Shorter Notes: A Quadrilateral-Free Arrangement of Sixteen Lines

1972 ◽  
Vol 34 (1) ◽  
pp. 317 ◽  
Author(s):  
G. J. Simmons
Keyword(s):  
Free Arrangement

scholarly journals Erratum to Free Arrangement and Rhombic Tilings

1997 ◽  
Vol 17 (3) ◽  
pp. 359-359
Author(s):  
P. H. Edelman ◽  
V. Reiner
Keyword(s):  
Free Arrangement

scholarly journals The freeness of Ish arrangements

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Takuro Abe ◽  
Daisuke Suyama ◽  
Shuhei Tsujie
Keyword(s):  
Catalan Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Free Arrangement ◽  
International Audience ◽  
New Interpretation ◽  
Necessary And Sufficient ◽  
Shi Arrangement

International audience The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q; t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free L’arrangement Ish a été introduit par Armstrong pour donner une nouvelle interprétation des nombres $q; t$-Catalan de Garsia et Haiman. Armstrong et Rhoades ont montré qu’il y avait des ressemblances frappantes entre l’arrangement Shi et l’arrangement Ish et ont posé des conjectures. L’une d’elles est de savoir si l’arrangement Ish est un arrangement libre ou pas. Dans cet article, nous vérifions que l’arrangement Ish est supersoluble et donc libre. De plus, on donne une condition nécessaire et suffisante pour que l’arrangement Ish réduit soit libre.

scholarly journals Supersolvable orders and inductively free arrangements

2017 ◽  
Vol 15 (1) ◽  
pp. 587-594
Author(s):  
Ruimei Gao ◽  
Xiupeng Cui ◽  
Zhe Li
Keyword(s):  
Free Arrangement ◽  
Supersolvable Arrangement

Abstract In this paper, we define the supersolvable order of hyperplanes in a supersolvable arrangement, and obtain a class of inductively free arrangements according to this order. Our main results improve the conclusion that every supersolvable arrangement is inductively free. In addition, we assert that the inductively free arrangement with the required induction table is supersolvable.

scholarly journals Nearly free curves and arrangements: a vector bundle point of view

2019 ◽  
pp. 1-24 ◽  
Author(s):  
Simone Marchesi ◽  
Jean Vallès
Keyword(s):  
Single Point ◽  
Point Of View ◽  
Line Bundles ◽  
Zero Section ◽  
Combinatorial Invariant ◽  
Counter Example ◽  
Free Arrangement ◽  
Finite Set ◽  
Interesting Family ◽  
Arrangement Of Lines

Abstract Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao’s conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.

scholarly journals The freeness of ideal subarrangements of Weyl arrangements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Takuro Abe ◽  
Mohamed Barakat ◽  
Michael Cuntz ◽  
Torsten Hoge ◽  
Hiroaki Terao
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Height Distribution ◽  
Free Arrangement ◽  
Positive Roots ◽  
International Audience

International audience A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula. Un arrangement de Weyl est défini par l’arrangement d’hyperplans du système de racines d’un groupe de Weyl fini. Quand un ensemble de racines positives est un idéal dans le poset de racines, nous appelons l’arrangement correspondant un sous-arrangement idéal. Notre théorème principal affirme que tout sous-arrangement idéal est un arrangement libre et que ses exposants sont donnés par la partition duale de la distribution des hauteurs, ce qui avait été conjecturé par Sommers-Tymoczko. En particulier, quand le sous-arrangement idéal est égal à l’arrangement de Weyl, notre théorème principal donne la célèbre formule par Shapiro, Steinberg, Kostant et Macdonald. La démonstration du théorème principal n’utilise pas de classification. Elle dépend fortement de la théorie des arrangements libres et diffère ainsi grandement des démonstrations précédentes de la formule.

scholarly journals The Module of Logarithmic p-Forms of a Locally Free Arrangement

2001 ◽  
Vol 241 (2) ◽  
pp. 699-719 ◽  
Author(s):  
Mircea Mustaţǎ ◽  
Henry K. Schenck
Keyword(s):  
Free Arrangement
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