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

TL;DR

This paper investigates the combinatorial invariants of pencils of cubic curves on the projective line over finite fields with characteristic greater than 3, focusing on orbit distributions under group actions.

Contribution

It classifies and determines the orbit distributions of lines related to cubics in projective space under specific group actions, extending understanding of geometric configurations over finite fields.

Findings

Determined point orbit distributions of lines in specific geometric configurations.

Computed plane orbit distributions for lines intersecting or tangent to the twisted cubic.

Classified orbits of lines as imaginary chords or axes of the twisted cubic.

Abstract

In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on $\mathrm{PG}(1,q)$, for $q$ odd and $q$ not divisible by 3. These equivalence classes are considered as orbits of lines in $\mathrm{PG}(3,q)$, under the action of the subgroup $G\cong \mathrm{PGL}(2,q)$ of $\mathrm{PGL}(4,q)$ which preserves the twisted cubic $\mathcal{C}$ in $\mathrm{PG}(3,q)$. 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 $\mathcal{C}$, have non-empty intersection with $\mathcal{C}$, or are imaginary chords or imaginary axes of $\mathcal{C}$.