2006/03/21 by Javier Fernandez de Bobadilla, de Bobadilla, Javier Fernandez
Mathematics · #14J17 #32S15 #32S50 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV #msc:14J17 #msc:32S15 #msc:32S50
paper · pdf · doi:10.48550/arxiv.math/0603508
52 pages, corrected typos, improved exposition, added abstract, added new theorem on singularity stems
arxiv created 2006/10/20 · arxiv updated 2009/12/01
We focus on topological equisingularity of families of holomorphic function germs with 1-dimensional critical set. We introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular. We show that if a family of germs with 1-dimensional critical set has constant generic Le numbers then it is equisingular at the critical set, and hence topologically equisingular (answering a question of D. Massey). We use this to modify the definition of singularity stem present in the literature, introducing and characterising topological stems (being this concept closely related with Arnold's series of singularities). We provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly. Finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non-flat deformation with singular special fibre and smooth generic fibre.