On the isotopism classes of finite semifields

Abstract

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 semifield plane. The ternary ring corresponding to a semifield plane can be made into a non-associative algebra called a semifield, and two semifield planes are isomorphic if and only if the corresponding semifields are isotopic. In [S. Ball, G. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (1) (2007) 117–129] it was shown that each finite semifield gives rise to a particular configuration of two subspaces with respect to a Desarguesian spread, called a 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.