On Fq-linear sets of PG(3,q3) 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,qn) 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 q4 with kernel Fq2 and center Fq, 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 Fi, i=0,…,5 (F3 might be empty), according to the different configurations of the associated linear sets of PG(3,q3). Also, we prove that to any semifield belonging to the family F5 is associated an Fq-pseudoregulus of PG(3,q3) and we characterize the known examples of semifields of the family F5 in terms of the associated Fq-pseudoregulus.