I am looking at Terry Tao's blog where he reviews Bombieri's proof of the Hasse Weil bound.
At some point he argues as follows. Let $C$ be a curve defined over $\mathbb{F_q}$ and let $\pi : C \to \mathbb{A}^1$ be a morphism where the fiber over each point is either empty or of size $d$. Then look at the lifted curve $C'$ consisting of points of the form $(x_1, \ldots, x_d)$ where the $x_i$ are all in $C$ and distinct. In other words the points of $C'$ are precisely the fibers. Note that $C'$ is defined over $\mathbb{F_q}$.
Then $S_d$ acts on $C'$ in a natural way. The claim is that it acts transitively on the irreducible components of $C'$, which need not be absolutely irreducible, even if $C$ is.
My questions are:
- Is there an easy example that shows that $C'$ need not be absolutely irreducible, even if $C$ is?
- Why does $S_d$ act transitively on the irreducible components?
- If $C$ is affine, must $C'$ be affine? If yes then is there an easy description of its ring of regular functions?
Here is the link to the blog post:
http://terrytao.wordpress.com/2014/05/02/the-bombieri-stepanov-proof-of-the-hasse-weil-bound/