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I have a question which seems a little obvious, but I don't know how to phrase the answer. Let $C$ be an abstract curve, and let $C_0$ be an embedding of $C$ into $\mathbb{P}^N$ such that $C_0$ is smooth. Does every automorphism of $C$ induce an automorphism of $C_0$? (I think that yes). What can be said about the opposite direction and about the case when $C_0$ is singular? Thank you!

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You don't know how to phrase the answer? Is the question homework, then? –  Henning Makholm Oct 7 '11 at 12:40
No, it is not homework... just a bad choice of words. sorry for that. –  Brooke Oct 10 '11 at 13:59

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