In this thesis, we will study the smooth linear section of the Grassmannian Gr(2, n). Explicitly, we give a criterion for the rationality of such linear section in terms of its codimension in the Plü ̈cker embedding in projective space. Moreover, to obtain a better understanding of the birational parametrization of these linear sections, we analyze their Hodge structures in the cases of even and odd codimensions. To be more precise, we provide numerous examples which suggest certain patterns of Hodge diamonds corresponding to even and odd cases and derive the proof of general patterns for codimension 3 smooth linear section of Gr(2, n) corresponding to odd and even n.