Abstract
The Bi-Lipschitz geometry is one of the main subjects in the modern approach of singularity theory. However, it rises from works of important mathematicians of the last century, especially Zariski. In this work we investigate the Bi-Lipschitz equisingularity of families of essentially isolated determinantal singularities inspired by the approach of Mostowski and Gaffney.
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References
Ament, D.A.H., Nuño-Ballesteros, J.J., Oréfice-Okamoto, B., Tomazella, J.N.: The Euler obstruction of a function on a determinantal variety and on a curve. Bull. Braz. Math. Soc. (N.S.) 47(3), 955–970 (2016)
Bruce, J.W.: Families of symmetric matrices. Mosc. Math. J. 3(2), 335–360 (2003)
Bruns, W., Vetter, U.: Determinantal Rings. Springer, New York (1998)
Damon, J.: The unfolding and determinancy theorems for subgoups of \({\cal{A}}\) and \({\cal{K}}\). Memoirs of the American Mathematical Society, Providence (1984)
Damon, J., Pike, B.: Solvable groups, free divisors and nonisolated matrix singularities II: vanishing topology. Geom. Topol. 18(2), 911–962 (2014)
Ebeling, W., Gusein-Zade, S.M.: On indices of 1-forms on determinantal singularities. Proc. Steklov Inst. Math. 267(1), 113–124 (2009)
Frühbis-Krüger, A., Neumer, A.: Simple Cohen–Macaulay codimension 2 singularities. Commun. Algebra 38(2), 454–495 (2010)
Frühbis-Krüger, A.: Classification of simple space curves singularities. Commun. Algebra 27(8), 3993–4013 (1999)
Gaffney, T.: Bi-Lipschitz equivalence, integral closure and invariants. In: Manoel, M. (ed.) Proceedings of the 10th International Workshop on Real and Complex Singularities. Universidade de São Paulo, M. C. Romero Fuster, Universitat de Valencia, Spain, C. T. C. Wall, University of Liverpool, London Mathematical Society Lecture Note Series (No. 380) (2010)
Gaffney, T., Grulha Jr., N.G., Ruas, M.A.S.: The local Euler obstruction and topology of the stabilization of associated determinantal varieties (2017). arXiv:1611.00749 [math.AG]
Greuel, G.M., Steenbrink, J.: On the Topology of Smoothable Singularities. In: Proceedings of Symposia in Pure Mathematics, vol. 40, Part 1, pp. 535–545 (1983)
Lejeune-Jalabert, M., Teissier, B.: Clôture intégrale des idéaux et équisingularité. (French) [Integral closure of ideals and equisingularity] With an appendix by Jean-Jacques Risler. Ann. Fac. Sci. Toulouse Math. (6) 17(4), 781–859 (2008)
Mostowski, T.: A criterion for Lipschitz equisingularity. Bull. Polish Acad. Sci. Math. 37(1—-6), 109–116 (1989). (1990)
Nuño-Ballesteros, J.J., Oréfice-Okamoto, B., Tomazella, J.N.: The vanishing Euler characteristic of an isolated determinantal singularity Israel. J. Math. 197(1), 475–495 (2013)
Pereira, M.S.: Variedades Determinantais e Singularidades de Matrizes. Tese de Doutorado, ICMC-USP (2010). http://www.teses.usp.br/teses/disponiveis/55/55135/tde-22062010-133339/en.php
Pham, F., Teissier, B.: Fractions lipschitziennes d’une algèbre analytique complexe et saturation de Zariski. Centre Math. l’École Polytech, Paris (1969)
Pham, F.: Fractions lipschitziennes et saturation de Zariski des algèbres analytiques complexes. Exposé d’un travail fait avec Bernard Teissier. Fractions lipschitziennes d’une algèbre analytique complexe et saturation de Zariski, Centre Math. lÉcole Polytech., Paris, 1969. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 649–654. Gauthier-Villars, Paris (1971)
Ruas, M.A.S., Pereira, M.S.: Codimension two determinantal varieties with isolated singularities. Math. Scand. 115(2), 161–172 (2014)
Schaps, M.: Deformations of Cohen–Macaulay schemes of codimension 2 and nonsingular deformations of space curves. Am. J. Math. 99, 669–684 (1977)
Wahl, J.: Smoothings of normal surface singularities. Topology 20, 219–246 (1981)
Zariski, O.: General theory of saturation and of saturated local rings. II. Saturated local rings of dimension 1. Am. J. Math. 93, 872–964 (1971)
Zhang, X.: Chern-Schwartz-MacPherson Class of Determinantal Varieties (2016). arXiv:1605.05380 [math.AG]
Acknowledgements
The authors are grateful to Terence Gaffney and Maria Aparecida Soares Ruas for the inspiration and support for this work, to David Trotman for his careful reading and valuable comments, to Anne Frühbis-Krüger, for her comments and suggestions which provided the improvement of this paper, mainly in Theorem 2.7 and by the remark that appears here as Remark 2.8 and to the referee for the excellent suggestions which improved this work. The first author was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo-FAPESP, Brazil, Grant 2013/22411-2. The second author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo-FAPESP, Brazil, Grant 2017/09620-2 and Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq, Brazil, Grant 303046/2016-3. The third author was supported by Proex ICMC/USP in a visit to São Carlos, where part of this work was developed.
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da Silva, T.F., Grulha, N.G. & Pereira, M.S. The Bi-Lipschitz Equisingularity of Essentially Isolated Determinantal Singularities. Bull Braz Math Soc, New Series 49, 637–645 (2018). https://doi.org/10.1007/s00574-017-0067-3
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DOI: https://doi.org/10.1007/s00574-017-0067-3
Keywords
- Bi-Lipschitz equisingularity
- Essentially isolated determinantal singularities
- 1-unfoldings
- Finite determinacy
- Canonical vector fields