18 problems

Batyrev–Popov conjecture. Cox rings of del Pezzo surfaces are presented by ideals of quadrics.

Let $M(Q,\mathbf{r})$ be a quiver flag variety, and let $p:M(Q,\mathbf{r})\hookrightarrow M(Q',\mathbf{1})$ be the multigraded Plücker embedding. Suppose that the induced map on Pi…

Let $\mathcal{F}$ be the projective bundle used to construct the threefold Mori dream space $X$, and let $f$ and $g$ be the defining equations of the two hypersurfaces whose comple…

Finiteness of Cox ring. The ring

Let $U'\subset Y'$ be an snc minimal model of an affine log Calabi–Yau variety with maximal boundary, let $Y\to Y'$ be the universal torsor, and let $U\subset Y$ be its restriction…

Cattani–Cox–Dickenstein's Codimension One conjecture. The critical-degree component of the quotient by the $F_i$ should satisfy

Let $\mathbb P=\mathbb P_\Sigma$ be a projective toric surface with Cox ring $\mathbb C[x_1,\dots,x_n]$ and lattice ideal $I_{\mathbb P}$. Let $e$ denote the identity of the torus…

Let a $(k,d)$--KD polytope be the polytope associated with the relevant Cox-ring presentation, and let the $(k,d)$--Mukai edge graph be the graph whose edges are forced by the corr…

Let $X_{k,d}:=\operatorname{Bl}_{k}\mathbb{P}^d$ be the blow-up of $\mathbb{P}^d$ at $k$ points, and let $\operatorname{Cox}(X_{k,d})$ denote its Cox ring. For $(k,d)\neq(8,4)$, se…

Spherical skeleton conjecture. We have

Let $X$ be the symplectic resolution under consideration, let $\Theta(\mathcal{R}(X))$ denote its Cox ring image, and set $\mathcal{P}=\mathbb{C}[V]^H$. Let $t_1,t_2$ be the Lauren…

Rationality–Cox ring conjecture. $E^1(X)$ is rational if and only if

Let $\mathcal{M}_\vartheta$ be a quiver flag variety, let $\lvert\underline{\det}(\mathscr{W})\rvert$ be the toric variety associated with the sequence of determinant line bundles,…

Cox-ring reconstruction conjecture. Equality holds in the displayed inclusion for generic $X_n$, and only two summands suffice on the right-hand side.

Let $X$ be one of the extremal elliptic threefolds obtained by blowing up the base locus of a nondegenerate net of quadrics in $\text{\bf P}^3$. The classification consists of fini…

Let $X_G$ be the del Pezzo surface of degree three associated with six generic linear forms, and let $\boldsymbol{\ell}=(\ell_1,\ell_2,\ldots,\ell_6)$ be generic. In the case $d=3$…

Let $X$ be a del Pezzo surface. Let $I_X$ be the defining ideal in a presentation ${\rm Cox}(X)\cong k[\mathcal G]/I_X$, where $\mathcal G$ is a minimal homogeneous generating set…

A log-Fano variety is a projective variety with mild singularities for which there exists an effective boundary divisor making the pair log Fano. Hu–Keel's conjecture. Every log-Fa…