Embeddings and hyperplanes of the Lie geometry \(A_{n,\{1,n\}}(\mathbb{F})\)

In this paper we consider a family of projective embeddings of the geometry \(A_{n,\{1,n\}}(\mathbb{F})\) of point-hyperplanes flags of \(\mathrm{PG}(n,\mathbb{F})\). The natural embedding \(\varepsilon_{\mathrm{nat}}\) is one of them. It maps every point-hyperplane flag \((p,H)\) onto the vector-line \(\langle {\bf x}\otimes\xi\rangle\), where \({\bf x}\) is a representative vector of \(p\) and \(\xi\) is a linear functional describing \(H\). The other embeddings have been discovered more than two decads ago by Thas and Van Maldeghem for the case \(n = 2\) and recently generalized to any \(n\) by De Schepper, Schillewaert and Van Maldeghem. They are obtained as twistings of \(\varepsilon_{\mathrm{nat}}\) by non-trivial automorphisms of \(\mathbb{F}\). Explicitly, for \(\sigma\in \mathrm{Aut}(\mathbb{F})\setminus\{\mathrm{id}_\mathbb{F}\}\), the twisting \(\varepsilon_\sigma\) of \(\varepsilon_{\mathrm{nat}}\) by \(\sigma\) maps \((p,H)\) onto \(\langle {\bf x}^\sigma\otimes \xi\rangle\). We shall prove that, when \(\mathrm{Aut}(\mathbb{F}) \neq \{\mathrm{id}_\mathbb{F}\}\), a geometric hyperplane \({\cal H}\) of \(A_{n,\{1,n\}}(\mathbb{F})\) arises from \(\varepsilon_{\mathrm{nat}}\) and at least one of its twistings or from at least two distinct twistings of \(\varepsilon_{\mathrm{nat}}\) if and only if \({\cal H} = \{(p,H)\in A_{n,\{1,n\}}(\mathbb{F}) \mid p\in A or a \in H\}\) for a possibly non-incident point-hyperplane pair \((a,A)\) of \(\mathrm{PG}(n,\mathbb{F})\). We call these hyperplanes quasi-singular hyperplanes. With the help of this result we shall prove that if \(|\mathrm{Aut}(\mathbb{F})| › 1\) then \(A_{n,\{1,n\}}(\mathbb{F})\) admits no absolutely universal embedding.Mathematics Subject Classifications: 51A45, 20F40, 15A69Keywords: Lie geometries, Segre varieties, embeddings, hyperplanes