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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

REFERENCES

1. Ahmed, I., Ruas, M. A. S. and Tomazella, J. N., Invariants of topological relative right equivalence, Math. Proc. Cambridge Philos. Soc. 155 (2) (2013), 307–315.Google Scholar
2. Bivià-Ausina, C. and Nuño-Ballesteros, J. J., Multiplicity of iterated Jacobian extensions of weighted homogeneous map germs, Hokkaido Math. J. 29 (2) (2000), 341–368.Google Scholar
3. Bruce, J. W. and Roberts, R. M., Critical points of functions on analytic varieties, Topology 27 (1) (1988), 57–90.Google Scholar
4. Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, UK, 1993).Google Scholar
5. Buchweitz, R. O. and Greuel, G. M., The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (3) (1980), 241–248.Google Scholar
6. Damon, J., Topological triviality and versality for subgroups of A and K: II. Suficient conditions and applications, Nonlinearity 5 (1992), 373–412.Google Scholar
7. Furuya, M. and Tomari, M., A characterization of semi-quasihomogeneous functions in terms of the Milnor number, Proc. Amer. Math. Soc. 132 (7) (2004), 1885–1890.Google Scholar
8. Grulha, N. G. Jr., The Euler obstruction and Bruce-Roberts' Milnor number, Q. J. Math. 60 (3) (2009), 291–302.CrossRefGoogle Scholar
9. Goryunov, V. V., Functions on space curves, J. London Math. Soc. (2) 61 (2000), 807–822.CrossRefGoogle Scholar
10. Greuel, G. M., Dualität in der lokalen kohomologie isolierter singularitäten, Math Ann. 250 (1980), 157–173.CrossRefGoogle Scholar
11. Hamm, H. A., Lokale topologische Eigenschaften komplexer Räume, Math. Ann. 191 (1971), 235–252.Google Scholar
12. Tráng, Lê Dũng and Ramanujam, C. P., The invariance of Milnor's number implies the invariance of the topological type, Amer. J. Math. 98 (1) (1976), 67–78.Google Scholar
13. Looijenga, E. J. N., Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series, vol. 77 (Cambridge University Press, Cambridge, UK, 1984).Google Scholar
14. Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, Cambridge, UK, 1989).Google Scholar
15. Milnor, J., Singular points of complex hypersurfaces, Annals of Math. Studies (Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968).Google Scholar
16. Milnor, J. and Orlik, P., Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385–393.Google Scholar
17. Mond, D. and van Straten, D., Milnor number equals Tjurina number for functions on space curves, J. London Math. Soc. (2) 63 (2001), 177–187.Google Scholar
18. Nuño-Ballesteros, J. J., Oréfice, B. and Tomazella, J. N., The Bruce-Roberts number of a function on a weighted homogeneous hypersurface, Q. J. Math. 64 (1) (2013), 269–280.Google Scholar
19. Nuño-Ballesteros, J. J., Oréfice-Okamoto, B. and Tomazella, J. N., The vanishing Euler characteristic of an isolated determinantal singularity, Israel J. Math. 197 (1) (2013), 475–495.Google Scholar
20. Nuño-Ballesteros, J. J. and Tomazella, J. N., The Milnor number of a function on a space curve germ, Bull. Lond. Math. Soc. 40 (1) (2008), 129–138.Google Scholar
21. Ruas, M. A. S. and Tomazella, J. N., Topological triviality of families of functions on analytic varieties, Nagoya Math. J., 175 (2004), 39–50.CrossRefGoogle Scholar
22. Saito, K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (2) (1980), 265–291.Google Scholar
23. Timourian, J. G., The invariance of Milnor's number implies topological triviality, Amer. J. Math. 99 (2) (1977), 437–446.Google Scholar
24. Varchenko, A. N., A lower bound for the codimension of the stratum μ-constant in terms of the mixed hodge structure, Vest. Mosk. Univ. Mat. 37 (1982), 29–31.Google Scholar