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 $${\mathbb {C}}$$ C and $$\mathbb {R}$$ R in 1906–1907. The analogous problem for finite fields $$\mathbb {F}_q$$ 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 $$\mathrm {PG}(5,q)$$ 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 $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) viewed as a subgroup of $$\mathrm {PGL}(6,q)$$ PGL ( 6 , q ) . Since q is odd, the orbits of lines in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) under this action correspond to the aforementioned pencils of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q ) . The main contribution of this paper is to determine the line-orbit distribution of a plane $$\pi $$ π corresponding to a net of rank one, namely, the number of lines in $$\pi $$ π belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $$\pi $$ 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 $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) , and hence the orbit sizes.

A classification of planes intersecting the Veronese surface over finite fields of even order

“…Alternatively, V(F q ) is the set of points (u 00 , u 01 , u 02 , u 11 , u 12 , u 22 ) ∈ PG(5, q) for which the rank of the symmetric matrix defined by [u ij ] is 1. We use the same notation as in [19,20], where a point P = (y 0 , y 1 , y 2 , y 3 , y 4 , y 5 ) of PG(5, q) is represented by a symmetric 3 × 3 matrix…”

“…Clearly, rank-distributions are K-invariants. The next definition introduces stronger K-invariants, first introduced in [20] and more formally in [1], which play a fundamental role in classifying subspaces of PG(5, q). Definition 2.1.…”

“…For example, if q is even and W is a solid of PG(5, q), then its line-orbit distribution completely determines its K-orbit [1]. In [20] it was shown that the line-orbit distributions are a complete invariant for the K-orbits of planes meeting the Veronese surface V(F q ), q odd. The line orbits themselves were determined (for all q) in [18], based on the canonical forms of tensors of format (2, 3, 3) from [21].…”

“…Indeed, K-orbits of points and hyperplanes are easily determined (see Section 2), lines were classified in [18], and solids in [1]. Moreover, planes in PG (5, q) intersecting the Veronese surface V(F q ) in at least one point, q odd, are classified in [19,20]. Therefore, to complete the classification of subspaces of PG (5, q), we need to classify planes which are disjoint from V(F q ) for all q and planes meeting V(F q ) for q even.…”

Solids in the space of the Veronese surface in even characteristic

“…Powerful combinatorial invariants (called orbit distributions, see Section 2 below) introduced in [16] and [3], were used to determine the number of the different types of conics contained in pencils of conics in PG (2, q). In [18] it was shown that, in the case of nets of rank one over finite field of odd order, these combinatorial invariants give in fact a complete invariant for the equivalence classes of these nets.…”