Equisingularity and EIDS

Abstract

We continue the study of the equisingularity of determinantal singularities for essentially isolated singularities (EIDS).These singularities are generic except at isolated points.Definition 2.1. A point x ∈ X = M −1 (Σ t ) is called essentially non-singular if, at the point x, the map M is transversal to the corresponding stratum of the variety Σ t .

The local Euler obstruction and topology of the stabilization of associated determinantal varieties

“…To do that they use the holomorphic triviality of the stratification of Hom(C n , C n+k ). The next result is Corollary 2.16 of [16], where…”

“…In [16] the authors work with the multiplicity of the pair (JM ( i X), N ( i X)), where JM ( i X) is the Jacobian module and N ( i X) is the module of infinitesimal first order deformation of i X, induced from the first order deformation of the presentation matrix of X. They showed that the multiplicity of that pair is well defined for EIDS.…”

“…One of the main ingredients to prove our main result of this section is the next proposition. As remarked in [16], in the EIDS context, instead of a smoothing, we have a stabilization -a determinantal deformation of X to the generic fiber. Then the multiplicity of the polar curve of JM z (X ) over the parameter space at the origin in a stabilization is the number of critical points that a generic linear form has on the complement of the singular set on a generic fiber.…”

“…Remark 2.9. In [16], applying P rop. 4 of [8] to EIDS case, Gaffney and Ruas obtained the equations below.…”

“…In fact, the original motivation for the paper came from two sources; it was noted in earlier work on [16] that for generic maps the Euler obstruction for generic determinantal singularities should be closely related to the Euler obstruction of the pullback. Further, the Euler obstruction of the generic determinantal singularities appeared in the formula for the Euler characteristic of the stabilization of an EIDS for sufficiently generic sections.…”