On a construction of affine Grassmannians and spine spaces

Abstract

In this paper we introduce the notion of spine space, generalizing the notion of affine grassmannians. We describe the set of strong subspaces of a spine space (Thm. 2.6, 2.7), construct the horizon of a spine space, and study the automorphisms of a spine space. (2000): 51A45, 51D25. Mathematics Subject Classification

Affine Geometry of Spine Spaces

“…Following the notation of [2], we have the set Tk,m(W) °f points U of such that dimf/ fl W = m, and the family Gk,m(W) of all at least two-element sections g = p fl ^^(IV), where p is a fc-pencil of For such a line g we use notation g = p, and denote by g°° the point of the set g\g, which in fact is at most one-element. The family Gk, m (W) is the union of the family -4fc )7n (W) of affine lines, i.e.…”

“…Note that <?i°, 6 II. Evidently, there is a line q through <?2° in Prom [2] it is known that the line q is the horizon of II. Since for every line g C X, g intersects q we are through.…”

“…The paper is a continuation of the theory of spine spaces originated in [2] and developed in [4] and [3]. It seems that there are two approaches to the geometry of spine spaces, the projective one, with no parallelity relation involved, and the affine, where the parallelity is defined.…”

Desarguesian Closure of Binomial Graphs

“…Note also that if 0 < k < n then max(0,2k -n) < k -1 < k and is the structure of set-theoretical pencils, as defined in [4], Finally, note a similarity between the construction of binomial graph, as a certain relation in the boolean lattice (P(V), C) and the construction of spine space (cf. [3]) and the "distance function" in the lattice of all vector subspaces of a vector space V.…”

Extended parallelity in spine spaces and its geometry

“…Now, let us fix an arbitrary subspace W of V , and a natural number m such that max(0, k − codim W ) =: m min ≤ m ≤ m max := min(k, dim W ). Following the notation of [11], we have F k,m (W ) the set of points U of P such that dim U ∩ W = m, and G k,m (W ) the family of all at least two-element sections g = p ∩ F k,m (W ), p being a k-pencil of P. For such a line g we use notation g = p, and denote by g ∞ the point of the set g \ g, which is either empty or one-element. If g ∞ exists we call it the improper point of g. The family G k,m (W ) is the union of the family A k,m (W ) of affine lines, i.e.…”