(d, σ) -Veronese variety and some applications

Let K be the Galois field Fqt of order q^t, q = p^e, p a prime, A = Aut(K) be the automorphism group of K and σ = (σ_0,...,σ_{d−1}) ∈ A_d , d ≥ 1. In this paper the following generalization of the Veronese map is studied: ν_{d,σ} : ∈ PG(n − 1, K) −→ ∈ PG(nd − 1, K). Its image will be called the (d, σ)-Veronese variety V_{d,σ} . For d = t, σ a generator of Gal(Fq^t |Fq ) and σ = (1, σ, σ^2,...,σ^{t−1}), the (t, σ)-Veronese variety V_{t,σ} is the variety studied in [9, 11, 13]. Such a variety is the Grassmann embedding of the Desarguesian spread of PG(nt −1, F_q ) and it has been used to construct codes [3] and (partial) ovoids of quadrics, see [9, 12]. Here, we will show that V_{d,σ} is the Grassmann embedding of a normal rational scroll and any d +1 points of it are linearly independent. We give a characterization of d +2 linearly dependent points of V_{d,σ} and for some choices of parameters, V_{p,σ} is the normal rational curve; for p = 2, it can be the Segre’s arc of PG(3, q^t); for p = 3 V_{p,σ} can be also a |V_{p,σ} |-track of PG(5, q^t). Finally, investigate the link between such points sets and a linear code C_{d,σ} that can be associated to the variety, obtaining examples of MDS and almost MDS codes.