We characterize the finite Veronesean
Hn Í PG(n(n+2),q){\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),
dH2 Í PG(8,q){\cal H}_2 \, \subseteq \, PG(8,q)
is characterized by the following properties: (1)
|H2|=q4+q2+1|{\cal H}_2|=q^4+q^2+1
; (2) each hyperplane of
PG(8,
q) meets
H2{\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
H2{\cal H}_2
shares exactly
q2+1 points with it.
Keywords Projective spaces - Hermitian Veronesean - ovoids
51E24