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

Abstract

Abstract. In [2] 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 [7] we proved that the configuration needed for the geometric construction given in [2] 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 [2]. 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. Finite semifieldsA finite semifield S is a finite division algebra, which is not necessarily associative, i.e., an algebra with at least two elements, and two binary operations + and •, satisfying the following axioms: (S1) (S, +) is a group with identity element 0;A finite field is of course a trivial example of a semifield. The first non-trivial examples of semifields were constructed by Dickon in [5]. One easily shows that the additive group of a semifield is elementary abelian, and the additive order of the elements of S is called the characteristic of S. Contained in a semifield are the following important substructures, all of which are isomorphic to a finite field. The left nucleus N l (S), the middle nucleus *

On BEL-configurations and finite semifields

“…As pointed out in the proof of Theorem 3, it follows from the BELconstruction (Theorem 2) that S(D r,n,q , U f , W g ) forms a semifield spread, and this was proven geometrically in [3]. It then follows that S f,g is a (pre)semifield, with multiplication defined in (13). However, with the explicit descriptions of the subspaces U f and W g , defined by the r-tuples of endomorphisms f i and g i , we are able to prove directly that the multiplication (13) defines a presemifield.…”

“…The concept of a BEL-configuration was introduced in [3], and further developed in [12] and [13]. In this section we will recall the definition, and how a BEL-configuration can be used to construct of a semifield spread.…”

“…In this section we will recall the definition, and how a BEL-configuration can be used to construct of a semifield spread. We restrict ourselves to BEL-configurations where both subspaces U and W are subspaces as in the original paper [3], and will not consider the generalization [13] to linear sets. We obtain an explicit formula for the pre-semifield multiplication obtained from a BEL-configuration in V (rn, q).…”

Finite semifields and nonsingular tensors

“…The general case. If we don't make any further assumptions on the sizes of the nuclei, then we have the following theorem from [47].…”

“…In this section we concentrate on a geometric construction of finite semifield spreads. The construction we give here is taken from [47], but the main idea is the slightly less general construction given in [9] (where L is a subspace, i.e. t = 1).…”