Finite semifields with a large nucleus and higher secant varieties to Segre varieties

Abstract

In [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007] a geometric construction was given of a finite semifield from a certain configuration of two subspaces with respect to a Desarguesian spread in a finite-dimensional vector space over a finite field. Moreover, it was proved that any finite semifield can be obtained in this way. In [Lavrauw, Finite Fields Appl. 14: 897–910, 2008] we proved that the configuration needed for the geometric construction given in [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007] for finite semifields is equivalent with an (n – 1)-dimensional subspace skew to a determinantal hypersurface in PG(n² – 1, q), and provided an answer to the isotopism problem in [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007]. In this paper we give a generalisation of the BEL-construction using linear sets, and then concentrate on this configuration and the isotopism problem for semifields with nuclei that are larger than its centre.