Enumerative geometry of skew lines in ${\mathbb{P}}^3$ with a given associated finite group

Abstract

For any finite set $\mathcal{L}$ of 3 or more skew lines in ${\mathbb{P}}_{\overline{K}}^3$ over an algebraically closed field $\overline{K}$ of arbitrary characteristic, there is a canonical associated subgroup $G_{\mathcal{L}}$ of ${\mathrm{PGL}}_2(\overline{K})$. Given a finite subgroup $G\subset {\mathrm{PGL}}_2(\overline{K})$ we study which configurations of lines have $G_{\mathcal{L}}=G$. We derive an upper bound on the number $|\mathcal{L}|$ of lines in terms of the order $|G|$ of the group $G$ and as an application we classify up to projective equivalence which sets $\mathcal{L}$ in ${\mathbb{P}}_{\mathbb{C}}^3$ have $G_{\mathcal{L}}=G$ for certain finite nonabelian groups $G$.