Finite subgroups of $\operatorname{PGL}_2(K)$ arising from configurations of skew lines in $\mathbb{P}^3_K$

Abstract

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.