Abstract
We classify all embeddings θ: PG(n, q) → PG(d, q), with \(d \geqslant \tfrac{{n(n + 3)}} {2}\), such that θ maps the set of points of each line to a set of coplanar points and such that the image of θ generates PG(d, q). It turns out that d = ½n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well.
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References
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J. A. Thas and H. Van Maldeghem: Characterizations of the finite quadric Veroneseans \(\mathcal{V}_n^{2^n }\), Quart. J. Math. 55 (2004), 99–113.
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Thas, J.A., Van Maldeghem, H. Generalized Veronesean embeddings of projective spaces. Combinatorica 31, 615–629 (2011). https://doi.org/10.1007/s00493-011-2651-2
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DOI: https://doi.org/10.1007/s00493-011-2651-2