10 problems

Let $k$ be a field and let $T$ be an algebraic $k$-torus. The torus $T$ is stably $k$-rational if its function field becomes purely transcendental over $k$ after adjoining finitely…

Let $k$ be a field, let $x_1,\dots,x_k$ be variables, and let $G$ be a finite group acting by permutation on these variables. Noether's rationality conjecture. The invariant field…

Non-rationality conjecture. A generic cubic fourfold is non-rational.

Le Bruyn's conjecture. The torus $T_n$ is never $k$-rational, except possibly when $n=6$.

Let $K/k$ be a finite separable extension with Galois closure $L/k$, let $T=R^{(1)}_{K/k}(\mathbbm{G}_m)$ be its norm one torus, and write … For an integer $d\geq 1$, consider…

Non-rationality conjecture. Then $X$ is never ${\mathsf{k}}$-rational.

Let $X$ be a cubic fourfold. Say that $X$ has an associated $K3$ surface when it has an associated $K3$ surface in the sense of Hassett. Harris–Hassett conjecture. If $X$ has no as…

Let $X$ be a cubic fourfold. Define its algebraicity index by … where … and $d_X$ is the determinant of $H^{2,2}(X)\cap H^{4}(X,{\mathbb Z})$. Algebraicity-index bound. Cubic fourf…

Let $X$ be a cubic fourfold, and let $T_X$ denote its transcendental cohomology. Say that $X$ has an associated $K3$ surface when $T_X$ can be realized as the transcendental cohomo…

Kuznetsov–Hassett rationality conjecture. A general cubic fourfold belonging to $\mathcal C_d$ is rational if and only if $d$ is admissible.