All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold at all? Or do I need to change hypothesis?

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isolated, the general line through one of them will not contain any of the others, so you obtain again a rational paramatrization. The (finite number of )lines containing two double points are entirely contained in the hypersurface, since they have at least four intersection with it (counted with multiplicity). – Francesco Polizzi Sep 28 '12 at 11:59