Equinormalizable theory for plane curve singularities with embedded points and the theory of equisingularity (Q713353)

The paper under review studies families of complex plane curve singularities, parametrized by \({\mathbb C}^k\), \(k \geq 1\), or an open set thereof. These are its main results: (1) Consider a family of germs \(f:(X,0) \to ({\mathbb C}^k,0)\) (\(k \geq 1\), \(f\) is flat), where we assume that the fibers have dimension \(\leq 1\), with at most finitely many singular points, but they may be non-reduced. Such a family is also called a (local) deformation of the germ of curve \((X_0,0)\), the fiber at \(0\). There is an induced family \(f^u:(X^u,0) \to ({\mathbb C}^k,0)\), where \((X^u,0)\) is the \textit{unmixed} subgerm (or part) of \((X,0)\) (i.e., the union of the irreducible components of maximum dimension of \((X_{\mathrm{red}},0)\).) Then it is proved that \(f^u\) is again flat and the deformation it defines is \(\delta\)-constant if and only if \(f^u\) is equinormalizable. The concepts of \textit{unmixed part}, \(\delta\)-\textit{invariant} and \textit{equinormalizable morphism} are reviewed by the author. \textit{G. M. Greuel} and the author have obtained (in an article in preparation) similar results for families of curves which are not necessarily planar, but the methods required are different. The similar problem in case all the fibers are reduced curves was studied by \textit{B. Teissier} in the 1970's, see [Résolution simultanée. I. II. Lect. Notes Math. 777, 71--146 (1980; Zbl 0464.14005)]. (2) The author proves theorems comparing different possible notions of equisingularity for deformations of plane curve singularities, reduced everywhere except possibly at finitely many points, parametrized by \(({\mathbb C}^k,0)\). Some of these notions involve certain associated weighted trees, whose vertices correspond to infinitiely near points of the singularity. The connection of these notions and the property that the family be \(\delta\)-constant is also studied. Some of these results were obtained by the reviewer in [\textit{A. Nobile}, Pac. J. Math. 170, No. 2, 543--566 (1995; Zbl 0903.14002)], but working with one-parameter families (i.e., \(k=1\).) The article contains several interesting examples.