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~~.

**References.**

[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).