Generalized Veronesean Embeddings of Projective Spaces, Part II. The lax case.

Abstract

We classify all embeddings \(\theta\) : \(PG(n,\mathbb{K}) \rightarrow PG(d,\mathbb{F})\), with \(d \geq \frac{n(n+1)}{2}\)
and \(\mathbb{K},\mathbb{F}\) skew fields with \(|\mathbb{K}| > 2\), such that \(\theta\) maps the set of points of each line of \(PG(n, \mathbb{K})\) to a set of coplanar points of \(PG(n, \mathbb{F})\), and such that the image of \(\theta\) generates \(PG(d, \mathbb{F})\). It turns out that \(d = \frac{1}{2}n(n + 3)\) and all examples “essentially” arise from a similar “full” embedding \(\theta’\) : \(PG(n, \mathbb{K}) \rightarrow PG(d, \mathbb{K})\) by identifying \(\mathbb{K}\) with subfields of F and embedding \(PG(d, \mathbb{K})\) into \(PG(d, \mathbb{F})\) by several ordinary field extensions. These “full” embeddings satisfy one more property and are classified in \([5]\). They relate to the quadric Verone-sean of \(PG(n, \mathbb{K})\) in \(PG(d, \mathbb{K})\) and its projections from subspaces of \(PG(n, \mathbb{K})\) generated by sub-Veroneseans (the point sets corresponding to subspaces of \(PG(n, \mathbb{K})\), if \(\mathbb{K}\) is commutative, and to a degenerate analogue of this, if \(\mathbb{K}\) is noncommutative.