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Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1263–1293 | Cite as

An observation concerning the vanishing topology of certain isolated determinantal singularities

  • Matthias ZachEmail author
Article
  • 31 Downloads

Abstract

We extend the results about the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities from a previous article by Frühbis-Krüger and Zach (On the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities. arXiv:1501.01915, 2015). Due to the Hilbert–Burch theorem, these singularities have a canonical determinantal structure and a well behaved deformation theory, which, in particular, yields a unique Milnor fiber. Studying the case of possibly non-isolated singularities in the Tjurina transform, as introduced in Frühbis-Krüger and Zach (2015), we reveal that in dimension 3 and 2 there is always exactly one special vanishing cycle in degree 2 closely related to the determinantal structure of the singularity.

Keywords

Determinantal singularity Non-isolated singularity Vanishing topology 

Mathematics Subject Classification

32S30 14B07 32S50 32S55 

Notes

Acknowledgements

The author wishes to thank A. Frühbis-Krüger for guidance and support, M. Tibăr for discussions during a visit in Hannover and the organization of the workshop on nonisolated singularities in Lille and D. Siersma for further discussions on the topic. Furthermore, he thanks M.A.S. Ruas and the ICMC for hospitality and a stimulating mathematical framework during a stay at the USP in São Carlos. Thanks also for conversations with T. Gaffney and M.A.S. Ruas on determinantal singularities, which significantly broadened the viewpoint of this paper. During the preparation of this article, the author was funded by the DFG program GRK 1463 and the DAAD program IP@Leibniz at the Leibniz Universität Hannover.

References

  1. 1.
    Brasselet, J.P., Chachapoyas, N., Ruas, M.A.S.: Generic sections of essentially isolated determinantal singularities. Int. J. Math. 28(11), 1750083 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burch, L.: On ideals of finite homological dimension in local rings. Proc. Camb. Phil. Soc. 64, 941–948 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Damon, J., Pike, B.: Solvable groups, free divisors and nonisolated matrix singularities II: vanishing topology. Geom. Topol. 18, 911–962 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    de Jong, T.: Some classes of line singularities. Math. Z. 198, 493–517 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ebeling, W., Gusein-Zade, S.: On indices of 1-forms on determinantal singularities. Proc. Steklov Inst. Math. 267, 113–124 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Frühbis-Krüger, A., Zach, M.: On the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities. arXiv:1501.01915 (2015)
  7. 7.
    Frühbis-Krüger, A., Neumer, A.: Simple Cohen–Macaulay codimension 2 singularities. Commun. Algebra 38(2), 454–495 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gaffney, T., Rangachev, A.: Pairs of modules and determinantal isolated singularities. arXiv:1501.00201v2 (2016)
  9. 9.
    Gaffney, T., Ruas, M.A.S.: Equisingularity and EIDS. arXiv:1602.00362 (2016)
  10. 10.
    Gibson, C.G., et al.: Topological stability of smooth mappings. Lecture notes in mathematics. Springer-Verlag, New York (1976)CrossRefGoogle Scholar
  11. 11.
    Greuel, G.-M., Steenbrink, J.: On the topology of smoothable singularities. In: Singularities, Part 1 (Arcata, Calif., 1981), Vol. 40, pp. 535–545. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, R.I. (1983)Google Scholar
  12. 12.
    Hamm, H.: Lokale topologische Eigenschaften komplexer Räume. Math. Ann. 191, 235–252 (1972)CrossRefzbMATHGoogle Scholar
  13. 13.
    Iomdin, I.N.: Local topological properties of complex algebraic sets. Sibirsk. Mat. Z. 15(4), 784–805 (1974)MathSciNetGoogle Scholar
  14. 14.
    Milnor, J.W.: Morse Theory. Princeton University Press, Princeton, NJ (1963)zbMATHGoogle Scholar
  15. 15.
    Møller Pedersen, H.: On Tjurina transform and resolution of determinantal singularities. arXiv:1604.06029v2 (2016)
  16. 16.
    Nuño-Ballesteros, J.J., Oréfice-Okamoto, B., Tomazella, J.N.: The vanishing Euler characteristic of an isolated determinantal singularity. Isr. J. Math. 197, 475–495 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nuño-Ballesteros, J.J., Oréfice-Okamoto, B., Tomazella, J.N.: Equisingularity of families of isolated determinantal singularities. Math. Z. 289, 1409–1425 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ruas, M.A.S., da Silva Pereira, M.: Codimension two determinantal varieties with isolated singularities. Math. Scand. 115, 161–172 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Schaps, M.: Deformations of Cohen–Macaulay schemes of codimension 2 and nonsingular deformations of space curves. Am. J. Math. 99, 669–684 (1977)CrossRefzbMATHGoogle Scholar
  20. 20.
    Siersma, D.: Isolated line singularities. In: Proc. of Sympos. Pure Mathematics, vol 40, Am. Math. Soc., Providence, RI Part 2, (Acarta, Calif., 1981), pp. 485–496 (1983)Google Scholar
  21. 21.
    Siersma, D.: Variation mappings on singularities with a 1-dimensional critical locus. Topology 30(3), 445–469 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Siersma, D., Tibăr, M.: Vanishing homology of projective hypersurfaces with 1-dimensional singularities. Eur. J. Math. 3, 565–586 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tráng, Lê Dũng: Some remarks on relative monodromy. In: Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo 1976, pp. 397–403 (1977)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Algebraische GeometrieLeibniz Universtät HannoverHannoverGermany

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