The complement of a subspace in a classical polar space

Abstract

In a polar space, embeddable into a projective space, we fix a subspace, that is contained in some hyperplane. The complement of that subspace resembles a slit space or a semiaffine space. We prove that under some assumptions the ambient polar space can be recovered in this complement. (2010): 51A15, 51A45. Mathematics Subject Classification

Affine semipolar spaces

“…Since Z is contained in a hyperplane of Q the next theorem follows from [16,Theorem 3.11] which says that the ambient, thick, nondegenerate, embeddable polar space of rank at least 3 can be recovered in the complement of its subspace that is contained in a hyperplane. We give an independent, not so complex proof based on the decomposition of the hyperbolic polar space Q.…”

“…Set D := {q ∈ Y : no line in G has direction q} (16) Note that when η u : V −→ V is onto for each nonzero u, then D = V × {θ}. Directly from (11) all the points collinear with p = [0, θ] form the set { [v, u] : v = 0} and all the points collinear with q = [0, u 0 ] form the set…”

“…To establish the automorphism group of the relation ∼ ∼ and U we need some additional assumption that the set of directions V × {θ} can be characterized in terms of the projective geometry of the horizon of A(Y) with the set D [defined in (16)] distinguished. It is hard to give a formal, precise formula stating that.…”

“…Now, assume that F is an affine automorphism of U and (**) is valid. Then, F leaves invariant the set D defined in (16) and it is a composition τ [v0,u0] • F 0 , where F 0 ∈ GL(Y) and [v 0 , u 0 ] ∈ Y . The map F 0 can be presented in the form F 0 ([v, u]) = [ψ 1 (v)+ψ 2 (u), ϕ 1 (u)+ϕ 2 (v)] for some linear maps…”