Finite subgroups of $\operatorname{PGL}_2(K)$ arising from configurations of skew lines in $\mathbb{P}^3_K$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914613645279232 |
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| author | Favacchio, Giuseppe |
| author_facet | Favacchio, Giuseppe |
| contents | We study finite groups arising from configurations of pairwise skew lines in $\mathbb{P}^3_K$. To such a configuration ${L}$ one associates a group $G_{L}\subset \mathrm{PGL}_2(K)$ acting on each line, and we investigate which finite subgroups of $\mathrm{PGL}_2(K)$ can occur in this way.
Our main tool is a matrix description of skew lines in $\mathbb{P}^3_K$, which gives explicit generators for $G_{L}$ in terms of matrices in $\mathrm{GL}_2(K)$. In the abelian case, we prove that the relevant matrices are simultaneously upper triangular and obtain explicit families realizing cyclic groups and elementary abelian $p$-groups. In the non-abelian case, we show that, in non-modular characteristic, no dihedral group $D_n$ with $n\ge 3$ can occur, while configurations realizing $A_4$, $S_4$, and $A_5$ are constructed explicitly.
These results also yield new examples of point sets whose general projection is a complete intersection. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19811 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite subgroups of $\operatorname{PGL}_2(K)$ arising from configurations of skew lines in $\mathbb{P}^3_K$ Favacchio, Giuseppe Algebraic Geometry Commutative Algebra Group Theory 14N20, 14L30, 20H20 We study finite groups arising from configurations of pairwise skew lines in $\mathbb{P}^3_K$. To such a configuration ${L}$ one associates a group $G_{L}\subset \mathrm{PGL}_2(K)$ acting on each line, and we investigate which finite subgroups of $\mathrm{PGL}_2(K)$ can occur in this way. Our main tool is a matrix description of skew lines in $\mathbb{P}^3_K$, which gives explicit generators for $G_{L}$ in terms of matrices in $\mathrm{GL}_2(K)$. In the abelian case, we prove that the relevant matrices are simultaneously upper triangular and obtain explicit families realizing cyclic groups and elementary abelian $p$-groups. In the non-abelian case, we show that, in non-modular characteristic, no dihedral group $D_n$ with $n\ge 3$ can occur, while configurations realizing $A_4$, $S_4$, and $A_5$ are constructed explicitly. These results also yield new examples of point sets whose general projection is a complete intersection. |
| title | Finite subgroups of $\operatorname{PGL}_2(K)$ arising from configurations of skew lines in $\mathbb{P}^3_K$ |
| topic | Algebraic Geometry Commutative Algebra Group Theory 14N20, 14L30, 20H20 |
| url | https://arxiv.org/abs/2512.19811 |