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A stratification of the space of cubic surfaces

Published online by Cambridge University Press:  24 October 2008

J. W. Bruce
J. W. Bruce
Affiliation:
University of Liverpool
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Extract

In (4) the classification of (complex, projective) cubic surfaces by the number and nature of their singularities is carried out. This gives a natural partition of the vector space of cubic surfaces (which we denote by H3(4, 1)). In this paper we investigate the differential geometric properties of this partition; we show that it provides a finite constructible stratification of H3(4,1) which, in the notation of (10), is Whitney (A) regular. In fact Whitney (B) regularity holds over each stratum other than E6, but this stratum of cubic cones has an exceptional (equianharmonic) orbit at which (B) regularity fails. It remains to be seen whether or not this is the only exceptional orbit.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 3 , May 1980 , pp. 427 - 441
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Arnol'd, V. I.Normal forms of functions near degenerate critical points. Russian. Math. Survey 29 (1974), 11–49.CrossRefGoogle Scholar
(2)Brocker, T. and Lander, L. Differentiable germs and catastrophes. London Math. Soc. Lecture Note Series, no. 17 (Cambridge University Press, 1975).Google Scholar
(3)Bruce, J. W.Stratifying jet spaces – some special cases. Math. Proc. Cambridge Philos. Soc. 85 (1979), 439–444.CrossRefGoogle Scholar
(4)Bruce, J. W. and Wall, C. T. C.On the classification of cubic surfaces. J. London Math. Soc. (2), 19 (1979), 245–256.CrossRefGoogle Scholar
(5)Gibson, C. G.Regularity of the Segre stratification. Math. Proc. Cambridge Philos. Soc. 80 (1976), 91–97.CrossRefGoogle Scholar
(6)Gibson, C. G. et al. Topological stability of smooth mappings. Lecture Notes in Maths. no. 552 (Berlin, Springer-Verlag, 1976).Google Scholar
(7)Kas, A. and Schlessinger, M.On the versal deformation of a complex space with an isolated singularity. Math. Ann. 196 (1972), 23–29.CrossRefGoogle Scholar
(8)Looijenga, E.On the semi-universal deformation of a simple-elliptic hypersurface singularity. I. Unimodularity. Topology 16 (1977), 257–262.CrossRefGoogle Scholar
(9)Lyaschko, O. V.Decomposition of simple singularities of functions. Functional Anal. Appl. 10 (1976), 122–127.CrossRefGoogle Scholar
(10)Wall, C. T. C. Regular stratifications. ‘Dynamical Systems – Warwick 1974’. Lecture Notes in Maths, no. 468 (Berlin, Springer-Verlag, 1975).Google Scholar