A combinatorial characterization of the Baer and the unital cone in $$PG(3,q^2)$$

The Characterization of Cones as Pointsets with 3 Intersection Numbers

“…In other words it is a set of √ q 3 + 1 points of type (1, √ q + 1) 1 in PG(2, q). The following result was also proved in [19].…”

“…For n = 1 and 2 it is called a Baer subline and a Baer subplane respectively. The following result was proved in [19].…”

“…Theorem 1. [19,Theorem 1.1. ] In PG(3, q), with q = p h a square prime power, a 2-blocking set of type (q + 1, q + √ q + 1, √ q 3 + q + 1) 2 is a Baer cone.…”

On Baer cones in $$\mathrm {PG}(3, q)$$

“…Let π and V / ∈ π be a plane and a point of PG(3, q 2 ), respectively, and let B be a Baer subplane of π. A Baer cone of PG(3, q 2 ) with vertex V and base B is the set of points of the union of the lines through V and any point of B [3]. A Baer cone C has non-empty intersection with the lines of the projective space and if α is a plane of PG(3, q 2 ) then |α ∩ C| ∈ {q 2 + 1, q 2 + q + 1, q 3 + q 2 + 1}.…”

“…In [3] one may find a characterization of Baer cones in terms of the sizes of their intersection with the planes of the projective space. To state this result, we need to recall the following definition.…”