Space Curve Singularities Research Articles

Deforming monomial space curves into set-theoretic complete intersection singularities

We deform monomial space curves in order to construct examples of set-theoretical complete intersection space curve singularities. As a by-product we describe an inverse to Herzog's construction of minimal generators of non-complete intersection numerical semigroups with three generators.

A Classiffier for Unimodular Isolated Complete Intersection Space Curve Singularities

Abstract C.T.C. Wall classified the unimodular complete intersection singularities. He indicated in the list only the μ-constant strata and not the complete classification in each case. In this article we give a complete list of space curve unimodular singularities and also the description of a classifier. Instead of computing the normal forms, the singularity is identified by certain invariants.

Simple curve singularities

We classify simple parametrisations of complex curve singularities. Simple means that all neighbouring singularities fall in finitely many equivalence classes. We take the neighbouring singularities to be the ones occurring in the versal deformation of the parametrisation. This leads to a smaller list than that obtained by looking at the neighbours in a fixed space of multi-germs. Our simple parametrisations are the same as the fully simple singularities of Zhitomirskii, who classified real plane and space curve singularities. The list of simple parametrisations of plane curves is the A-D-E list. Also for space curves the list coincides with the lists of simple curves of Giusti and Fr\"uhbis-Kr\"uger, in the sense of deformations of the curve. For higher embedding dimension no classification of simple curves is available, but we conjecture that even there the list is exactly that of curves with simple parametrisations.

A classifier for simple space curve singularities

The classification of Bruce and Gaffney respectively Gibson and Hobbs for simple plane curve singularities respectively simple space curve singularities is characterized in terms of invariants. This is the basis for the implementation of a classifier in the computer algebra system singular.

LOCAL ANALYSIS OF GRAUERT–REMMERT-TYPE NORMALIZATION ALGORITHMS

Normalization is a fundamental ring-theoretic operation; geometrically it resolves singularities in codimension one. Existing algorithmic methods for computing the normalization rely on a common recipe: successively enlarge the given ring in form of an endomorphism ring of a certain (fractional) ideal until the process becomes stationary. While Vasconcelos' method uses the dual Jacobian ideal, Grauert–Remmert-type algorithms rely on so-called test ideals. For algebraic varieties, one can apply such normalization algorithms globally, locally, or formal analytically at all points of the variety. In this paper, we relate the number of iterations for global Grauert–Remmert-type normalization algorithms to that of its local descendants. We complement our results by a study of ADE singularities. All intermediate singularities occurring in the normalization process are determined explicitly. Besides ADE singularities the process yields simple space curve singularities from the list of Frühbis-Krüger.

Using a bihomogeneous resultant to find the singularities of rational space curves

We provide a new technique to detect the singularities of rational space curves. Given a rational parametrization of a space curve, we first compute a μ-basis for the parametrization. From this μ-basis we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, we construct a new sparse resultant matrix for these three bivariate polynomials. We then compute the parameter values corresponding to the singularities by applying Gaussian elimination to this resultant matrix. Let νQ denote the multiplicity of the singular point Q, and let n be the degree of the curve. We find that when ∑νQ⩽2n−3, the last nonzero row after Gaussian elimination represents a univariate polynomial whose roots are exactly the parameter values of the singularities with the correct multiplicity. Otherwise the last two nonzero rows represent two bivariate polynomials whose common roots provide the parameter values of the singularities. We also show that if R is this resultant matrix, then size(R)−rank(R) gives the total multiplicity ∑νQ(νQ−1) of all the singular points including the infinitely near singular points of a rational space curve and we provide bounds on the expression ∑νQ(νQ−1) for the total multiplicity of all the singular points of a rational space curve. To verify our results, we present several examples to illustrate our methods.

Simple Cohen–Macaulay Codimension 2 Singularities

In this article, we provide a complete list of simple isolated Cohen–Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities.

Axial moving planes and singularities of rational space curves

Relationships between the singularities of rational space curves and the moving planes that follow these curves are investigated. Given a space curve C with a generic 1–1 rational parametrization F ( s , t ) of homogeneous degree d, we show that if P and Q are two singular points of orders k and k ′ on the space curve C , then there is a moving plane of degree d − k − k ′ with axis P Q ↔ that follows the curve. We also show that a point P is a singular point of order k on the space curve C if and only if there are two axial moving planes L 1 and L 2 of degree d − k such that: (1) the axes of L 1 , L 2 are orthogonal and intersect at P, and (2) the intersection of the moving planes L 1 and L 2 is the cone through the curve C with vertex P together with d − k copies of the plane containing the axes of L 1 and L 2 . In addition, we study relationships between the singularities of rational space curves and generic moving planes that follow these curves. In particular, we show that if p ( s , t ) , q ( s , t ) , r ( s , t ) are a μ-basis for the moving planes that follow a rational space curve F ( s , t ) , then P is a singular point of F ( s , t ) of order k if and only if deg ( gcd ( p ( s , t ) ⋅ P , q ( s , t ) ⋅ P , r ( s , t ) ⋅ P ) ) = k . Moreover, the roots of this gcd are the parameters, counted with proper multiplicity, that correspond to the singularity P. Using these results, we provide straightforward algorithms for finding all the singularities of low degree rational space curves. Our algorithms are easy to implement, requiring only standard techniques from linear algebra. Examples are provided to illustrate these algorithms.

Fully simple singularities of plane and space curves

In this wor we introduce the definition of fully simple singularities of parameterized curves and explain that this definition is more natural than the definition of simple singularities. The set of fully simple singularities is much smaller than the set of simple ones. We determine and classify all fully simple singularities of plane and space curves, with any number of components. Our classification results imply that any fully simple singularity of a plane or a space curve is quasi-homogeneous (whereas there is a number of non-quasi-homogeneous simple singularities). Another outcome of our classification results is a one-to-one correspondence between the fully simple singularities of plane curves and the classical A-D-E singularities of functions.

Coxeter-Dynkin diagrams of some space curve singularities

The so called wedge singularities, that consist of a plane curve singularity C and a line transverse to the plane of C, are the simplest space curve singularities which are not a complete intersection. We show that for every wedge singularity X there is an isolated complete intersection singularity Y related to X and we describe the discriminant of X in terms of Y. We also show that the monodromy group of X corresponds to the one of Y.

Algorithm for the Semigroup of a Space Curve Singularity

The semigroup of values of irreducible space curve singularities is the set of intersection multiplicities among hypersurfaces and the given curve. It is an invariant of the singularity, and for plane curves it characterizes the equisingularity type considered by Zariski. For space curve singularities the semigroup of values is a numerical semigroup and it can not be computed by means of the exponents of any Puiseux parametrization, as in the plane case. We obtain an algorithm for calculating the semigroup of values of a space curve singularity, which determines the generators of the semigroup and the valuation ideals associated with the semigroup. We give a Maple version of the algorithm.

Modular Deformations and Space Curve Singularities

We investigate different concepts of modular deformations of germs of isolated singularities (infinitesimal, Artinian, formal). An obstruction calculus based on the graded Lie algebra structure of the tangent cohomology for modular deformations is introduced. As the main result the characterisation of the maximal infinitesimally modular subgerm of the miniversal family as flattening stratum of the relative Tjurina module is extended from ICIS to space curve singularities.

Computing moduli spaces for space curve singularities

In this article, an algorithm for explicit computation of moduli spaces for semiquasihomogeneous space curve singularities is presented. The main tool for this are standard basis computations.

The Structure of the Discriminant of Some Space-Curve Singularities

The Structure of the Discriminant of Some Space-Curve Singularities

Milnor Number Equals Tjurina Number for Functions on Space Curves

The equality of the Milnor number and Tjurina number for functions on space curve singularities, as conjectured recently by V. Goryunov, is proved. As a consequence, the discriminant in such a situation is a free divisor.

Classification of simple space curve singularities

(1999). Classification of simple space curve singularities. Communications in Algebra: Vol. 27, No. 8, pp. 3993-4013.

The deformations of the simple space curve singularities

In [4] and [5] Marc Giusti classified the simple isolated complete intersection singularities of positive dimension. He showed that besides the simple hypersurface singularities there are only simple singularities of curves in complex 3-space. He gave a list of adjacencies between these singularities. Later V. V. Goryunov (see [6]) found some additional adjacencies, but the complete list remained unknown (see [1, p. 23]). In this note we prove the following theorem: Theorem The following diagram shows the complete list of adjacencies between the simple space curve singularities which are not hypersurface singularities. Here Giusti’s list is completed by the deformations Z10 → T9 and Z10 → U9. S5 S6 S7 S8 S9 · · ·

Combinatorial aspects of sequences of point blowing ups

We study morphisms given by composition of a sequence of point blow ups of smoothd-dimensional varieties in terms of combinatorial information coming from thed-ary intersection form on divisors with exceptional support. Some application to the classification of space curve singularities are derived.

Simple singularities of space curves

The genesis of this paper lies in theoretical questions in kinematics where a central role is played by naturally occurring families of rigid motions of 3-space. The resulting trajectories are parametrized families of space curves, and it is important to understand the generic singularities they can exhibit. For practical purposes one seeks to classify germs of space curves of fairly small Ae-codimension. It is however little harder to list the A-simple germs, which includes all germs of Ae-codimension ≤11: that then is the principal objective of this paper.

A generalisation of Noether's formula for the number of virtual double points to space curve singularities

A generalisation of Noether's formula for the number of virtual double points to space curve singularities