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

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 similar or equal to 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. (C) 2021 Elsevier Inc. All rights reserved.