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Let $ k$ be a field. The proof that $ Aut(\mathbb{P}_k^n)=PGL(n+1:k)$ relies on the fact for any automorphism $ \alpha$ of $ \mathbb{P}_k^n$ , $ \alpha^*(\mathcal{O}_{\mathbb{P}_k^n}(1)) = \mathcal{O}_{\mathbb{P}_k^n}(1)$ . It is not necessarily true that $ \pi^*O_{\mathbb{P}_A}(1)\simeq O_{\mathbb{P}_A}(1)$ if $ \pi:\mathbb{P}_A^{n}\to\mathbb{P}_A^{n}$ is an automorphism. See counterexample: https://math.stackexchange.com/questions/979958/automorphism-of-the-projective-space-mathbbp-an In particular, we can’t use the same typeRead more