Maximal linear sections of Grassmannians

Consider a finite dimensional vector space over a field and the collection of all its subspaces of a fixed dimension. It is well-known that this constitutes a nice geometric object, namely, the Grassmannian with its canonical Plucker embedding. Linear sections of Grassmanians, of which Schubert varieties in Grassmannians are special cases, are interesting objects from algebraic, topological and combinatorial viewpoints. We consider the following question: Among the sections of Grassmannians by linear subspaces of a fixed dimension of the Plucker projective space, which are ``maximal'' ? The
term `maximal' can be interpreted in several ways and we will be particularly interested in maximality with respect to the number of points, when working over a finite field.  In general, this
is an open problem. ..........(contd) An attempt will be made to keep the prerequisites at a minimum.