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NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES

Published online by Cambridge University Press:  20 March 2017

J. J. NUÑO-BALLESTEROS ,
B. ORÉFICE-OKAMOTO  and
J. N. TOMAZELLA
J. J. NUÑO-BALLESTEROS
Affiliation:
Departament de Geometria i Topologia, Universitat de València, Campus de Burjassot, 46100 BurjassotSpain e-mail: Juan.Nuno@uv.es
B. ORÉFICE-OKAMOTO
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, 13560-905, São Carlos, SP, Brazil e-mails: bruna@dm.ufscar.br, tomazella@dm.ufscar.br
J. N. TOMAZELLA
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, 13560-905, São Carlos, SP, Brazil e-mails: bruna@dm.ufscar.br, tomazella@dm.ufscar.br
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Abstract

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We consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

Keywords

Primary 32S15Secondary 58K6032S30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 1 , January 2018 , pp. 175 - 185
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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