On -linear sets of and semifields

Any finite semifield 2-dimensional over its left nucleus and 2n-dimensional over its center defines a linear set of rank 2n of PG ( 3 , q n ) disjoint from a hyperbolic quadric and conversely [G. Lunardon, Translation ovoids, J. Geom. 76 (2003) 200–215]. Using this connection, semifields 2-dimensional over their left nucleus and 4-dimensional over their center were classified [I. Cardinali, O. Polverino, R. Trombetti, Semifield planes of order q 4 with kernel F q 2 and center F q , European J. Combin. 27 (2006) 940–961]. In this paper we give a characterization result in the case n = 3 , proving that there exist five or six non-isotopic families of such semifields, the families F i , i = 0 , … , 5 ( F 3 might be empty), according to the different configurations of the associated linear sets of PG ( 3 , q 3 ) . Also, we prove that to any semifield belonging to the family F 5 is associated an F q -pseudoregulus of PG ( 3 , q 3 ) and we characterize the known examples of semifields of the family F 5 in terms of the associated F q -pseudoregulus.