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

2021/04/10 by Gülizar Günay, Günay, Gülizar, Michel Lavrauw +1 · 2 citations
Computer Science · Engineering · Mathematics · #Algebraic Geometry (math.AG) #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2104.04756

openalex publication_date 2021/04/10 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

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.

Cited by

• Incidence of lines, points, and planes in PG(3,q) with respect to the twisted cubic
• On the PGL₂(q)-orbits of lines of PG(3,q) and binary quartic forms in characteristic three

Related

• Incidence of lines, points, and planes in PG(3,q) with respect to the twisted cubic
• Twisted cubic and orbits of lines in PG(3,q), II
• Twisted cubic and orbits of lines in PG(3,q)
• Further results on orbits and incidence matrices for the class O₆ of lines external to the twisted cubic in PG(3,q)
• Orbits of the class O₆ of lines external with respect to the twisted cubic in PG(3,q)