论文标题
Cayley代数在某些无限田地上产生$ Q $ -FANO飞机，而$ Q $覆盖的设计超过了其他田地
Cayley algebras give rise to $q$-Fano planes over certain infinite fields and $q$-covering designs over others
论文作者
论文摘要
令$ f $为一个字段。 A $2$-$(7, 3, 1)_F$-subspace design, or $q$-Fano plane, over $F$, is a $7$-dimensional vector space $V$ over $F$ together with a collection $\mathfrak{B}$ of three-dimensional subspaces of $V$ such that every two-dimensional subspace of $V$ is contained in exactly one element $B$ of $ \ mathfrak {b} $。自1970年代以来，任何领域的存在$ Q $ -FANO飞机的存在问题都已开放，并且在特殊情况下引起了极大的关注。在这里，我们显示了$ 2 $ - $（7、3、1）_f $ -subspace在某些无限字段$ f $上的设计，包括（包括（其他）$ \ Mathbb {q}，\ Mathbb {r} $和$ \ $ \ MATHBB {f} _Q（x，y，y，z） Space $ v $是$ f $上的Cayley Division代数$ o $的7维元素的7维空间，$ \ Mathfrak {B} $由$ o $ $ o $的所有4维次级主管的$ V $组成。我们将以独立的方式介绍Cayley代数的必要背景。 接下来，我们研究如果将相同的程序应用于分裂（而不是分裂）Cayley代数会发生什么。通过识别所有这些的所有四维亚总代构，我们表明，在这种情况下，我们的构建仍然产生最小的$（7、3、2）$ $ $ $ $ $ $ $ $ $ $ - 覆盖的设计。也就是说：$ v $的每个二维子空间都包含在最终的$ \ mathfrak $ \ mathfrak {b} $的三维子空间的一个元素中，$ v $的三维子空间，没有$ \ mathfrak {b} $的适当子集。但是，这些$ Q $覆盖的设计都不是$ Q $ -FANO飞机。在$ f $是有限的情况下，我们计算$ \ mathfrak {b} $的元素数量。 我们还通过识别$ \ f $ f $ -f $ -vector $ v $的$ Q $ -FANO飞机和$ Q $覆盖设计的设计，通过识别$ \ mathfrak {b} $作为Grassmanian $ gr_3（v）$ extiral the Classical Pano的范围。
Let $F$ be a field. A $2$-$(7, 3, 1)_F$-subspace design, or $q$-Fano plane, over $F$, is a $7$-dimensional vector space $V$ over $F$ together with a collection $\mathfrak{B}$ of three-dimensional subspaces of $V$ such that every two-dimensional subspace of $V$ is contained in exactly one element $B$ of $\mathfrak{B}$. The question of existence of $q$-Fano planes over any field has been open since the 1970s and has attracted considerable attention in the special case that $F$ is finite. Here we show the existence of $2$-$(7, 3, 1)_F$-subspace designs over certain infinite fields $F$, including (among others) $\mathbb{Q}, \mathbb{R}$ and $\mathbb{F}_q(x, y, z)$ for $q$ odd. The space $V$ is the 7-dimensional space of imaginary elements in a Cayley division algebra $O$ over $F$ and $\mathfrak{B}$ consists of the intersections with $V$ of all 4-dimensional subalgebras of $O$. We will present the required background on Cayley algebras in a self-contained fashion. Next we study what happens if we apply the same procedure to split (rather than division) Cayley algebras. By identifying all four-dimensional subalgebras of these, we show that in that case our construction still yields an inclusion minimal $(7, 3, 2)$ $q$-covering design. That is: every two-dimensional subspace of $V$ is contained in at least one element of the resulting set $\mathfrak{B}$ of three-dimensional subspaces of $V$ and no proper subset of $\mathfrak{B}$ has this property. However none of these $q$-covering designs are $q$-Fano planes. In the case that $F$ is finite we compute the number of elements of $\mathfrak{B}$. We also give a purely combinatorial construction of our $q$-Fano planes and $q$-covering designs for an abstract 7-dimensional $F$-vector space $V$ by identifying the collection $\mathfrak{B}$ as a subvariety of the Grassmanian $Gr_3(V)$ defined entirely in terms of the classical Fano plane.
