Tree Singularities: Limits, Series and Stability

  • Duco Van Straten
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 23)


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.


Base Space Surface Singularity Deformation Theory Tree Singularity Singular Locus 
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  1. [1]
    M. Andrée, Homologie des Algèbres Commutatives, Grundlehren der math. Wissenschaften 206, Springer, Berlin (1976).Google Scholar
  2. [2]
    J. Arndt, Verselle deformationen zyklischer Quotientensingularitäten, Dissertation, Universität Hamburg, 1988.MATHGoogle Scholar
  3. [3]
    V. Arnol’d, S. Guzein-Zade, A. Varchenko, Singularities of Differentiable Maps, Vol. I & II, Birkhäuser, Boston (1988).CrossRefMATHGoogle Scholar
  4. [4]
    M. Artin, On isolated rational singularities of surfaces, Am. J. of Math., 88 (1966), 129–136.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    M. Artin, On the Solutions of Analytic Equations, Inv. Math., 5 (1968), 277–291.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    M. Artin, Algebraic construction of Brieskorns resolutions, J. of Alg., 29 (1974), 330–348.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    M. Artin, Deformations of Singularities, Lecture Notes on Math., 54 Tata Institute of Fundamental Research, Bombay (1976).Google Scholar
  8. [8]
    J. Bingener, Modulräume in der analytischen Geometrie 1 & 2, Aspekte der Mathematik, Vieweg, (1987).Google Scholar
  9. [9]
    E. Brieskorn, Rationale Singularitäten komplexer Flächen, Inv. Math., 4 (1967), 336–358.CrossRefMathSciNetGoogle Scholar
  10. [10]
    R.-O. Buchweitz, Contributions à la Théorie des Singularités, Thesis, Université Paris VII, (1981).Google Scholar
  11. [11]
    K. Behnke and J. Christophersen, Hypersurface sections and Obstructions (Rational Surface Singularities), Comp. Math., 77 (1991), 233–258.MATHMathSciNetGoogle Scholar
  12. [12]
    R. Buchweitz and G.-M. Greuel, The Milnor number and deformations of complex curve singularities, Inv. Math., 58 (1980), 241–281.CrossRefMATHMathSciNetGoogle Scholar
  13. [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. [14]
    G. Fisher, Complex Analytic Geometry, SLNM 538, Springer, Berlin, (1976).Google Scholar
  15. [15]
    H. Flenner, Über Deformationen holomorpher Abbildungen, Osnabrücker Schriften zur Mathematik, Reihe P Preprints, Heft, 8 (1979).Google Scholar
  16. [16]
    H. Grauert, Über die Deformationen isolierter Singularitäten analytischer Mengen, Inv. Math., 15 (1972), 171–198.CrossRefMATHMathSciNetGoogle Scholar
  17. [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. [18]
    H. Grauert and R. Remmert, Analytische Stellenalgebren, Grundlehren d. math. Wissens, Bd. 176, Springer, Berlin, (1971).Google Scholar
  19. [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.MATHMathSciNetGoogle Scholar
  20. [20]
    L. Illusie, Complex Cotangente et Déformations 1, 2, SLNM, 239 (1971), SLNM, 289 (1972), Springer, Berlin.Google Scholar
  21. [21]
    T. de Jong and D. van Straten, A Deformation Theory for Non-Isolated Singularities, Abh. Math. Sem. Univ. Hamburg, 60 (1990), 177–208.CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    T. de Jong and D. van Straten, Deformations of the Normalization of Hypersurfaces, Math. Ann., 288 (1990), 527–547.CrossRefMATHMathSciNetGoogle Scholar
  23. [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.CrossRefMATHMathSciNetGoogle Scholar
  24. [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. [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.MATHGoogle Scholar
  26. [26]
    T. de Jong and D. van Straten, Deformation Theory of Sandwiched Singularities, Duke Math. J., Vol 95, No. 3, (1998), 451–522.CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    B. Kaup, Über Kokerne und Pushouts in der Kategorie der komplexanalytischen Räume, Math. Ann., 189 (1970), 60–76.CrossRefMATHMathSciNetGoogle Scholar
  28. [28]
    J. Kollár and N. Shepherd-Barron, Threefolds and deformations of surface singularities, Inv. Math., 91 (1988), 299–338.CrossRefMATHGoogle Scholar
  29. [29]
    A. Laudal, Formal Moduli of Algebraic Structures, SLNM 754, Springer, Berlin, (1979).Google Scholar
  30. [30]
    H. Laufer, On Minimally Elliptic Singularities, Amer. J. Math., 99 (1977), 1257–1295.CrossRefMATHMathSciNetGoogle Scholar
  31. [31]
    H. Laufer, Ambient Deformations for Exceptional Sets in Two-manifolds, Inv. Math., 55 (1979), 1–36.CrossRefMATHMathSciNetGoogle Scholar
  32. [32]
    S. Lichtenbaum and M. Schlesinger, The cotangent Complex of a Morphism, Trans. AMS, 128 (1967), 41–70.CrossRefMATHGoogle Scholar
  33. [33]
    P. Mazet, Analytic Sets in Locally Convex Spaces, Math. Studies 89, North-Holland, (1984).Google Scholar
  34. [34]
    D. Mond, On the classification of germs of maps from2 to3, Proc. London Math. Soc., 50 (1985), 333–369.CrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    V. Palomodov, Moduli and Versal Deformations of Complex Spaces, Soviet Math. Dokl., 17 (1976), 1251–1255.Google Scholar
  36. [36]
    R. Pellikaan, Hypersurface Singularities and Resolutions of Jacobi Modules, Thesis, Rijksuniversiteit Utrecht, (1985).Google Scholar
  37. [37]
    R. Pellikaan, Finite Determinacy of Functions with Non-Isolated Singularities, Proc. London Math. Soc. (3), 57 (1988), 357–382.CrossRefMathSciNetGoogle Scholar
  38. [38]
    R. Pellikaan, Deformations of Hypersurfaces with a One-Dimensional Singular Locus, J. Pure Appl. Algebra, 67 (1990), 49–71.CrossRefMATHMathSciNetGoogle Scholar
  39. [39]
    R. Pellikaan, Series of Isolated Singularities, Contemp. Math., 90 Proc. Iowa, R. Randell (ed.), (1989).Google Scholar
  40. [40]
    R. Pellikaan, On Hypersurfaces that are Sterns, Comp. Math., 71 (1989), 229–240.MATHMathSciNetGoogle Scholar
  41. [41]
    H. Pinkham, Deformations of algebraic Varieties with G m-action, Astérisque, 20 1974.Google Scholar
  42. [42]
    O. Riemenschneider, Deformations of rational singularities and their resolutions, in: Complex Analysis, Rice University Press 59 (1), 1973, 119–130.MathSciNetGoogle Scholar
  43. [43]
    O. Riemenschneider, Deformationen von Quotientensingularitten (nach zyklischen Gruppen), Math. Ann., 209 (1974), 211–248.CrossRefMATHMathSciNetGoogle Scholar
  44. [44]
    D. S. Rim, Formal deformation theory, SGA 7(1), Exp. VI, Springer, Berlin, (1972).Google Scholar
  45. [45]
    G. Ruget, Déformations des germes d’espaces analytiques, Sém. Douady-Verdier 1971–72, Astérisque, 16 (1974), 63–81.MATHMathSciNetGoogle Scholar
  46. [46]
    M. Schlessinger, Functors of Artin Rings, Trans. Am. Math. Soc., 130 (1968), 208–222.Google Scholar
  47. [47]
    M. Schlessinger, Rigidity of quotient singularities, Inv. Math., 14 (1971), 17–26.CrossRefMATHMathSciNetGoogle Scholar
  48. [48]
    M. Schlessinger, On Rigid Singularities, in: Complex Analysis, Rice University Press 59(1), 1973.Google Scholar
  49. [49]
    R. Schrauwen, Series of Singularities and their Topology, Thesis, Utrecht, (1991).Google Scholar
  50. [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. [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. [52]
    D. Siersma, Singularities with Critical Locus a One-Dimensional Complete Intersection and transverse A 1, Topology and its Applications, 27, 51–73 (1987).CrossRefMATHMathSciNetGoogle Scholar
  53. [53]
    D. Siersma, The Monodromy of a Series of Singularities, Comm. Math. Helv., 65 (1990), 181–197.CrossRefMATHMathSciNetGoogle Scholar
  54. [54]
    J. Stevens, Improvements of Non-isolated Surface Singularities, J. London Soc. (2), 39 (1989), 129–144.CrossRefMATHMathSciNetGoogle Scholar
  55. [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. [56]
    J. Stevens, Partial Resolutions of Rational Quadruple Points, Indian J. of Math., (1991).Google Scholar
  57. [57]
    J. Stevens, The Versal Deformation of Universal Curve Singularities, ESP-preprint no. 5.Google Scholar
  58. [58]
    D. van Straten, Weakly Normal Surface Singularities and their Improvements, Thesis, Leiden, (1987).Google Scholar
  59. [59]
    G. Tjurina, Locally semi-universal flat deformations of isolated singularities of complex spaces, Math. USSR Izvestia, 3(5) (1969), 967–999.CrossRefGoogle Scholar
  60. [60]
    G. Tjurina, Absolute isolatedness of rational singularities and triple rational points, Func. Anal. Appl., 2 (1968), 324–332.CrossRefMathSciNetGoogle Scholar
  61. [61]
    G. Tjurina, Resolutions of singularities of plane deformations of double rational points, Func. Anal. Appl., 4 (1970), 68–73.CrossRefMathSciNetGoogle Scholar
  62. [62]
    J. Wahl, Equisingular deformations of normal surface singularities 1, Ann. Math., 104 (1976), 325–356.CrossRefMATHMathSciNetGoogle Scholar
  63. [63]
    J. Wahl, Equations defining Rational Surface Singularities, Ann. Sci. Ec. Nor. Sup., 4e série, t. 10 (1977), 231–264.MATHMathSciNetGoogle Scholar
  64. [64]
    J. Wahl, Simultaneous Resolution of Rational Singularities, Comp. Math., 38 (1979), 43–54.MATHMathSciNetGoogle Scholar
  65. [65]
    J. Wahl, Simultaneous Resolution and Discriminant Loci, Duke Math. J., 46(2) (1979), 341–375.CrossRefMATHMathSciNetGoogle Scholar
  66. [66]
    J. Wahl, Elliptic deformations of Minimally elliptic Singularities, Math. Ann., 253 (1980), 241–262.CrossRefMATHMathSciNetGoogle Scholar
  67. [67]
    J. Wahl, Smoothings of Normal Surface Singularities, Topology, 20 (1981), 219–246.CrossRefMATHMathSciNetGoogle Scholar
  68. [68]
    I. Yomdin, Complex Surfaces with a 1-dimensional Singular Locus, Siberian Math. J., 15(5) (1974), 1061–1082.MathSciNetGoogle Scholar

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© János Bolyai Mathematical Society and Springer-Verlag 2013

Authors and Affiliations

  • Duco Van Straten
    • 1
  1. 1.Fachbereich Physik, Mathematik und InformatikJohannes Gutenberg-UniversitätMainzGermany

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