311 problems

Terao's conjecture. Freeness is determined solely by the intersection lattice $\mathcal{L}$; equivalently, freeness is constant on $\mathcal{R}(\mathcal{L})$.

Let $\mathcal{A}$ be a real hyperplane arrangement, with magnitude homology groups $MH_{k,\ell}(\mathcal{A})$ and intersection lattice $L(\mathcal{A})$. Magnitude homology conjectu…

Let ${\mathcal{A}}$ be a rank $r$ central multiarrangement in $\mathbb{C}^n$ with $|{\mathcal{A}}|=m$. Define … Here $\Omega_n$ is the superspace algebra, $\lambda_L$ is the linear…

Let ${\mathcal{A}}$ be a free linear hyperplane arrangement and let $H\in{\mathcal{A}}$. Write ${\mathcal{A}}^H$ for the restriction of ${\mathcal{A}}$ to $H$. Orlik's conjecture.…

Let ${\mathcal A}$ be a central hyperplane arrangement, let $M({\mathcal A})$ be its complement, and let ${\mathcal R}^1({\mathcal A})$ denote the first resonance variety of…

Let $\mathcal{B}(n,k)$ denote the Manin–Schechtman arrangement for fixed $n$ and $k$, and let $P(n,k)$ be the combinatorial poset proposed by Bayer and Brandt to describe its large…

Let $G$ be the fundamental group of a complex hyperplane arrangement, let $\theta_n(G)$ denote its Chen ranks, and let $\mathcal{R}^1(G;\mathbb{C})\subseteq H^1(G;\mathbb{C})$ be i…

Let a $(4,d)$-multinet be a multinet on a hyperplane arrangement with four classes and weight $d$, and call it non-trivial when it is not the trivial multinet. Yuzvinsky's conjectu…

Suciu's conjecture. For every hyperplane arrangement $A$, this Chen rank formula holds for all sufficiently large $q$. The source records this as a conjectural extension of the sta…

Divisibility conjecture. For any $m,i\geq 0$, the polynomial $\widetilde{f^m_i}(x,y)+\widetilde{f^m_i}(y,x)$ is divisible by

Duval–Klivans–Martin conjecture. For every graph $G$, the Pak–Stanley labeling is a surjection from the regions of the $G$-Shi arrangement onto the $G$-parking functions.

Let $\mathcal{A}$ be a central arrangement in $\mathbb{K}^{n+1}$. Write $\operatorname{rgin}(J(\mathcal{A}))$ for the generic initial ideal of its Jacobian ideal, and set … A minim…

Let $\Phi$ be an irreducible root system with Coxeter number $h$, and let $V$ be its ambient vector space. For integers $a\leq b$, define the truncated affine Weyl arrangement … wh…

Let ${\mathcal D}$ be an arrangement of hyperplanes on a projective space, and let its combinatorics mean the incidence data of the arrangement. The arrangement is free when the du…

Let $\mathcal{A}$ be a hyperplane arrangement with complement $M$, let $\Phi_{\lambda}$ be its master function for weights $\lambda$, and let $\chi(M)$ denote the Euler characteris…

Let $V$ be a finite set, let $G=(V,E)$ be a directed graph, and let … where $\varepsilon_G(i,j)=1$ if $(i,j)\in E$ and $0$ otherwise, and let $c\mathscr{A}_m(G)$ be its cone. A tot…

Mu–Stanley conjecture. The $\psi$-graphical arrangement $\mathcal{A}_{G,\psi}$ is free if and only if it is supersolvable.

Solomon–Terao order-degree conjecture. If $\eta\in U_{d+1}({\mathcal{A}})$, then $ST_{d+1}({\mathcal{A}};x)$ is monic of degree

Let the Grünbaum–Cuntz catalogue be the catalogue of known three-dimensional simplicial hyperplane arrangements, consisting of 95 sporadic arrangements and three infinite families.…

Let $\mathcal{H}=\{H_i\}$ be a hyperplane arrangement in $\mathbb{CP}^n$, let $C^{\circ}$ be its stable cone, and let $Q$ be the associated quadratic form. For a weight vector…

Let $k\in\mathbb{Z}_{>0}$, let $\Phi$ be a root system, let $\Phi^+$ be its set of positive roots, and let $\delta\in\Phi^+$. Write $\delta^c$ for the complement of $\delta$ in the…

Let $\mathcal{H}(d,n)$ be an arrangement of $n$ (pseudo)hyperplanes in $\mathbb{P}^d$, and call a maximal cell complete if it is bounded by every hyperplane of the arrangement. Rou…

Let $m\geq 1$ and $k\geq 1$ be integers. For $d=\Delta(m;k)$, let $\Delta(m;k)$ denote the minimum dimension such that any $m$ masses on $\mathbb{R}^d$ can be equipartitioned by…

Freeness conjecture for $G(n,n,3)$. The arrangement $\mathcal{A}$ is free if and only if $n$ is not prime, or $n$ is prime and the characteristic of $k$ is not $n$.

Let $r$ be a positive integer. Given $r$ blue, $r$ red, and $r$ green lines in the plane in general position, where a family is in general position when the normal vectors of any t…