Non-isolated Hypersurface Singularities Research Articles

Computing Holonomic D-Modules Associated to a Family of Non-isolated Hypersurface Singularities via Comprehensive Gröbner Systems of PBW Algebra

Computing Holonomic D-Modules Associated to a Family of Non-isolated Hypersurface Singularities via Comprehensive Gröbner Systems of PBW Algebra

Nondegenerate locally tame complete intersection varieties and geometry of nonisolated hypersurface singularities

We give a criterion to test geometric properties such as Whitney equisingularity and Thom’s a f a_f condition for new families of (possibly nonisolated) hypersurface singularities that “behave well” with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.

Algebraic analysis of Siersma's non-isolated hypersurface singularities

We examine isolated line singularities, transversal $A_1$-type non-isolated hypersurface singularities studied by D. Siersma, in the context of algebraic analysis. By using Poincaré-Birkhoff-Witt algebra, we explicitly compute a Gröbner basis of the annihilator $\text{Ann}_{D_X[s]}f^s$ in a non-commutative ring associated with these singularities. We compute local cohomology solutions of the associated holonomic $D$-modules by utilizing the Gröbner basis of $\text{Ann}_{D_X[s]}f^s$ and determine in particular the monodromy structure of the local cohomology solutions along a singular stratum of hypersurfaces. As a byproduct, we obtain micro-local $b$-functions of isolated line singularities in an explicit manner.

Holonomic <i>D</i>-Modules Associated with a Simple Line Singularity and the Vertical Monodromy

Holonomic D-modules associated with certain kind of non-isolated hypersurface singularities, introduced by T. de Jong, are considered. Structures of these holonomic D-modules such as characteristic variety, monodromy are explicitly determined. The key of our approach is the use of the notion of algebraic local cohomology. An application to Betti numbers of local Milnor fibers is also discussed.

Milnor monodromies and mixed Hodge structures for non-isolated hypersurface singularities

We study the Milnor monodromies of non-isolated hypersurface singularities and show that the reduced cohomology groups of the Milnor fibers are concentrated in the middle degree for some eigenvalues of the monodromies. As an application of this result, we give an explicit formula for some parts of their Jordan normal forms.

The boundary of the Milnor fibre of certain non-isolated singularities

Let $$ \Phi : (\mathbb {C}^2, 0) \rightarrow ( \mathbb {C}^3, 0) $$ be a finitely determined complex analytic germ and let $$(\{f=0\},0)$$ be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary of the Milnor fibre of f from the double-point-geometry of $$\Phi $$ .

Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

We prove that if [Formula: see text] is a family of line singularities with constant Lê numbers and such that [Formula: see text] is a homogeneous polynomial, then [Formula: see text] is equimultiple. This extends to line singularities a well-known theorem of Gabrièlov and Kušnirenko concerning isolated singularities. As an application, we show that if [Formula: see text] is a topologically [Formula: see text]-equisingular family of line singularities, with [Formula: see text] homogeneous, then [Formula: see text] is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated hypersurface singularities.

Nonisolated forms of rational triple point singularities of surfaces and their resolutions

We give a list of nonisolated hypersurface singularities of which normalizations are the rational triple point singularities of surfaces and construct their minimal resolution graphs by a method introduced by Oka for isolated complete intersection singularities. We also show that nonisolated forms of rational triple point singularities and their normalizations are both Newton non-degenerate.

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

AbstractWe show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

AbstractWe show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families withisolatedsingularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

Carrousel in family and\break non-isolated hypersurface singularities in ℂ3

Abstract We study the boundary L t ${L_{t}}$ of the Milnor fiber for the reduced holomorphic germs f : ( ℂ 3 , 0 ) → ( ℂ , 0 ) ${f:(\mathbb{C}^{3},0)\rightarrow(\mathbb{C},0)}$ having a non-isolated singularity at 0. We prove that L t ${L_{t}}$ is a graph manifold by using a new technique of carrousels depending on one parameter. Our results enable us to compare the topology of L t ${L_{t}}$ and of the link of the normalization of f - 1 ⁢ ( 0 ) ${f^{-1}(0)}$ .

Equisingularity of families of hypersurfaces and applications to mappings

In the study of equisingularity of isolated singularities we have the classical theorem of Briancon, Speder and Teissier which states that a family of iso- lated hypersurface singularities is Whitney equisingular if and only if the µ - sequence for a hypersurface is constant in the family. In this paper we general- ize to non-isolated hypersurface singularities. By assuming non-contractibility of strata of a Whitney stratification of the non-isolated singularities outside the origin we show that Whitney equisingularity of a family is equivalent to constancy of a certain selection of invariants from two distinct generalizations of the µ -sequence. Applications of this theorem to equisingularity of more general mappings are given. AMS Mathematics Subject Classification 2000 : 32S15, 32S30, 32S60.

On nonisolated hypersurface singularities

We study germs of holomorphic functions whose singular sets are hypersurfaces with isolated singularity in the cases where the transversal singularity is A 1. For these singularities, we completely describe the homotopy structure of the Milnor fibers.

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 z2. 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.

Brieskorn modules and Gauss-Manin systems for non-isolated hypersurface singularities

We study the Brieskorn modules associated to a germ of a holomorphic function with non-isolated singularities and show that the Brieskorn module has naturally the structure of a module over the ring of microdifferential operators of non-positive degree, and that the kernel of the morphism to the Gauss–Manin system coincides with the torsion part for the action of t and also with that for the action of the inverse of the Gauss–Manin connection. This torsion part is not finitely generated in general, and a sufficient condition for the finiteness is given here. A Thom–Sebastiani-type theorem for the sheaf of Brieskorn modules is also proved when one of two functions has an isolated singularity.

Lê modules and traces

We show how some of our recent results clarify the relationship between the Lê numbers and the cohomology of the Milnor fiber of a non-isolated hypersurface singularity. The Lê numbers are actually the ranks of the free Abelian groups—the Lê modules—appearing in a complex whose cohomology is that of the Milnor fiber. Moreover, the Milnor monodromy acts on the Lê module complex, and we describe the traces of these monodromy actions in terms of the topology of the critical locus.