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…

Spanning conjecture. If $k eleq r eleq 2k-n-2$, then $(Rn/(E+(Q-1)))r$ is spanned by the expressions in the generating family for which $i1<(n-1)+(r-k)$; if $k-n<r<k$, the expressi…

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…

Let $\Phi$ be an irreducible crystallographic root system of rank $\ell$, let $m\geq 1$, and let ${\mathcal A}^m_\Phi$ be its generalized Catalan hyperplane arrangement. Let…

Let ${\mathcal A}$ be a complex hyperplane arrangement, let $A=H^*(X({\mathcal A}),\Bbbk)$ be its Orlik-Solomon algebra over a field of characteristic $0$, and let $E$ be the exter…

Let $W$ be a finite Coxeter group of rank $n$, with $R$ its representation on the set of roots. Let $G'$ be the virtual $W$-module obtained by evaluating at $t=1$ the graded quotie…

Edelman–Reiner conjecture. The following arrangements are free:

Annihilator-generator conjecture. For any central arrangement $Q$,

Orlik–Randell conjecture. For the fiber $X_1=\operatorname{Var}(Q-1)$, there is a finite-dimensional homogeneous vector space $U\subset R_n$ such that

MLS LCS formula. If ${\mathcal A}$ is an MLS arrangement with group $G$, then, for all $k\ge 2$,

Points-and-lines conjecture. The underlying matroid of a free or $K(\pi,1)$ arrangement is determined by its points and lines.

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

Let $\mathcal{A}$ be a complex hyperplane arrangement with arrangement group $G=G(\mathcal{A})$. Let $R_1(\mathcal{A})=\bigcup_i L_i$ be its resonance variety, and set…

Let $\mathcal{A}$ be a complex hyperplane arrangement with arrangement group $G=G(\mathcal{A})$. Write the resonance variety as $R_1(\mathcal{A})=\bigcup_i L_i$, where the $L_i$ ar…

Modular-flat freeness conjecture. If $X$ is a modular flat and both ${\mathcal A}_X$ and $c{\mathcal A}_{{\overline{v}}}$ are free arrangements, then ${\mathcal A}$ is a free arran…

Let $\mathcal{A}$ and $\mathcal{B}$ be real central hyperplane arrangements, and let $R$ be an integral domain with $\operatorname{char}R\neq 2$. Write…

Let $k$ be a field, let $R=k[x_1,\ldots,x_n]$, and let $\mathcal A$ be a central hyperplane arrangement with defining linear forms whose product is $f$. Let $I\subseteq R$ be the i…