13 problems

Let $K$ be a finitely generated field and let $X$ be a smooth projective variety over $K$. The variety $X$ has potential WHP if $X_L$ has the weak Hilbert property for some finite…

Let $K$ be a number field. A $K$-variety $X$ satisfies weak weak approximation (WWA) if there is a finite set $S_0$ of places of $K$ such that, for every finite set $S_1$ of places…

Néron-model WHP conjecture. Then $E$ has WHP over $K$.

Weak Hilbert property conjecture for geometric fibrations. In the situation above, $Y$ has WHP over $K$.

Integral weak Hilbert property conjecture. If $\mathcal{Y}(S)^{(1)}$ is Zariski dense in $Y$, then $\mathcal{Y}$ has WHP over $S$ potentially.

Fibration conjecture for WHP. (a) Let $\Xi$ be the set of all $z\in Z(K)$ such that $Y_z$ has WHP over $K$. If $\Xi$ is not strongly thin in $Z$, then $Y$ has WHP over $K$. (b) If…

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

Toric integral-model Hilbert property conjecture. If $N_M=N$, then $(\mathcal{X},\mathcal{M})$ satisfies the $\mathcal{M}$-Hilbert property over every open $B\subsetneq C$. If furt…

Let $K$ be a number field and let $X$ be a variety over $K$. Assume that $X(K)$ is Zariski dense and that $X$ has a smooth, projective model over $K$ which is algebraically simply…

Let $X$ be a special smooth projective geometrically connected variety over a number field $k$. Campana–Corvaja–Zannier conjecture. There is a number field $L/k$ such that $X_L$ sa…

Primitivity conjecture for the over-exceptional lattice. If $X$ has infinite automorphism group, then the embedding

Let $X$ be a smooth proper geometrically connected variety over a finitely generated field $k$ of characteristic zero. Campana–Corvaja–Zannier conjecture. The following are equival…

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…