Unexpected properties of the Klein configuration of $60$ points in ${\mathbb P}^3$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866929411970826240 |
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| author | Pokora, Piotr Szemberg, Tomasz Szpond, Justyna |
| author_facet | Pokora, Piotr Szemberg, Tomasz Szpond, Justyna |
| contents | 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. \ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_08863 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Unexpected properties of the Klein configuration of $60$ points in ${\mathbb P}^3$ Pokora, Piotr Szemberg, Tomasz Szpond, Justyna Algebraic Geometry Commutative Algebra Combinatorics 14C20 and 14N20 and 13A15 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. \ |
| title | Unexpected properties of the Klein configuration of $60$ points in ${\mathbb P}^3$ |
| topic | Algebraic Geometry Commutative Algebra Combinatorics 14C20 and 14N20 and 13A15 |
| url | https://arxiv.org/abs/2010.08863 |