13 problems

Hassett–Tschinkel's conjecture. These are all the extremal rays of the cone of curves.

Mori cone generation conjecture. The Mori cone $M$ of $X_\psi$ is generated by the classes of the curves $\ell_{\mathbf{t}}$, $\sigma_{\mathbf{s},\mathbf{t}}$, and…

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ sufficiently general points, and let $\mathcal{Q}_n$ and $\Delta_n^{\preccurlyeq}$ be as defined in the source. Strong Delta-conjectu…

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ sufficiently general points. Let $\mathcal{Q}_n$ be the nonnegative cone and let $\Delta_n^{\preccurlyeq}$ be the cone defined in the…

Boundedness conjecture. Let $Y$ be an irreducible holomorphic symplectic manifold birational to $X$. The set

$-1$-curves conjecture. The Mori cone of $S$ is generated by the light cone and the $-1$-curves. Dually,

Strong $\Delta$-Conjecture. If $s\geqslant 11$ and $R$ is a rational de Fernex non-positive ray with $R^2=0$, then $R$ is not effective and therefore is a good ray.

Strong $\Delta$-Conjecture. If $s\geqslant 11$ and $R$ is a rational De Fernex non-positive ray with $R^2=0$, then $R$ is not effective, and therefore is a good ray.

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ sufficiently general points, and let $\operatorname{NE}(X)$ denote its Mori cone. Let $\mathcal{Q}$ be the nonnegative cone and let…

Let $S$ be a finite set with $|S|\geq 3$, and let $\overline{M}_{0,S}$ be the moduli space of stable genus-zero curves with marked points indexed by $S$. Its boundary curve classes…

Numerical-dimension conjecture. If

Interior-of-the-Mori-cone conjecture. The class $[C]$ lies in the interior of

Let $(X,g)$ be a polarized variety deformation equivalent to a four-dimensional generalized Kummer variety. Let $(R,R)$ denote the Beauville–Bogomolov form on homology. The general…