$(d,\text{σ})$-Veronese variety and some applications

Abstract

Let $\mathbb{K}$ be the Galois field ${\mathbb{F}}_{q^t}$ of order $q^t,q=p^e,p$ a prime, $A=\mathrm{Aut}(\mathbb{K})$ be the automorphism group of $\mathbb{K}$ and $\mathbf{\sigma }=({\sigma }_0,\dots ,{\sigma }_{d-1})\in A^d$, $d\ge 1$. In this paper the following generalization of the Veronese map is studied: $${\nu }_{d,\mathbf{\sigma }}:⟨v⟩\in \mathrm{PG}(n-1,\mathbb{K})⟶⟨v^{{\sigma }_0}\otimes v^{{\sigma }_1}\otimes \cdots \otimes v^{{\sigma }_{d-1}}⟩\in \mathrm{PG}(n^d-1,\mathbb{K}).$$ Its image will be called the $(d,\mathbf{\sigma })$-$Veronese$ $variety$ ${\mathcal{V}}_{d,\mathbf{\sigma }}$. Here, we will show that ${\mathcal{V}}_{d,\mathbf{\sigma }}$ 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 ${\mathcal{V}}_{d,\mathbf{\sigma }}$ and for some choices of parameters, ${\mathcal{V}}_{p,\mathbf{\sigma }}$ is the normal rational curve; for $p=2$, it can be the Segre's arc of $\mathrm{PG}(3,q^t)$; for $p=3$ ${\mathcal{V}}_{p,\mathbf{\sigma }}$ can be also a $|{\mathcal{V}}_{p,\mathbf{\sigma }}|$-track of $\mathrm{PG}(5,q^t)$. Finally, investigate the link between such points sets and a linear code ${\mathcal{C}}_{d,\mathbf{\sigma }}$ that can be associated to the variety, obtaining examples of MDS and almost MDS codes.