Versal Deformations in Spaces of Polynomials of Fixed Weight

Abstract

This work was largely inspired by a paper of Shustin, in which he proves that for a plane curve of given degree n whose singularities are not too complicated the singularities are versally unfolded by embedding the curve in the space of all curves of degree n; however, our methods are very different. The main result gives fairly explicit lower bounds on the sum of the Tjurina numbers at the singularities of a deformation of a weighted-homogeneous hypersurface, when the deformation is the fibre over an unstable point of an appropriate unfolding. The result is sufficiently flexible to cover a variety of applications, some of which we describe. In particular, we will deduce a generalisation of Shustin‘s result. Properties of discriminant matrices of unfoldings of weighted-homogeneous functions are crucial to the arguments; the parts of the theory needed are described.