35 problems

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 $V$ 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…

Let $\Phi$ be a reduced root system with positive roots $\Phi^+$, and let $\mathcal I$ be an ideal in $\Phi^+$. The arrangement of ideal type $\mathscr A_{\mathcal I}$ is the subar…

Edelman–Reiner conjecture. The following arrangements are free:

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 ${\mathscr H}_n$ be the hyperpolygonal arrangement in ${\mathbb R}^n$, and let a complexified arrangement mean the arrangement obtained by extending its defining real hyperplan…

Ziegler's nonfreeness conjecture. For ${\mathcal{A}}'$ and $X$ as in Ziegler's generic restriction conjecture, the arrangement ${\mathcal{A}}'\cap X$ is never free; equivalently, a…

Let ${\mathcal{A}}'$ be an essential and free arrangement in $V={\mathbb{K}}^\ell$ with $\ell\geq 4$. Let $X$ be a generic subspace with $\dim X>2$, and define the restriction … He…

Let ${\mathcal{A}}$ be a hyperplane arrangement in a vector space of dimension $\ell$, and let $\Psi({\mathcal{A}};x,t)$ denote its Solomon–Terao polynomial. A polynomial is palind…

Let an arrangement be free if its module of logarithmic derivations $D(\mathcal{A})$ is a free module over the polynomial ring, and let it be formal if all relations among its defi…

Orlik–Solomon–Terao conjecture. For every $1\leq d\leq\ell$, the first $d$ exponents of $\mathscr{A}$ are realized as the exponents of a free restriction of $\mathscr{A}$ to some i…

Deletion double-point conjecture. If ${\mathcal A}\setminus\{H\}$ is not free, then

Nonfree line-arrangement double-point conjecture. If ${\mathcal A}$ is not free, then

Let ${\mathscr A}$ be a hyperplane arrangement and let ${\mathscr A}'$ be a deletion of ${\mathscr A}$. Suppose that the characteristic polynomials $\chi({\mathscr A},t)$ and…

Class-inclusion conjecture. The following strict inclusion and non-inclusion relations should hold:

Addition–deletion conjecture. The following assertions should hold:

A Dirichlet arrangement is a hyperplane arrangement arising from a directed graph and a distinguished set of vertices; let its cone be the central arrangement obtained by homogeniz…

Root-difference conjecture. Then $d=d_i$ for some $i$, or $d>d_\ell$. The source notes that this conclusion was established under the additional assumption that…

Let ${\mathcal{A}}$ be a hyperplane arrangement, and let $L({\mathcal{A}})$ denote its intersection lattice. A plus-one generated arrangement is one whose derivation module has the…

Solomon–Terao Hilbert-series conjectures. The arrangement ${\mathcal{A}}$ is free if and only if

Let $\Phi$ be a reduced root system, let $\Phi^+$ be its set of positive roots, and let $\mathcal{I}$ be an ideal in $\Phi^+$. Denote by $\mathcal{A}_{\mathcal{I}}$ the arrangement…

Modified Orlik's conjecture. The restriction ${\mathcal{A}}^H$ is free.

Tohăneanu's low-exponent supersolvability conjecture. Then $\mathcal{A}$ is supersolvable. The conjecture concerns free arrangements with exactly one exponent equal to $3$ and is v…

Classification conjecture. The multi-braid arrangement $(A_\ell,\mathbf{m})$ is free if and only if it is one of the multi-braid arrangements constructed in Corollary. Equivalently…

Let $\Phi$ be a reduced root system with Weyl arrangement $\mathscr A=\mathscr A(\Phi)$, and let $\mathcal I$ be an ideal in $\Phi^+$. Write $\mathscr A_{\mathcal I}$ for the subar…