11 problems

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…

Regularity conjecture. Then

Spherical skeleton conjecture. We have

Let $S_r$ be a del Pezzo surface of degree $(-K_{S_r})^2=9-r$ with $4\leq r\leq 8$. Let $G_r$ be a basis of the vector space … for $r\leq 7$, and for $r=8$ let $G_8$ be a basis of…

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…

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,…

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…