Versality Properties of Projective Hypersurfaces
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 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.
KeywordsVersal Deformation Versality Property Gorenstein Algebra Saturated Ideal Projective Hypersurface
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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