Combinatorial invariants for nets of conics in PG (2 , q)

Official URL: https://dx.doi.org/10.1007/s10623-021-00881-9

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 C and R in 1906–1907. The analogous problem for finite fields Fq 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 (for q odd) 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 PG (5 , q) , q odd, 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 PGL (3 , q) viewed as a subgroup of PGL (6 , q). Since q is odd, the orbits of lines in PG (5 , q) under this action correspond to the aforementioned pencils of conics in 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 PGL (3 , q) , and hence the orbit sizes.