I've met a curious question when I did the exercise on page 20 of Serre's Topics in Galois Theory.
Let $K$ be a field of characteristic $0$, $W \to \mathbb{P}^1_K$ be a finite dominant morphism of geometrically irreducible curves, and $V$ be an affine geometrically irreducible variety over $K$. Serre states that there exists a morphism $f: V \to \mathbb{P}^1_K$, s.t. $f^{-1}(W)$ is geometrically irreducible. When $dim(V)>1$, it's easy to show by an argument at generic point. But things become strange when dim(V)=1. It's also easy to show the Hilbert property of $V$ implies that of $\mathbb{P}^1_K$, and find the $f$ s.t. every irreducible component of $f^{-1}(W)$ is geometrically irreducible. But I can't see the irreducibility of $f^{-1}(W)$. Even I use the Bertini theorem, I just vaguely think there is a map from a finite covering of $V$ to $\mathbb{P}^1_K$, s.t. the pull back to this covering is irreducible.
The question can be formulated in another way. Give a finite extention $K(x,y_0)$ of pure transcendental field $K(x)$, which are geometrically irreducible over $K$. Give a finitely generated algebra $K[x_1,z_1,...,z_m]$ geometrically irreducible over $K$ whose function field is $K(x_1,y_1)$, where $x,x_1$ are transcendental, others are algebraic over $K(x)$ or $K(x_1)$. It's equivalent to show that there exists a ring map $f: K[x] \to K[x_1,z_1,...,z_m]$, s.t. $K(x,y_0) \otimes K(x_1,y_1)$ is a field. The key point is to determine the image of $x$.
I have a conjecture to imply the question above. But I can't prove or disprove it. Let $k$ be an algebraically closed field of characteristic $0$. Let $f_0(Y)=Y^n+g_1(x)Y^{n-1}+...+g_n(x)$ be an irreducible polynomial on $k(x)$, which corresponds to an $n-$dimensional Galois extention $L_0$ and $n>1$. One can prove that $f_c(Y)=Y^n+g_1(x+c)Y^{n-1}+...+g_n(x+c)$ is irreducible and the corresponding field extention $L_c$ is Galois for all $c \in k$. Define $\mathcal{I}=\{T \subseteq k| |T|=\infty \}$. I suspect that $\forall S \in \mathcal{I}$, $\bigcap_{c \in S} L_c=k(x)$. Or equivalently, $\not\exists S \in \mathcal{I}$, $\exists L \neq k(x)$ finite galois over $k(x)$, s.t. $\forall c_1\neq c_2 \in S$, $L_{c_1}\cap L_{c_2}＝L$. Or equivalently, $\not\exists S \in \mathcal{I}$, s.t. $\forall c_1, c_2 \in S$, $L_{c_1}＝L_{c_2}$. Or equivalently, $\forall c_1\neq c_2 \in k$, $L_{c_1} \neq L_{c_2}$.
I've been entangled in these problems for a few days. I appreciate your help if you give an advise on whichever aspect.