Pairs of modules and determinantal isolated singularities

Gaffney, Terence; Rangachev, Antoni

doi:10.48550/arxiv.1501.00201

Cited by 9 publications

(15 citation statements)

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“…For the following Corollary, it is convenient to change our notation a little since the main tool is based on [15], so we match the notation there. We let Σ r denote Σ r+1 , that is we let Σ r denote the matrices of kernel rank r.…”

Section: Proof By Definition We Have Eumentioning

confidence: 99%

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

2018

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This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern-Schwartz-MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney's m d multiplicity.

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Section: Proof By Definition We Have Eumentioning

confidence: 99%

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

2018

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“…Following [GR16] we show that we can control the presence of vertical components of D for X → Y by computing the restricted local volumes from generic one-parameter families connecting X y 0 and X y to fibers X w for which the local volume stabilizes. This gives an extension of the LVF to good base spaces of arbitrary dimension.…”

Section: Introductionmentioning

confidence: 99%

“…Gaffney ([Gaf92] and [Gaf96] based on ideas conceived in [Gaf93]) and Gaffney and Kleiman ([GK99]) treated the case of isolated completeintersection singularities using the Buchsbaum-Rim (BR) multiplicity. More recently, Gaffney and Rangachev [GR16] adressed the case of families of isolated determinantal singularities using Gaffney's Multiplicity-Polar Theorem for the relative BR multiplicity. In all this cases the base space W of miniversal deformations of X 0 is smooth, X w is smooth for generic w, and ε(J rel (X , f )(w) = 0 which as we show in Rmk.…”

Section: Introductionmentioning

confidence: 99%

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Local volumes, equisingularity, and generalized smoothability

Rangachev

2021

Preprint

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We introduce the restricted local volume of a relatively very ample invertible sheaf as an invariant of equisingularity by determining its change across families. We apply this result to give numerical control of Whitney-Thom (differential) equisingularity for families of isolated complex analytic singularities. The characterization of the vanishing of the local volume gives rise to the class of deficient conormal (dc) singularities. We introduce a notion of generalized smoothability by considering the class of singularities that deform to dc singularities. Using Whitney stratifications and the functoriality properties of conormal spaces we show that fibers of conormal spaces are well-behaved under transverse maps. Then by Thom's transversality, the structure theorems of Hilbert-Burch and Buchsbaum-Eisenbud, we show that all smoothable singularities of dimension at least 2, Cohen-Macaulay codimension 2, Gorenstein codimension 3, and more generally determinantal and Pfaffian singularities deform to dc singularities. Contents1. Introduction 1 2. The Excess-Degree Formula and applications 7 3. Computing and vanishing of local volumes 13 4.

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“…Recently, significant progress has been made for this class, e.g. in [26], [2], [24], [8], [17]. In [15] the use of Tjurina modifications made it possible to relate a given determinantal singularity to an often singular variety, which happens to be an ICIS under rather mild conditions.…”

Section: Introductionmentioning

confidence: 99%

On discriminants, Tjurina modifications and the geometry of determinantal singularities

Frühbis–Krüger

2018

Topology and its Applications

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We describe a method for computing discriminants for a large class of families of isolated determinantal singularities -more precisely, for subfamilies of G-versal families. The approach intrinsically provides a decomposition of the discriminant into two parts and allows the computation of the determinantal and the non-determinantal loci of the family without extra effort; only the latter manifests itself in the Tjurina transform. This knowledge is then applied to the case of Cohen-Macaulay codimension 2 singularities putting several known, but previously unexplained observations into context and explicitly constructing a counterexample to Wahl's conjecture on the relation of Milnor and Tjurina numbers for surface singularities.where J F denotes the submodule generated by the k matrices, each holding the partial derivatives of the entries of F w.r.t. one of the variables, and J op = AF + F B | A ∈ Mat(m, m; C{x}), Mat(n, n; C{x})Remark 2.10 The group G is a subgroup of the group K of Mather and coincides with it e.g. for complete intersections and Cohen-Macaulay codimension 2 singularities. It also appears as the subgroup K V of K in the literature, where

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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.

Cited by 9 publications

References 16 publications

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

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

Local volumes, equisingularity, and generalized smoothability

On discriminants, Tjurina modifications and the geometry of determinantal singularities

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