论文标题
$ {\ mathbb p}^3 $的$ 60 $点的Klein配置的意外属性
Unexpected properties of the Klein configuration of $60$ points in ${\mathbb P}^3$
论文作者
论文摘要
费利克斯·克莱因（Felix Klein）在研究常规二十面体及其对称性的过程中遇到了高度对称的配置，$ {\ mathbb p}^3 $中的$ 60 $点。这种配置已经出现在各种措施中，也许尤其是作为$ 60 $反射平面的点的配置，$ g_ {31} $在Shephard-todd列表中。 在本说明中，我们表明，从最近启动的两种研究途径的角度来看，$ 60 $的积分表现出有趣的属性。首先，它们产生了两个完全不同的意外程度的$ 6 $的意外表面。 Cook II，Harbourne，Migliore，Nagel于2018年引入了意外的高度。其次，Chiantini和Migliore在2020年观察到，$ {\ Mathbb p}^3 $中有一组非平凡的点，具有令人惊讶的属性，即他们对$ {\ Mathbb p}^2 $的一般投影是一个完整的交叉点。他们找到了一个这样的家族，他们称之为网格。他们论文的附录描述了$ {\ mathbb p}^3 $中的$ 24 $点的异国配置，这不是网格，但具有出色的属性，即其一般投影是完整的交叉点。我们表明，klein配置也不是网格，它将其投影到完整的交集。我们还确定其适当的子集，享受相同的属性。 \ \
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. \
