The geometric field of linearity of linear sets

TL;DR

This paper investigates the geometric properties of linear sets in projective spaces, introducing the concept of the geometric field of linearity to understand when the weights of points determine the linearity over a superfield.

Contribution

It introduces the notion of the geometric field of linearity and establishes conditions under which the converse relationship between point weights and linearity holds.

Findings

The geometric field of linearity helps characterize linear sets without points of weight 1.

The main theorem links the absence of weight-1 points to the linearity over the geometric field.

Conditions are identified where the converse of the weight-size relation remains valid.

Abstract

If an Fq-linear set LU in a projective space is defined by a vector subspace U which is linear over a proper superfield of Fq, 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 PG(1,q^h) but for linear sets of rank 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.