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?

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?
ag.algebraicgeometry arithmeticgeometry birationalgeometry rationalfunctions projectivegeometry


