Linear sets in the projective line over the endomorphism ring of a finite field
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arXiv
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| Natura: | Preprint |
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2016
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| _version_ | 1866914661587222528 |
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| author | Havlicek, Hans Zanella, Corrado |
| author_facet | Havlicek, Hans Zanella, Corrado |
| contents | Let $\mathrm{PG}(1,E)$ be the projective line over the endomorphism ring $E=End_q({\mathbb F}_{q^t})$ of the $\mathbb F_q$-vector space ${\mathbb F}_{q^t}$. As is well known there is a bijection $Ψ:\mathrm{PG}(1,E)\rightarrow{\cal G}_{2t,t,q}$ with the Grassmannian of the $(t-1)$-subspaces in $\mathrm{PG}(2t-1,q)$. In this paper along with any $\mathbb F_q$-linear set $L$ of rank $t$ in $\mathrm{PG}(1,q^t)$, determined by a $(t-1)$-dimensional subspace $T^Ψ$ of $\mathrm{PG}(2t-1,q)$, a subset $L_T$ of $\mathrm{PG}(1,E)$ is investigated. Some properties of linear sets are expressed in terms of the projective line over the ring $E$. In particular the attention is focused on the relationship between $L_T$ and the set $L'_T$, corresponding via $Ψ$ to a collection of pairwise skew $(t-1)$-dimensional subspaces, with $T\in L'_T$, each of which determine $L$. This leads among other things to a characterization of the linear sets of pseudoregulus type. It is proved that a scattered linear set $L$ related to $T\in\mathrm{PG}(1,E)$ is of pseudoregulus type if and only if there exists a projectivity $φ$ of $\mathrm{PG}(1,E)$ such that $L_T^φ=L'_T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1603_02232 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Linear sets in the projective line over the endomorphism ring of a finite field Havlicek, Hans Zanella, Corrado Combinatorics 51E20, 51C05, 51A45, 51B05 Let $\mathrm{PG}(1,E)$ be the projective line over the endomorphism ring $E=End_q({\mathbb F}_{q^t})$ of the $\mathbb F_q$-vector space ${\mathbb F}_{q^t}$. As is well known there is a bijection $Ψ:\mathrm{PG}(1,E)\rightarrow{\cal G}_{2t,t,q}$ with the Grassmannian of the $(t-1)$-subspaces in $\mathrm{PG}(2t-1,q)$. In this paper along with any $\mathbb F_q$-linear set $L$ of rank $t$ in $\mathrm{PG}(1,q^t)$, determined by a $(t-1)$-dimensional subspace $T^Ψ$ of $\mathrm{PG}(2t-1,q)$, a subset $L_T$ of $\mathrm{PG}(1,E)$ is investigated. Some properties of linear sets are expressed in terms of the projective line over the ring $E$. In particular the attention is focused on the relationship between $L_T$ and the set $L'_T$, corresponding via $Ψ$ to a collection of pairwise skew $(t-1)$-dimensional subspaces, with $T\in L'_T$, each of which determine $L$. This leads among other things to a characterization of the linear sets of pseudoregulus type. It is proved that a scattered linear set $L$ related to $T\in\mathrm{PG}(1,E)$ is of pseudoregulus type if and only if there exists a projectivity $φ$ of $\mathrm{PG}(1,E)$ such that $L_T^φ=L'_T$. |
| title | Linear sets in the projective line over the endomorphism ring of a finite field |
| topic | Combinatorics 51E20, 51C05, 51A45, 51B05 |
| url | https://arxiv.org/abs/1603.02232 |