13 problems

Colliot-Thélène's conjecture. Weak approximation with Brauer–Manin obstruction is the only obstruction to weak approximation for smooth rationally connected varieties over number f…

Let $B$ be a smooth affine connected curve over a field of characteristic $0$, and let $X_B$ be a $B$-flat projective variety whose geometric generic fiber…

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

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

Let $X$ be a smooth projective rationally connected variety over the function field of a curve. Let $B$ be the base curve and let $\pi:\mathcal X\rightarrow B$ be a regular model o…

Let $G$ be an adjoint algebraic group over an arbitrary infinite field. Platonov's conjecture. The variety underlying $G$ should be rational, and consequently $G$ should satisfy we…

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…

Weak Approximation conjecture. Every possible intersection profile can be realized by some section.

Weak Approximation conjecture. Since the Brauer group of $k(\mathbb{P}^{1})$ is trivial, the condition that a section can intersect only components of fibers occurring with multipl…

Let $K=\mathbb{F}_q(t)$, and let $X$ be a smooth complete intersection of multidegree $(d_1,\ldots,d_R)$ in projective space of dimension parameter $n$. Weak approximation means th…

Weak locally split values conjecture. There exists $t_0\in k$ satisfying condition (2) of the strong conjecture, while condition (1) is replaced by: $t_0$ is arbitrarily close to…

Let $k$ be an algebraic number field, let $\mathfrak{p}$ be a prime of $k$, and let $A$ be an abelian variety over $k$. For a local Weil–Châtelet class…

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…