Problem in Schubert calculus

Let F be a finite field.
Let G be the plucker embedding of the Grassmanian G(k,n) to P^N
Let P be the set of F-points in G and L the set of F-lines (through a
pair of F-points) in P^N which are contained in G.
I would like an incidence matrix for P and L.

I am not familiar with Schubert calculus, so I cant formulate the
problem in those terms.  If this were done, one could dramatically
generalize the question above.

John Voight wrote Magma code for the construction of the Grassmanian
and I added a very brute force search for lines.  This chokes on all but the
smallest examples.

Grassmanian := function(S,k,n);
  R := PolynomialRing(S, n*k);
  M := Matrix(R,n,k,[R.i : i in [1..n*k]]);
  mins := Minors(M, k);
  X := Proj(R);
  Y := ProjectiveSpace(S, #mins-1);
  phi := map Y | mins>;
  return X, Y, phi;
end function;

R1 := PolynomialRing(F,2);
P1 := Proj(R1);
Lns := {@ @};

for i in [1..N] do
  for j in [i+1..N] do
    pt := Pts[i]; qt := Pts[j];
    ln := map < P1 -> Y | [ R1.1*pt[i] + R1.2*qt[i]
                              : i in [1..Dimension(Y)+1] ] >;
    if Points(Image(ln)) subset Points(Gr) then
        Lns := Lns join {@ IndexedSetToSet(Points(Image(ln))) @};
    end if;
  end for;
end for;