9 problems

Persistence Conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X$ is absolutely arithmetically hyperbolic.

Let $k$ be an algebraically closed field of characteristic zero, and let $X$ be a variety over $k$. Persistence Conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X$ i…

Persistence conjecture. If $X$ is arithmetically hyperbolic over $k$, then its base change $X_L$ is arithmetically hyperbolic over $L$. This asserts that finiteness of integral poi…

Let $k$ be an algebraically closed field of characteristic zero. For a projective variety $X$ over $k$, consider pseudo-Mordellicity, pseudo-arithmetic hyperbolicity, pseudo-groupl…

Hassett–Tschinkel puncture problem. There should exist a number field $K$, a finite set $S$ of finite places of $K$, a model $\mathcal A$ for $A$ over $\mathcal O_{K,S}$, and a mod…

Arithmetic Persistence Conjecture. The base change $X_L$ is arithmetically hyperbolic over $L$.

Demailly–Green–Griffiths–Lang–Vojta conjecture. The following statements are equivalent: $X$ is algebraically hyperbolic over $k$; $X$ is bounded over $k$; for all $n\geq 1$ and…

Throughout, $k$ is an algebraically closed field of characteristic zero, and a variety over $k$ is a finite type separated integral $k$-scheme. A variety $X$ over $k$ is arithmetic…

Let $k$ be an algebraically closed field of characteristic zero. A finite type separated scheme $X$ over $k$ is arithmetically hyperbolic if, for every $\mathbb{Z}$-finitely genera…