arXiv subjects
Publications and source records attributed to Terence Gaffney.
In this work, we investigate the projectivized analytic spectrum of the double of a module, establishing some general properties, and we apply these results to $\mbox{Projan}(\cR((JM(X))_D))$ over the origin in $C\times C$, where $C$ is an irreducible curve in a hypersurface $X$.
We study equisingularity of families of reduced curves over smooth parameter spaces of arbitrary positive dimension, using the difference between two analytic invariants of a curve singularity: the multiplicity of its Jacobian ideal and its complete intersection discrepancy. This difference provides a fiberwise multiplicity criterion for Whitney equisingularity. We prove that Whitney equisingularity (equivalently, strong simultaneous resolution) is characterized by equidimensionality of the fibers of the exceptional locus of either the relative conormal space or the relative Nash blowup. We further show that this condition is equivalent to the emptiness of the relative polar variety of smallest dimension. In addition, we establish that the Milnor number of a reduced curve is Zariski upper semicontinuous. As an application, we show that the constancy of the Milnor number in a family of reduced curves is equivalent to its topological equisingularity.
In this work, we extend the concept of the Lipschitz saturation of an ideal to the context of modules in some different ways, and we prove they are generically equivalent.
This paper is the second part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a generic deformation of a singularity to control the Whitney equisingularity type of these curves.
In this work we extend the concept of the Lipschitz saturation of an ideal defined in [5] to the context of modules in some different ways, and we prove they are generically equivalent.
This paper is the first part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a generic deformation of a singularity to control the Whitney equisingularity type of these curves.
The study of Essentially Isolated Determinantal Singularites or EIDS was initiated by Ebeling and Gusein-Zade. They are non-smoothable as determinantal singularities, and in general have non-isolated singularities. Their singularities are generic in a deleted neighborhood of the origin, hence their description as "essentially isolated". In this paper we study these singularities from the "landscape" point of view introduced in MathArxiv 1501.00201. Using this point of view we show a connection between invariants coming from the topology of the stabilization and invariants coming from the infinitesimal geometry of the singularity. This gives us a criterion for the Whitney equisingularity of EIDS families.
In this work, we extend the concept of the double of an ideal defined in \cite{G2}, to the context of modules. We also obtain the genericity of the infinitesimal Lipschitz condition A for an enlarged class of analytic spaces.
We introduce a join construction as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom's A_f condition.
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.
Two blow-analytically equivalent real analytic plane function germs are sub-analytically bi-Lipschitz contact equivalent
We introduce two different notions of infinitesimal bi-Lipschitz equivalence for functions, one related to bi-Lipschitz triviality of families of functions, one related to homeomorphisms which are bi-Lipschitz on the fibers of the functions in the family. We show that the first is not a generic condition, and that the second is.
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 $X$.
We continue the development of the theory of infinitesimal Lipschitz equivalence, showing the genericity of the condition for families of hypersurfaces with isolated singularities.
In this work we show how the Multiplicity Polar Theorem can be used to calculate Chern numbers for a collection of 1-forms.
We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a geometric interpretation of the Reid-Roberts-Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal $I$ of a ring $A$ an ideal $I_>$, which consists of all elements of $A$ such that $v(a)>v(I)$, for all Rees valuations $v$ of $I$. The ideal $I_>$ plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition $A_f$ and is contained in every reduction of $I$. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative closure.
Given a family of pairs of modules parametrised by a smooth space Y, the Multiplicity-Polar Theorem relates the multiplicity of the pair of modules at a special point of the parameter to the multiplicity of the pair at a generic point. This theorem is proved in this paper (Corollary 1.4). The applications of the Multiplicity-Polar theorem are of two types. In the first we construct a deformation so that we can understand the significance of the multiplicity of the pair, or show we can use the multiplicity of the pair to describe some geometric behavior. Applications of this type are the geometric significance of the BR-multiplicity (Theorem 2.1), the geometric significance of multiplicity of the pair (J(f), I) where f defines a hypersurface with non-isolated singularities, J(f) is the jacobian ideal of f and I is the ideal of the singular locus of the hypersurface. We do the cases where I defines a complete intersection (Theorem 2.3), or the hypersurface is the image of a finitely determined map-germ F: C^2,0\to C^3,0 (Theorem 2.6). We then show how the multiplicity of the pair can be used to calculate the index of a differential 1-form with an isolated singularity on a germ of an isolated singularity (Theorem 2.8). In the second kind of application, we use the generalization of the Principle of Specialization of Integral Dependence to prove results in equisingularity. The particular condition we study in this paper is the W_f condition (Theorems 2.10 and 2.11).
The basic examples of functions defining non-isolated hypersurface singularities are the A(d) singularities and the D(q,p) singularities. The A(d) singularities, up to analytic equivalence, are the product of a Morse function and the zero map, while the simplest D(q,p) singularity is the Whitney umbrella. These are the basic examples, because they correspond to stable germs of functions in the study of germs of functions with non-isolated singularities. Given a germ of a function which defines a non-isolated hypersurface singularity at the origin, which in the appropriate sense, has finite codimension in the set of such germs, the singularity type of such germs away from the origin is A(d) or D(q,p). In this note we calculate the homotopy type of the Milnor fiber of germs of type D(q,p), as well as their Lê numbers. The calculation of the Lê numbers involves the use of an incidence variety which may be useful for studying germs of finite codimension. The calculation shows that the set of symmetric matrices of kernel rank greater than or equal to 1 is an example of a hypersurface singularity with a Whitney stratification (given by the rank of the matrices) in which only one singular stratum gives a component of top dimension of the singular set of the conormal.