Semicontinuity for families of power series II

Source abstract

The present article is a generalization of our paper \cite{GP21}, together with new applications. The original problem came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants cannot increase if we deform a given singularity. In \cite{GP21} we proved this for the completed fiber dimension under deformations with section. In general the nearby fiber contains however several singularities and in this article we prove that also the sum of the completed fiber dimensions behaves semicontinuous. This is well known for analytic or algebraic families. However, for families of power series the problem is more difficult, since the the modules defining the invariants are quasi-finite but not finite over the base space. In fact, in general the usual fibre dimension is not semicontinuous. However, if we pass to the completed fibres in a family of rings or modules over arbitrary Noetherian rings, we can prove that their dimension is semicontinuous. The proof is different from that given in \cite{GP21} and, though more general, even easier. Finally we apply this to prove the semicontinuity of several singularity invariants, such as the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections. We end with a problem concerning the Milnor number of an isolated complete intersection singularity.