Gosset polytopes in Picard groups of del Pezzo surfaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1268.14038
DOI: 10.4153/CJM-2011-063-6
author: Jae-Hyouk Lee
published in: Canadian Journal of Mathematics
publication date: 2012-04-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.3025
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 22E99
Mathematics Subject Classification ID: 51F15
zbMATH DE Number: 6023029
zbMATH Keywords: Gosset polytope
zbMATH Keywords: Picard group
zbMATH Keywords: des Pezzo surface
MaRDI profile type: Publication
arXiv ID: 0905.3025
OpenAlex ID: W2963664828
title: Gosset polytopes in Picard groups of del Pezzo surfaces
review text: The correspondence between the geometry of a del Pezzo surface \(S_r\) and the geometry of \(r\)-dimensional Gosset polytopes \((r-4)_{21}\) is studied. The Gosset polytopes are in \({\text{ Pic}}~S_r\otimes{\mathbb Q}\) whose vertices are lines. It is shown that the divisor classes in \({\text{ Pic}}~ S_r\) correspond to skew lines, exceptional systems, and rulings. As an application, the monoidal transform for lines is used in order to study the local geometry of the polytope \((r-4)_{21}\). Furthermore, it is shown that the Gieser transform and the Bertini transform induce a symmetry of the polytopes \(3_{21}\) and \(4_{21}\).
reviewed by: Boris Odehnal
description: scientific article; zbMATH DE number 6023029