Unexpected properties of the Klein configuration of $60$ points in ${\mathbb P}^3$

TL;DR

This paper explores the Klein configuration of 60 points in projective 3-space, revealing unexpected degree 6 surfaces, and shows that this configuration and its subsets project to complete intersections in the plane, connecting to recent research on special point sets.

Contribution

It demonstrates that the Klein configuration produces two novel unexpected surfaces and that it and its subsets project to complete intersections, extending recent findings on special point configurations.

Findings

Two unexpected degree 6 surfaces associated with the Klein configuration.

The Klein configuration and its subsets project to complete intersections in ${f P}^2$.

The configuration is not a grid, contrasting with previously studied sets.

Abstract

Felix Klein in 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 post notably as the configuration of points dual to the $60$ reflection planes in the group $G_{31}$ 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 to 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 non-trivial 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. \