$$(d,\varvec{\sigma })$$-Veronese variety and some applications

Abstract

AbstractLet $${\mathbb K}$$ K be the Galois field $${\mathbb F}_{q^t}$$ F q t of order $$q^t, q=p^e, p$$ q t , q = p e , p a prime, $$A={{\,\mathrm{{Aut}}\,}}({\mathbb K})$$ A = Aut ( K ) be the automorphism group of $${\mathbb K}$$ K and $$\varvec{\sigma }=(\sigma _0,\ldots , \sigma _{d-1}) \in A^d$$ σ = ( σ 0 , … , σ d - 1 ) ∈ A d , $$d \ge 1$$ d ≥ 1 . In this paper the following generalization of the Veronese map is studied: $$\begin{aligned} \nu _{d,\varvec{\sigma }} : \langle v \rangle \in {{\,\mathrm{{PG}}\,}}(n-1,{\mathbb K}) \longrightarrow \langle v^{\sigma _0} \otimes v^{\sigma _1} \otimes \cdots \otimes v^{\sigma _{d-1}} \rangle \in {{\,\mathrm{{PG}}\,}}(n^d-1,{\mathbb K}). \end{aligned}$$ ν d , σ : ⟨ v ⟩ ∈ PG ( n - 1 , K ) ⟶ ⟨ v σ 0 ⊗ v σ 1 ⊗ ⋯ ⊗ v σ d - 1 ⟩ ∈ PG ( n d - 1 , K ) . Its image will be called the $$(d,\varvec{\sigma })$$ ( d , σ ) -Veronese variety$$\mathcal V_{d,\varvec{\sigma }}$$ V d , σ . For $$d=t$$ d = t , $$\sigma $$ σ a generator of $$\textrm{Gal}({\mathbb F}_{q^t}|{\mathbb F}_{q})$$ Gal ( F q t | F q ) and $$\varvec{\sigma }=(1,\sigma ,\sigma ^2,\ldots ,\sigma ^{t-1})$$ σ = ( 1 , σ , σ 2 , … , σ t - 1 ) , the $$(t,\varvec{\sigma })$$ ( t , σ ) -Veronese variety $$\mathcal V_{t,\varvec{\sigma }}$$ V t , σ is the variety studied in [9, 11, 13]. Such a variety is the Grassmann embedding of the Desarguesian spread of $${{\,\mathrm{{PG}}\,}}(nt-1,{\mathbb F}_q)$$ PG ( n t - 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 $$\mathcal V_{d,\varvec{\sigma }}$$ V d , σ is the Grassmann embedding of a normal rational scroll and any $$d+1$$ d + 1 points of it are linearly independent. We give a characterization of $$d+2$$ d + 2 linearly dependent points of $$\mathcal V_{d,\varvec{\sigma }}$$ V d , σ and for some choices of parameters, $$\mathcal V_{p,\varvec{\sigma }}$$ V p , σ is the normal rational curve; for $$p=2$$ p = 2 , it can be the Segre’s arc of $${{\,\mathrm{{PG}}\,}}(3,q^t)$$ PG ( 3 , q t ) ; for $$p=3$$ p = 3 $$\mathcal V_{p,\varvec{\sigma }}$$ V p , σ can be also a $$|\mathcal V_{p,\varvec{\sigma }}|$$ | V p , σ | -track of $${{\,\mathrm{{PG}}\,}}(5,q^t)$$ PG ( 5 , q t ) . Finally, investigate the link between such points sets and a linear code $${\mathcal C}_{d,\varvec{\sigma }}$$ C d , σ that can be associated to the variety, obtaining examples of MDS and almost MDS codes.