Related papers: MAT-free reflection arrangements
Only few categories of free arrangements are known in which Terao's conjecture holds. One of such categories consists of $3$-arrangements with unbalanced Ziegler restrictions. In this paper, we generalize this result to arbitrary…
Let W be a finite complex reflection group acting on the complex vector space V and let A(W) = (A(W), V) be the associated reflection arrangement. In an earlier paper by the last two authros, we classified all inductively free reflection…
We prove the addition-deletion theorems for the Solomon-Terao polynomials, which have two important specializations. Namely, one is to the characteristic polynomials of hyperplane arangements, and the other to the Poincar\`{e} polynomials…
We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present…
This paper studies the algebraic structure of a new class of hyperplane arrangement $A$ obtained by deleting two hyperplanes from a free arrangement. We provide information on the minimal free resolutions of the logarithmic derivation…
We study two-dimensional integrable $N=1$ supersymmetric theories (without topological charges) in the presence of a boundary. We find a universal ratio between the reflection amplitudes for particles that are related by supersymmetry and…
In the category of free arrangements, inductively and recursively free arrangements are important. In particular, in the former, the conjecture by Terao asserting that freeness depends only on combinatorics holds true. A long standing…
In this note we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. This addition theorem allows us to find various relationships between non-free arrangements,…
In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $\mathbb{Q}$.
We introduce a new class of arrangements of hyperplanes, called (strictly) plus-one generated arrangements, from algebraic point of view. Plus-one generatedness is close to freeness, i.e., plus-one generated arrangements have their…
Extending earlier work by Sommers and Tymoczko, in 2016 Abe, Barakat, Cuntz, Hoge, and Terao established that each arrangement of ideal type $\mathcal{A}_\mathcal{I}$ stemming from an ideal $\mathcal{I}$ in the set of positive roots of a…
Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a…
We study the free path problem, i.e., if we are given two free arrangements of hyperplanes, then we can connect them by free arrangements or not. We prove that if an arrangement $\mathcal{A}$ and $\mathcal{A} \setminus \{H,L\}$ are free,…
In this paper, we show how to compute using Fitting ideals the nonfree locus of the moduli space of arrangements of a rank $3$ simple matroid, i.e., the subset of all points of the moduli space which parametrize nonfree arrangements. Our…
We prove the torsion freeness of the decomposable Orlik--Solomon algebra of a simple matroid on ground set $[n]$. In the class of hypersolvable \& non-supersolvable complex hyperplane arrangements, the torsion freeness, in a certain degree,…
Tame arrangements were informally introduced by Orlik and Terao for the study of Milnor fibers of hyperplane arrangements. After that, tame arrangements have been applied to a lot of researches on arrangements including freeness, master…
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…
In 2006, Gao and Zeng \cite{GZ} gave the free field realizations of highest weight modules over a class of extended affine Lie algebras. In the present paper, applying the technique of localization to those free field realizations, we…
In this article we show that any free hyperplane arrangement with exponents 1's and 2's is a supersolvable arrangement. We conjecture that any free arrangement with exponents 1's, 2's and exactly one 3, is also supersolvable, and we show…
The main result of the paper establishes the irreducibility of a large family of nonzero central charge induced modules over Affine Lie algebras for any non standard parabolic subalgebra. It generalizes all previously known partial results…