13 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…

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…

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…

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, 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…