14 problems

Artin–Shioda conjecture. Every supersingular $K3$ surface is unirational.

Bogomolov–Karzhemanov–Kuyumzhiyan conjecture. Every unirational algebraic variety $X$ is birationally stably flexible.

Let $\mathcal{V}_{\bf r}$ be the Severi variety of unicuspidal rational curves of type $({\rm S},{\bf r})$, under the hypotheses and notation of the codimension conjecture above. U…

Artin–Rudakov–Shafarevich–Shioda conjecture. A K3 surface over $k$ is supersingular if and only if it is unirational.

Let $X$ be a unirational variety over a number field. The Colliot-Thélène–Sansuc conjecture. $X$ has the Hilbert property. This conjecture connects the abundance of rational points…

Artin--Shioda--Rudakov--Safarevic conjecture. A K3 surface is supersingular if and only if it is unirational.

Let $X$ be a unirational variety over an algebraically closed field of characteristic zero, and let $H^i_{nr}$ denote unramified cohomology. Unramified-cohomology finiteness conjec…

Let $X$ be a unirational variety over a field $k$. A rational tower is a sequence of dominant rational maps … over $k$, with rational source $X_n$ and geometrically rational irredu…

Let $X$ be a supersingular K3 surface over an algebraically closed field. Artin's conjecture. The surface $X$ is unirational. The paper proves this conjecture, resolving Artin's qu…

Let $k$ be a number field and let $K/k$ be a cyclic extension of degree $3$. For a polynomial $f\in k[t]$ of degree $6$, let $\mathcal{S}_{f}$ denote the variety defined by the nor…

Let $X$ be a unirational algebraic variety over a field $\mathbf{k}$. The field $\mathbf{k}(X)(y_1,\ldots,y_m)$ is said to admit an infinitely transitive model if it has a model th…

Let $\mathcal{K}_{36}$ denote the moduli space of primitively polarized smooth $\mathrm{K3}$ surfaces of genus $36$. Unirationality conjecture. The moduli space $\mathcal{K}_{36}$…

Shioda–Artin supersingularity conjectures. For K3 surfaces, Shioda-supersingularity implies unirationality; Artin-supersingularity implies unirationality; and Artin-supersingularit…

Shioda's conjecture. A K3 surface is unirational if and only if it is Shioda-supersingular.