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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:
 5.1e+21 Combinatorial structures in finite projective spaces 5.1e+22 Blocking sets, ovals, $$k$$-arcs
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