Abstract
The aim of the paper is the study of the orbits of the action of PGL4 on the space ℙ3 of the cubic surfaces of ℙ3, i.e., the classification of cubic surfaces up to projective motions. A varietyQ⊂ℙ19 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.
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Brundu, M., Logar, A. Parametrization of the orbits of cubic surfaces. Transformation Groups 3, 209–239 (1998). https://doi.org/10.1007/BF01236873
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DOI: https://doi.org/10.1007/BF01236873