Parametrization of the orbits of cubic surfaces

The aim of the paper is the study of the orbits of the action of PGL₄ on the space ³ of the cubic surfaces of ³, i.e., the classification of cubic surfaces up to projective motions. A varietyQ¹⁹ is explicitely constructed as the union of 22 disjoint irreducible components which are either points or open subsets of linear spaces. More precisely, each orbit of the above action intersects one componentX ofQ in a finite number of points and the action of PGL₄ restricted on each componentX is equivalent to the action of a finite groupG _X onX which can be explicitely computed. Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms.The results are obtained with computational techniques and with the aid of some computer algebra systems like CoCoA, Macaulay and Maple.Partially supported by MURSTPartially supported by MURST and CNR