Minimal Intransigent Hypersurfaces
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We give examples of hypersurfaces of degree d in P n(ℂ), whose singularities are not versally deformed by the family H d(n) of all hypersurfaces of degree d in P n(ℂ), and which are of minimal codimension with this property.
In the three cases (n, d) = (2, 6), (3, 4) and (5, 3), such hypersurfaces necessarily have one-parameter symmetry. We list the possibilities. The singularities of these hypersurfaces are not all simple, and they are simultaneously topologically versally deformed by H d(n).
In less degenerate cases the examples we give are hypersurfaces with only simple singularities. The failure of versality can be expected to show itself in the geometry of H d(n), either because the μ-constant stratum S containing the hypersurface is of codimension less than μ in H d(n), or because S is not smooth. We will see elsewhere that this is the case for the examples we consider here. In particular, the singularities of these hypersurfaces are not topologically versally deformed by H d(n).
KeywordsPolynomial Vector Generic Choice Simple Singularity Gorenstein Algebra Cusp Singularity
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