Abstract
In the projective space \(\textrm{PG}(3,q)\), we consider the orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of \(\textrm{PG}(3,q)\) are partitioned into classes, each of which is a union of line orbits. In this paper, all classes of lines consisting of a unique orbit are found. For the remaining line types, with one exception, it is proved that they consist exactly of two or three orbits; sizes and structures of these orbits are determined. Also, the subgroups of the stabilizer group of the twisted cubic fixing lines of the orbits are obtained. Problems which remain open for one type of lines are formulated and, for \(5\le q\le 37\) and \(q=64\), a solution is provided.
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The research of A. A. Davydov was supported in part by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INDAM) (Contract No. U-UFMBAZ-2019-000160, 11.02.2019). The research of S. Marcugini and F. Pambianco was supported in part by University of Perugia (Project No. 98751: Strutture Geometriche, Combinatoria e loro Applicazioni, Base Research Fund 2017–2019).
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Davydov, A.A., Marcugini, S. & Pambianco, F. Orbits of Lines for a Twisted Cubic in \(\textrm{PG}(3,q)\). Mediterr. J. Math. 20, 132 (2023). https://doi.org/10.1007/s00009-023-02279-4
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DOI: https://doi.org/10.1007/s00009-023-02279-4