有限域自同态环上射影直线中的线性集
Linear sets in the projective line over the endomorphism ring of a finite field
AI总结：
该研究聚焦有限域自同态环上射影直线中的线性集，借助环上射影直线相关对应关系探究线性集性质，重点刻画伪正则型散射线性集的等价条件，丰富了线性集的理论研究。
摘要：
设$\mathrm{PG}(1,E)$为有限域$\mathbb F_q$上向量空间${\mathbb F}_{q^t}$的自同态环$E=End_q({\mathbb F}_{q^t})$上的射影直线。众所周知，它与$\mathrm{PG}(2t-1,q)$中$(t-1)$维子空间构成的格拉斯曼流形之间存在双射$Ψ:\mathrm{PG}(1,E)\rightarrow{\cal G}_{2t,t,q}$。本文针对$\mathrm{PG}(1,q^t)$中秩为$t$的任意$\mathbb F_q$-线性集$L$（由$\mathrm{PG}(2t-1,q)$的一个$(t-1)$维子空间$T^Ψ$确定），研究了$\mathrm{PG}(1,E)$的一个子集$L_T$。部分线性集的性质通过环$E$上的射影直线来表述。研究重点尤其关注$L_T$与集合$L'_T$之间的关系，后者通过$Ψ$对应一族两两斜交的$(t-1)$维子空间，其中$T\in L'_T$，每个子空间都确定$L$。这一研究除其他结论外，还得到了伪正则型线性集的一种刻画。本文证明，与$T\in\mathrm{PG}(1,E)$相关的散射线性集$L$为伪正则型，当且仅当存在$\mathrm{PG}(1,E)$的一个射影变换$φ$，使得$L_T^φ=L'_T$成立。
英文摘要：
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 $L_T^φ=L'_T$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
