On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms in characteristic three
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912539155103744 |
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| author | Kaipa, Krishna Pradhan, Puspendu |
| author_facet | Kaipa, Krishna Pradhan, Puspendu |
| contents | We consider the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $\mathbb{F}_q$ into orbits of the group $PGL_2(q)$ 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 $PGL_2(q)$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms in characteristic three Kaipa, Krishna Pradhan, Puspendu Combinatorics 51E20, 51N35, 14N10, 05B25, 05E10, 05E14, 05E18 We consider the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $\mathbb{F}_q$ into orbits of the group $PGL_2(q)$ 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 $PGL_2(q)$-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. |
| title | On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms in characteristic three |
| topic | Combinatorics 51E20, 51N35, 14N10, 05B25, 05E10, 05E14, 05E18 |
| url | https://arxiv.org/abs/2508.11229 |