A combinatorial characterization of the Baer and the unital cone in $$PG(3,q^2)$$ P G ( 3 , q 2 )
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Journal of Geometry
A combinatorial characterization of the Baer and the unital cone in P G(3, q 2) Stefano Innamorati and Fulvio Zuanni Abstract. Bruen, see [5], and Bruen and Thas, see [7], proved that in P G(2, q 2 ) a blocking set of type (1, q + 1)1 is either a Baer subplane or a unital. In this paper a cone-generalization of this result in P G(3, q 2 ) is provided. Mathematics Subject Classification. 51E20, 51E21. Keywords. Three character sets, Sets of type (q 2 +1, q 2 +q+1, q 3 +q 2 +1)2 , Sets of type (q 2 +1, q 3 +1, q 3 +q 2 +1)2 , Baer cones, Unital cones, Blocking sets.
1. Introduction and motivation An algebraic variety K can be thought as a set of points in a projective space that has a certain behavior with respect to subspaces. In a finite projective space P G(r, q) the intersection varieties have a finite number of points. By combinatorial characterization of a variety we mean the classification of those sets of P G(r, q) which, for axiom, possess a certain number of incidence properties of the given variety. The variety is intended to be characterized if it proves that a set satisfying of the axiomatized properties is the variety. Characterizations are solid findings when the required axioms are few, essential and significant, see [10,11], and [16]. In a Galois projective space P G(r, q), of dimension r ≥ 2 and order q = ph a prime power, let K be a k-set, i.e. a set of k points of P G(r, q). Let d be an d integer such that 1 ≤ d ≤ r − 1 and let us denote by θd := i=0 q i the number of points of a d-subspace of P G(r, q). If we put s := r−d, then a set K is called a s-blocking set if every d-dimensional subspace has non-empty intersection with K. In a projective plane a 1-blocking set K is simply called blocking set, and trivial if K contains a line. For each integer i such that 0 ≤ i ≤ min{k, θd }, 0123456789().: V,-vol
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S. Innamorati, F. Zuanni
J. Geom.
let us denote by tdi = tdi (K) the number of d-subspaces meeting the set K in exactly i points. The numbers tdi are called the characters of K with respect to the d-subspaces, see [17]. Now let m1 , m2 , . . . , ms be s integers such that 0 ≤ m1 < m2 < · · · < ms ≤ min{k, θd }. A set K is said to be of class [m1 , m2 , . . . , ms ]d if tdi > 0 only if i ∈ {m1 , m2 , . . . , ms }. Moreover, K is said to be of type (m1 , m2 , . . . , ms )d if tdi > 0 if and only if i ∈ {m1 , m2 , . . . , ms }. The integers m1 , m2 , . . . , ms are called intersection numbers with respect to the d-subspaces. A fundamental question in finite geometry is to recognize geometric structures from intersection numbers, see [1,4,15] and [20]. A large number of papers is devoted to sets with two intersection numbers, see [2,9,13,18] and [19], and not much seems to be known in the general case of sets with more than two, see [3,8,12] and [14]. A unital is a 2-(q 3 + 1, q + 1, 1) design. An embedded unital is a unital whose points are those of a projective plane and whose blocks are the sets of q + 1 of these points on a line, see [11], chap. 12. Bruen, see
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