Some Characterizations of Finite Hermitian Veroneseans

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.