Corvaja–Zannier weak Hilbert property conjecture

Let $K$ be a finitely generated field of characteristic zero, and let $Y$ be a smooth connected $K$-variety. If $Y(K)$ is Zariski dense in $Y$, then $Y$ has weak Hilbert property (WHP) potentially over $K$.

Corvaja–Zannier conjecture. Under these hypotheses, $Y$ has WHP potentially over $K$.

This conjecture is implied by a central conjecture of Corvaja and Zannier. It is known for abelian varieties and connected algebraic groups, as well as in other cases, but remains open in general.