Real and Complex Singularities pp 299-310 | Cite as

# Minimal Intransigent Hypersurfaces

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## Abstract

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*).

## Keywords

Polynomial Vector Generic Choice Simple Singularity Gorenstein Algebra Cusp Singularity## Preview

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