MFO Scientific ProgramOWLF 2022
Tran, Tan Nhat
MetadataShow full item record
In [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper. More specifically, let $\mathscr A$ be a free arrangement of rank $\ell$. Suppose that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ - when listed in increasing order - are realized as the exponents of a free restriction of $\mathscr A$ to some intersection of reflecting hyperplanes of $\mathscr A$ of dimension $d$. Following [MR21], we call such an arrangement $\mathscr A$ with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call $\mathscr A$ flag-accurate. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.
The following license files are associated with this item: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.
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.
Showing items related by title, author, creator and subject.
[SNAP-2014-005-EN] Harbourne, Brian; Szemberg, Tomasz (Mathematisches Forschungsinstitut Oberwolfach, 2014)We discuss certain open problems in the context of arrangements of lines in the plane.
[OWP-2018-21] Douglass, J. Matthew; Pfeiffer, Götz; Röhrle, Gerhard (Mathematisches Forschungsinstitut Oberwolfach, 2018-10-22)We refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group W. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit ...
[OWP-2017-14] Hoge, Torsten; Röhrle, Gerhard (Mathematisches Forschungsinstitut Oberwolfach, 2017-04-30)Let $A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $A''$ of $A$ to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger ...