Collinearly complete sets and finite subgroups from configurations of skew $n$-planes in $\mathbb P^{2n+1}_K$

Abstract

We study collinearly complete finite sets of points arising from configurations of pairwise skew $n$-planes in $\mathbb {P}^{2n+1}_K$. To such a configuration we associate a groupoid generated by the natural collinearity correspondences between the $n$-planes, and we investigate the geometry of its finite orbits. In characteristic zero, we prove a rigidity result for the case in which the associated group is finite cyclic. After a suitable normalization, the matrices defining the configuration are simultaneously diagonalizable and admit a common two-block decomposition. Consequently, the orbit of a general point meets each $n$-plane in a collinear set, and the full orbit is contained in a distinguished projective $3$-space. This reduces the geometry of such orbits to the classical case of skew lines in $\mathbb {P}^3_{\mathbb C}$: inside the distinguished $\mathbb{P}^3_{\mathbb C}$, the orbit is geproci. We also prove that finite unions of general orbits are cut out set-theoretically from the union of the $n$-planes by a reducible surface. Finally, we show that this characteristic-zero rigidity fails in positive characteristic. In characteristic $2$, we construct cyclic examples whose orbit slices are Fano plane configurations, and we exhibit a genuinely higher-dimensional finite non-cyclic example in projective $5$-space with associated group ${\rm PGL}_3(\mathbb F_2)\cong {\rm PSL}_2(\mathbb F_7)$.