The geometric field of linearity of linear sets

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