Lax embeddings of the Hermitian unital
A lax Veronesian embedding of a linear space \(({\mathcal P},{\mathcal L})\) in a projective space \(\mathrm{PG}(d,{\mathbb F})\) is a map \(\theta\) such that the image of \({\mathcal P}\) generates \(\mathrm{PG}(d,{\mathbb F})\) and the image of the points of any line is a plane oval. If \({\mathcal U}=H(2,{\mathbb L}),\) (\({\mathbb L}\) is a quadratic extension of \({\mathbb K})\) is an Hermitian unital of \(\mathrm{PG}(2,{\mathbb L}),\) then it has an Veronesian embedding in \(\mathrm{PG}(7,{\mathbb F})\) (\({\mathbb F} \simeq {\mathbb K})\) in which the plane ovals are conics (standard Veronesian embedding). In this paper, the authors prove that if \(\theta\) is a lax generalized Veronesian embedding of the Hermitian unital of \(\mathrm{PG}(2,{\mathbb L}),\) where \({\mathbb L}\) is a quadratic extension of \({\mathbb K},\) \(|{\mathbb K}|>2,\) in a \(\mathrm{PG}(d,{\mathbb F})\) \((d\geq 7),\) such that disjoint blocks span disjoint subspaces of \(\mathrm{PG}(d,{\mathbb F}),\) then either (a) \(d=7\) and the embedding is the standard Veronesian embedding in a canonical subgeometry \(\mathrm{PG}(7,{\mathbb K}')\) of \(\mathrm{PG}(7,{\mathbb F})\), or (b) the embedding is the projection from a point \(p \in {\mathcal U}\) of \({\mathcal U} \setminus \{p\}\) from a subgeometry \(\mathrm{PG}(7, {\mathbb K}')\) into a hyperplane \(\mathrm{PG}(6,{\mathbb K}')\) together with an arbitrary point \(p'\) lying outside the \({\mathbb F}-\)span of \(\mathrm{PG}(6, {\mathbb K}')\) where \({\mathbb K}'\simeq{\mathbb K}\).
scientific article; zbMATH DE number 6168452