The author considers the action of the small affine group \(Aff_7\) (the seven-dimensional group of orientation-preserving linear transformations of the three-dimensional affine space that preserve a fixed half-space) on the small space of 3-jets \(J_3\) of all 3-jets of surfaces tangent to the plane that defines the above half-space at a fixed point. Under this action of the group \(Aff_7\), the space \(J_3\) is decomposed into 22 orbits. Hence a 3-jet of a co-oriented surface lying in the three-dimensional affine space can be reduced to one of 22 forms by an affine transformation and thus the author distinguishes 22 types of points on a surface embedded in the space and a natural stratification is obtained. This stratification refines the well-known classification of surface points into four types given by the second fundamental form (elliptic, hyperbolic, parabolic and degenerate points).
The author studies the complex associated to the above stratification and establishes two linear homology relations. These results are used to obtain geometrical properties of generic cubic surfaces, in the three-dimensional projective space, diffeomorphic to the projective plane. The paper finishes with an interesting conjecture on the existence of a minimal number of these special points on a cubic surface.