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

2021/03/23 by Davydov, Alexander A., Marcugini, Stefano, Pambianco, Fernanda
#51E21 #51E22 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.12655

Abstract

In the projective space PG(3,q), we consider the orbits of lines under the stabilizer group of the twisted cubic. It is well known that the lines can be partitioned into classes every of which is a union of line orbits. All types of lines forming a unique orbit are found. For the rest of the line types (apart from one of them) it is proved that they form exactly two or three orbits; sizes and structures of these orbits are determined. Problems remaining open for one type of lines are formulated. For 5≤ q≤37 and q=64, they are solved.

Related

• Twisted cubic and orbits of lines in PG(3,q), II
• Orbits of the class O₆ of lines external with respect to the twisted cubic 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)
• Incidence of lines, points, and planes in PG(3,q) with respect to the twisted cubic
• Twisted cubic and plane-line incidence matrix in PG(3,q)