Euler characteristics of Boardman strata and the Milnor number of an isolated singularity of a complete intersection

1977 ◽  
Vol 10 (3) ◽  
pp. 231-233
Author(s):  
I. N. Iomdin
Keyword(s):  
Complete Intersection ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Euler Characteristics

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.

scholarly journals The jump of the Milnor number in the X 9 singularity class

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Szymon Brzostowski ◽  
Tadeusz Krasiński
Keyword(s):  
Milnor Number ◽  
Isolated Singularity ◽  
Milnor Numbers

AbstractThe jump of the Milnor number of an isolated singularity f 0 is the minimal non-zero difference between the Milnor numbers of f 0 and one of its deformations (f s). We prove that for the singularities in the X 9 singularity class their jumps are equal to 2.

scholarly journals On the topology of full non-degenerate complete intersection variety

1991 ◽  
Vol 121 ◽  
pp. 137-148 ◽  
Author(s):  
Mutsuo Oka
Keyword(s):  
Complete Intersection ◽  
Exceptional Divisor ◽  
Isolated Singularity ◽  
Laurent Polynomials ◽  
Intersection Variety

Let h1(u),…, hk(u) be Laurent polynomials of m-variables and letbe a non-degenerate complete intersection variety. Such an intersection variety appears as an exceptional divisor of a resolution of non-degenerate complete intersection varieties with an isolated singularity at the origin (Ok4]).

scholarly journals The Bruce–Roberts Number of A Function on A Hypersurface with Isolated Singularity

2020 ◽  
Vol 71 (3) ◽  
pp. 1049-1063
Author(s):  
J J Nuño-Ballesteros ◽  
B Oréfice-Okamoto ◽  
B K Lima-Pereira ◽  
J N Tomazella
Keyword(s):  
Complete Intersection ◽  
Hypersurface Singularity ◽  
Isolated Singularity ◽  
Characteristic Variety ◽  
Intersection Singularity ◽  
Complete Intersection Singularity ◽  
Isolated Complete Intersection Singularity

Abstract Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi \colon ({\mathbb{C}}^n,0)\to ({\mathbb{C}},0)$ and $f\colon ({\mathbb{C}}^n,0)\to{\mathbb{C}}$ such that the Bruce–Roberts number $\mu _{BR}(f,X)$ is finite. We first prove that $\mu _{BR}(f,X)=\mu (f)+\mu (\phi ,f)+\mu (X,0)-\tau (X,0)$, where $\mu $ and $\tau $ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen–Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

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.

scholarly journals Generic sections of essentially isolated determinantal singularities

2017 ◽  
Vol 28 (11) ◽  
pp. 1750083 ◽  
Author(s):  
Jean-Paul Brasselet ◽  
Nancy Chachapoyas ◽  
Maria A. S. Ruas
Keyword(s):  
Hyperplane Section ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Dimension Formula ◽  
Determinantal Singularities ◽  
Hyperplane Sections

We study the essentially isolated determinantal singularities (EIDS), defined by Ebeling and Gusein-Zade [S. M. Guseĭn-Zade and W. Èbeling, On the indices of 1-forms on determinantal singularities, Tr. Mat. Inst. Steklova 267 (2009) 119–131], as a generalization of isolated singularity. We prove in dimension [Formula: see text], a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing the previous results by Snoussi in dimension [Formula: see text]. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of Lê concerning the constancy of the Milnor number.

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