7 problems

Let $K$ be a number field and let $\mathscr{X}/\mathcal{O}_K$ be a model for a hyperbolic curve. Let $\mathcal{L}$ be an allowable family of extensions $L/K$. Write…

Let $m\geq 2$, let $k$ be a field finitely generated over the prime field, and let $U_1$ be a hyperbolic curve over $k$. Write $\pi_1^{\operatorname{tame}}(U_i)^{(m)}$ for the quot…

Let $k$ be a field of characteristic $p>0$, let $U_1$ and $U_2$ be hyperbolic curves over $k$, and let $G_k$ denote the absolute Galois group of $k$. For each $m\geq 0$, write…

Let $K$ be a field of characteristic $0$, let $X$ be a hyperbolic curve over $K$, and choose a geometric point $\ast$ of $X_{\overline{K}}$. Write…

Let $K$ be finitely generated over $\mathbb{Q}$, let $Y$ be a normal variety over $K$, and let $X$ be a hyperbolic curve over $K$. For geometric base points, write…

Let $k$ be finitely generated over $\mathbb{Q}$. Let $T/k$ be a smooth variety, and let $X/k$ be a smooth, proper, geometrically connected hyperbolic curve. Grothendieck's Hom conj…

Let $X/k$ be a smooth, geometrically connected hyperbolic curve, let $K$ be its function field, and let $\overline X$ be its smooth compactification. A section of $\pi_1(X/k)$ is c…