Lines, surfaces and duality

J.W. Bruce

    Research output: Contribution to journalArticlepeer-review

    2 Citations (Scopus)

    Abstract

    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.)
    Original languageEnglish
    Pages (from-to)53-61
    JournalMathematical Proceedings of the Cambridge Philosophical Society
    Volume112
    Issue number1
    DOIs
    Publication statusPublished - 1992

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