Summary
We describe a new construction to obtain a simple hypersurface singularity from the corresponding simple complex Lie-groupG. LetX be the closed orbit in the projective space attached to the Lie algebra\(\mathfrak{g}\) ofG. Consider a regular nilpotent element\(y_0 \in \mathfrak{g}\) and denote byH y 0 the hyperplane orthogonal toy 0 with respect to the Killing form. Then the hyperplane sectionX⊃H y 0, has exactly one singularity which is simple of desired type. By variation of the pointy 0 we obtain a versal deformation. The construction generalizes with minor modifications to any characteristicp of the basefield. Even in bad characteristic we recover at least the positive part of the semiuniversal deformation. We prove that forp=2 a simple, quasihomogeneous singularity of type A7 resp. D8 is adjacent to E7 resp. E8 provided its dimension is even. Furthermore A8 is adjacent to E8 forp=3.
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Knop, F. Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten. Invent Math 90, 579–604 (1987). https://doi.org/10.1007/BF01389179
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DOI: https://doi.org/10.1007/BF01389179