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A combinatorial characterization of the Baer and the unital cone in \(\mathrm{PG}(3,q^2)\). (English) Zbl 1452.51001
The characterisation of point sets by their intersection numbers is a classical problem in finite geometry.
A set of points \(S\) in \(\mathrm{PG}(n,q)\) is said to be of type \((m_1,m_2,\dots,m_s)_d\) if the size of the intersection of any \(d\)-space with \(S\) is contained in \(\{m_1,m_2,\dots,m_s\}\) and all possibilities \(m_i\) occur as an intersection for some \(d\)-space.
It is a well-known result of A. Bruen [Bull. Am. Math. Soc. 76, 342–344 (1970; Zbl 0207.02601)] and A. A. Bruen and J. A. Thas [Geom. Dedicata 6, 193–203 (1977; Zbl 0367.05009)] that a set of type \((1,q+1)_1\) in \(\mathrm{PG}(2,q^2)\) is either a Baer subplane or a unital.
In this paper, the authors deal with two particular examples of point sets with three intersection numbers with respect to planes. The first example is a Baer cone (a cone with vertex a point and base a Baer subplane), the second is a unital cone (a cone with vertex a point and base a unital). Both examples form \(2\)-blocking sets in \(\mathrm{PG}(3,q^2)\). A 2-blocking set in \(\mathrm{PG}(3,q^2)\) is a blocking set with respect to lines.
The two main results of this paper show that a 2-blocking set of type \((q^2 +1,q^2 +q+1,q^3 +q^2 +1)_2\) is a Baer cone and a 2-blocking set of type \((q^2 +1,q^3 +1,q^3 +q^2 +1)_2\) is a unital cone.
The proofs are combinatorial. The hypothesis that the point sets form 2-blocking sets enables the authors to make use of the known results about planar blocking sets in \(\mathrm{PG}(2,q^2)\).
MSC:
51E20 Combinatorial structures in finite projective spaces
51E21 Blocking sets, ovals, \(k\)-arcs
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