6 problems

Strong rational connectedness conjecture. The smooth locus $X^{sm}$ of a log Del Pezzo surface $(X,\Delta)$ is strongly rationally connected.

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…

Colliot-Thélène–Sansuc–Kato–Saito conjecture. The complex

Let $k$ be a number field, and let $\{X_\omega\}_\omega$ be the family of all smooth, proper, geometrically integral varieties over $k$. The Brauer–Manin obstruction to weak approx…

Hassett–Tschinkel's weak approximation conjecture. Proper rationally connected varieties defined over $F$ satisfy weak approximation.

Let $k$ be an algebraically closed field of characteristic zero, let $B$ be a curve over $k$, and let $k(B)$ be its function field. Let $X$ be a rationally connected variety over…