Let $\Delta$ be a thick dual polar space of rank $n \geq 2$ and let $e$ be a full polarized embedding of $\Delta$ into a projective space $\Sigma$. For every point $x$ of $\Delta$ and every $i \in \{ 0,\ldots,n \}$, let $T_i(x)$ denote the subspace of $\Sigma$ generated by all points $e(y)$ with $\d(x,y) \leq i$. We show that $T_i(x)$ does not contain points $e(z)$ with $\d(x,z) \geq i+1$. We also show that there exists a well-defined map $e_i^x$ from the set of $(i-1)$-dimensional subspaces of the residue $Res_\Delta(x)$ of $\Delta$ at the point $x$ (which is a projective space of dimension $n-1$) to the set of points of the quotient space $T_i(x)/T_{i-1}(x)$. In this paper we study the structure of the maps $e_i^x$ and the subspaces $T_i(x)$ for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature.
Cardinali, I., De, B. (2008). The structure of full polarized embeddings of symplectic and hermitian dual polar spaces. ADVANCES IN GEOMETRY, 8, 111-137 [10.1515/advgeom.2008.008].
The structure of full polarized embeddings of symplectic and hermitian dual polar spaces
CARDINALI, ILARIA;
2008-01-01
Abstract
Let $\Delta$ be a thick dual polar space of rank $n \geq 2$ and let $e$ be a full polarized embedding of $\Delta$ into a projective space $\Sigma$. For every point $x$ of $\Delta$ and every $i \in \{ 0,\ldots,n \}$, let $T_i(x)$ denote the subspace of $\Sigma$ generated by all points $e(y)$ with $\d(x,y) \leq i$. We show that $T_i(x)$ does not contain points $e(z)$ with $\d(x,z) \geq i+1$. We also show that there exists a well-defined map $e_i^x$ from the set of $(i-1)$-dimensional subspaces of the residue $Res_\Delta(x)$ of $\Delta$ at the point $x$ (which is a projective space of dimension $n-1$) to the set of points of the quotient space $T_i(x)/T_{i-1}(x)$. In this paper we study the structure of the maps $e_i^x$ and the subspaces $T_i(x)$ for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature.File | Dimensione | Formato | |
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https://hdl.handle.net/11365/37962
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