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Low-dimensional Singularities with Free Divisors as Discriminants

Ragnar-Olaf Buchweitz·Wolfgang Ebeling·Hans-Christian Graf v. Bothmer
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Abstract

We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in C^5. It is shown that such free divisors often admit a "fast normalization", obtained by a single application of the Grauert-Remmert normalization algorithm. For a particular Gorenstein surface singularity in C^5, namely the simple elliptic singularity of type \tilde A_4, we exhibit an explicit discriminant matrix and show that the slice of the discriminant for a fixed j-invariant is the cone over the dual variety of an elliptic curve.

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Ragnar-Olaf Buchweitz, Wolfgang Ebeling, Hans-Christian Graf v. Bothmer. 2006-12-20. Low-dimensional Singularities with Free Divisors as Discriminants. https://arxiv.org/abs/math/0612119

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