Non-projective embeddings in the grassmann variety

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