Pairs of modules and determinantal isolated singularities

Abstract

We continue the development of the study of the equisingularity of isolated singularities, in the determinantal case. This version of the paper includes a substantial amount of new material (76% larger). The new material introduces the idea of the landscape of singularity, which includes the allowable deformations of the singularity and associated structure useful for equisingularity questions. Fixing a presentation matrix M of a determinantal singularity means viewing the singularity as a section via M of the set of matrices of a given or smaller rank. Varying M gives the allowable deformations of X. This version also includes a description of the conormal varieties of the rank singularities, which is applied to our machinery. There is also an example of a determinantal singularity which is a member of two Whitney equisingular families, whose generic elements have topologically distinct smoothings. This example shows that it is impossible to find an invariant which depends only on an analytic space $X$ with an isolated singularity, whose value is independent of parameter for all Whitney equisingular deformations of $X$, and which is determined by the geometry of a smoothing of $X$.

37 pages. An earlier version of version 1 was the basis of a talk given at the International Workshop on Real and Complex Singularities at Sao Carlos in the summer of 2014 by the first author

References in corpus (2)

• The Multiplicity-Polar Theorem
• Genus of numerical semigroups generated by three elements

Cited by in corpus (5)

• On the Vanishing Topology of Isolated Cohen-Macaulay Codimension 2 Singularities
• The local Euler obstruction and topology of the stabilization of associated determinantal varieties
• Lipschitz Normal Embeddings in the Space of Matrices
• Local volumes, equisingularity, and generalized smoothability
• Two Formulas for the BR Multiplicity