Polar Multiplicities and Euler obstruction of the stable types in weighted homogeneous map germs from ℂn to ℂ3, n ≥ 3

2007 ◽  
Author(s):  
E. C. Rizziolli ◽  
M. J. Saia
Keyword(s):  
Euler Obstruction ◽  
Homogeneous Map

Polar multiplicities and Euler obstruction for ruled surfaces

2012 ◽  
Vol 43 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Nivaldo G. Grulha ◽  
Marcelo E. Hernandes ◽  
Rodrigo Martins
Keyword(s):  
Ruled Surfaces ◽  
Euler Obstruction

The deformation multiplicity of a map germ with respect to a Boardman symbol

2001 ◽  
Vol 131 (5) ◽  
pp. 1003-1022 ◽  
Author(s):  
C. Bivià-Ausina ◽  
J. J. Nuño-Ballesteros
Keyword(s):  
Geometrical Interpretation ◽  
Algebraic Multiplicity ◽  
Homogeneous Map

We define the deformation multiplicity of a map germ f: (Cn, 0) → (Cp, 0) with respect to a Boardman symbol i of codimension less than or equal to n and establish a geometrical interpretation of this number in terms of the set of Σi points that appear in a generic deformation of f. Moreover, this number is equal to the algebraic multiplicity of f with respect to i when the corresponding associated ring is Cohen-Macaulay. Finally, we study how algebraic multiplicity behaves with weighted homogeneous map germs.

Alternative algorithms for computing generic μ∗-sequences and local Euler obstructions of isolated hypersurface singularities

2019 ◽  
Vol 18 (08) ◽  
pp. 1950156
Author(s):  
Katsusuke Nabeshima ◽  
Shinichi Tajima
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Rational Functions ◽  
Isolated Singularity ◽  
Euler Obstruction ◽  
Standard Bases ◽  
Hypersurface Singularities ◽  
Previous Algorithm ◽  
Computation Speed

A new algorithm is introduced for computing [Formula: see text]-sequences of isolated hypersurface singularities. It is shown that the new algorithm results in better performance, compared to our previous algorithm that utilizes parametric local cohomology systems, in computation speed. Furthermore, it can be used to compute local Euler obstruction of a hypersurface with an isolated singularity. The key idea of the new algorithm is computing standard bases in a local ring over a field of rational functions.

Unfolding plane curves with cusps and nodes

2015 ◽  
Vol 145 (1) ◽  
pp. 161-174
Author(s):  
Juan J. Nuño Ballesteros
Keyword(s):  
Finite Codimension ◽  
Plane Curves ◽  
Singular Set ◽  
One Dimensional ◽  
Double Points ◽  
Euler Obstruction ◽  
Type Singularity ◽  
Transverse Slice

Given an irreducible surface germ (X, 0) ⊂ (ℂ3, 0) with a one-dimensional singular set Σ, we denote by δ1 (X, 0) the delta invariant of a transverse slice. We show that δ1 (X, 0) ≥ m0 (Σ, 0), with equality if and only if (X, 0) admits a corank 1 parametrization f :(ℂ2, 0) → (ℂ3, 0) whose only singularities outside the origin are transverse double points and semi-cubic cuspidal edges. We then use the local Euler obstruction Eu(X, 0) in order to characterize those surfaces that have finite codimension with respect to -equivalence or as a frontal-type singularity.

scholarly journals Computing Euler Obstruction Functions Using Maximum Likelihood Degrees

2020 ◽  
Vol 2020 (20) ◽  
pp. 6699-6712
Author(s):  
Jose Israel Rodriguez ◽  
Botong Wang
Keyword(s):  
Algebraic Geometry ◽  
Maximum Likelihood ◽  
Numerical Algorithm ◽  
Algebraic Statistics ◽  
Computational Algebraic Geometry ◽  
Euler Obstruction

Abstract We give a numerical algorithm computing Euler obstruction functions using maximum likelihood degrees. The maximum likelihood degree is a well-studied property of a variety in algebraic statistics and computational algebraic geometry. In this article we use this degree to give a new way to compute Euler obstruction functions. We define the maximum likelihood obstruction function and show how it coincides with the Euler obstruction function. With this insight, we are able to bring new tools of computational algebraic geometry to study Euler obstruction functions.

On the polar varieties of ruled hypersurfaces

2016 ◽  
Vol 164 (2) ◽  
pp. 193-204
Author(s):  
M.E. HERNANDES ◽  
R. MARTINS ◽  
M.E. RODRIGUES HERNANDES
Keyword(s):  
Positive Integer ◽  
Euler Obstruction ◽  
Base Curve

AbstractThe aim of this work is to characterise the k-polar variety Pk(X) of an (n − 1)-ruled hypersurface X ⊂ ℂn+1. More precisely, we prove that Pk(X) is empty for all k >1 and the first polar variety is empty or it is an (n − 2)-ruled variety in ℂn+1, whose multiplicity is obtained by the multiplicity of the base curve and the multiplicity of one directrix of X. As a consequence we obtain the Euler obstruction Eu0(X) of X and, in addition, we exhibit (n − 1)-ruled hypersurfaces such that Eu0(X) = m, for any prescribed positive integer m.

scholarly journals Global Euler obstruction and polar invariants

2005 ◽  
Vol 333 (2) ◽  
pp. 393-403 ◽  
Author(s):  
José Seade ◽  
Mihai Tibăr ◽  
Alberto. Verjovsky
Keyword(s):  
Euler Obstruction

EULER OBSTRUCTION, POLAR MULTIPLICITIES AND EQUISINGULARITY OF MAP GERMS IN $\mathcal{O}(n,p), n < p$

2006 ◽  
Vol 17 (08) ◽  
pp. 887-903 ◽  
Author(s):  
V. H. JORGE PÉREZ ◽  
M. J. SAIA
Keyword(s):  
The Other ◽  
Holomorphic Map ◽  
Stable Type ◽  
Euler Obstruction ◽  
Whitney Equisingularity ◽  
The Relationship

There are two main goals in this article, one of them is to minimize the number of invariants needed to obtain Whitney equisingular one parameter families of finitely determined holomorphic map germs ft:(ℂn,0) → (ℂp,0), with n < p. The other is to show how to compute the local Euler obstruction of the stable types which appear in a finitely determined map germ in these dimensions. The polar multiplicities of all stable types and the 0-stable singularities are the invariants that guarantee the Whitney equisingularity of such families and the polar multiplicities are the numbers that also allow us to compute the local Euler obstruction. Therefore our first step is to describe all stable types which appear when n < p and show the relationship between the polar multiplicities in each stable type. Using the fact that these polar multiplicities are upper semi-continuous we minimize the number of invariants that guarantee Whitney equisingularity of such a family. We also apply the relationship between the polar multiplicities in each stable type and a result of Lê and Teissier to show how to compute the local Euler obstruction of the stable types which appear in these dimensions.

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