Geodesic
Invariants of cubic forms in four variables
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (1982), pp. 42-49 
N. D. Beklemishev. Invariants of cubic forms in four variables. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (1982), pp. 42-49. http://geodesic.mathdoc.fr/item/VMUMM_1982_2_a10/
@article{VMUMM_1982_2_a10,
     author = {N. D. Beklemishev},
     title = {Invariants of cubic forms in four variables},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {42--49},
     year = {1982},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1982_2_a10/}
}
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We give a new proof of the completeness of the classical system of invariants of cubic forms in four variables. We prove that the orbit-space of the space of cubic forms in four variables with respect to the action of $\mathrm{SL}(4)$ is a hypersurface and that the projectivization of the factor with respect to the natural action of $\mathbf{C}^*$ is a rational variety.