On Affine Homogeneous Real Hypersurfaces of General position in $\Bbb C^3$

The article develops a coefficients approach to the problem of the description of affine homogeneous real hypersurfaces of 3-dimensional complex space. The connections are studied and used between the coefficients of affine canonical equations for strictly pseudoconvex surfaces of $ \Bbb C^3 $ space and Lie algebras of affine vector fields on such varieties. The previously developed scheme for obtaining a list of all algebras, as well as the Homogeneous manifolds, has shown its effectiveness in four of the seven cases determined by the second-order coefficients of the canonical equations of such surfaces. In the case of surfaces of general position studied in this paper, this scheme leads to the complicated system of quadratic equations. Here the authors build only several families of matrix Lie algebras corresponding to homogeneous manifolds from the class under consideration. Two families of affine homogeneous surfaces are obtained by integrating the constructed Lie algebras. It is shown that some of the obtained “different” families of the Lie algebras lead to the affine-equivalent homogeneous surfaces. It means that the scheme used requires refinement and simplification. Within the framework of the modification of the described scheme, two theorems about the relations between the coefficients of the canonical equations of affine-homogeneous surfaces of general position are proved. Theorem 1 identifies a basic set of Taylor coefficients of the canonical equation of affine-homogeneous surface in general position, uniquely defining any such surface. Under the condition of a nonzero polynomial of the third degree from this equation this set contains, generally speaking, 92 real coefficient (of the second, third and fourth degrees). A number of statements have been received decreasing this rough estimate of the dimension of the moduli space for the family of homogeneous surfaces under consideration. Four types of polynomials of the third degree are distinguished, for each of which the specific simplifications are possible of the canonical equations as well as the whole homogeneity problem. In Theorem 2 the absence of homogeneous surfaces is proved in one of these four cases. The proof is carried out by means of sufficiently large symbolic computations, implemented in the Maple package.