Automorphism group of $\mathbb{P}^{1}_{\mathbb{C}}$

Asked Dec 18 '19 at 21:00

Active Dec 18 '19 at 21:00

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Show that every automorphism of $\mathbb{P}^{1}_{\mathbb{C}}$ is given by a linear automorphism of $\mathbb{C}^{2}$.

Equivalently: Show that $\operatorname{Aut}(\mathbb{P}^{1}_{\mathbb{C}})=\operatorname{PGL}_{2}(\mathbb{C})$.

I have no idea where even to start, and I am feeling a little bit ashamed about that. I know this is not true for $\mathbb{P}^{n}_{\mathbb{C}}$ if $n\geq 2$. Can somebody please give me a hint?

asked Dec 18 '19 at 21:00

The Thin Whistler

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Automorphism as what? – Dec 18 '19 at 21:11
What do you mean? $Aut(\mathbb P^n) \cong PGL_{n+1}$ for all $n$. – Tabes Bridges Dec 18 '19 at 21:45
Is there a proof that does not rely on twisted sheaves? – The Thin Whistler Dec 18 '19 at 23:17

Automorphism group of $\mathbb{P}^{1}_{\mathbb{C}}$

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