We give a characterization of Fermat cubic hypersurfaces of dimension greater than 2 in characteristic 2 in terms of the property, called (GMRZ), that a projective variety admits an embedding whose Gauss map is of rank 0. In contrast to the higher dimensional case, for cubic surfaces the above characterization is no longer true. Moreover, we prove that the process of blowing up at points preserves the property(GMRZ), and that every smooth rational surface in fact satisfies (GMRZ)in the characteristic 2 case.
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