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

Suppose Gamma is a Lie incidence geometry defined over some field F having a Lie incidence geometry Gamma (0) of the same type but defined over a subfield F-0 less than or equal to 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 Gamma (0) in order to obtain a generating set for Gamma? We note that if Gamma is generated by the points of an apartment, then no additional points axe 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 and O-2n(+)(F). It turns out that in these cases the maximum number of points one needs to add to Gamma (0) in order to generate Gamma equals the maximal number of roots one needs to adjoin to F-0 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 Gamma (0) does not generate Gamma. As a byproduct we determine the generating rank of the line grassmannian of the polar geometry associated to Sp(2n) (F) (n greater than or equal to 3), if F is a prime field of odd characteristic.