7 problems

Let $G$ be a finite simple group, let $V$ be a faithful representation of $G$ over $\text{\bf C}$, and let $V/G$ denote the quotient variety. Bogomolov's stable rationality conject…

Let $X\subset \mathbb P^{n+1}$ be a smooth quartic hypersurface, with $n>0$. Stable rationality means that $X\times \mathbb P^m$ is rational for some integer $m\geq 0$. Stable irra…

Let $n\geq 3$ and let $X\subset \mathbb P^{n+1}$ be a very general cubic hypersurface. Stable irrationality conjecture. $X$ is not stably rational. The conjecture asks whether the…

Let $k$ be the ground field, and let $X\subset \mathbb{P}^{n+1}_k$ be a smooth quartic hypersurface of dimension $n>0$. Stable irrationality conjecture. Every such $X$ is stably ir…

Let $A$ be a finite abelian group, let $F^c(A)$ be its free central extension, and let $V$ be a faithful linear representation of $F^c(A)$ over an algebraically closed field $k$. F…

Let $\pi \colon X \to S$ be an $n$-dimensional standard conic bundle with $n \ge 3$, and let $\Delta$ denote the discriminant divisor of $\pi$. Higher-dimensional discriminant conj…

Stable rationality conjecture. This quotient is always stably rational.