On the isotopism classes of finite semifields

A projective plane is called a translation plane if there exists a line $L$ such that the group of elations with axis $L$ acts transitively on the points not on $L$. A translation plane whose dual plane is also a translation plane is called a {\it semifield plane}. The ternary ring corresponding to a semifield plane can be made into a non-associative algebra called a {\it semifield}, and two semifield planes are isomorphic if and only if the corresponding semifields are {\it isotopic}. In [Ball et al., A geometric construction of finite semifields. J. Algebra, 311 (1): 117--129, 2007] it was shown that each finite semifield gives rise to a particular configuration of two subspaces with respect to a Desarguesian spread, called a {\it BEL-configuration}, and vice versa that each BEL-configuration gives rise to a semifield. In this manuscript we investigate the question when two BEL-configurations determine isotopic semifields. We show that there is a one-to-one correspondence between the isotopism classes of finite semifields and the orbits of the action a subgroup of index two of the automorphism group of a Segre variety on subspaces of maximum dimension skew to a determinantal hypersurface.