Let $Y\to X$ be a finite surjective morphism of smooth projective geometrically connected varieties over $\mathbb{Q}$. Let $k$ be a number field and consider the induced morphism

$$f:Res_{k/\mathbb{Q}} Y_k\to Res_{k/\mathbb{Q}} X_k.$$

Let $\Delta\to Res_{k/\mathbb{Q}}X$ be the diagonal embedding of $X$ into the Weil restriction $Res_{k/\mathbb{Q}} X_k$ of $X_k$ to $\mathbb{Q}$.

What are the irreducible components of the scheme-theoretic inverse image $f^{-1}(\Delta)$? Is $f^{-1}(\Delta)$ not just equal to $Y$?

Here $f^{-1}(\Delta) $ really is just the fibre product of $\Delta$ with $Res_{k/\mathbb{Q}} Y_k$ over $Res_{k/\mathbb{Q}} X_k$.

Let $Y\to X$ be a finite surjective morphism of smooth projective geometrically connected varieties over $\mathbb{Q}$. Let $k$ be a number field and consider the induced morphism

$$f:Res_{k/\mathbb{Q}} Y_k\to Res_{k/\mathbb{Q}} X_k.$$

Let $\Delta\to Res_{k/\mathbb{Q}}X$ be the diagonal embedding of $X$ into the Weil restriction $Res_{k/\mathbb{Q}} X_k$ of $X_k$ to $\mathbb{Q}$.

What are the irreducible components of the scheme-theoretic inverse image $f^{-1}(\Delta)$? Is $f^{-1}(\Delta)$ equal to $Y$, and is the morphism $f^{-1}(\Delta)\to X$ the morphism $Y\to X$ we started with?

Here $f^{-1}(\Delta) $ is defined to be the fibre product of $\Delta$ with $Res_{k/\mathbb{Q}} Y_k$ over $Res_{k/\mathbb{Q}} X_k$.