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 D Q ( 4 , F ) of an orthogonal quadrangle Q ( 4 , F ) and the dual D H ( 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 D Q ( 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 D Q ( 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 D H ( 4 , F ) , we prove that the dimension of its Grassmann embedding is 10 and the image of D H ( 4 , F ) under the Grassmann embedding is a 2-dimensional algebraic subvariety of the Grassmannian of lines of a 4-dimensional projective space.