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

Abstract

We study finite groups arising from configurations of pairwise skew lines in ${\mathbb{P}}_K^3$. 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 ${\mathbb{P}}_K^3$, 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.

Revised version. Fixed the hypothesis of Theorem 4.3 and reformulated Lemma 2.5, Theorem 2.6, and Corollary 2.7 to clarify the group generators. Minor exposition changes