We consider the point-line geometries that arise as a shadow space in a spherical building with a diagram of type A n , B n , C n , D n or E n , and determine in which cases the geometry is spanned by the set of points on an apartment. It turns out that this happens precisely in the cases corresponding to a minimal weight.
Let Γ be the dual of a classical polar space and let e be a projective embedding of Γ, defined over a commutative division ring. We shall prove that, if e is homogeneous, then it is polarized.MSC 2000: 51B25, 51E24, 51A45, 51A50.
We prove that the Grassmannian of totally isotropic k-spaces of the polar space associated to the unitary group SU 2n (F) (n ∈ N) has generating rank 2n k when F = F 4 . We also reprove the main result of Blok [4], namely that the Grassmannian of totally isotropic k-spaces associated to the symplectic group Sp 2n (F) has generating rank 2n k − 2n k−2 , when Char(F) = 2.
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