On the conflict variety of an isolated hypersurface singularity
Let \(f:(\mathbb{C}^ n,0)\to(\mathbb{C},0)\) be a holomorphic germ with isolated singularity and \(F:(\mathbb{C}^ k\times\mathbb{C}^ n,0)\to(\mathbb{C}^ k\times\mathbb{C},0)\) its versal unfolding. The discriminant \(\Delta(F)\) of this unfolding is a branched cover of \((\mathbb{C}^ k,0)\). The branch locus of this cover \(D(F)\) contains as one irreducible component the bifurcation variety \(\text{Bif}(F)\). The union of all the other components is called the conflict variety \(\text{Con}(F)\). In the article the reduced hypersurfaces \(\text{Con}(F)\) and \(\text{Bif}(F)\) in \((\mathbb{C}^ k,0)\) are studied. For instance it is proved: \(\text{Con}(F)\) is irreducible except the cases of \(A_ 1\), \(A_ 2\) and \(D_ n\)- singularities. The singularity is uniquely determined (modulo stably right --- left equivalence) by \(\text{Con}(F)\cup\text{Bif}(F)\).
scientific article; zbMATH DE number 108920