8 problems

Finite generation conjecture. The ring $R_{\mathsf{zip}}$ is a finitely generated $k$-algebra; equivalently, the stack \mathop{\text{G-\tt ZipFlag}}\nolimits^\mu is a Mori drea…

Mori-surface conjecture. The surface $\mathbb{F}_r$ is a Mori surface.

Low-point Mori dream space conjecture. The Hilbert scheme $\mathbb{F}_r^{[r-k]}$ is a Mori dream space, and the decomposition of…

Let $X$ be a Mori dream space, and let $Y_\bullet$ be a flag for which the global semigroup $\Gamma_{Y_\bullet}(X)$ is defined. The Cox ring of $X$ is denoted by…

Let $\mathbb G(r,n)_1$ be the blow-up of the Grassmannian $\mathbb G(r,n)$ at one point, let $H$ denote the pullback of the hyperplane class, and let $E$ denote the exceptional div…

Monin–Rana's conjecture. The ideal $I_n$ is the unique $B$-saturation of $J_n$; it is minimally generated by $\binom{n-1}{d+1}$ polynomials of degree $d$ for $d=3,4,5,\ldots,n-2$,…

Log canonical Fano Mori dream space conjecture. Then $X$ is a Mori dream space.

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…