Special Points of Surfaces in the Three-Dimensional Projective Space

In the paper, $3$-jets of two-dimensional surfaces in the three-dimensional affine space are classified. It is shown that there are exactly $22$ types of co-oriented $3$-jets of surfaces. The action of the group of affine transformations on the space of $3$-jets is studied. We calculate a universal complex of singularities that is related to the orbits of the group action. Two linear homology relations for the numbers of special elliptic, hyperbolic, and parabolic points of a compact two-dimensional surface embedded in $\mathbb{R}^3$ are indicated. The stratification of some real cubic surfaces with respect to the types of $3$-jets is described.