Geometric meaning of the splitting field over a function field

Question

Let $K$ be a field and consider the ring of polynomial in two variables $K[x,t]$. Now take a polynomial $f(x)\in K[x]$ of positive degree and consider it in the bigger ring $K(t)[x]$. Suppose that $f(x)-t\in K(t)[x]$ has distinct roots in $\overline{K(t)}$. The quotient $K(t)[x]/(f(x)-t)$ corresponds to an irreducible curve $C$ over $K$. The question now is: consider the splitting field $E$ of the polynomial $f(x)-t$ over $K(t)$. This will be a finite extension of $K(t)$, so it will correspond to a curve $C'$ over $K$. Moreover, there is an embedding $K(t)[x]/(f(x)-t)\to E$. What is the geometric meaning (if there is one) of the curve $C'$ and of the corresponding map of curves $C'\to C$?

Answer 1

More generally what you're talking about is the the geometric meaning of base-change, and the resulting map to the original curve.

So, let's assume that $X$ is a curve over $K$, and $E$ an extension of $K$. One then obtains another curve $X_E$ by so-called base change. This comes equipped with a map $X_E\to X$.

I would posit that it's actually easier to think about what the process from getting to $X_E$ to $X$ is, then the other way around.

Let's take a very specific example. Let's let $X=\mathbb{A}^1_\mathbb{R}$, $K=\mathbb{R}$, and $E=\mathbb{C}$. Then, $X_\mathbb{C}=\mathbb{A}^1_\mathbb{C}$. So, what is the map $\mathbb{A}^1_\mathbb{C}\to\mathbb{A}^1_\mathbb{R}$ doing? Let's recall what $\mathbb{A}^1_\mathbb{C}$ looks like. Essentially from the fact that $\mathbb{C}[T]$ is a PID, we know that the elements of $\mathbb{A}^1_\mathbb{C}$ are of the form $(T-a)$, for $a\in\mathbb{C}$, and the generic point $\xi_\mathbb{C}:=(0)$. So, $\mathbb{A}^1_\mathbb{C}$ looks like $\mathbb{C}$, and some fuzzy point $\xi_\mathbb{C}$. Similarly, since $\mathbb{R}[T]$ is a PID, the elements of $\mathbb{A}^1_\mathbb{R}$ look like $(T-a)$, for $a\in\mathbb{R}$, $(p(T))$ where $p(T)$ is an irreducible quadratic, and $\xi_\mathbb{R}:=(0)$. So, $\mathbb{A}^1_\mathbb{R}$ looks like $\mathbb{R}$, the set of irreducible quadratics, and a fuzzy point.

The map $\mathbb{A}^1_\mathbb{C}\to\mathbb{A}^1_\mathbb{R}$ does the following, it takes $\xi_\mathbb{C}\mapsto\xi_\mathbb{R}$, and it takes $(T-a)\mapsto (m_a(T))$, where $m_a(T)$ is the minimal polynomial of $a$ over $\mathbb{R}$. So, for example, $(T-\pi)\mapsto (T-\pi)$, but $(T-i)\mapsto (T^2+1)$. Essentially, the map respects points of $\mathbb{C}$ that are real, and identifies points of $\mathbb{C}$ which aren't in $\mathbb{R}$.

Another way of thinking about this is the following. Even though $\mathbb{A}^1_\mathbb{C}$ contains this fuzzy point $\xi_\mathbb{C}$, let's ignore that for a second and think of $\mathbb{A}^1_\mathbb{C}$ as $\mathbb{C}$. We then have an action of $G:=\text{Gal}(\mathbb{C}/\mathbb{R})=\{\text{id},\text{conj.}\}$ on $\mathbb{A}^1_\mathbb{C}$. We can then think about the map $\mathbb{A}^1_\mathbb{C}\to\mathbb{A}^1_\mathbb{R}$ as $(T-a)\mapsto Ga$, where $Ga$ denotes the Galois orbit of $a$. So, in other words' we can think of $\mathbb{A}^1_\mathbb{R}$ as being $\mathbb{C}/G$, and the map $\mathbb{A}^1_\mathbb{C}\to\mathbb{A}^1_\mathbb{R}$ as being the quotient map $\mathbb{C}\to\mathbb{C}/G$.

Now, back to the general case. We have a curve $X/K$, which we will denote $X_K$, and some finite extension $E\supseteq K$. We can think of $X_K$ and $X_E$ as being quotients of $X_{\overline{K}}$. Namely, using the same logic as above, we can think of $X_{\overline{K}}$ as being the actual solutions in $\overline{K}^n$ (where $n$ is the number of variables your polynomials are in) to whatever polynomials define $X$. Then, the maps $X_{\overline{K}}\to X_E$ and $X_{\overline{K}}\to X_K$ are both quotient maps, identifying $X_E$ with $X_{\overline{K}}/\text{Gal}(\overline{K}/K)$ and $X_E$ with $X_{\overline{K}}/\text{Gal}(\overline{K}/E)$. The map $X_E\to X_K$ is then obvious, it's the quotient map

$$X_{\overline{K}}/\text{Gal}(\overline{E}/K)\to (X_{\overline{K}}/\text{Gal}(\overline{K}/E))/(\text{Gal}(\overline{K}/K)$$

So, thinking about it in sequence, $X_E$ is $X_{\overline{K}}$ with some points identified, and similarly $X_K$ is $X_{\overline{K}}$ with more points identified. The, map $X_E\to X_K$ is then just the map which identifies in $X_E$ the points of $X_{\overline{K}}$ which are identified under the $\text{Gal}(\overline{K}/K)$ action, but weren't identified udner the $\text{Gal}(\overline{K}/E)$ action.