Smooth irreducible curves over $\mathbb Q$

Question

Let $C$ be a smooth, projective, irreducible curve in $\mathbb C\mathbb P^n$ defined over $\mathbb Q$.

1. Is it true that if its genus is $0$ then $C(\mathbb Q)$ is isomorphic with $\mathbb Q\mathbb P^1$ (over $\mathbb Q$)?

2. I understand that Mordell's theorem says that if genus is $1$ then $C(\mathbb Q)$ is either empty or finitely generated, and

3. Falting's theorem says that for $g\geq 2$, $C(\mathbb Q)$ is finite.

Is there a (fairly elementary) textbook discussing these statements?

Answer 1

The curve $X^2+Y^2+Z^2=0$ in $\Bbb P^2$ is of genus zero but not isomorphic over $\Bbb Q$ to $\Bbb P^1(\Bbb Q)$ (since it has no points over $\Bbb Q$).

Good luck finding an elementary book discussing Faltings' theorem beyond simply mentioning it. (Although Bombieri found a simpler proof than Faltings' that's far from easy).