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

TL;DR

This paper studies projective embeddings of the point-hyperplane flag geometry of a projective space, classifies hyperplanes arising from these embeddings, and shows the non-existence of an absolutely universal embedding when the field has non-trivial automorphisms.

Contribution

It classifies hyperplanes associated with embeddings of the geometry and proves the non-existence of an absolutely universal embedding for fields with automorphisms.

Findings

Hyperplanes correspond to point-hyperplane pairs or twistings of the natural embedding.

When automorphisms of the field exist, certain hyperplanes are characterized as quasi-singular.

No absolutely universal embedding exists if the field has non-trivial automorphisms.

Abstract

In this paper we consider a family of projective embeddings of the geometry $\Gamma = A_{n,\{1,n\}}(F)$ of point-hyperplanes flags of the projective geometry $\Sigma = PG(n,F)$. The natural embedding $\varepsilon_{mathrm{nat}}$ is one of them. It maps every point-hyperplane flag $(p,H)$ of $\Sigma$ onto the vector-line $\langle x\otimes\xi\rangle$, where $x$ is a representative vector of $p$ and $\xi$ is a linear functional describing $H$. The other embeddings have been discovered by Thas and Van Maldeghem (2000) for the case $n = 2$ and later generalized to any $n$ by De Schepper, Schillewaert and Van Maldeghem (2023). They are obtained as twistings of $\varepsilon_{\mathrm{nat}}$ by non-trivial automorphisms of $F$. Explicitly, for $\sigma\in Aut(F)\setminus\{\mathrm{id}_F\}$, the twisting $\varepsilon_\sigma$ of $\varepsilon_{\mathrm{nat}}$ by $\sigma$ maps $(p,H)$ onto $\langle x\sigma\otimes \xi\rangle$. We shall prove that, when $|Aut(F)| > 1$ a geometric hyperplane $\cal H$ of $\Gamma$ arises from $\varepsilon_{\mathrm{nat}}$ and one of its twistings or from two distinct twistings of $\varepsilon_{\mathrm{nat}}$ if and only if ${\cal H} = \{(p,H)\in \Gamma \mid p\in A \mbox{ or } a \in H\}$ for a possibly non-incident point-hyperplane pair $(a,A)$ of $\Sigma$. We call these hyperplanes quasi-singular hyperplanes. With the help of this result we shall prove that if $|Aut(F)| > 1$ then $\Gamma$ admits no absolutely universal embedding.