I've convinced myself that the projective Fermat curve $V(X^m + Y^m - Z^m) \subset \mathbb{P}^2$ is isomorphic to a projective line if and only if $m =1$ or $m = 2$, but I'm not sure how to prove this fact.
Are there proofs of this fact available online? Can anyone provide a proof?
You can find the explicit isomorphisms in the case when $m = 1$ or $m = 2$. Then if $m \geq 3$ you could use the degree genus formula, since the Fermat curves are smooth, to see that the genus of the Fermat curves for $m \geq 3$ is at least $1$, but any projective line has genus $0$.
Off course, you also need to know that two isomorphic curves have the same genus.