10 problems

Let $d$ be a natural number and let $\epsilon>0$ be a real number. Let $X$ be an $\epsilon$-lc, $\mathbb{Q}$-factorial variety of dimension $d$, and let $f\colon X\to Z$ be a Mori…

Let $X/S$ be a threefold Mori fibre space, and let $\mathcal{P}(X/S)$ denote its pliability. Topological pliability conjecture. The pliability $\mathcal{P}(X/S)$ is, in a natural w…

Let $X/S$ and $Y/T$ be threefold Mori fibre spaces, and let rigidity mean birational rigidity of the Mori fibre space. Topological rigidity conjecture. Whether a threefold Mori fib…

Let $X/S$ and $Y/T$ be threefold Mori fibre spaces, with total spaces $X$ and $Y$ diffeomorphic. Topological rationality conjecture. If $X$ is rational, then $Y$ is also rational.…

Let $X\to S$ be a Mori fibre space, and let $\mathcal{P}(X/S)$ denote its pliability. Pliability algebraicity conjecture. The pliability $\mathcal{P}(X/S)$ is, in a natural way, an…

Let $X$ be a smooth uniruled variety. K-stable-fibre conjecture. The variety $X$ is birational to a Mori fibre space whose general fibre is K-stable. This conjecture links biration…

A central model is a model in the birational geometry of a proper morphism, with its Picard number measuring the complexity of the associated birational data. Mori fibrations are c…

McKernan's conjecture. There is $\delta\in\mathbb R^{>0}$ such that if $f\colon X\to Z$ is a Mori fibre space, where $X$ is an $\epsilon$-lc $\mathbb Q$-factorial variety of dimens…

Cheltsov–Shramov conjecture. The varieties

Let $Y_4\subset \mathbb{P}^4$ be a factorial quartic hypersurface, or let $Y_{2,3}\subset \mathbb{P}^5$ be a generic complete intersection, in each case with no worse than terminal…