LINES, SURFACES AND DUALITY

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.) Now the dual surface X* is a subset of the corresponding dual projective space. We wish to consider the contact of this surface with lines also. The main result we establish here is that the contact between the surface X and a tangent line is the same as the contact between the dual surface, at the dual point, and the dual line, namely the pencil of planes whose axis is the initial tangent line. (Contact here is in the usual sense of ‘k-point' contact, i.e. infinitesimally k points in common.) In particular the flecnodal curves of X and X* correspond (one of the results of Shcherbak). Of course for this result to be true we need to avoid parabolic points on the surface where the Gauss curvature vanishes, since the corresponding dual points are non-smooth. © 1992, Cambridge Philosophical Society. All rights reserved.