The largest intersection lattice of a discriminantal arrangement
type: Item
instance of: scholarly article
title: The largest intersection lattice of a discriminantal arrangement
zbMATH Open document ID: 0965.52008
author: Christos A. Athanasiadis
published in: Beiträge zur Algebra und Geometrie
publication date: 2000-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/228055
full work available at URL: 1.html
review text: Let \(\mathcal A\) be an essential arrangement of \(n\) linear hyperplanes in \({\mathbb R}^k\), with normal vectors \(\alpha_1,\ldots,\alpha_n\). The discriminantal arrangement \(B({\mathcal A})\) associated with \(\mathcal A\) is the arrangement in \({\mathbb R}^n\) with normals the nonzero vectors of the form \[ \alpha_S=\sum_{i=1}^{k+1} (-1)^i\det(\alpha_{s_1},\ldots,\hat{\alpha}_{s_i},\ldots,\alpha_{s_{k+1}})e_i, \] where \(S=\{s_1<\cdots<s_{k+1}\}\) runs over all \((k+1)\)-subsets of \([n]\), \(\hat{\mathstrut}\) denoting an omitted vector. The complement of \(B({\mathcal A})\) can be identified with the set of all affine arrangements in \({\mathbb R}^k\) consisting of generic parallel translates of the hyperplanes of \(\mathcal A\). The construction of \(B({\mathcal A})\) is dual to the construction of fiber polytope from a given zonotope. The discriminantal arrangement \(B({\mathcal A})\) was introduced by \textit{Yu. I. Manin} and \textit{V. V. Shekhtman} [Adv. Stud. Pure Math. 17, 289-308 (1989; Zbl 0759.20002)] in the case where \(\mathcal A\) is itself a generic arrangement. Even in this special case the combinatorial type (esp. the intersection lattice) of \(B({\mathcal A})\) is not uniquely determined by that of \({\mathcal A}\). In fact the matroid of \(B({\mathcal A})\) is the dual of the adjoint of that of \({\mathcal A}\) determined by the realization \({\mathcal A}\). Different realizations of the same matroid may yield non-isomorphic adjoints. On the other hand, the combinatorial type of \(B({\mathcal A})\) is constant for \({\mathcal A}\) ranging over a proper Zariski-open set \({\mathcal O}(n,k)\) of ``sufficiently generic'' arrangements. \textit{M. M. Bayer} and \textit{K. A. Brandt} [J. Algebr. Comb. 6, No. 3, 229-246 (1997; Zbl 0884.52013)] conjectured that arrangements in \({\mathcal O}(n,k)\) are very generic, in the sense that their discriminantals have the largest number of flats in each rank among all discriminantal arrangements of esse…
Mathematics Subject Classification ID: 52B35
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 1460277
zbMATH Keywords: arrangement
zbMATH Keywords: discriminantal
zbMATH Keywords: generic
zbMATH Keywords: matroid
zbMATH Keywords: Dilworth completion
zbMATH Keywords: matroid of circuits
zbMATH Keywords: intersection lattice
reviewed by: Michael Falk
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1460277