Low-dimensional Singularities with Free Divisors as Discriminants

Details Low-dimensional Singularities with Free Divisors as Discriminants Low-dimensional Singularities with Free Divisors as Discriminants

2006-12-05

arxiv.org/abs/math/0612119v2

J. Algebraic Geom. 18 (2009), 371-406

Mathematics

Algebraic Geometry

29 pages, misprints and references corrected

Scientific paper

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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