Characterising ovoidal cones by their intersection numbers

2024/02/08 by Bart De Bruyn, De Bruyn, Bart, Geertrui Van de Voorde +1
Engineering · Computer Science · Mathematics · #graph theory and CDMA systems #Graph Labeling and Dimension Problems #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2402.05666

Abstract

In this paper, we characterise ovoidal cones by their intersection numbers. We first show that a set of points of PG(4,q) which intersects planes in 1, q+1 or 2q+1 points is either an ovoidal cone or a parabolic quadric, unless q=3, in which case also a sporadic example with automorphism group M₁₁ exists. We then show that a set of points of PG(4,q) which blocks all planes and intersects solids in q+1, q²+1 or q²+q+1 points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted.

Related

• The characterization of cones as pointsets with 3 intersection numbers
• A characterization of the planes meeting a hyperbolic quadric of \PG(3,q) in a conic
• Flocks of Cones: Herds and Herd Spaces
• A family of m-ovoids of parabolic quadrics
• Sets of points which project to complete intersections