22 problems

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

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

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…

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 ${\mathcal A}$ and ${\mathcal B}$ be line arrangements with isomorphic intersection lattices … For an arrangement ${\mathcal A}$, let $r({\mathcal A})$ denote the minimal degre…

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

Addition–deletion conjecture. The following assertions should hold:

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…

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

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…

Let ${\mathcal{A}}$ be an $\ell$-arrangement and let $H\in{\mathcal{A}}$. Write ${\mathcal{A}}'={\mathcal{A}}\setminus\{H\}$, and let $\pi({\mathcal{A}};t)$ denote the Poincaré pol…

Let $\mathcal{RF}^*$ denote the class of recursively free arrangements satisfying the additional condition denoted by a star in the source, let $\mathcal{DF}$ denote the class of d…

Let an arrangement be called free if its module of logarithmic derivations is free, inductively free if it belongs to the inductively free class, and rigid if its moduli space is z…

Let $\mathcal{A}_{G,\psi}$ be a $\psi$-graphical arrangement, and call it free when its cone is a free arrangement. Recall that an arrangement is supersolvable when the intersectio…

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