Advertisement

Free deformations of hypersurface singularities

  • A. G. Aleksandrov
  • J. Sekiguchi
Article
  • 73 Downloads

Abstract

The article is devoted to the study of the classification problem for Saito free divisors making use of the deformation theory of varieties. In particular, in the quasihomogeneous case, we describe an approach for computation of free deformations of quasicones over quasismooth varieties based on properties of deformations of varieties with \( {\mathbb{G}_m} \)-action. We also discuss some applications including the problem of compactification of modular spaces and computation of free deformations for certain simple, unimodal, and unimodular singularities.

Keywords

Modular Space Deformation Theory Modular Deformation Weighted Projective Space Hypersurface Singularity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    A. G. Aleksandrov, “Normal forms of one-dimensional quasihomogeneous complete intersections,” Mat. Sb. (N.S.), 117(159), No. 1, 3–31 (1982).Google Scholar
  2. 2.
    A. G. Aleksandrov, “Cohomology of a quasihomogeneous complete intersection,” Izv. Akad. Nauk SSSR Ser. Mat., 49, No. 3, 467–510 (1985).MathSciNetGoogle Scholar
  3. 3.
    A. G. Aleksandrov, “Euler-homogeneous singularities and logarithmic differential forms,” Ann. Global Anal. Geom., 4, No. 2, 225–242 (1986).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. G. Aleksandrov, “The Milnor numbers of nonisolated Saito singularities,” Funct. Anal. Appl., 21, 1–9 (1987).CrossRefGoogle Scholar
  5. 5.
    A. G. Aleksandrov, “Nonisolated Saito singularities,” Mat. Sb. (N.S.), 137(179), No. 4, 554–567 (1988).Google Scholar
  6. 6.
    A. G. Aleksandrov, “Nonisolated hypersurface singularities,” In: Theory of Singularities and its Applications, 211–246, Adv. in Soviet Math. 1, Amer. Math. Soc., Providence, RI (1990).Google Scholar
  7. 7.
    A. G. Aleksandrov, “Modular space for complete intersection curve singularities,” In: Finite or Infinite Dimensional Complex Analysis (Fukuoka, 1999), 1–19, Lect. Notes Pure Appl. Math. 214, Dekker, New York (2000).Google Scholar
  8. 8.
    A. G. Aleksandrov, “Stratum of freeness for deformations of singularities,” Sci. Bull. Belgorod State Univ., Ser. Math. Phys. 13(68), No. 17/2, 6–18 (2009).Google Scholar
  9. 9.
    V. I. Arnol’d, “Normal forms of functions in the neighborhood of degenerate critical points,” Usp. Mat. Nauk, 29, No. 2(176), 11–49 (1974).MATHGoogle Scholar
  10. 10.
    P. Cartier, “Les arrangements d’hyperplans: un chapitre de géométrie combinatoire,” Bourbaki Seminar, Vol. 1980/81, 1–22, Lecture Notes in Math., 901, Springer, Berlin-New York (1981).Google Scholar
  11. 11.
    I. V. Dolgačhev, “Weighted projective varieties,” Lect. Notes Math., 956, 34–71, Springer-Verlag (1982).Google Scholar
  12. 12.
    G.-M. Greuel, G. Pfister, H. Schönemann, SINGULAR 3.0. – A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2005).Google Scholar
  13. 13.
    B. Martin, “Algorithmic computation of flattenings and of modular deformations, J. Symbol. Comput., 34, No. 3, 199–212 (2002).CrossRefMATHGoogle Scholar
  14. 14.
    V. P. Palamodov, “Moduli and versal deformations of complex spaces,” Dokl. Akad. Nauk SSSR, 230, No. 1, 34–37 (1976).MathSciNetGoogle Scholar
  15. 15.
    H. C. Pinkham, “Deformations of algebraic varieties with \( {\mathbb{G}_m} \)-action,” Astérisque, No. 20, 1–131, Société Mathématique de France, Paris (1974).Google Scholar
  16. 16.
    K. Saito, “Quasihomogene isolierte Singularitäten von Hyperflächen,” Invent. Math. 14, 123–142 (1971).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    K. Saito, “On the uniformization of complements of discriminant loci,” In: Hyperfunctions and Linear Partial Differential Equations, RIMS Kōkyūroku, 287, 117–137 (1977).Google Scholar
  18. 18.
    K. Saito, “Theory of logarithmic differential forms and logarithmic vector fields,” J. Fac. Sci. Univ. Tokyo, ser. IA, 27, No. 2, 265–291 (1980).MATHGoogle Scholar
  19. 19.
    J. Sekiguchi, “Some topics related with discriminant polynomials,” In: Algebraic Analysis and Number Theory, RIMS Kōkyūroku, 810, 85–94 (1992).Google Scholar
  20. 20.
    J. Sekiguchi, “A classification of weighted homogeneous Saito free divisors,” J. Math. Soc. Jpn., 61, No. 4, 1071–1095 (2009).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    J. Sekiguchi, “Three dimensional Saito free divisors and deformations of singular curves,” J. Siberian Federal Univ., Math. Phys., 1, 33–41 (2008).Google Scholar
  22. 22.
    H. Terao, “Arrangements of hyperplanes and their freeness I.,” J. Fac. Sci. Univ. Tokyo, Ser. I A, 27, No. 2, 293–312 (1980).MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Institute for Control Sciences, Russian Academy of SciencesMoscowRussia
  2. 2.Tokyo University of Agriculture and TechnologyTokyoJapan

Personalised recommendations