关于Kelly关于Sylvester Gallai设计的$(1,3)$ -表示的一个猜想
On a Conjecture of Kelly on $(1,3)$-representation of Sylvester Gallai Designs
摘要：
我们给出了L.M.Kelly猜想成立的一个精确判据，如下所述。如果在$\mathbb{R}^l,l\geq 4$中存在有限族$Σ$，使得$Σ$中每两条线的三维仿射跨度（壳）至少包含$Σ$的另一条线，则我们有$Σ$完全包含在三维空间中当且仅当仿射包的排列是中心的。最后，本文将对高维斜仿射空间，即$\mathbb{R}^6$中sylvester-gallai设计的$(2,5)$ -表示，引出一个类似的问题，这将在最后一节中回答。
英文摘要：
We give an exact criterion of a conjecture of L.M.Kelly to hold true which is stated as follows. If there is a finite family $Σ$ of mutually skew lines in $\mathbb{R}^l,l\geq 4$ such that the three dimensional affine span (hull) of every two lines in $Σ$, contains at least one more line of $Σ$, then we have that $Σ$ is entirely contained in a three dimensional space if and only if the arrangement of affine hulls is central. Finally, this article leads to an analogous question for higher dimensional skew affine spaces, that is, for $(2,5)$-representations of sylvester-gallai designs in $\mathbb{R}^6$, which is answered in the last section.
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。
