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Reading a book of Algebraic Geometry, I find the following passage: "... And this determine $C^*$ up to an automorphism of $(\mathbb{P}^{g-1})^*$."

Where $C^*$ is a subset of $(\mathbb{P}^{g-1})^*$.

How could I understand this? As are the automorphism of $(\mathbb{P}^{g-1})^*$? What it means to determine a subset in $(\mathbb{P}^{g-1})^*$ up to an automorphism?

Thanks

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An automorphism of $X$ is an isomorphism $f : X \to X$ (in the appropriate categeory). Incidentally for the projective space is just an element of $\rm{PGL}_n$.

This is proved in "A first course in algebraic geometry" by Harris. I can't remember the precise chapter but the proof goes like this : $f : \Bbb P^n \to \Bbb P^n$ induces an isomorphism $Pic(\Bbb P^n) \to Pic(\Bbb P^n)$, so it should send any hyperplane to another hyperplane. From this it is not too hard to deduce that $f$ should comes from a linear automorphism.

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