Non-projective embeddings in the Grassmann variety

We investigate properties of the grassmann embedding of dual classical thick generalized quadrangles focusing on the Grassmann embedding of the dual DQ(4,F) of an orthogonal quadrangle Q(4,F) and the dual DH(4,F) of a hermitian quadrangle H(4,F). We prove that, if the characteristic of the field F is different from 2 then the dimension of the grassmann embedding of DQ(4,F) is 10 and its image is isomorphic to the quadratic veronese variety of a 3-dimensional projective space. If F is a perfect field of characteristic 2 then the dimension of the grassmann embedding of DQ(4,F) is proved to be 9 and its image is a 3-dimensional algebraic subvariety of the Grassmannian of lines of a 4-dimensional projective space. Moving to consider the dual quadrangle DH(4,F), we prove that the dimension of its Grassmann embedding is 10 and the image of DH(4,F) under the Grassmann embedding is a 2-dimensional algebraic subvariety of the Grassmannian of lines of a 4-dimensional projective space.