7 problems

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 $U$ be the double space considered in the source, and let ${\mathcal S}$ denote the family of curves used there to construct the fibrations $V_l/\mathbb{P}^1$. Khashin's conjec…

Laza's algebraicity-index conjecture. If $\kappa_Y\leq 1$, then $Y$ is irrational.

Shokurov's conjecture. If

Rationality conjecture for $\mathbb{Q}$-factorial nodal quartic double solids. Every $\mathbb{Q}$-factorial nodal quartic double solid is irrational.

Let $\pi:X\to\mathbb{P}^1$ be a generic relative complete intersection of two four-dimensional quadric fibrations, and let $T\to S$ be the associated discriminant cover. Let…

Let $X \to \mathbb{P}^1$ be a fibration in complete intersections of two four-dimensional quadrics over an algebraically closed field of characteristic zero. Write $\mathcal{A}_X$…