标题： 圆锥曲线网的组合不变量在$\text{PG}(2,q)$中 显示英文标题

标题： Combinatorial invariants for nets of conics in $\text{PG}(2,q)$

摘要： 在射影平面上对二次曲线线性系统的分类问题可以追溯到Jordan，他在1906-1907年对 over $\mathbb{C}$ 和 $\mathbb{R}$ 的圆锥曲线束（一维系统）进行了分类。 对于有限域 $\mathbb{F}_q$ ，其中 $q$ 为奇数的情况，该类似问题由Dickson在1908年解决。 1914年，Wilson试图对奇特征有限域上的网（二维系统）进行分类，但他的分类不完整且包含一些错误。 在最近的一篇文章中，我们完成了Wilson对秩一网的分类，即那些包含重直线的网。 本文的目的是引入并计算这些网的一些组合不变量，我们预计这些不变量将在各种应用中有所帮助。 我们的方法是几何的，因为我们把一个秩一网看作是 $\text{PG}(5,q)$ 中的一个平面，它至少与二次曲面Veronesean相交于一点；然后，如果相应的平面在 $\text{PGL}(3,q)$ 作为 $\text{PGL}(6,q)$ 的子群作用下属于同一轨道，则两个这样的网是等价的。 我们之前已经确定了在该作用下$\text{PG}(5,q)$中直线的轨道，这些轨道对应于$\text{PG}(2,q)$中上述的圆锥曲线束。 本文的主要贡献是确定对应于一个秩为一的网的平面$\pi$中直线轨道的分布，即$\pi$中属于每个直线轨道的直线数量。 结果表明，这一不变量列表完全决定了$\pi$的轨道，我们将利用这一事实在未来的工作中开发一种计算秩为一的网轨道的高效算法。 作为更直接的应用，我们还确定了$\text{PGL}(3,q)$中秩为一的网的稳定子，从而确定了轨道的大小。 显示英文摘要

摘要： 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 $\pi$ corresponding to a net of rank one, namely, the number of lines in $\pi$ belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $\pi$, 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.