149 problems

Let $\mathcal{A}$ be a finite simplicial hyperplane arrangement of rank $r$, and let $W(\mathcal{A})$ be its poset of regions with the standard region order. Is…

Line-closed matroid–quadratic Orlik–Solomon conjecture. $G$ is line-closed if and only if $A(G)$ is quadratic.

Let $M$ be a matroid, or let $\mathcal{A}$ be a hyperplane arrangement, and let $A(M)$, respectively $A(\mathcal{A})$, denote its Orlik–Solomon algebra. A matroid or arrangement is…

Let ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ be central hyperplane arrangements defined over the same field. Their intersection lattices $L({\mathcal{A}}_1)$ and…

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…

Ramos's conjecture. The triple $(d,k,m)$ is a solution if and only if

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

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 $\Phi$ be a root system of rank $\ell$ with Coxeter number $h$. For integers $a,b$ with $a\leq 1\leq b$, let $\mathcal{A}_{\Phi}^{[a,b]}$ be the truncated affine Weyl arrangeme…

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 $G$ be a connected graph, and let ${\mathscr A}_G$ denote its connected subgraph arrangement. A free arrangement is flag-accurate if it is accurate and the tuple of flats witne…

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…

Arrangement Bernstein–Sato conjecture.

Athanasiadis's freeness conjecture. The coning $c\mathcal{A}_G$ is free if and only if $G$ satisfies (A1) and (A2).

Spanning conjecture. If $k eleq r eleq 2k-n-2$, then $(R_n/(E+(Q-1)))_r$ is spanned by the expressions in the generating family for which $i_1<(n-1)+(r-k)$; if $k-n<r<k$, the expre…

Let $\mathcal{H}$ be a Coxeter arrangement on a finite-dimensional inner product space $V$, let $a\in V$, and let $R$ satisfy $\|a\|\leq R$. Assume that the number of hyperplanes o…

For a simple graph $G$, let $\mathscr{A}_G$ denote its graphic arrangement. A graph is strongly chordal if it has no induced $n$-sun for $n\geq 3$. An arrangement is flag-accurate…

Freeness conjecture for $\mathscr{D}^r_\ell(a)$. The cone $\mathbf{c}\mathscr{D}^r_\ell(a)$ is free with exponents

For a root system of Dynkin type, let $\operatorname{Cat}^m$ denote the corresponding extended Catalan arrangement. An arrangement is flag-accurate when it admits a flag satisfying…

Let ${\pazocal{A}}$ be an arrangement of rank at least $3$. For each prime $p$ and integer $s\ge 1$, let ${e}_{p^s}({\pazocal{A}})$ denote the corresponding elementary divisor mult…

Generic local-system cohomology conjecture. For every $p$ and almost all $a\in A_1^*$ among those with

Let ${\mathcal A}$ be a central hyperplane arrangement with connected underlying matroid, and let $C({\mathcal A})$ denote its complement. Homotopy-type conjecture. The homotopy ty…

Let $R$ be an irreducible root system in $\mathbb{R}^n$, and let $a$ and $b$ be integers with $a+b\geq 2$. Define the truncated $R$-affine arrangement ${\cal A}^{ab}(R)$ by the hyp…

Let an arrangement of finitely many complex central hyperplanes have complement $M$. Let the zero-divisors-cup-length of $H^*(M,\mathbf C)$ mean the largest number of zero-divisors…