The contact of a cubic surface with an analytic surface

Abstract

The projective differential geometry of a surface in the neighborhood of one of its points has been enriched by the consideration of various quadrics covariant to the surface. Among these are the quadrics of Darboux, the quadric of Lie, and the canonical quadric of Wilczynski. All of these are members of the three-parameter family of quadrics having contact of the second order with the surface at the point considered. There is no nonsingular quadric having contact of the third order at an ordinary point of an unrestricted surface. In fact, if there exists a non-singular quadric having contact of the third order, at a general point of a surface, then the surface itself is a quadric, and there is a, pencil of quadrics having contact of the third order at each point. All of these pencils contain, of course, the surface itself. It is the purpose of this paper to investigate the contact of a cubic surface with an analytic surface and to determine necessary and sufficient conditions that a surface be a cubic. Since a cubic is determined by nineteen points, and since it is necessary to impose (n+1)(n+2)/2 conditions to make an algebraic surface have contact of order n with an analytic surface, it follows that there is a four-parameter family of cubics having contact of order four at a point of an analytic surface. There is no non-composite cubic with contact of order five at a general point of a surface, unless the surface is restricted to be itself a cubic. There is a pencil of cubics with contact of the fifth order at each point of a cubic surface. These remarks suffice to indicate the trend of the following investigation.