Skip to main content
SpringerLink
Log in

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

  • Research Articles
  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

In this paper we study relationships between some topological and analytic invariants of zero-dimensional germs, or multiple points. Among other things, it is shown that there exist no rigid zero-dimensional Gorenstein singularities and rigid almost complete intersections. In the proof of the first result we exploit the canonical duality between homology and cohomology of the cotangent complex, while in the proof of the second we use a new method which is based on the properties of the torsion functor. In addition, we obtain highly efficient estimates for the dimension of the spaces of the first lower and upper cotangent functors of arbitrary zero-dimensional singularities, including the space of derivations. We also consider examples of nonsmoothable zero-dimensional noncomplete intersections and discuss some properties and methods for constructing such singularities using the theory of modular deformations, as well as a number of other applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. G. Aleksandrov, “Deformations of bouquets of quasihomogeneous one-dimensional singularities”, Funkts. Anal. Prilozhen., 15:1 (1981), 67–68; English transl.: Funct. Anal. Appl., 15:1 (1981), 53–55.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. G. Aleksandrov, “Cohomology of a quasihomogeneous complete intersection”, Izv. Akad. Nauk SSSR, Ser. Mat.,, 49:3 (1985), 467–510; English transl.: Math. USSR Izv., 26:3 (1986), 437–477.

    MathSciNet  Google Scholar 

  3. A. G. Aleksandrov, “Duality, derivations and deformations of zero-dimensional singularities”, Zero-dimensional schemes. Proceedings of the International conference held in Ravello, Italy, June 8–13, 1992, de Gruyter, Berlin–New York, 1994 (reprint 2016), 11–31.

    Google Scholar 

  4. A. G. Aleksandrov, “Differential forms on zero-dimensional singularities”, Funkts. Anal. Prilozhen., 52:4 (2018), 3–22; English transl.: Funct. Anal. Appl., 52:4 (2018), 241–257.

    Article  MathSciNet  Google Scholar 

  5. A. G. Aleksandrov, “Analytic invariants of multiple points”, Methods Appl. Anal., 25:3 (2018), 167–204.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, Princeton, New Jersey, 1956.

    MATH  Google Scholar 

  7. D. A. Cartwright, D. Erman, M. Velasco, and B. Viray, “Hilbert schemes of \(\,8\) points”, Algebra Number Theory, 3:7 (2009), 763–795.

    MathSciNet  MATH  Google Scholar 

  8. H. Grauert and H. Kerner, “Deformationen von Singularitäten komplexer Räume”, Math. Ann., 153 (1964), 236–260.

    Article  MathSciNet  MATH  Google Scholar 

  9. G.-M. Greuel, “Der Gauß-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten”, Math. Ann., 214:3 (1975), 235–266.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Herzog (ed.) and E. Kunz (ed.), Der kanonische Module eines Cohen–Macaulay Rings, Lecture Notes in Math., 238 Springer–Verlag, Berlin–Heidelberg–New York, 1971.

    Book  MATH  Google Scholar 

  11. J. Herzog, “Eindimensionale fast-vollständige Durchschnitte sind nicht starr”, Manuscripta Math., 30:1 (1979), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. A. Iarrobino, “The number of generic singularities”, Rice Univ. Stud., 59:1 (1973), 49–51.

    MathSciNet  MATH  Google Scholar 

  13. A. A. Iarrobino, “Punctual Hilbert schemes”, Mem. Amer. Math. Soc., 10:188 (1977), 1–124.

    MathSciNet  MATH  Google Scholar 

  14. A. Iarrobino, J. Emsalem, “Some zero-dimensional generic singularities; finite algebras having small tangent space”, Compositio Math., 36:2 (1978), 145–188.

    MathSciNet  MATH  Google Scholar 

  15. A. A. Iarrobino, “Deforming complete intersection Artin algebras. Appendix: Hilbert function of \(\mathbb C[x,y]/I\)”, Singularities (Summer Inst. Arcata-1981), Proc. Symp. Pure Math., Amer. Math. Soc., Providence, RI, 1983, 593–608.

    Google Scholar 

  16. E. Kunz, “Almost complete intersections are not Gorenstein rings”, J. Algebra, 28:11 (1974), 1–15.

    MathSciNet  MATH  Google Scholar 

  17. A. G. Kushnirenko, “Newton’s polyhedron and the number of solutions to a system of \(k\) equations in \(k\) unknowns”, Uspekhi Mat. Nauk, 30:2 (1975), 266–267.

    Google Scholar 

  18. S. Lichtenbaum and M. Schlessinger, “The cotangent complex of a morphism”, Trans. Amer. Math. Soc., 128:1 (1967), 41–70.

    Article  MathSciNet  MATH  Google Scholar 

  19. V. P. Palamodov, “Multiplicity of holomorphic mappings”, Funkts. Anal. Prilozhen., 1:3 (1967), 54–65; English transl.: Funct. Anal. Appl., 1 (1968), 218–226.

    Google Scholar 

  20. V. P. Palamodov, “Moduli in versal deformations of complex spaces”, Variétiés Analytique Compactes (Colloq., Nice 1977), Lect. Notes in Math., 683 Springer-Verlag, Berlin, 1978, 74–115.

    Chapter  Google Scholar 

  21. H. C. Pinkham, Deformations of algebraic varieties with \(\mathbb G_m\)action, Astérisque, 20 Société Mathématique de France, Paris, 1974.

    MATH  Google Scholar 

  22. D. S. Rim, “Torsion differentials and deformation”, Trans. Amer. Math. Soc., 169 (442):1 (1972), 257–278.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Schaps, Astérisque, Société Mathématique de France, Paris, 1973.

    Google Scholar 

  24. M. Schlessinger, “Rigidity of quotient singularities”, Invent. Math., 14:1 (1971), 17–26.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Schlessinger, “On rigid singularities”, Rice Univ. Stud., 59:1 (1973), 147–162.

    MathSciNet  MATH  Google Scholar 

  26. P. Seibt, “Infinitesimal extensions of commutative algebras”, J. Pure Appl. Algebra, 16:2 (1980), 197–206.

    Article  MathSciNet  MATH  Google Scholar 

  27. G. N. Tyurina, “Locally semiuniversal flat deformations of isolated singularities of complex spaces”, Izv. Akad. Nauk SSSR, Ser. Mat.,, 33:5 (1969), 1026–1058; English transl.: Math. USSR Izv., :3 (1971), 967–999.

    MathSciNet  Google Scholar 

  28. R. Waldi, “Deformation von Gorenstein-Singularitäten der Kodimension 3”, Math. Ann., 242:1 (1979), 201–208.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author thanks Tony Iarrobino and Claus Hertling for helpful discussions of many ideas related to this topic, as well as the referee for valuable advice and remarks.

Author information

Authors and Affiliations

  1. Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    A. G. Aleksandrov

Authors
  1. A. G. Aleksandrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Aleksandrov.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 3–25 https://doi.org/10.4213/faa3886.

Translated by A. G. Aleksandrov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aleksandrov, A.G. On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities. Funct Anal Its Appl 56, 1–18 (2022). https://doi.org/10.1134/S0016266322010014

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0016266322010014

Keywords

Search

Navigation