Deformationen isolierter Kurvensingularitäten mit eingebetteten Komponenten. (Deformations of isolated curve singularities with embedded components)
\textit{R.-O. Buchweitz} and the second author studied deformations of reduced isolated curve singularities in their paper in Invent. Math. 58, 241-281 (1980; Zbl 0458.32014). The main invariants in that paper are the Milnor number \(\mu\) and the number of virtual double points \(\delta\). In families of complex spaces it is quite common, that special fibres contain embedded components. This has the effect that the generic fibre can consist of (isolated) fat points and disjoint 1-dimensional components. This is the main theme in the paper under review. The authors define a new invariant \(\epsilon\), which measures the non-reduced structure and they generalize the definitions of \(\mu\) and \(\delta\). The new definitions behave very well with respect to the topological properties of the singularities. As an example: the jump in \(\mu\) is equal to the change in the Euler characteristic of the fibre. This is similar to the reduced case, but in the non-reduced case also may new phenomena occur. The paper is well-written and contains all the technical details, which are necessary to understand the situation.
scientific article; zbMATH DE number 4202700
scientific article