Semicontinuity of Singularity Invariants in Families of Formal Power Series

TL;DR

This paper investigates the semicontinuity of singularity invariants like Milnor and Tjurina numbers in families of formal power series, providing new conditions under which these invariants vary predictably in algebraic families.

Contribution

It introduces a general framework using completed and Henselian tensor products to prove semicontinuity of fiber dimensions of modules in families of formal power series.

Findings

Fiber dimension is semicontinuous under mild conditions.

Applicable to Milnor and Tjurina numbers in hypersurfaces.

Provides bounds for invariants in algebraic families.

Abstract

The problem we are considering came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal power series parametrized by some algebraic variety. In contrast to the case of analytic or algebraic families, where such a bound is well known, the problem is rather subtle, since the modules defining the invariants are quasi-finite but not finite over the base space. In fact, in general the fibre dimension is not semicontinuous and the quasi-finite locus is not open. However, if we pass to the completed fibers in a family of modules we can prove that their fiber dimension is semicontinuous under some mild conditions. We prove this in a rather general framework by introducing and using the completed and the Henselian tensor product, the proof being more involved as one might think. Finally we apply this to the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections and to the determinacy in a family of ideals.