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

TL;DR

This paper introduces combinatorial invariants for nets of conics of rank one in finite projective planes, aiding classification and orbit determination under group actions, with applications to stabilizer and orbit size calculations.

Contribution

It computes the line-orbit distribution of nets of rank one in PG(2,q), providing a complete invariant for their classification under PGL(3,q).

Findings

Line-orbit distribution uniquely determines the net's orbit.

Stabilizers of nets of rank one are explicitly determined.

Results facilitate efficient orbit computation algorithms.

Abstract

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.