Let \(K := GF(q)\) be a finite field with \(q\) elements \(n \in {\mathbb N},\) \(N_n := 1/2n(n + 3)\). The Veronesean map \(v : K^{n+1} \to K^{N_n + 1}\), \[ (x_0,x_1,\dots x_n) \mapsto (x_0^2,x_1^2,\dots ,x_n^2,x_0x_1,\dots ,x_0x_n,x_1x_2,\dots x_1x_n,\dots ,x_{n-1}x_n) \] induces a map \(\bar{v}\) from the projective space \(PG(n,q)\) into the projective space \(PG(N_n,q)\) and the image \(V_n := \bar{v}(PG(n,q))\) is a quadratic Veronesean.
Let \({\mathcal H} \) be the set of all hyperplanes of \(PG(n,q)\), let “\(\langle\;\rangle\)” denote the incidence closure in \(PG(N_n,q)\) and let \({\mathcal S}_n := \{ \langle \bar{v}(H)\rangle \mid H \in {\mathcal H}\}\). Then \({\mathcal S}_n\) has the properties (VS1)-(VS5):
{(VS1)} \(\forall \{A,B\} \in { {\mathcal S}_n \choose 2} : \langle A \cup B\rangle \) is a hyperplane of \(PG(N_n,q)\),
{(VS2)} \(\forall \{A,B,C\} \in {{\mathcal S}_n \choose 3} : \langle A \cup B \cup C\rangle = PG(N_n,q)\),
{(VS3)} No point of \(PG(N_n,q)\) is contained in every member of \({\mathcal S}_n\),
{(VS4)} The intersection of any non-empty collection of members of \({\mathcal S}_n\) is a subspace of dimension \(N_i := i(i + 3) /2\) for some \(i \in \{-1, 0, 1, \dots , n - 1 \}\),
{(VS5)} \(\exists \{A,B,C\} \in { {\mathcal S}_n \choose 3}\) with \(A \cap B = B \cap C = C \cap A.\)
For the characterizations of these finite quadratic Veroneseans the authors start from a collection \({\mathcal S}\) of \(q^n + q^{n-1} + \dots + q + 1\) subspaces of dimension \(N_{n-1}\) of the projective space \(PG(N_n,q)\) with \(n \geq 2\) satisfying all or some of the properties (VS1)-(VS5). If all properties are satisfied (Theorem 1.1) then either \({\mathcal S}\) coincides with the set \({\mathcal S}_n\) of a Veronesean \(V_n\) or \(q\) is even and some other conditions are satisfied. If \(q \geq n\) and \({\mathcal S}\) satisfies (VS1)-(VS3) then also (VS4) (Theorem 1.2). Also the case where (VS1)-(VS4) but not (VS5) are satified is discussed (Theorem 1.3). From Theorem 1.1 the authors obtain the following characterization of finite quadratic Veroneseans (Theorem 1.6): Let \(\theta : PG(n,q) \to PG(m,q)\) be an injection, \(m \geq N_n\) with \(n \geq 2\) and \(q > 2\), such that for each line \(L\) of \(PG(n,q)\), \(\theta (L)\) is a plane oval in \(PG(m,q)\) and such that \(\langle \theta (PG(n,q))\rangle = PG(m,q)\). Then \(\theta (PG(n,q))\) is a quadratic Veronesean \(V_n\).