9 problems

Let $X$ be a $K3$ surface over a field of positive characteristic. A $K3$ surface is called supersingular if it has maximal possible Picard rank, namely $22$, and it is unirational…

Unirationality conjecture. Unirationality characterizes supersingularity for symplectic varieties.

Geometric characterization conjecture. Second Artin supersingularity is equivalent to unirationality, and it is also equivalent to rational chain connectedness.

Fully supersingular Bloch--Beilinson conjecture. Numerical and algebraic equivalence coincide on ${\rm CH}^{i}(X)_{\mathbb{Q}}$, so $\operatorname{Griff}^{i}(X)_{\mathbb{Q}}=0$; th…

Representability conjecture. All rational Chow groups of $X$ are representable; more strongly, their homologically trivial parts are represented by abelian varieties.

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

Motivic conjecture. The Chow motive of $X$ is of supersingular abelian type, the rational Chow ring is representable, and it satisfies the Bloch--Beilinson conjecture and Beauville…

Artin's Frobenius-pullback conjecture. The elliptic fibration on $X$ arises via Frobenius pullback from a rational elliptic surface.

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