arXiv subjects

Terence Gaffney

Publications and source records attributed to Terence Gaffney.

24 records · Page 2Linked to original sources

arXiv2005

The Multiplicity Polar Theorem and Isolated Singularities

Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength. In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool is the multiplicity-polar theorem which applies to families of pairs of modules, linking the multiplicity at a general pair of the family to the multiplicity of the pair at the special fiber. The specific problem we work on is: Suppose X,0 is an equidimensional complex analytic set in C^n, and T^*_X(C^n) is the closure of the conormal vectors to the smooth part of X, s, a section of T^*(C^n), the cotangent bundle of C^n restricted to X. Suppose the intersection of the image of s, denoted im(s), with T^*_X(C^n) is isolated. Then we calculate the intersection multiplicity im(s)\cdot T^*_X(C^n) using the multiplicity of pairs. It is easy to see that if X=C^n and s(x)=df(x), where f has an isolated singularity at the origin, then this number is the Milnor number of f. Variations of this number appear in several different situations, such as, the theory of differential forms on singular spaces, the theory of D-modules, the Milnor fiber of a function with an isolated singularity, the A_f stratification condition of Thom, and the theory of the Euler invariant. We discuss the application of our result to all of these situations.

math.AG↗

arXiv2005

The multiplicity of pairs of modules and hypersurface singularities

This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a submodule of finite length of a free module F over the local ring of an equidimensional complex analytic germ, that the number of points at which a generic perturbation of a minimal reduction of M is not equal to F, is the multiplicity of M. In this paper we apply this result to the study of stratification conditions on families of hypersurfaces, obtaining the first set of invariants giving necessary and sufficient conditions for the A_f condition for hypersurfaces with non-isolated singularities.

math.AG↗

arXiv2005

Equisingularity of sections, $(t^r)$ condition, and the integral closure of modules

This paper uses the theory of integral closure of modules to study the sections of both real and complex analytic spaces. The stratification conditions used are the (t^) conditions introduced by Thom and Trotman. Our results include a new simple proof showing how the (t^r) conditions improve under Grassman modification, and a characterization of the (t^r) conditions using the multiplicity of a submodule of the Jacobian module of the singularity. This gives numerical criteria for Verdier Equisingularity of families of sections of the space.

math.AG↗

arXiv2005

The Lê numbers of the square of a function and their applications

Lê numbers were introduced by Massey with the purpose of numerically controlling the topological properties of families of non-isolated hypersurface singularities and describing the topology associated with a function with non-isolated singularities. They are a generalization of the Milnor number for isolated hypersurface singularities. In this note the authors investigate the composite of an arbitrary square-free f and $z^2$. They get a formula for the Lê numbers of the composite, and consider two applications of these numbers. The first application is concerned with the extent to which the Lê numbers are invariant in a family of functions which satisfy some equisingularity condition, the second is a quick proof of a new formula for the Euler obstruction of a hypersurface singularity. Several examples are computed using this formula including any X defined by a function which only has transverse D(q,p) singularities off the origin.

math.CV↗

arXiv2005

Nilpotents, Integral Closure and Equisingularity conditions

In earlier work, the author described various stratification conditions for a complex analytic set X in terms of the theory of integral closure of modules. However, even if an analytic set has a reduced structure, often geometric operations like intersection give a non-reduced structure on the resulting analytic space. In this note we show that it is not necessary for X to have a reduced structure to apply the earlier integral closure results. As an application we give a simple proof of a necessary and sufficient condition for the Whitney conditions to pass to the intersection of X with a hyperplane containing the smooth stratum Y, and for this intersection to be "typical". In the case where X is a family of ICIS singularities, we show the condition on H at a point y of Y depends only on the fiber of the family over y.

math.CV↗

arXiv1996

Segre Numbers and Hypersurface Singularities

We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the Rees-Böger theorem, and the formula for the multiplicity of the product of two ideals. These results are applied to the study of various equisingularity conditions, such as Verdier's condition W, and conditions $A_f$ and $W_f$.

alg-geom↗