Minimal Intransigent Hypersurfaces

  • Andrew A. du Plessis
Conference paper
Part of the Trends in Mathematics book series (TM)


We give examples of hypersurfaces of degree d in P n(ℂ), whose singularities are not versally deformed by the family H d(n) of all hypersurfaces of degree d in P n(ℂ), and which are of minimal codimension with this property.

In the three cases (n, d) = (2, 6), (3, 4) and (5, 3), such hypersurfaces necessarily have one-parameter symmetry. We list the possibilities. The singularities of these hypersurfaces are not all simple, and they are simultaneously topologically versally deformed by H d(n).

In less degenerate cases the examples we give are hypersurfaces with only simple singularities. The failure of versality can be expected to show itself in the geometry of H d(n), either because the μ-constant stratum S containing the hypersurface is of codimension less than μ in H d(n), or because S is not smooth. We will see elsewhere that this is the case for the examples we consider here. In particular, the singularities of these hypersurfaces are not topologically versally deformed by H d(n).


Polynomial Vector Generic Choice Simple Singularity Gorenstein Algebra Cusp Singularity 
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  1. [1]
    Arnold, V.I., S.M. Gusein-Zade and A.N. Varchenko, Singularities of differentiable maps II, Monographs in Mathematics 83, Birkhäuser, 1985.Google Scholar
  2. [2]
    Goryunov, V.V., Symmetric quartics with many nodes, pp. 147–161 in Singularities and curves, ed. V.I. Arnol’d, Advances in Soviet Mathematics 21, AMS, Providence, RI, 1994.Google Scholar
  3. [3]
    Greuel, G.-M., and U. Karras, Families of varieties with prescribed singularities, Compositio Math. 69 (1989), 83–110.MathSciNetMATHGoogle Scholar
  4. [4]
    Greuel, G.-M., C. Lossen and E. Shustin, New asymptotics in the geometry of equisingular families of curves, International Mathematics Research Notices 13 (1997), 595–611.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Luengo, I., On the existence of complete families of projective plane curves, which are obstructed, J. London Math. Soc. (2) 36 (1987), 33–43.MathSciNetMATHGoogle Scholar
  6. [6]
    du Plessis, A. A., Versality properties of projective hypersurfaces, this volume.Google Scholar
  7. [7]
    du Plessis, A.A., and C.T.C. Wall, The geometry of topological stability, London Math. Soc. monographs new series 9, Oxford University Press, 1995.Google Scholar
  8. [8]
    du Plessis, A.A. and C.T.C. Wall, Curves in P 2(ℂ) with one-dimensional symmetry, Revista Mat. Complutense 12 (1999), 117–132.MATHGoogle Scholar
  9. [9]
    du Plessis, A.A. and C.T.C. Wall, Singular hypersurfaces, versality, and Gorenstein algebras, J. Alg. Geom. 9 (2000), 309–322.MATHGoogle Scholar
  10. [10]
    du Plessis, A.A. and C.T.C. Wall, Hypersurfaces in P n(ℂ) with 1-parameter symmetry groups, Proc. Roy. Soc. Lond. A 456 (2000), 2515–2541.CrossRefMATHGoogle Scholar
  11. [11]
    Segre, B., Esistenza e dimensione di sistemi continui di curve piane algebriche con dati caraterri, Atti. Acad. naz. Lincei Rend. (ser. 6) 10 (1929), 31–38.Google Scholar
  12. [12]
    Severi, F., Vorlesung über algebraische Geometrie, Teubner, 1921, and Johnson, 1968.Google Scholar
  13. [13]
    Shustin, E., Smoothness of equisingular families of plane algebraic curves, International Math. Research Notices 2 (1997), 67–82.CrossRefMathSciNetGoogle Scholar
  14. [14]
    van Straten, D., A quintic hypersurface in P 4 with 130 nodes, Topology 32 (1993), 857–864.CrossRefMathSciNetMATHGoogle Scholar
  15. [15]
    Teissier, B. The hunting of invariants in the geometry of discriminants, pp. 565–678 in Real and complex singularities, ed. P. Holm, Sijthoff & Noordhoff International Publishers, 1977.Google Scholar
  16. [16]
    Wall, C.T.C., Notes on trimodal singularities, Manuscripta Math. 100 (1999), 131–157.CrossRefMathSciNetMATHGoogle Scholar
  17. [17]
    Wirthmüller, K., Universell topologisch triviale Deformationen, doctoral thesis, Universit ät Regensburg, 1979.Google Scholar
  18. [18]
    Zariski, O., Algebraic Surfaces (2nd ed.), Springer Verlag, 1971.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Andrew A. du Plessis
    • 1
  1. 1.Department of Mathematical SciencesUniversity of AarhusAarhus CDenmark

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