Complete Intersection Singularities Research Articles

APPROXIMATIONS OF NON-ISOLATED SINGULARITIES OF FINITE CODIMENSION WITH RESPECT TO AN ISOLATED COMPLETE INTERSECTION SINGULARITY

It is shown that a function whose critical locus is an isolated complete intersection singularity of arbitrary dimension, and that has finite codimension (in the sense of R. Pellikaan, Proc. London Math. Soc. (3) 57 (1998) 357–382) with respect to the ideal defining the isolated complete intersection singularity, can be approximated by a function whose critical locus is a finite number of Morse points together with the Milnor fibre of the isolated complete intersection singularity, having there well-known types of singularities. 2000 Mathematics Subject Classification 32S30, 32S55 (primary).

On the equations defining toric l.c.i.-singularities

Based on Nakajima’s Classification Theorem we describe the precise form of the binomial equations which determine toric locally complete intersection (“l.c.i.”) singularities.

Indices of 1-forms on an Isolated Complete Intersection Singularity

There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on $R^n$ for vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the setting, for an icis $V=f^{-1}(0)$, $f:(C^n, 0) \to (C^k, 0)$, $f$ is real, we define a complex analytic family of quadratic forms parameterized by the points $\epsilon$ of the image $(C^k, 0)$ of the map $f$, which become for $\epsilon$ and in this case their signatures defer from the real index by $\chi(V_\epsilon)-1$, where $\chi(V_\epsilon)$ is the Euler characteristic of the corresponding smoothing $V_\epsilon=f^{-1}(\epsilon)\cap B_\delta$ of the icis $V$.

Algorithmic Computation of Flattenings and of Modular Deformations

This paper presents an algorithm for the computation of any jet of the flattening stratum of a module over a local algebra based on an obstruction theory for lifting flatness. It is applied to modular deformations of singular germs. Infinitesimal modular deformations of isolated complete intersection singularities are characterized as flattening strata of its first tangent cohomology. Some examples are discussed that indicate relations with moduli spaces of certain classes of singularities. Implementations are done in Singular.

A characterization of quasi-homogeneous complete intersection singularities

It is well-known that quasi-homogeneity is characterized by equality of the Milnor and Tjurina numbers for isolated complex analytic hypersurface singularities and for certain low-dimensional singularities. We prove that this characterization extends to any complex isolated complete intersection singularity of positive dimension.

Transformée de Mellin des intégrales-fibres associées aux singularités isolées d'intersection complète quasihomogènes. (Mellin Transform of Fibre Integrals Associated to Isolated Complete Intersection Singularities)

The Mellin transform of the fibre integral is calculated for certain quasihomogeneous isolated complete intersection singularities (above all, unimodal singularities of the list by Giusti and Wall). We show the symmetry property of the Gauss–Manin spectra (Theorem 3.1) and shed light on the lattice structure of the poles of the Mellin transform that are expressed by means of some topological data of the singularities (Theorem 4.3, Theorem 5.2). As an application of these results, we express the Hodge number of the fibre in terms of the Gauss–Manin spectra.

Sur le nombre de Milnor d'une singularité semi-quasi-homogène

Let f=( f 1,…, f p ) be a semi-quasi-homogeneous family of holomorphic functions in a neighborhood of the origin in C n . We prove that f defines an isolated complete intersection singularity, and we express the Milnor number of this singularity as the colength of an ideal naturally associated to f. This generalizes a formula due to G.M. Greuel. To cite this article: J. Briançon, H. Maynadier-Gervais, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 317–320.

Variation of the Milnor Fibration in Pencils of Hypersurface Singularities

Let Φ = (f, g):(Cn+ 1,0) → (C2, 0) be a pair of holomorphic germs with no blowing up in codimension 0. (Two examples are the following: Φ defines an isolated complete intersection singularity; g = lN where ℓ is a generic linear form with respect to f and N> 0.) We study how the Milnor fibrations of the germs ϕ(α:β) = α g-β f are related to each other when (α:β) varies in P1. More precisely, we construct isotopic subfibrations or subfibres of the Milnor fibrations of any two such germs. The proofs are based on the precise study of the subdiscs of complex lines meeting a fixed complex plane curve germ transversally, generalizing Lê's work on the Cerf diagram. 2000 Mathematical Subject Classification: 32S55, 32S15, 32S30.

All toric local complete intersection singularities admit projective crepant resolutions

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric local complete intersection singularities. Our strikingly simple proof makes use of Nakajima's classification theorem and of some techniques from toric and discrete geometry.

Differential Forms on Free and Almost Free Divisors

We introduce a variant of the usual Kahler forms on singular free divisors, and show that it enjoys the same depth properties as Kahler forms on isolated hypersurface singularities. Using these forms it is possible to describe analytically the vanishing cohomology, and the Gauss–Manin connection, in families of free divisors, in precise analogy with the classical description for the Milnor fibration of an isolated complete intersection singularity, due to Brieskorn and Greuel. This applies in particular to the family Formula of discriminants of a versal deformation Formula of a singularity of a mapping.

Strange Duality and Polar Duality

A relation is described between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M. Kobayashi. It is shown that this relation continues to hold for the extension of Arnold's strange duality found by C. T. C. Wall and the author. By a method of Ehlers–Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. This method is generalized to the isolated complete intersection singularities embraced in the extended duality. This is used to explain the duality of characteristic polynomials of the monodromy discovered by K. Saito for Arnold's original strange duality and extended by the author to the other cases.

A Characterization of Quasi-Homogeneous Purely Elliptic Complete Intersection Singularities

It is well-known that quasi-homogeneity is characterized by equality of the Milnor and Tjurina numbers for isolated complex analytic hypersurface singularities and for certain low-dimensional singularities. In this paper we prove that this characterization extends to isolated purely elliptic complete intersection singularities, with bounds on neither the embedding codimension nor the dimension of the singularity.

On geometry of fronts in wave propagations

We give a geometric descriptions of (wave) fronts in wave propagation processes. Concrete form of defining function of wave front issued from initial algebraic variety is obtained by the aid of Gauss-Manin systems associated with certain complete intersection singularities. In the case of propagations on the plane, we get restrictions on types of possible cusps that can appear on the wave front. 0. Introduction. In this note, we intend to develop a preparatory theory toward asymptotic analysis around singular loci that appear in wave propagation process with initial data whose C∞ singularities are located on certain smooth algebraic variety. For this purpose, we study the geometry of the (wave) front, i.e. the singular support S.S. of the solutions to the Cauchy problem where it loses C∞ smoothness. We give a concrete expression of defining function of the wave front in terms of the initial data (Theorem 10). We impose conditions on the initial data so that they are defined by a quasihomogeneous polynomial with isolated singularities. The key trick is to understand the (wave) front as a discriminantal set for a deformation of complete intersection singularity. This point of view has been exposed in [20]. As is well known, the fundamental solutions of hyperbolic operators are expressed by means of certain kinds of Gel′fand-Leray integrals. Thus the geometry of singular support (i.e. front) of the solutions is reduced to the study of singular support of these types of integrals. Homological and cohomological approach to the analysis of sharp-diffuse type of these integrals has been initiated by [3]. V. A. Vassiliev [21] uses F. Pham’s approach [15] analyzing homology classes to describe the ramification character of certain kinds of Gel′fand-Leray integrals, consequently to clarify the sharp-diffuse type (see Section 4 for 1991 Mathematics Subject Classification: Primary 35L25, 58G17, 33C75; Secondary 32S40, 78A05, 33C20. The paper is in final form and no version of it will be published elsewhere.

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 · · ·

Flips from 4-folds with isolated complete intersection singularities

We investigate a flipping contraction g : X → Y from a 4-fold X with at worst isolated complete intersection singularities. If Y has an anti-bicanonical divisor (= bi-elephant ) with only rational singularities, then g carries an inductive structure involving a chain of blow-ups (which we call La Torre Pendente ), and in particular, the flip exists. This naturally contains Reid's " Pagoda " as an anticanonical divisor (= elephant ) and its proper transforms.

Parallelism in MuPAD

S INGULAR is a computer algebra system designed to be a power-house for polynomial computations with special features for algebraic geometry and singularity theory.The development of S INGULAR started in 1986, motivated by the question of the existence of complete intersection singularities which are not quasi-homogeneous, but whose Poincare-complex is exact. The existence of such singularities could be shown in 1987 with an implementation of Mora's algorithm on an Atari in Modula-2 [11]. In 1990, S INGULAR was ported to Unix, and a first public-domain distribution followed. The code of S INGULAR was rewritten in C/C++. An interpreted user-level programming language and user-written libraries were added in 1993. With the release of version 1.0 in May 1997, many important features were added, including:• Polynomial factorization• Factorizing Buchberger algorithm• FGLM algorithm• Primary decomposition of ideals• MP links for general and efficient storage and inter-process communicationsThe goal of the poster is to discuss and demonstrate these new features of S INGULAR where, in combination with the posters on MP and MuPAD, special emphasize will be paid to S INGULAR 's MP interface and its resulting possibilities and problems.

Deforming Calabi-Yau threefolds

This paper first generalises the Bogomolov-Tian-Todorov unobstructedness theorem to the case of Calabi-Yau threefolds with canonical singularities. The deformation space of such a Calabi-Yau threefold is no longer smooth, but the general principle is that the obstructions to deforming such a threefold are precisely the obstructions to deforming the singularities of the threefold. Secondly, these results are applied to smoothing singular Calabi-Yau threefolds with crepant resolutions. Any such Calabi-Yau threefold with isolated complete intersection singularities which are not ordinary double points is smoothable. A Calabi-Yau threefold with non-complete intersection isolated singularities is proved to be smoothable under much stronger hypotheses.

Du Bois invariants of isolated complete intersection singularities

Du Bois invariants of isolated complete intersection singularities

Graded Lie Algebras and Quasihomogeneous Complete Intersection Singularities

Graded Lie Algebras and Quasihomogeneous Complete Intersection Singularities

Computing Gauss–Manin Systems for Complete Intersection Singularities Sμ

Abstract The Gauss–Manin systems with coefficients having logarithmic poles along the discriminant sets of the principal deformations of complete intersection quasihomogeneous singularities Sμ are calculated. Their solutions in the form of generalized hypergeometric functions are presented.