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

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 (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 mathrm {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 mathrm {PGL}(3,q) viewed as a subgroup of mathrm {PGL}(6,q). Since q is odd, the orbits of lines in mathrm {PG}(5,q) under this action correspond to the aforementioned pencils of conics in mathrm {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 mathrm {PGL}(3,q), and hence the orbit sizes.