dc.contributor.author Pokora, Piotr dc.contributor.author Szemberg, Tomasz dc.contributor.author Szpond, Justyna dc.date.accessioned 2020-10-07T13:11:26Z dc.date.available 2020-10-07T13:11:26Z dc.date.issued 2020-10-07 dc.identifier.uri http://publications.mfo.de/handle/mfo/3799 dc.description.abstract Felix Klein in course of his study of the regular 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. en_US 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. dc.language.iso en_US en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2020,19 dc.subject Base loci en_US dc.subject Cones en_US dc.subject Complete intersections en_US dc.subject Finite Heisenberg group en_US dc.subject Klein configuration en_US dc.subject Projections en_US dc.subject Reflection group en_US dc.subject Special linear systems en_US dc.subject Unexpected hypersurfaces en_US dc.title Unexpected Properties of the Klein Configuration of 60 Points in $\mathbb{P}^3$ en_US dc.type Preprint en_US dc.rights.license Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. de dc.rights.license This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties. en dc.identifier.doi 10.14760/OWP-2020-19 local.scientificprogram Research in Pairs 2020 en_US local.series.id OWP-2020-19 en_US local.subject.msc 14 en_US local.subject.msc 13 en_US dc.identifier.urn urn:nbn:de:101:1-2020101510090616060079 dc.identifier.ppn 1738643948
﻿