Abstract
In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital \({\mathcal{U}}\) of \({\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}\) a quadratic extension of the field \({\mathbb{K}}\) and \({|\mathbb{K}| \geq 3}\), in a \({\mathsf{PG}(d,\mathbb{F})}\), with \({\mathbb{F}}\) any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry \({\mathsf{PG}(7,\mathbb{K}^{\prime})}\) of \({\mathsf{PG}(7,\mathbb{F})}\) (and d = 7) or it consists of the projection from a point \({p \in \mathcal{U}}\) of \({\mathcal{U}{\setminus} \{p\}}\) from a subgeometry \({\mathsf{PG}(7,\mathbb{K}^{\prime})}\) of \({\mathsf{PG}(7,\mathbb{F})}\) into a hyperplane \({\mathsf{PG}(6,\mathbb{K}^{\prime})}\). In order to do so, when \({|\mathbb{K}| >3 }\) we strongly use the linear representation of the affine part of \({\mathcal{U}}\) (the line at infinity being secant) as the affine part of the generalized quadrangle \({\mathsf{Q}(4,\mathbb{K})}\) (the solid at infinity being non-singular); when \({|\mathbb{K}| =3}\), we use the connection of \({\mathcal{U}}\) with the generalized hexagon of order 2.
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This is one of several papers published together in Designs, Codes and Cryptography on the “Special Issue on Finite Geometries”.
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Pepe, V., Van Maldeghem, H. Lax embeddings of the Hermitian unital. Des. Codes Cryptogr. 68, 325–347 (2013). https://doi.org/10.1007/s10623-011-9571-4
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DOI: https://doi.org/10.1007/s10623-011-9571-4