4 problems

Let $R$ be a finitely generated smooth commutative algebra over $\mathbb Z$, let $g\in\operatorname{Aut}(P_{n,R})$, and for each sufficiently large prime $p$ let $M_{g,p}$ be a bim…

Let $V$ be an $n$-dimensional projective simple normal crossing variety over an algebraically closed field of characteristic zero, and let $A$ be a finitely generated $\mathbb{Z}$-…

Let $X$ be a smooth, irreducible $n$-dimensional projective variety defined over an algebraically closed field $k$ of characteristic zero. If $X_A$ is a model of $X$ defined over a…

Let $Y$ be a smooth, irreducible variety over an algebraically closed field $k$ of characteristic zero, and let $\mathfrak{a}$ be a nonzero ideal on $Y$. Given any model $Y_A$ and…