(d,varvec{sigma })-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 varvec{sigma }=(sigma _0,ldots , sigma _{d-1}) in A^d, d ge 1. In this paper the following generalization of the Veronese map is studied: νd,σ:⟨v⟩∈PG(n-1,K)⟶⟨vσ0⊗vσ1⊗⋯⊗vσd-1⟩∈PG(nd-1,K).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ u _{d,\\varvec{\\sigma }} : \\langle v \\rangle \\in {{\\,\\mathrm{{PG}}\\,}}(n-1,{\\mathbb K}) \\longrightarrow \\langle v^{\\sigma _0} \\otimes v^{\\sigma _1} \\otimes \\cdots \\otimes v^{\\sigma _{d-1}} \\rangle \\in {{\\,\\mathrm{{PG}}\\,}}(n^d-1,{\\mathbb K}). \\end{aligned}$$\\end{document}Its image will be called the (d,varvec{sigma })-Veronese varietymathcal V_{d,varvec{sigma }}. For d=t, sigma a generator of textrm{Gal}({mathbb F}_{q^t}|{mathbb F}_{q}) and varvec{sigma }=(1,sigma ,sigma ^2,ldots ,sigma ^{t-1}), the (t,varvec{sigma })-Veronese variety mathcal V_{t,varvec{sigma }} is the variety studied in [9, 11, 13]. Such a variety is the Grassmann embedding of the Desarguesian spread of {{,mathrm{{PG}},}}(nt-1,{mathbb F}_q) and it has been used to construct codes [3] and (partial) ovoids of quadrics, see [9, 12]. Here, we will show that mathcal V_{d,varvec{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,varvec{sigma }} and for some choices of parameters, mathcal V_{p,varvec{sigma }} is the normal rational curve; for p=2, it can be the Segre’s arc of {{,mathrm{{PG}},}}(3,q^t); for p=3mathcal V_{p,varvec{sigma }} can be also a |mathcal V_{p,varvec{sigma }}|-track of {{,mathrm{{PG}},}}(5,q^t). Finally, investigate the link between such points sets and a linear code {mathcal C}_{d,varvec{sigma }} that can be associated to the variety, obtaining examples of MDS and almost MDS codes.