5 problems

Nonemptiness conjecture. $Z_n(L)$ is not empty for any $g\leq 2n-2$.

Let $(X,L)$ be a polarized K3 surface of genus $g\geqslant 3$. Let $C_{en}$ denote the curve of elliptic nodes of $(X,L)$, namely the curve swept out by the singular points of the…

Recognizability conjecture. If $R$ is a constant cycle curve, then it is recognizable.

Let $X$ be a complex K3 surface, and let $c_X\in\operatorname{CH}^2(X)$ be the distinguished Beauville–Voisin class. A constant cycle curve is a curve on $X$ whose points have a co…

Let $X$ be a K3 surface. A constant cycle curve is a curve on $X$ whose points all have the same class in the appropriate Chow group; its order is the integer used to measure the a…