On Veldkamp Lines
نویسندگان
چکیده
One says that Veldkamp lines exist for a point-line geometry Γ if, for any three distinct (geometric) hyperplanes A, B and C (i) A is not properly contained in B and (ii) A ∩ B ⊆ C implies A ⊂ C or A ∩ B = A ∩ C. Under this condition, the set V of all hyperplanes of Γ acquires the structure of a linear space – the Veldkamp space – with intersections of distinct hyperplanes playing the role of lines. It is shown here that an interesting class of strong parapolar spaces (which includes both the half-spin geometries and the Grassmannians) possess Veldkamp lines. Combined with other results on hyperplanes and embeddings, this implies that for most of these parapolar spaces, the corresponding Veldkamp spaces are projective spaces. The arguments incorporate a model of partial matroids based on intersections of sets.
منابع مشابه
On the Veldkamp Space of GQ(4, 2)
The Veldkamp space, in the sense of Buekenhout and Cohen, of the generalized quadrangle GQ(4, 2) is shown not to be a (partial) linear space by simply giving several examples of Veldkamp lines (V-lines) having two or even three Veldkamp points (V-points) in common. Alongside the ordinary V-lines of size five, one also finds V-lines of cardinality three and two. There, however, exists a subspace...
متن کاملA Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line
It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of a combinatorial Grassmannian of type G2(7), V(G2(7)). The lines of the ambient symplectic polar space are those lines of V(G2(7)) whose cores feature an odd number of points of G2(7). After introducing the basic properties of three different types of points and seven distinct types of lines...
متن کاملThe Veldkamp Space of the Smallest Slim Dense Near Hexagon
We give a detailed description of the Veldkamp space of the smallest slim dense near hexagon. This space is isomorphic to PG(7, 2) and its 2 − 1 = 255 Veldkamp points (that is, geometric hyperplanes of the near hexagon) fall into five distinct classes, each of which is uniquely characterized by the number of points/lines as well as by a sequence of the cardinalities of points of given orders an...
متن کاملVeldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties
Let S(N) ≡ PG(1, 2)×PG(1, 2)×· · ·×PG(1, 2) be a Segre variety that is N -fold direct product of projective lines of size three. Given two geometric hyperplanes H ′ and H ′′ of S(N), let us call the triple {H , H , H ∆H } the Veldkamp line of S(N). We shall demonstrate, for the sequence 2 ≤ N ≤ 4, that the properties of geometric hyperplanes of S(N) are fully encoded in the properties of Veldka...
متن کاملA Classification of the Veldkamp Lines of the Near Hexagon L 3 × GQ ( 2 , 2 )
Using a standard technique sometimes (inaccurately) known as Burnside’s Lemma, it is shown that the Veldkamp space of the near hexagon L3×GQ(2, 2) features 158 different types of lines. We also give an explicit description of each type of a line by listing the types of the three geometric hyperplanes it consists of and describing the properties of its core set, that is the subset of points of L...
متن کامل