Abstract
We characterize the finite Veronesean \({\cal H}_n \, \subseteq \, PG(n(n+2),q)\) of all Hermitian varieties of PG(n,q2) as the unique representation of PG(n,q2) in PG(d,q), d≥ n(n+2), where points and lines of PG(n,q2) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q2) generates PG(d,q). Using this result for n=2, we show that \({\cal H}_2 \, \subseteq \, PG(8,q)\) is characterized by the following properties: (1) \(|{\cal H}_2|=q^4+q^2+1\); (2) each hyperplane of PG(8,q) meets \({\cal H}_2\) in q2+1, q3+1 or q3+q2+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with \({\cal H}_2\) shares exactly q2+1 points with it.
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Thas, J.A., Van Maldeghem, H. Some Characterizations of Finite Hermitian Veroneseans. Des Codes Crypt 34, 283–293 (2005). https://doi.org/10.1007/s10623-004-4860-9
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DOI: https://doi.org/10.1007/s10623-004-4860-9