Classification of automorphisms of projective space

Solution 1:

An automorphism of $\mathbb{P}^n_k$ induces an automorphism of the Picard group $\text{Pic}(\mathbb{P}^n_k) \cong \mathbb{Z}$. Such an automorphism must send the generator $\mathcal{O}(1)$ to a generator. Since the only two generators of $\mathbb{Z}$ are $1$ and $-1$, $f^*(\mathcal{O}(1))$ must be $\mathcal{O}(1)$ or $\mathcal{O}(-1)$. But $\mathcal{O}(-1)$ has no nonzero global sections, so it cannot be the pullback of $\mathcal{O}(1)$ (recall that $\mathcal{O}(1)$ pulls back to a line bundle together with $n+1$ global sections which have no common zero).

Solution 2:

Well, $f^*(\mathcal{O}(1))$ must be a line bundle on $\mathbb{P}^n$. In fact, $f^*$ gives a group automorphism of $\text{Pic}(\mathbb{P}^n) \cong \mathbb{Z}$, with inverse $(f^{-1})^*$. Thus, $f^*(\mathcal{O}(1))$ must be a generator of $\text{Pic}(\mathbb{P}^n)$, either $\mathcal{O}(1)$ or $\mathcal{O}(-1)$. But $f^*$ is also an automorphism on the space of global sections, again with inverse $(f^{-1})^*$. Since $\mathcal{O}(1)$ has an $(n+1)$-dimensional vector space of global sections, but $\mathcal{O}(-1)$ has no non-zero global sections, it is impossible for $f^*(\mathcal{O}(1))$ to be $\mathcal{O}(-1)$.