The geometric field of linearity of linear sets

Abstract

AbstractIf an $${\mathbb {F}}_q$$ F q -linear set $$L_U$$ L U in a projective space is defined by a vector subspace U which is linear over a proper superfield of $${\mathbb {F}}_{q}$$ F q , then all of its points have weight at least 2. It is known that the converse of this statement holds for linear sets of rank h in $$\mathrm {PG}(1,q^h)$$ PG ( 1 , q h ) but for linear sets of rank $$k<h$$ k < h the converse of this statement is in general no longer true. The first part of this paper studies the relation between the weights of points and the size of a linear set, and introduces the concept of the geometric field of linearity of a linear set. This notion will allow us to show the main theorem, stating that for particular linear sets without points of weight 1, the converse of the above statement still holds as long as we take the geometric field of linearity into account.