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.

Cardinali, I. (2013). Non-projective embeddings in the grassmann variety. ELECTRONIC NOTES IN DISCRETE MATHEMATICS, 40, 53-57 [10.1016/j.endm.2013.05.011].

### Non-projective embeddings in the grassmann variety

#### Abstract

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.
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2013
Cardinali, I. (2013). Non-projective embeddings in the grassmann variety. ELECTRONIC NOTES IN DISCRETE MATHEMATICS, 40, 53-57 [10.1016/j.endm.2013.05.011].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11365/45496`