There is the following geometric interpretation of Noether's normalization theorem:
Let $X$ be an $n$-dimensional affine variety. Then there is a surjective morphism $\varphi : X \to \mathbb{A}^n$ with finite fibers, i.e. $|\varphi^{-1}(p)| < \infty$ for all $p \in \mathbb{A}^n$.
I wonder if there could be also a surjective morphism $X \to \mathbb{A}^n$ with an infinite fiber. Can you name some example?
Thank you in advance!
Added later: After writing this question I realized that a fiber is an intersection of $n$ hypersurfaces of $X$ so it is likely that their intersection is zero-dimensional and hence finite. Is this true? In this case how can we show that the dimension actually drops with each intersection, i.e. why are the irreducible components of the intersection of the first $i$ hypersurfaces not contained in the ($i+1$)-th one?
Take the blowup of $\mathbb{A}^2$ at a point $p$; this gives a birational morphism $\mbox{Bl}_p(\mathbb{A}^2)\to\mathbb{A}^2$ whose fiber over $p$ is infinite.
Edit: For an affine example, consider $\phi:\mathbb{A}^2\to\mathbb{A}^2$ where $(x,y)\mapsto(xy,x+x^2+xy)$. If $(z_0,w_0)\in\mathbb{A}^2$ and $x_0$ is a root of $x^2+x=w_0-z_0$ ($x_0\neq0$ when $z_0=w_0$), then $\phi(x_0,z_0/x_0)=(z_0,w_0)$. Over $(0,0)$, however, we get the fiber $\{(0,y):y\in k\}\cup\{(-1,0)\}$.