相关论文: Divisionally free arrangements of hyperplanes
We call a poset factorable if its characteristic polynomial has all positive integer roots. Inspired by inductive and divisional freeness of a central hyperplane arrangement, we introduce and study the notion of inductive posets and their…
We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane and that the property of being nearly free is combinatorial for line arrangements of up to 12…
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 introduce and prove the $n$-dimensional Pizza Theorem: Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{R}^{n}$. If $K$ is a measurable set of finite volume, the {pizza quantity} of $K$ is the alternating sum of the volumes of…
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…
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for…
Inspired by Terao's freeness conjecture, we examine Ziegler pairs, which are pairs of hyperplane arrangements that share the same underlying matroid but have different modules of logarithmic derivations. In this paper, we present a general…
A reduced divisor on a nonsingular variety defines the sheaf of logarithmic 1-forms. We introduce a certain coherent sheaf whose double dual coincides with this sheaf. It has some nice properties, for example, the residue exact sequence…
The extended Shi and Catalan arrangements are well investigated arrangements. In this paper, we prove that the cone of the extended Catalan arrangement of type A is always hereditarily free, while we determine the dimension in which the…
In this article we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual-partition formula. Then it immediately follows that there exists a saturated free filtration of the cone of any affine…
A topological hyperplane is a subspace of R^n (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in R^n is a finite set H such that k topological…
Let $\mathscr A$ be a Coxeter arrangement of rank $\ell$. In 1987 Orlik, Solomon and Terao conjectured that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ -- when listed in increasing order -- are realized as the…
In this note, we give two applications of \cite[Theorem 3.1]{Hwang}. We first study the free family $\mathcal{K}$ of hyperplane sections of the smooth hypersurface $X\subset\mathbb{P}^{n+1}$ of degree $d\ge 3$. We prove that $X$ is…
Holm introduced $m$-free $\ell$-arrangements which is a generalization of free arrangements, while he asked whether all $\ell$-arrangements are $m$-free for $m$ large enough. Recently Abe and the author verified that this question is in the…
Let $\mathcal C :f=0$ be a curve arrangement in the complex projective plane. If $\mathcal C$ contains a curve subarrangement consisting of at least three members in a pencil, then one obtains an explicit syzygy among the partial…
Let $\mathcal{A} = \mathcal{A}(W)$ be the reflection arrangement of the finite complex reflection group $W$. By Terao's famous theorem, the arrangement $\mathcal{A}$ is free. In this paper we classify all reflection arrangements which…
Athanasiadis studied arrangements obtained by adding shifted hyperplanes to the braid arrangement. Similarly, Bailey studied arrangements obtained by adding tilted hyperplanes to the braid arrangement. These two kinds of arrangements are…
We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the…
Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A = A(W) be the associated hyperplane arrangement of W. Terao has shown that each such reflection arrangement A is free. There is the stronger…
In this paper, we study the class of free multiarrangements of hyperplanes. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over the rationals.