8 problems

Let $(A,\Theta)\in\mathcal A_g$ be a principally polarized abelian variety, and let a $d$-dimensional subvariety of $A$ have minimal cohomology class, meaning the class is the mini…

Birational distinction conjecture. The category $\mathcal{A}_X$ allows one to distinguish the birational type of $X$, or equivalently, the category $\mathcal{A}_Y$ allows one to di…

Let $A\in{\cal A}_4$ be a principally polarized abelian variety, let $\Gamma_{00}$ be the van Geemen–van der Geer linear system on $A$, and let $X\subset\Gamma_{00}$ be the cubic t…

Let $X\subset\mathbb{P}^4$ be a cubic threefold, let $Y\subset X$ be a hyperplane, and let $C\subset X$ be a smooth rational curve of degree $d$, with $X$, $Y$, and $C$ all general…

Nonlinearizability conjecture. The $G$-action is not linearizable.

Linearizability conjecture. The $G$-action is linearizable if and only if $G$ fixes a node.

Let $Y$ be a smooth cubic threefold, let $S$ be a singular hyperplane section, and let $D$ be a Weil divisor on $S$ such that $\mathcal{O}_S(D)$ is a stable object parameterized by…

Let $V_3\subset\mathbb{P}^4$ be a smooth cubic threefold over the complex numbers. Clemens–Griffiths conjecture. The variety $V_3$ is not rational. The statement was a longstanding…