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

Piotr Pokora, Tomasz Szemberg, Justyna Szpond

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. \

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

Abstract

Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of points in . This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the reflection planes in the group in the Shephard-Todd list. In the present note we show that the 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 . Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of points is a cone with a single singularity of multiplicity and the other has three singular points of multiplicities and . Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in with the surprising property that their general projection to 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 points in 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. \

Paper Structure

This paper contains 6 sections, 9 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Klein's arrangement $(60_{15})$ -- a historical outline
  3. A finite Heisenberg group and a group of order $80$
  4. Unexpected hypersurfaces associated with $Z_{60}$
  5. Projections of $Z_{60}$
  6. More subsets of $Z_{60}$ with the geproci property

Key Result

Lemma 3.1

The set $Z_{60} \subset \mathbb{P}^{3}$ contains all the intersection points of lines in $\mathbb L_{30}$. Moreover, for every point $P\in Z_{60}$ there are exactly three lines from $\mathbb L_{30}$ passing through $P$.

Figures (1)

  • Figure 1: Points from $Z_{60}$ in the $w=0$ plane

Theorems & Definitions (25)

  • Lemma 3.1: Line-point incidences
  • Lemma 3.2: Generators of $I(Z_{60})$
  • proof
  • Definition 4.1
  • Definition 4.2: Unexpected cone property
  • Theorem 4.3: Unexpected cone property of $Z_{60}$
  • proof
  • Theorem 4.4: Unexpected surface with $3$ general points
  • proof
  • Remark 4.5
  • ...and 15 more