scholarly journals Codimension Two Determinantal Varieties with Isolated Singularities

2014 ◽  
Vol 115 (2) ◽  
pp. 161 ◽  
Author(s):  
Maria Aparecida Soares Ruas ◽  
Miriam Da Silva Pereira
Keyword(s):  
Betti Number ◽  
Normal Forms ◽  
Milnor Number ◽  
Isolated Singularities ◽  
Generic Fiber ◽  
Linear Projection ◽  
Codimension Two ◽  
Determinantal Varieties ◽  
Surface Singularities ◽  
Generic Section

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in $\mathsf{C}^4$, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the $1$-form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last section we compute the Milnor number of some normal forms from Frühbis-Krüger and Neumer [7] list of simple determinantal surface singularities.

Quartic curves in characteristic 2

1995 ◽  
Vol 117 (3) ◽  
pp. 393-414 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Finite Deformation ◽  
Normal Forms ◽  
Positive Characteristic ◽  
Surface Singularities ◽  
The Subject ◽  
Characteristic 2 ◽  
Simple Singularities ◽  
Module Type ◽  
Quartic Curves ◽  
Article 5

Simple singularities in positive characteristicSimple singularities in positive characteristic have been discussed by many authors, and the article [5] in particular establishes the subject on a firm footing. In it a simple, or ‘ADE’ singularity is defined by a list of normal forms and it is shown that the following conditions on a singularity are equivalent: (i) it is simple, (ii) it has finite deformation type, (iii) it has finite Cohen-Macaulay module type. Moreover, the normal forms for surface singularities coincide with the earlier list of Artin [1] and those for curves with the list of [9]: in those papers further characterizations were obtained.

scholarly journals Normal forms for CR singular codimension-two Levi-flat submanifolds

2015 ◽  
Vol 275 (1) ◽  
pp. 115-165 ◽  
Author(s):  
Xianghong Gong ◽  
Jiří Lebl
Keyword(s):  
Normal Forms ◽  
Codimension Two ◽  
Flat Submanifolds

scholarly journals Invariants of topological relative right equivalences

2013 ◽  
Vol 155 (2) ◽  
pp. 307-315 ◽  
Author(s):  
IMRAN AHMED ◽  
MARIA APARECIDA SOARES RUAS ◽  
JOÃO NIVALDO TOMAZELLA
Keyword(s):  
Analytic Function ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Function Germ ◽  
Analytic Variety ◽  
Relative Version ◽  
Topological Invariance

AbstractLet (V,0) be the germ of an analytic variety in $\mathbb{C}^n$ and f an analytic function germ defined on V. For functions with isolated singularity on V, Bruce and Roberts introduced a generalization of the Milnor number of f, which we call Bruce–Roberts number, μBR(V,f). Like the Milnor number of f, this number shows some properties of f and V. In this paper we investigate algebraic and geometric characterizations of the constancy of the Bruce–Roberts number for families of functions with isolated singularities on V. We also discuss the topological invariance of the Bruce–Roberts number for families of quasihomogeneous functions defined on quasihomogeneous varieties. As application of the results, we prove a relative version of the Zariski multiplicity conjecture for quasihomogeneous varieties.

A NOTE ON THE TRIPLE-ZERO LINEAR DEGENERACY: NORMAL FORMS, DYNAMICAL AND BIFURCATION BEHAVIORS OF AN UNFOLDING

2002 ◽  
Vol 12 (12) ◽  
pp. 2799-2820 ◽  
Author(s):  
E. FREIRE ◽  
E. GAMERO ◽  
A. J. RODRÍGUEZ-LUIS ◽  
A. ALGABA
Keyword(s):  
Periodic Orbits ◽  
Normal Forms ◽  
Zero Eigenvalue ◽  
Codimension Two ◽  
Local Bifurcations ◽  
Linear Degeneracy

This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,…) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the Rössler equation.

scholarly journals HOMOLOGICAL INDEX FOR 1-FORMS AND A MILNOR NUMBER FOR ISOLATED SINGULARITIES

2004 ◽  
Vol 15 (09) ◽  
pp. 895-905 ◽  
Author(s):  
W. EBELING ◽  
S. M. GUSEIN-ZADE ◽  
J. SEADE
Keyword(s):  
Complete Intersection ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Analytic Variety ◽  
Curve Singularities ◽  
Complex Analytic Variety ◽  
Arbitrary Curve

We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gómez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.

Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

2015 ◽  
Vol 25 (06) ◽  
pp. 1550091 ◽  
Author(s):  
Kie Van Ivanky Saputra
Keyword(s):  
Dynamical System ◽  
Invariant Manifold ◽  
Normal Forms ◽  
Volterra Model ◽  
Infection Model ◽  
Global Bifurcations ◽  
Codimension Two ◽  
Codimension One ◽  
Variation Of Parameters ◽  
Local And Global Bifurcations

We investigate a dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analyzed and yield the transcritical bifurcation as the codimension-one bifurcation while the saddle-node–transcritical interaction and the Hopf–transcritical interactions as the codimension-two bifurcations. The unfolding of this degeneracy is also analyzed and reveal global bifurcations such as homoclinic and heteroclinic bifurcations. We apply our results to a modified Lotka–Volterra model and to an infection model in HIV diseases.

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