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

In \cite{BEL} 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 \cite{Lavrauw2008} we proved that the configuration needed for the geometric construction given in \cite{BEL} for finite semifields is equivalent with an $(n-1)$-dimensional subspace skew to a determinantal hypersurface in $\PG(n^2-1,q)$, and provided an answer to the isotopism problem in \cite{BEL}. 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 center.