Zero-Dimensional Schemes for Singularities

  • Gert-Martin GreuelEmail author
  • Christoph Lossen
  • Eugenii Shustin
Part of the Springer Monographs in Mathematics book series (SMM)


This section is devoted to the study of zero-dimensional schemes in a smooth projective surface \(\varSigma \), associated to and concentrated in the (finite) set of singular points of a reduced curve C on \(\varSigma \). In this chapter, a curve (singularity) we will always mean a reduced curve (singularity), unless we explicitly say the opposite. We introduce the notion of cluster schemes (cf. Sect. 1.1.2) which allows us to encode the topological type of the singularities of C and, in Sect. 1.1.4, a class of zero-dimensional schemes encoding the analytic type of the singularities.


  1. [GLS6]
    Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer, Berlin (2007)Google Scholar
  2. [EnC]
    Enriques, F., Chisini, O.: Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. Libro IV (1915)Google Scholar
  3. [Wae1]
    van der Waerden, B.L.: Infinitely near points. Ind. Math. 12, 401–410 (1950)zbMATHGoogle Scholar
  4. [Cas2]
    Casas-Alvero, E.: Singularities of Plane Curves. LMS, vol. 276. Cambridge University Press, Cambridge (2000)Google Scholar
  5. [BHPV]
    Barth, W., Hulek, K., Peters, C., van de Ven, A.: Compact Complex Surfaces. Springer, Berlin (1984)Google Scholar
  6. [HaR2]
    Hartshorne, R.: Algebraic Geometry. Graduate Text in Mathematics, vol. 52. Springer, Berlin (1977)CrossRefGoogle Scholar
  7. [Dra]
    Draper, R.N.: Intersection theory in analytic geometry. Math. Ann. 180, 175–204 (1969)MathSciNetCrossRefGoogle Scholar
  8. [Cas1]
    Casas-Alvero, E.: Infinitely near imposed singularities and singularities of polar curves. Math. Ann. 287, 429–454 (1990)MathSciNetCrossRefGoogle Scholar
  9. [CGSL]
    Campillo, A., Gonzalez-Sprinberg, G., Lejeune-Jalabert, M.: Clusters of infinitely near points. Math. Ann. 306, 169–194 (1996)MathSciNetCrossRefGoogle Scholar
  10. [Lip]
    Lipman, J.: Proximity inequalities for complete ideals in regular two-dimensional local rings. Contemp. Math. 159, 293–306 (1994)MathSciNetCrossRefGoogle Scholar
  11. [NoV]
    Nobile, A., Villamayor, O.E.: Equisingular stratifications associated to families of planar ideals. J. Algebr. 193(1), 239–259 (1997)MathSciNetCrossRefGoogle Scholar
  12. [Bra]
    Brauner, K.: Zur Geometrie der Funktionen zweier Veränderlicher. Abh. Math. Sem. Hamburg 6, 8–54 (1928)CrossRefGoogle Scholar
  13. [Zar3]
    Zariski, O.: Algebraic Surfaces, 2nd edn. Springer, Berlin (1971)zbMATHGoogle Scholar
  14. [BrK]
    Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Birkhäuser, Basel (1986)CrossRefGoogle Scholar
  15. [Zar1]
    Zariski, O.: Polynomial ideals defined by infinitely near base points. Am. J. Math. 60, 151–204 (1938)MathSciNetCrossRefGoogle Scholar
  16. [GrP]
    Greuel, G.-M. Pfister, G.: A Singular Introduction to Commutative Algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schnemann. 2nd extended ed. Springer, Berlin (2007)Google Scholar
  17. [DeL]
    Decker, W., Lossen, C.: Computing in Algebraic Geometry. A Quick Start Using Singular. ACM, vol. 16. Springer, Berlin (2006)Google Scholar
  18. [Wah1]
    Wahl, J.: Equisingular deformations of plane algebroid curves. Trans. Am. Math. Soc. 193, 143–170 (1974)MathSciNetCrossRefGoogle Scholar
  19. [Mil]
    Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, Princeton (1968)Google Scholar
  20. [HiH]
    Hironaka, H.: On the arithmetic genera and the effective genera of algebraic curves. Mem. Coll. Sci. Univ. Kyoto 30, 177–195 (1957)MathSciNetCrossRefGoogle Scholar
  21. [HaR1]
    Hartshorne, R.: Connectedness of the Hilbert scheme. Publ. Math. IHES 29, 261–304 (1966)zbMATHGoogle Scholar
  22. [Fog]
    Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math. 90, 511–521 (1968)MathSciNetCrossRefGoogle Scholar
  23. [Iar2]
    Iarrobino, A.: Hilbert scheme of points: overview of last ten years. In: Proceedings of the Summer Research Institute on Algebraic Geometry, Brunswick/Maine 1985, Part 2, Proceedings of Symposia in Pure Mathematics, vol. 46, pp. 297–320 (1987)Google Scholar
  24. [Dou2]
    Douady, A.: Le problème des modules locaux pour les espaces \({\mathbb{C}}\)-analytiques compactes. Ann. Sci. ENS 7, 569–602 (1974)zbMATHGoogle Scholar
  25. [Dou1]
    Douady, A.: Le problème des modules pour les sous-espaces analytiques compacts d’une espace analytique donné. Ann. Inst. Fourier 16, 1–95 (1966)MathSciNetCrossRefGoogle Scholar
  26. [Bin]
    Bingener, J.: Darstellbarkeitskriterien für analytische Funktoren. Ann. Sci. École Norm. Sup. 13, 317–347 (1980)MathSciNetCrossRefGoogle Scholar
  27. [Mum2]
    Mumford, D.: Geometric Invariant Theory. Springer, Berlin (1965)CrossRefGoogle Scholar
  28. [Bri]
    Briançon, J.: Description de Hilb\(^{n} {\mathbb{C}}\{x, y\}\). Invent. Math. 41, 45–89 (1977)MathSciNetCrossRefGoogle Scholar
  29. [Ris1]
    Risler, J.-J.: Sur les déformations équisingulières d’idéaux. Bull. Soc. Math. Fr. 101, 3–16 (1973)CrossRefGoogle Scholar
  30. [Ser2]
    Serre, J.-P.: Algèbre Locale. Multiplicités. SLNM, vol. 11. Springer, Berlin (1965)Google Scholar
  31. [AK]
    Altman, A., Kleiman, S.: Introduction to Grothendieck Duality Theory. Lecture Notes in Mathematics, vol. 146. Springer, Berlin (1970)Google Scholar
  32. [CGL1]
    Campillo, A., Greuel, G.-M., Lossen, C.: Equisingular deformations of plane curves in arbitrary characteristic. Compos. Math. 143, 829–882 (2007)MathSciNetCrossRefGoogle Scholar
  33. [BiF]
    Bingener, J., Flenner, H.: On the fibers of analytic mappings. In: Ancona, V., et al. (eds.) Complex Analysis and Geometry. The University Series in Mathematics, pp. 45–101. Plenum Press, New York (1993)CrossRefGoogle Scholar
  34. [MaY]
    Mather, J.N., Yau, S.-T.: Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69, 243–251 (1982)MathSciNetCrossRefGoogle Scholar
  35. [Lic]
    Lichtin, B.: Estimates and formulae for the \(C^0\) degree of sufficiency of plane curves. In: Singularities, Part 2 (Arcata 1981). Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 155–160 (1983)Google Scholar
  36. [DiH]
    Diaz, S., Harris, J.: Ideals associated to deformations of singular plane curves. Trans. Am. Math. Soc. 309, 433–468 (1988)MathSciNetCrossRefGoogle Scholar
  37. [Shu10]
    Shustin, E.: Smoothness of equisingular families of plane algebraic curves. Int. Math. Res. Not. 2, 67–82 (1997)MathSciNetCrossRefGoogle Scholar
  38. [Sai]
    Saito, K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14, 123–142 (1971)MathSciNetCrossRefGoogle Scholar
  39. [GuS]
    Gudkov, D.A., Shustin, E.: On the intersection of the close algebraic curves. In: Dold, A., Eckmann, B. (eds.) Topology (Leningrad, 1982). SLN, vol. 1060, pp. 278–289 (1984)CrossRefGoogle Scholar
  40. [Iar1]
    Iarrobino, A.: Punctual Hilbert Schemes. Memoirs of the American Mathematical Society, No. 188 (1977)Google Scholar
  41. [KPT]
    Kleiman, S., Piene, R., Tyomkin, I.: Enriques diagrams, arbitrarily near points, and Hilbert schemes. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 22(4), 411–451 (2011)Google Scholar
  42. [Pop]
    Popescu-Pampu, P.: Le cerf-volant dune constellation. Enseign. Math. (2) 57, 303–347 (2011)MathSciNetCrossRefGoogle Scholar
  43. [Zar2]
    Zariski, O.: Studies in equisingularity I–III. Am. J. Math. 87, 507–536; 972–1006 (1965), respectively Am. J. Math. 90, 961–1023 (1968)Google Scholar
  44. [CGL2]
    Campillo, A., Greuel, G.-M., Lossen, C.: Equisingular calculations for plane curve singularities. J. Symb. Comput. 42(1–2), 89–114 (2007)MathSciNetCrossRefGoogle Scholar
  45. [GrK]
    Greuel, G.-M., Karras, U.: Families of varieties with prescribed singularities. Compos. Math. 69, 83–110 (1989)MathSciNetzbMATHGoogle Scholar
  46. [GrL1]
    Greuel, G.-M., Lossen, C.: Equianalytic and equisingular families of curves on surfaces. Manuscr. Math. 91, 323–342 (1996)MathSciNetCrossRefGoogle Scholar
  47. [GLS3]
    Greuel, G.-M., Lossen, C., Shustin, E.: Plane curves of minimal degree with prescribed singularities. Invent. Math. 133, 539–580 (1998)MathSciNetCrossRefGoogle Scholar
  48. [Los]
    Lossen, C.: The geometry of equisingular and equianalytic families of curves on a surface. Ph.D. thesis, Universitt Kaiserslautern (1998)Google Scholar
  49. [Arn]
    Arnold, V.I.: Singularities of smooth mappings. (Russian) Uspehi Mat. Nauk 23(1), 3–44 (1968)Google Scholar
  50. [AGV]
    Arnol’d, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuser, Basel (1985)Google Scholar
  51. [Tou]
    Tougeron, J.C.: Idéaux de fonctions différentiables. Ann. Inst. Fourier (Grenoble) 18(1), 177–240 (1968)MathSciNetCrossRefGoogle Scholar
  52. [Mat]
    Mather, J.N.: Stability of \(C^\infty \) mappings. III. Finitely determined map-germs. Publ. Math. Inst. Hautes Études Sci. 35, 279–308 (1968)Google Scholar
  53. [Gaf]
    Gaffney, T.: A note on the order of determination of a finitely determined germ. Invent. Math. 52(2), 127–130 (1979)MathSciNetCrossRefGoogle Scholar
  54. [Wal]
    Wall, C.T.C.: Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13(6), 481–539 (1981)MathSciNetCrossRefGoogle Scholar
  55. [Dam]
    Damon, J.: The unfolding and determinacy theorems for subgroups of A and K. Mem. Am. Math. Soc. 50(306), x+88 pp. (1984)Google Scholar
  56. [BDW]
    Bruce, J.W., du Plessis, A.A., Wall, C.T.C.: Determinacy and unipotency. Invent. Math. 88(3), 521–554 (1987)MathSciNetCrossRefGoogle Scholar
  57. [Shu4]
    Shustin, E.: On manifolds of singular algebraic curves. Selecta Math. Sov. 10, 27–37 (1991)zbMATHGoogle Scholar
  58. [Shu2]
    Shustin, E.: New M-curve of the 8th degree. Math. Notes Acad. Sci. USSR 42, 606–610 (1987)zbMATHGoogle Scholar
  59. [GLS4]
    Greuel, G.-M., Lossen, C., Shustin, E.: Castelnuovo function, zero-dimensional schemes and singular plane curves. J. Algebr. Geom. 9(4), 663–710 (2000)MathSciNetzbMATHGoogle Scholar
  60. [Los1]
    Lossen, C.: New asymptotics for the existence of plane curves with prescribed singularities. Commun. Algebr. 27, 3263–3282 (1999)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  • Gert-Martin Greuel
    • 1
    Email author
  • Christoph Lossen
    • 1
  • Eugenii Shustin
    • 2
  1. 1.Fachbereich MathematikTU KaiserslauternKaiserslauternGermany
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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