论文标题
$ \ text {pg}（2，q）$中的圆锥网的组合不变性
Combinatorial invariants for nets of conics in $\text{PG}(2,q)$
论文作者
论文摘要
在19066----1907中，将圆锥形线性系统分类的问题至少可以追溯到约旦，后者将圆锥形的铅笔（一维系统）分类为$ \ mathbb {c} $和$ \ Mathbb {r} $在1906-----1907中。迪克森（Dickson）在1908年解决了有限领域的类似问题$ \ mathbb {f} _q $，$ q $奇数。1914年，威尔逊（Wilson）试图对圆锥（二维系统）进行分类，以奇特的特征性，但他的分类不足，但他的分类不足，并且包含了一些inccuracies。在最近的一篇文章中，我们完成了威尔逊对第一等级的网络分类，即包含重复线的净数。本文的目的是介绍和计算这些网的某些组合不变式，我们希望这些不变性在各种应用中使用。我们的方法是几何的，从某种意义上说，我们将排名第一的网作为$ \ text {pg}（5，q）$中的平面，至少在一个点上符合四边形Veronesean；当且仅当相应的平面属于$ \ text {pgl}（3，q）$的诱导动作下，被视为$ \ text {pgl}（6，q）$的诱导动作时，两个这样的网是等效的。我们以前已经在此操作下确定了$ \ text {pg}（5，q）$中的线条轨道，该行与上述圆锥形铅笔相对应，$ \ text {pg}（2，q）$。本文的主要贡献是确定对应于等级一的净净的平面$π$的线路分布，即，属于每个线路轨道的$π$中的线数。事实证明，此不变性列表完全确定了$π$的轨道，我们将在即将到来的工作中使用这一事实来开发一种有效的算法来计算给定等级一号净的轨道的轨道。作为一个更直接的应用，我们还确定了$ \ text {pgl}（3，q）$中排名第一的净稳定器，因此还确定了轨道尺寸。
The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $\mathbb{C}$ and $\mathbb{R}$ in 1906--1907. The analogous problem for finite fields $\mathbb{F}_q$ with $q$ odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson's classification of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in $\text{PG}(5,q)$ that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of $\text{PGL}(3,q)$ viewed as a subgroup of $\text{PGL}(6,q)$. We have previously determined the orbits of lines in $\text{PG}(5,q)$ under this action, which correspond to the aforementioned pencils of conics in $\text{PG}(2,q)$. The main contribution of this paper is to determine the line-orbit distribution of a plane $π$ corresponding to a net of rank one, namely, the number of lines in $π$ belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $π$, and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in $\text{PGL}(3,q)$, and hence the orbit sizes.
