Geometry and equisingularity of finitely determined map germs from $${\mathbb {C}}^n$$ C n to $${\mathbb {C}}^3$$ C 3 , $$n >2$$ n > 2

Abstract

In this article we describe the geometry and the Whitney equisingularity of finitely determined map germs \(f{:} ({\mathbb {C}}^n,0) \rightarrow ( {\mathbb {C}}^3,0)\) with \(n \ge 3\). In the study of the geometry, we first investigate the critical locus \(\Sigma (f)\) of the germ, which is in the source. Then the discriminant \(\Delta (f)\), the image of the critical locus by the germ f, is studied. Last, but not least we investigate the set X(f), which is the inverse image by f of the discriminant. If the critical locus is not empty, the set X(f) is an hypersurface in the source that has nonisolated singularity at the origin. Concerning the Whitney equisingularity of families, we use some of the properties of the strata to prove that the Whitney equisingularity of an unfolding F is equivalent to the constancy of the Le numbers of the hypersurfaces \(\Delta (f)\) and X (f). From this study we describe some relationship among the invariants needed to describe the Whitney equisingularity of families in these dimensions, we reduce the number of invariants needed to a total of \(2n+2\), which improves substantially the number required by Gaffney’s theorem.