Skip to main content
SpringerLink
Log in

Tree Singularities: Limits, Series and Stability

  • Chapter
Deformations of Surface Singularities

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 23))

  • 980 Accesses

Abstract

A tree singularity is a surface singularity that consists of smooth components, glued along smooth curves in the pattern of a tree. Such singularities naturally occur as degenerations of certain rational surface singularities. To be more precise, they can be considered as limits of certain series of rational surface singularities with reduced fundamental cycle. We introduce a general class of limits, construct series deformations for them and prove a stability theorem stating that under the condition of finite dimensionality of T 2 the base space of a semi-universal deformation for members high in the series coincides up to smooth factor with the “base space of the limit”. The simplest tree singularities turn out to have already a very rich deformation theory, that is related to problems in plane geometry. From this relation, a very clear topological picture of the Milnor fibre over the different components can be obtained.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Andrée, Homologie des Algèbres Commutatives, Grundlehren der math. Wissenschaften 206, Springer, Berlin (1976).

    Google Scholar 

  2. J. Arndt, Verselle deformationen zyklischer Quotientensingularitäten, Dissertation, Universität Hamburg, 1988.

    MATH  Google Scholar 

  3. V. Arnol’d, S. Guzein-Zade, A. Varchenko, Singularities of Differentiable Maps, Vol. I & II, Birkhäuser, Boston (1988).

    Book  MATH  Google Scholar 

  4. M. Artin, On isolated rational singularities of surfaces, Am. J. of Math., 88 (1966), 129–136.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Artin, On the Solutions of Analytic Equations, Inv. Math., 5 (1968), 277–291.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Artin, Algebraic construction of Brieskorns resolutions, J. of Alg., 29 (1974), 330–348.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Artin, Deformations of Singularities, Lecture Notes on Math., 54 Tata Institute of Fundamental Research, Bombay (1976).

    Google Scholar 

  8. J. Bingener, Modulräume in der analytischen Geometrie 1 & 2, Aspekte der Mathematik, Vieweg, (1987).

    Google Scholar 

  9. E. Brieskorn, Rationale Singularitäten komplexer Flächen, Inv. Math., 4 (1967), 336–358.

    Article  MathSciNet  Google Scholar 

  10. R.-O. Buchweitz, Contributions à la Théorie des Singularités, Thesis, Université Paris VII, (1981).

    Google Scholar 

  11. K. Behnke and J. Christophersen, Hypersurface sections and Obstructions (Rational Surface Singularities), Comp. Math., 77 (1991), 233–258.

    MATH  MathSciNet  Google Scholar 

  12. R. Buchweitz and G.-M. Greuel, The Milnor number and deformations of complex curve singularities, Inv. Math., 58 (1980), 241–281.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Christophersen, On the Components and Discriminant of the Versall Base Space of Cyclic Quotient Singularities, in: “Singularity Theory and its Applications”, Warwick 1989, D. Mond and J. Montaldi (eds.), SLNM 1462, Springer, Berlin, (1991).

    Google Scholar 

  14. G. Fisher, Complex Analytic Geometry, SLNM 538, Springer, Berlin, (1976).

    Google Scholar 

  15. H. Flenner, Über Deformationen holomorpher Abbildungen, Osnabrücker Schriften zur Mathematik, Reihe P Preprints, Heft, 8 (1979).

    Google Scholar 

  16. H. Grauert, Über die Deformationen isolierter Singularitäten analytischer Mengen, Inv. Math., 15 (1972), 171–198.

    Article  MATH  MathSciNet  Google Scholar 

  17. G.-M. Greuel, On deformations of curves and a formula of Deligne, in: “Algebraic Geometry”, Proc., La Rabida 1981, SLNM 961, Springer, Berlin, (1983).

    Google Scholar 

  18. H. Grauert and R. Remmert, Analytische Stellenalgebren, Grundlehren d. math. Wissens, Bd. 176, Springer, Berlin, (1971).

    Google Scholar 

  19. H. Hauser, La Construction de la Déformation semi-universelle d’un germe de variété analytique complexe, Ann. Scient. Éc. Norm. sup. 4 série, t. 18 (1985), 1–56.

    MATH  MathSciNet  Google Scholar 

  20. L. Illusie, Complex Cotangente et Déformations 1, 2, SLNM, 239 (1971), SLNM, 289 (1972), Springer, Berlin.

    Google Scholar 

  21. T. de Jong and D. van Straten, A Deformation Theory for Non-Isolated Singularities, Abh. Math. Sem. Univ. Hamburg, 60 (1990), 177–208.

    Article  MATH  MathSciNet  Google Scholar 

  22. T. de Jong and D. van Straten, Deformations of the Normalization of Hypersurfaces, Math. Ann., 288 (1990), 527–547.

    Article  MATH  MathSciNet  Google Scholar 

  23. T. de Jong and D. van Straten, On the Base Space of a Semi-universal Deformation of Rational Quadruple Points, Ann. of Math., 134 (1991), 653–678.

    Article  MATH  MathSciNet  Google Scholar 

  24. T. de Jong and D. van Straten, Disentanglements, in: “Singularity Theory and its Applications”, Warwick 1989, D. Mond and J. Montaldi (eds.), SLNM 1462, Springer, Berlin, (1991).

    Google Scholar 

  25. T. de Jong and D. van Straten, On the Deformation Theory of Rational Surface Singularities with Reduced Fundamental Cycle, J. of Alg. Geometry, 3 (1994), 117–172.

    MATH  Google Scholar 

  26. T. de Jong and D. van Straten, Deformation Theory of Sandwiched Singularities, Duke Math. J., Vol 95, No. 3, (1998), 451–522.

    Article  MATH  MathSciNet  Google Scholar 

  27. B. Kaup, Über Kokerne und Pushouts in der Kategorie der komplexanalytischen Räume, Math. Ann., 189 (1970), 60–76.

    Article  MATH  MathSciNet  Google Scholar 

  28. J. Kollár and N. Shepherd-Barron, Threefolds and deformations of surface singularities, Inv. Math., 91 (1988), 299–338.

    Article  MATH  Google Scholar 

  29. A. Laudal, Formal Moduli of Algebraic Structures, SLNM 754, Springer, Berlin, (1979).

    Google Scholar 

  30. H. Laufer, On Minimally Elliptic Singularities, Amer. J. Math., 99 (1977), 1257–1295.

    Article  MATH  MathSciNet  Google Scholar 

  31. H. Laufer, Ambient Deformations for Exceptional Sets in Two-manifolds, Inv. Math., 55 (1979), 1–36.

    Article  MATH  MathSciNet  Google Scholar 

  32. S. Lichtenbaum and M. Schlesinger, The cotangent Complex of a Morphism, Trans. AMS, 128 (1967), 41–70.

    Article  MATH  Google Scholar 

  33. P. Mazet, Analytic Sets in Locally Convex Spaces, Math. Studies 89, North-Holland, (1984).

    Google Scholar 

  34. D. Mond, On the classification of germs of maps from ℝ2 to ℝ3, Proc. London Math. Soc., 50 (1985), 333–369.

    Article  MATH  MathSciNet  Google Scholar 

  35. V. Palomodov, Moduli and Versal Deformations of Complex Spaces, Soviet Math. Dokl., 17 (1976), 1251–1255.

    Google Scholar 

  36. R. Pellikaan, Hypersurface Singularities and Resolutions of Jacobi Modules, Thesis, Rijksuniversiteit Utrecht, (1985).

    Google Scholar 

  37. R. Pellikaan, Finite Determinacy of Functions with Non-Isolated Singularities, Proc. London Math. Soc. (3), 57 (1988), 357–382.

    Article  MathSciNet  Google Scholar 

  38. R. Pellikaan, Deformations of Hypersurfaces with a One-Dimensional Singular Locus, J. Pure Appl. Algebra, 67 (1990), 49–71.

    Article  MATH  MathSciNet  Google Scholar 

  39. R. Pellikaan, Series of Isolated Singularities, Contemp. Math., 90 Proc. Iowa, R. Randell (ed.), (1989).

    Google Scholar 

  40. R. Pellikaan, On Hypersurfaces that are Sterns, Comp. Math., 71 (1989), 229–240.

    MATH  MathSciNet  Google Scholar 

  41. H. Pinkham, Deformations of algebraic Varieties with G m-action, Astérisque, 20 1974.

    Google Scholar 

  42. O. Riemenschneider, Deformations of rational singularities and their resolutions, in: Complex Analysis, Rice University Press 59 (1), 1973, 119–130.

    MathSciNet  Google Scholar 

  43. O. Riemenschneider, Deformationen von Quotientensingularitten (nach zyklischen Gruppen), Math. Ann., 209 (1974), 211–248.

    Article  MATH  MathSciNet  Google Scholar 

  44. D. S. Rim, Formal deformation theory, SGA 7(1), Exp. VI, Springer, Berlin, (1972).

    Google Scholar 

  45. G. Ruget, Déformations des germes d’espaces analytiques, Sém. Douady-Verdier 1971–72, Astérisque, 16 (1974), 63–81.

    MATH  MathSciNet  Google Scholar 

  46. M. Schlessinger, Functors of Artin Rings, Trans. Am. Math. Soc., 130 (1968), 208–222.

    Google Scholar 

  47. M. Schlessinger, Rigidity of quotient singularities, Inv. Math., 14 (1971), 17–26.

    Article  MATH  MathSciNet  Google Scholar 

  48. M. Schlessinger, On Rigid Singularities, in: Complex Analysis, Rice University Press 59(1), 1973.

    Google Scholar 

  49. R. Schrauwen, Series of Singularities and their Topology, Thesis, Utrecht, (1991).

    Google Scholar 

  50. N. Shepherd-Barron, Degenerations with Numerically Effective Canonical Divisor, in: “The Birational Geometry of Degenerations”, Progress in Math. 29, Birkhäuser, Basel, (1983).

    Google Scholar 

  51. D. Siersma, Isolated Line Singularities, in: “Singularities“, Arcata 1981, P. Orlik (ed.), Proc. Sym. Pure Math. 40(2), (1983), 405–496.

    Google Scholar 

  52. D. Siersma, Singularities with Critical Locus a One-Dimensional Complete Intersection and transverse A 1, Topology and its Applications, 27, 51–73 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  53. D. Siersma, The Monodromy of a Series of Singularities, Comm. Math. Helv., 65 (1990), 181–197.

    Article  MATH  MathSciNet  Google Scholar 

  54. J. Stevens, Improvements of Non-isolated Surface Singularities, J. London Soc. (2), 39 (1989), 129–144.

    Article  MATH  MathSciNet  Google Scholar 

  55. J. Stevens, On the Versal Deformation of Cyclic Quotient Singularities, in: “Singularity Theory and Applications“, Warwick 1989, Vol. I, D. Mond, J. Montaldi (eds.), SLNM 1426, Springer, Berlin, (1991).

    Google Scholar 

  56. J. Stevens, Partial Resolutions of Rational Quadruple Points, Indian J. of Math., (1991).

    Google Scholar 

  57. J. Stevens, The Versal Deformation of Universal Curve Singularities, ESP-preprint no. 5.

    Google Scholar 

  58. D. van Straten, Weakly Normal Surface Singularities and their Improvements, Thesis, Leiden, (1987).

    Google Scholar 

  59. G. Tjurina, Locally semi-universal flat deformations of isolated singularities of complex spaces, Math. USSR Izvestia, 3(5) (1969), 967–999.

    Article  Google Scholar 

  60. G. Tjurina, Absolute isolatedness of rational singularities and triple rational points, Func. Anal. Appl., 2 (1968), 324–332.

    Article  MathSciNet  Google Scholar 

  61. G. Tjurina, Resolutions of singularities of plane deformations of double rational points, Func. Anal. Appl., 4 (1970), 68–73.

    Article  MathSciNet  Google Scholar 

  62. J. Wahl, Equisingular deformations of normal surface singularities 1, Ann. Math., 104 (1976), 325–356.

    Article  MATH  MathSciNet  Google Scholar 

  63. J. Wahl, Equations defining Rational Surface Singularities, Ann. Sci. Ec. Nor. Sup., 4e série, t. 10 (1977), 231–264.

    MATH  MathSciNet  Google Scholar 

  64. J. Wahl, Simultaneous Resolution of Rational Singularities, Comp. Math., 38 (1979), 43–54.

    MATH  MathSciNet  Google Scholar 

  65. J. Wahl, Simultaneous Resolution and Discriminant Loci, Duke Math. J., 46(2) (1979), 341–375.

    Article  MATH  MathSciNet  Google Scholar 

  66. J. Wahl, Elliptic deformations of Minimally elliptic Singularities, Math. Ann., 253 (1980), 241–262.

    Article  MATH  MathSciNet  Google Scholar 

  67. J. Wahl, Smoothings of Normal Surface Singularities, Topology, 20 (1981), 219–246.

    Article  MATH  MathSciNet  Google Scholar 

  68. I. Yomdin, Complex Surfaces with a 1-dimensional Singular Locus, Siberian Math. J., 15(5) (1974), 1061–1082.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Physik, Mathematik und Informatik, Johannes Gutenberg-Universität, 55099, Mainz, Germany

    Duco Van Straten

Authors
  1. Duco Van Straten
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    András Némethi  & ágnes Szilárd  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2013 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

Van Straten, D. (2013). Tree Singularities: Limits, Series and Stability. In: Némethi, A., Szilárd, á. (eds) Deformations of Surface Singularities. Bolyai Society Mathematical Studies, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39131-6_7

Download citation

Publish with us

Policies and ethics

Search

Navigation