Complements of Grassmann substructures in projective Grassmannians

Abstract

Abstract. We prove that a projective Grassmannian can be recovered from the complement of one of its Grassmann substructures. Even more, the underlying projective space with the interval of its distinguished subspaces can be recovered. Mathematics Subject Classification (2010). 51A15, 51A45.Keywords. Slit space, Grassmannian, projective space, affine space. IntroductionThe idea of deleting a hyperplane from a geometric structure is not new. It has been applied in various contexts starting from an affine space as a result of deleting a hyperplane from a projective space. In [3] the complement of a geometric hyperplane, i.e. a proper subspace that either meets each line in a unique point or contains that line, in a Grassmann space, also known as a space of pencils, is investigated.Instead of a hyperplane any subspace can be deleted to obtain a more interesting structure, which in the case of projective geometry is called a slit space (cf. [5,6]). Just to mention [11] which deals with the so called partial geometry that consists of points of a finite projective space P not contained in a fixed secundum (a subspace of codimension 2) W of P and lines of P which do not intersect W. In [7] configurations that arise from finite affine planes by removing a pencil of lines are investigated. The authors generalise the result of [13] that the complement of a line in a finite affine plane can be embedded into a projective plane of the same order. The next example is the affine space of rectangular matrices, introduced in [12, Ch. 3], which resembles the structure of subspaces in a projective space that are complementary to a distinguished subspace. Two different approaches to this structure are delivered by [1] and [10].In both papers it is shown that the structure of complements is embeddable into an affine space and not all lines of that affine space arise as lines of this structure.Another good reason (frankly the real reason) to play with complements or slit spaces is as follows. Consider a finite dimensional projective space P coordinatized by an even characteristic field and endowed with a non-symplectic projective polarity ⊥, a pseudopolarity. Let W be the set of all self-conjugate points of P w.r.t. ⊥. Then W is known to be a proper subspace of P. If p is a point in P we write p ⊥ for the set of all points conjugate to p; for a subspace U we have then U ⊥ := p∈U p ⊥ . We say that a subspace U is regular when its radical U ∩ U ⊥ is void. Note that U ∩ U ⊥ ⊂ W and those subspaces that are skew (disjoint) to W are regular. So, our present result moves us closer to solving a similar problem of recovering the ambient projective Grassmannian from the Grassmannian of regular subspaces w.r.t. a pseudopolarity.In spaces of pencils interval subspaces, i.e. those induced by intervals in the lattice of subspaces of the underlying space, form a very important class. Each of them carries the structure of a space of pencils and any subspace with this property is an interval subspace (see [15]). This is the reason to call such subsp...

The complement of a point subset in a projective space and a Grassmann space

“…Such substructures are unique in that they bear the structure of Grassmannians and only those have this property. In Grassmann spaces interval subspaces play an analogous role, as it was shown in [26], and this is the reason to call them Grassmann subspaces. So, consider a projective Grassmann space M = P k (V ) and its interval subspace W := [Z, Y ] k for some subspaces Z, Y of V .…”

“…This is not a completely new question and there are some papers devoted to such recovery problem. In [19] projective Grassmannians are successfully recovered from complements of their Grassmann substructures. The concept of two-hole slit space is introduced in [13].…”

“…a point-line space with k-subspaces of V as points and k-pencils as lines (see [10], [19] for a more general definition, see also [9]). For 0 < k < n it is a partial linear space.…”

“…Note that in the case where W is a hyperplane, the complement D is an affine space, while if W is not necessarily a hyperplane but a proper subspace of P, then D is a slit space (cf. [6], [7], [19]).…”

Geometry on the lines of spine spaces

“…with k-subspaces of V as points and k-pencils as lines (see [7], [17] for a more general definition, see also [6]). For 0 < k < n it is a partial linear space.…”