24 problems

Let $S$ be a quasi-smooth, well-formed del Pezzo hypersurface of degree $d$ in the weighted projective space … where $S$ is defined over an algebraically closed field $\Bbbk$ of ch…

Let $X$ be a Fano variety with at most Kawamata log terminal singularities. An anti-canonical polar cylinder means a cylinder polarized by $-K_X$. The anti-canonical cylinder conje…

Let $S$ be a log del Pezzo surface. The cylindrical ample cone conjecture. If the cylindrical ample cone … contains the anticanonical class, then … Equivalently, the cylindrical am…

Consider unit-radius infinite cylinders in three-dimensional Euclidean space that touch a unit-radius ball. Kuperberg's conjecture. The maximum number of such cylinders is six. Six…

Let $S$ be the del Pezzo surface under consideration, and let $\mathbb P$ denote the ambient projective threefold in which it is embedded. The automorphism conjecture for del Pezzo…

Let $X$ be a smooth Fano threefold such that $\rho(X)=1$, $\iota(X)=1$, and $\operatorname{g}(X)=7$. The non-cylindricity conjecture. Then $X$ is not cylindrical. The source states…

Let $X$ be a very general smooth Fano threefold such that $\rho(X)=1$, $\iota(X)=1$, and $\operatorname{g}(X)=9$ or $\operatorname{g}(X)=10$. The non-cylindricity conjecture. Then…

For a projective variety $X$, let … be its cone of cylindrical ample divisors. The cylindrical ample cone conjecture. If $X$ is a Fano variety, then … The problem of determining th…

Let $X$ be a Fano variety with at most Kawamata log terminal singularities. The K-polystability conjecture. If $X$ does not contain $(-K_X)$-polar cylinders, then $X$ is K-polystab…

Let $X$ be a del Pezzo surface with quotient singularities and Picard rank $1$. Miyanishi's covering conjecture. The smooth locus of $X$ admits a finite unramified covering that co…

Uniqueness conjecture. The level $\mathcal{C}_{4}(1+\sqrt{2})$ has no points other than the known circle of four parallel cylinders modulo $SO(3)$; equivalently, it is a circle mod…

Let $C_\mathfrak{m}$ be the extremal configuration of six cylinders, and let $\mathcal{Z}(\mathsf{\geq 1})$ be the configuration space, modulo $SO(3)$, of six cylinders of radius a…

Let $C_\mathfrak{m}^+$ and $C_\mathfrak{m}^-$ be, respectively, the configuration $C_\mathfrak{m}$ and its mirror image. Let $\mathcal{Z}(\mathsf{>1})$ be the configuration space,…

Let $S$ be a del Pezzo surface of anticanonical degree at most $2$ with at worst DuVal singularities. The affine Fano variety $\mathbb{P}\setminus S$ is the complement of $S$ in th…

Small-radius maximal-section conjecture. If $r$ is sufficiently small, then the section orthogonal to $a$ is maximal.

Let $S$ be a log del Pezzo surface. Let $\mathrm{Amp}(S)$ be its ample cone in $\operatorname{Pic}(S)\otimes\mathbb{R}$, and let $\mathrm{Amp}^{cyl}(S)$ be the set of ample classes…

Let $G$ be a graph constructed from $K_5-e$, $H_1$, or $H_2$ by a sequence of $1$-extensions and edge additions, and let $(G,p)$ be a generic framework on a family of concentric cy…

Let $G$ be a graph, and let $K_5-e$, $H_1$, and $H_2$ be the three specified base graphs. The operations under consideration are edge addition, $1$-extension, and $1$-, $2$- and…

Let $B^n$ be the Euclidean unit ball, and let $Z_1,\ldots,Z_N$ be $m$-dimensional Euclidean cylinders covering $B^n$. Write $\sigma_{n-m}(Z_i)$ for the $(n-m)$-dimensional volume o…

Let $\mathcal{Y}$ denote the cylinder. Let $S$ be either the group $\mathcal{C}_s$ or the group $\mathcal{C}_i$, and let $(G,p)$ be an $S$-generic framework on $\mathcal{Y}$. The c…

Let $\mathcal{Y}$ denote the cylinder. Let $S$ be one of the groups $\mathcal{C}_{mh}$, $\mathcal{C}_{mv}$ or $\mathcal{S}_{2m}$, with the rotational axis equal to the cylinder axi…

Let $\mathcal{Y}$ denote the cylinder, and let $S$ be the cyclic group $\mathcal{C}_2$ representing a two-fold rotation about an axis perpendicular to the $z$-axis. Let $(G,p)$ be…

Let $t$ be fixed, and let $\mathscr{C}$ be a packing of $\mathbb{R}^3$ by congruent $t$-cylinders, with density denoted by $\rho^+(\mathscr{C})$. A collection of cylinders has axes…

Let $\mathscr{C}(t)$ be a packing of congruent circular cylinders of radius $1$ and length $t$ in $\mathbb{R}^3$, and let $\rho^+(\mathscr{C}(t))$ denote its packing density. Wilke…