On pencils of cubics on the projective line over finite fields of characteristic >3

doi:10.1016/j.ffa.2021.101960

Finite Fields and Their Applications

Volume 78, February 2022, 101960

https://doi.org/10.1016/j.ffa.2021.101960 Get rights and content

Abstract

In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on $PG (1, q)$ , for q odd and q not divisible by 3. These equivalence classes are considered as orbits of lines in $PG (3, q)$ , under the action of the subgroup $G ≅ PGL (2, q)$ of $PGL (4, q)$ which preserves the twisted cubic $C$ in $PG (3, q)$ . In particular we determine the point orbit distributions and plane orbit distributions of all G-orbits of lines which are contained in an osculating plane of $C$ , have non-empty intersection with $C$ , or are imaginary chords or imaginary axes of $C$ .

Section snippets

Introduction and motivation

Let $PG (n, q)$ denote the n-dimensional projective space over a finite field $F_{q}$ . We assume that q is an odd prime power and not divisible by 3. The projective linear group, denoted by $PGL (n + 1, q)$ , is defined as the group of projectivities induced by the action of the general linear group $GL (n + 1, q)$ on $PG (n, q)$ .

The cubic forms on $PG (1, q)$ form a four-dimensional vector space W, and subspaces of the projective space $PG (W)$ are called linear systems of cubics. One-dimensional linear systems are called

Preliminaries

In this section we review some theory and definitions used in our study. They are well known and can be extracted from standard textbooks on finite projective geometry.

A projective algebraic variety which is defined by the homogeneous polynomials $f_{1}, \dots, f_{t} \in F_{q} [X_{0}, X_{1}, . ., X_{n}]$ , is denoted by $Z (f_{1}, \dots, f_{t})$ . Given an algebraic variety $X$ defined (by homogeneous polynomials) over $F_{q}$ , the set of $F_{q^{n}}$ -rational points of $X$ in $PG (n, q^{n})$ is denoted by $X (F_{q^{n}})$ . For reasons of simplicity, these notations will

Lines contained in osculating planes

We start our classification with the lines which are the intersection of two osculating planes of the twisted cubic. The fact that these lines form one G-orbit is well known. For the sake of completeness, we have included an argument for this fact in the proof of the following lemma. The main contribution of the lemma is the determination of the point and plane orbit distributions of such lines.

Lemma 4

There is one orbit $L_{1} = O_{1}^{⊥}$ of external lines contained in the intersection of osculating planes of the

Lines meeting the twisted cubic

Since the polar of a line meeting the twisted cubic is a line contained in an osculating plane of the twisted cubic, the orbit distributions determined in the previous sections imply the orbit distributions of all lines meeting the twisted cubic. In this section we collect the results for the G-orbits which consist of lines meeting the twisted cubic, but not contained in an osculating plane.

The polar of a line belonging to the G-orbit $L_{1}$ is a chord of the twisted cubic $C$ , and so the orbit

Orbits of imaginary chords and imaginary axes

We start with the imaginary chords.

Lemma 18

There is one orbit $L_{9} = O_{3}$ of imaginary chords of the twisted cubic $C$ in $PG (3, q)$ with plane orbit distribution $[0, 0, 0, q + 1, 0]$ . The point orbit distribution of an imaginary chord ℓ is $O D_{0} (ℓ) = [0, 0, \frac{(q + 1)}{3}, 0, \frac{2 (q + 1)}{3}],$ if −3 is a non-square in $F_{q}$ and $O D_{0} (ℓ) = [0, 0, 0, q + 1, 0],$ if −3 is a square in $F_{q}$ . (See Fig. 10.)

Proof

Consider an imaginary chord ℓ. Let P, $P^{τ}$ denote the two points of $ℓ (F_{q^{2}}) \cap C (F_{q^{2}})$ (here τ denotes the Frobenius collineation of $PG (3, q^{2})$ ). Since a plane $Π = 〈 ℓ, P (t) 〉$

Conclusions

In this paper we determined the point orbit distributions and plane orbit distributions of ten orbits of lines in $PG (3, q)$ with respect to the twisted cubic, for q odd and not divisible by three. Our results are summarised in Table 1.

The remaining G-orbits of lines partition the class $O_{6}$ consisting of external lines to $C$ which are not imaginary chords and which are not contained in osculating planes. There are in total $q (q - 1) (q^{2} - 1)$ such lines. As far as we know the G-orbits of these lines are

Acknowledgement

The authors would like to thank Aart Blokhuis and Ruud Pellikaan for interesting discussions on the topic.

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^☆: The authors were supported by The Scientific and Technological Research Council of Turkey, TÜBİTAK (project no. 118F159).

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On pencils of cubics on the projective line over finite fields of characteristic >3☆

Abstract

Section snippets

Introduction and motivation

Preliminaries

Lines contained in osculating planes

Lines meeting the twisted cubic

Orbits of imaginary chords and imaginary axes

Conclusions

Acknowledgement

Discrete Math.

Linear Algebra Appl.

Arcs in finite projective spaces

EMS Surv. Math. Sci.

On the classification of semifield flocks

Adv. Math.

On planes through points off the twisted cubic in PG(3,q) and multiple covering codes

The extended coset leader weight enumerator of a twisted cubic code

Applications of line geometry over finite fields. I. The twisted cubic

Geom. Dedic.

Twisted cubic and orbits of lines in PG(3,q)

Twisted cubic and plane-line incidence matrix in PG(3,q)

On planes through points off the twisted cubic in $PG (3, q)$ and multiple covering codes

Twisted cubic and orbits of lines in $PG (3, q)$

Twisted cubic and plane-line incidence matrix in $PG (3, q)$