2024/08/05 by Sira Busch, Hendrik Van Maldeghem, Busch, Sira +1 · 1 citation
Mathematics · #51A50 (Secondary) #51E24 (Primary) 51A05 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2408.02413
openalex publication_date 2024/08/05 · openalex created_date 2024/10/14 · openalex updated_date 2026/07/28
We consider any classical Grassmannian geometry Γ; that is, any projective or polar Grassmann space. Suppose every line in Γ contains s+1 points. Then we classify all sets of points in Γ of cardinality s+1, with the property, that no object of opposite type in the corresponding building, is opposite every point of the set. It turns out that such sets are either lines, or hyperbolic lines in symplectic residues, or ovoids in large symplectic subquadrangles of rank 2 residues in characteristic 2. This is a far-reaching extension of a famous and fundamental result of Bose & Burton from the 1960s. We describe a new way to classify geometric lines in finite classical geometries and how our results correspond to blocking sets.