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Lines, surfaces and duality

Published online by Cambridge University Press:  24 October 2008

J. W. Bruce
J. W. Bruce
Affiliation:
The Department of Pure Mathematics, The University, PO Box 147, Liverpool, L69 3BX
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Extract

In the paper [12] Shcherbak studied some duality properties of projective curves and applied them to obtain information concerning central projections of surfaces in projective three space. He also states some interesting results relating the contact of a generic surface with lines and the contact of its dual with lines in the dual space. In this paper we extend this duality to cover non-generic surfaces. Our proof is geometric, and uses deformation theory. The basic idea is the following. Given a surface X in projective 3-space we can consider the lines tangent to X, and measure their contact. The points on the surface with a line yielding at least 4-point contact are classically known as the flecnodal. (The reason is that the tangent plane meets the surface in a nodal curve, one branch of which has an inflexion at the point in question; see Proposition 7 below. The line in question is the inflexional tangent, which is clearly asymptotic.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 1 , July 1992 , pp. 53 - 61
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Arnold, V. I.. Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singularities of projections of smooth surfaces. Russian Math. Surveys 34 (1979), 1–2.CrossRefGoogle Scholar
[2]Banchoff, T., Gaffney, T. J. and McCrory, C.. Cusps of Gauss Mappings (Pitman, 1982).Google Scholar
[3]Bruce, J. W.. The duals of generic hypersurfaces. Math. Scand. 49 (1981), 36–69.CrossRefGoogle Scholar
[4]Bruce, J. W.. On contact of hypersurfaces. Bull. London Math. Soc. 13 (1981), 51–54.CrossRefGoogle Scholar
[5]Bruce, J. W.. Envelopes duality and contact structures. Proc. Sympos. Pure Math. 40 (1983), 195–202.CrossRefGoogle Scholar
[6]Bruce, J. W.. Geometry of singular sets. Math. Proc. Cambridge Philos. Soc. 106 (1989), 495–509.CrossRefGoogle Scholar
[7]Bruce, J. W. and Romero-Fuster, M. C.. Duality and projections of curves and surfaces in 3-space. Quart. J. Math. Oxford Ser. (2), 42 (1991), 433–441.CrossRefGoogle Scholar
[8]Bruce, J. W.. Generic geometry, transversality and projections. Preprint (1991).Google Scholar
[9]Gaffney, T. J.. The structure of , classification and an application to differential geometry. Proc. Sympos. Pure Math. 40 (1983), 409–428.CrossRefGoogle Scholar
[10]Montaldi, J. A.. On contact between submanifolds. Michigan Math. J. 33 (1986), 195–199.CrossRefGoogle Scholar
[11]Montaldi, J. A.. On generic composites of mappings. Bull. London Math. Soc. 23 (1991), 81–85.CrossRefGoogle Scholar
[12]Shcherbak, O. P.. Projectively dual space curves and Legendre singularities. Trudy Tbiliss. Univ. 232–233 (1982), 280–336.Google Scholar
[13]Wall, C. T. C., finite determinacy of smooth map germs. Bull. London Math. Soc. 13 (1981), 481–539.CrossRefGoogle Scholar
[14]Wall, C. T. C.. Geometric properties of generic differentiable manifolds. In Geometry and Topology, Rio de Janeiro 1976, Lecture Notes in Math. vol. 597 (Springer-Verlag, 1977), pp. 707–774.Google Scholar