On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

In this paper we study relationships between some topological and analytic invariants of zero-dimensional germs, or multiple points. Among other things, it is shown that there exist no rigid zero-dimensional Gorenstein singularities and rigid almost complete intersections. In the proof of the first result we exploit the canonical duality between homology and cohomology of the cotangent complex, while in the proof of the second we use a new method which is based on the properties of the torsion functor. In addition, we obtain highly efficient estimates for the dimension of the spaces of the first lower and upper cotangent functors of arbitrary zero-dimensional singularities, including the space of derivations. We also consider examples of nonsmoothable zero-dimensional noncomplete intersections and discuss some properties and methods for constructing such singularities using the theory of modular deformations, as well as a number of other applications.