Some Characterizations of Finite Hermitian Veroneseans

We characterize the finite Veronesean Hn ⊆ 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 H2 ⊆PG(8, q) is characterized by the following properties: (1) |H2| = q4 + q2 + 1; (2) each hyperplane of PG(8, q) meets H2 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 H2 shares exactly q2 +1 points with it.