Cubic surfaces in AG(3, q) and projective planes of order q 3

Spread sets of projective planes of order q³ are represented as sets of q³ points in A AG(3, q³). A line through the origin in A can be interpreted as a space A₀ AG(3, q), and the spread set induces a cubic surface L in A₀. If the projective plane is a semifield plane of dimension 3 over its kernel, then L has the property that it misses a plane of A₀. Determining all such surfaces L leads to a complete classification of the semifield planes of order q³, whose spread sets are division algebras of dimension 3.An alternative proof of a result due to Menichetti, that finite division algebras of dimension 3 are associative or are twisted fields, follows with the classification.