On the maximum field of linearity of linear sets
Bence Csajbók, Giuseppe Marino, Valentina Pepe · Bulletin of the London Mathematical Society · 2024
Abstract Let denote an ‐dimensional ‐vector space. For an ‐dimensional ‐subspace of , assume that for each nonzero vector . If , then we prove the existence of an integer such that the set of one‐dimensional ‐subspaces generated by nonzero vectors of is the same as the set of one‐dimensional ‐subspaces generated by nonzero vectors of . If we view as a point set of , it means that and determine the same set of directions. We prove a stronger statement when . In terms of linear sets, it means that an ‐linear set of has maximum field of linearity only if it has a point of weight one. We also present some consequences regarding the size of a linear set.