This is a cute proof that I ran into of the simple connectedness of . It does not use Riemann–Hurwitz or differentials, and instead relies on a purely geometric argument.
Lemma. Let be an algebraically closed field. Then is simply connected.
Proof. Let be a finite étale Galois cover with Galois group . We have to show that is an isomorphism. The diagonal is ample, so the same goes for the pullback to [Hart, Exc. III.5.7(d)]. In particular, is connected [Hart, Cor. III.7.9].
But is isomorphic to copies of because the action
is an isomorphism. If is connected, this forces , so is an isomorphism.
The proof actually shows that if is a smooth projective variety such that is a set-theoretic complete intersection of ample divisors, then is simply connected.
Example. For a smooth projective curve of genus , the diagonal cannot be ample, as . We already knew this by computing the self-intersection ,
but the argument above is more elementary.
[Hart] Hartshorne, Algebraic geometry. GTM 52, Springer, 1977.
The example at the end is fine but I want to mention that it is much harder to see that a curve C of genus 1 or more has nontrivial finite etale coverings than it is to see that the projective line is simply connected. I don’t know a simple argument (that you can explain after teaching students about Riemann-Roch and Riemann-Hurewitz) for the non-simply-connectedness of C. Best, Johan
Good point! So if you want to show that is not ample, it’s probably still easier to compute its self-intersection (contrary to what I claim).