as

# Versality Properties of Projective Hypersurfaces

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## Abstract

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 *r*≥*n*(*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.

## Keywords

Versal Deformation Versality Property Gorenstein Algebra Saturated Ideal Projective Hypersurface## Preview

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