(d, σ) -Veronese variety and some applications

Let K be the Galois field of order q^t , A = Aut(K) be the automorphism group of K and σ = (σ0, . . . , σd−1) in A^d , d ≥ 1. In this paper the following generalization of the Veronese map is studied: νd,σ : 〈v〉 ∈ PG(n − 1, K) −→ 〈v ^σ0 ⊗ v^ σ1 ⊗ · · · ⊗ v^ σd−1 〉 ∈ PG(n d − 1, K). Its image will be called the (d, σ )-Veronese variety Vd,σ . We will show that Vd,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 Vd,σ and for some choices of parameters, Vp,σ is the normal rational curve; for p = 2, it can be the Segre’s arc of PG(3, q^t ); for p = 3 Vp,σ can be also a |Vp,σ |-track of PG(5, q^t ). Finally, investigate the link between such points sets and a linear code Cd,σ that can be associated to the variety, obtaining examples of MDS and almost MDS codes.