Point-line geometries with a generating set that depends on the underlying field

Suppose Γ is a Lie incidence geometry defined over some field F having a Lie incidence geometry Γ0 of the same type but defined over a subfield F0 ≤ F as a subgeometry. We investigate the following question: how many points (if any at all) do we have to add to the point-set of Γ0 in order to obtain a generating set for Γ? We note that if Γ is generated by the points of an apartment, then no additional points are needed. We then consider the long-root geometry of the group SLn+1(F) and the line-grassmannians of the polar geometries associated to the groups O2n+1(F), Sp2n(F) and O + 2n(F). It turns out that in these cases the maximum number of points one needs to add to Γ0 in order to generate Γ equals the maximal number of roots one needs to adjoin to F0 in order to generate F. We prove that in the case of the long-root geometry of the group SLn+1(F) the point-set of Γ0 does not generate Γ. As a byproduct we determine the generating rank of the line grassmannian of the polar geometry associated to Sp2n(F) (n ≥ 3), if F is a prime field of odd characteristic.