Real and Complex Singularities pp 299-310 | Cite as

# Minimal Intransigent Hypersurfaces

## Abstract

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*).

## Keywords

Polynomial Vector Generic Choice Simple Singularity Gorenstein Algebra Cusp Singularity## Preview

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## References

- [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]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]Greuel, G.-M., and U. Karras, Families of varieties with prescribed singularities, Compositio Math.
**69**(1989), 83–110.MathSciNetMATHGoogle Scholar - [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]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]du Plessis, A. A., Versality properties of projective hypersurfaces,
*this volume*.Google Scholar - [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]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]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]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]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]Severi, F.,
*Vorlesung über algebraische Geometrie*, Teubner, 1921, and Johnson, 1968.Google Scholar - [13]Shustin, E., Smoothness of equisingular families of plane algebraic curves, International Math. Research Notices
**2**(1997), 67–82.CrossRefMathSciNetGoogle Scholar - [14]van Straten, D., A quintic hypersurface in
*P*^{4}with 130 nodes, Topology**32**(1993), 857–864.CrossRefMathSciNetMATHGoogle Scholar - [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]Wall, C.T.C., Notes on trimodal singularities, Manuscripta Math.
**100**(1999), 131–157.CrossRefMathSciNetMATHGoogle Scholar - [17]Wirthmüller, K.,
*Universell topologisch triviale Deformationen*, doctoral thesis, Universit ät Regensburg, 1979.Google Scholar - [18]Zariski, O.,
*Algebraic Surfaces*(2nd ed.), Springer Verlag, 1971.Google Scholar