On the PGL₂(q)-orbits of lines of PG(3,q) and binary quartic forms in characteristic three

2025/08/15 by Kaipa, Krishna, Pradhan, Puspendu · 1 citation
#05B25 #05E10 #05E14 #05E18 #14N10 #51E20 #51N35 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.11229

Abstract

We consider the problem of classifying the lines of the projective 3-space PG(3,q) over a finite field \mathbbF_q into orbits of the group PGL₂(q) of linear symmetries of the twisted cubic C. The problem has been solved in literature in characteristic different from 3, and in this work, we solve the problem in characteristic 3. We reduce this problem to another problem, which is the classification of binary quartic forms into PGL₂(q)-orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.

Citations

• On the PGL₂(q)-orbits of lines of PG(3,q) and binary quartic forms
• On pencils of cubics on the projective line over finite fields of characteristic >3
• The extended coset leader weight enumerator of a twisted cubic code

Cited by

• Incidence of lines, points, and planes in PG(3,q) with respect to the twisted cubic

Related

• On the PGL₂(q)-orbits of lines of PG(3,q) and binary quartic forms
• 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)
• Twisted cubic and orbits of lines in PG(3,q), II
• Further results on orbits and incidence matrices for the class O₆ of lines external to the twisted cubic in PG(3,q)