Low-dimensional singularities with free divisors as discriminants

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 \mathbb {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 \mathbb {C}^5 , namely the simple elliptic singularity of type A ~ 4 \widetilde A_{4} , we exhibit an explicit discriminant matrix and show that the slice of the discriminant for a fixed j j -invariant is the cone over the dual variety of an elliptic curve.