On the Freeness of Equisingular Deformations of Plane Curve Singularities

We consider surface singularities in C3 arising as the total space of an equisingular deformation of an isolated curve singularity of the form f(x, y)+zg(x, y) with f and g weighted homogeneous. We give a criterion that such a surface is a free divisor in the sense of Saito. We deduce that the Hessian deformation defines a free divisor for nonsimple weighted homogeneous singularities, and that the failure of this property “almost” characterizes the simple singularites. The criterion also yields distinct deformations of the same curve singularity, exactly one of which is free, showing that freeness is not a topological property.