Unexpected Properties of the Klein Configuration of 60 Points in P3

Abstract

Felix Klein in the course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of 60 points in ${\mathbb{P}}^3$. This configuration has appeared in various guises, perhaps most notably as the configuration of points dual to the 60 reflection planes in the group ${\mathit{G}}_{31}$ in the Shephard–Todd list.

In the present note we show that the 60 points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree 6. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated with the configuration of 60 points is a cone with a single singularity of multiplicity 6 and the other has three singular points of multiplicities $4,2$ and 2. Secondly, Chiantini and Migliore observed in 2020 that there are nontrivial sets of points in ${\mathbb{P}}^3$ with the surprising property that their general projection to ${\mathbb{P}}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of 24 points in ${\mathbb{P}}^3$, which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.