For regular cubic surfaces over any field, a closed point of degree coprime to 3 forces a closed point of degree 1, 4, or 10; for cubic 3-folds and 4-folds, symmetric products are stably birational.
L'\'equivalence rationnelle sur les points ferm\'es des surfaces rationnelles fibr\'ees en coniques
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Closed points on cubic hypersurfaces
For regular cubic surfaces over any field, a closed point of degree coprime to 3 forces a closed point of degree 1, 4, or 10; for cubic 3-folds and 4-folds, symmetric products are stably birational.